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CHAPTER TEN
PART III
Arbitrage Pricing Theory
and Multifactor Models of
Risk and Return
THE EXPLOITATION OF security mispricing in
such a way that risk-free profits can be earned
is called arbitrage. It involves the simultaneous purchase and sale of equivalent securities in order to profit from discrepancies in
their prices. Perhaps the most basic principle
of capital market theory is that equilibrium
market prices are rational in that they rule
out arbitrage opportunities. If actual security prices allow for arbitrage, the result will
be strong pressure to restore equilibrium.
Therefore, security markets ought to satisfy
a “no-arbitrage condition.” In this chapter,
we show how such no-arbitrage conditions
together with the factor model introduced in
Chapter 8 allow us to generalize the security
market line of the CAPM to gain richer insight
into the risk–return relationship.
We begin by showing how the decomposition of risk into market versus firm-specific
influences that we introduced in earlier
chapters can be extended to deal with the
multifaceted nature of systematic risk. Multifactor models of security returns can be
used to measure and manage exposure to
each of many economywide factors such as
business-cycle risk, interest or inflation rate
bod61671_ch10_324-348.indd 324
risk, energy price risk, and so on. These models
also lead us to a multifactor version of the
security market line in which risk premiums
derive from exposure to multiple risk sources,
each with their own risk premium.
We show how factor models combined
with a no-arbitrage condition lead to a simple
relationship between expected return and
risk. This approach to the risk–return tradeoff is called arbitrage pricing theory, or APT.
In a single-factor market where there are no
extra-market risk factors, the APT leads to a
mean return–beta equation identical to that
of the CAPM. In a multifactor market with
one or more extra-market risk factors, the
APT delivers a mean-beta equation similar
to Merton’s intertemporal extension of the
CAPM (his ICAPM). We ask next what factors
are likely to be the most important sources
of risk. These will be the factors generating
substantial hedging demands that brought
us to the multifactor CAPM introduced in
Chapter 9. Both the APT and the CAPM therefore can lead to multiple-risk versions of the
security market line, thereby enriching the
insights we can derive about the risk–return
relationship.
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10.1 Multifactor Models: An Overview
The index model introduced in Chapter 8 gave us a way of decomposing stock variability
into market or systematic risk, due largely to macroeconomic events, versus firm-specific
or idiosyncratic effects that can be diversified in large portfolios. In the single-index model,
the return on a broad market-index portfolio summarized the impact of the macro factor.
In Chapter 9 we introduced the possibility that asset-risk premiums may also depend on
correlations with extra-market risk factors, such as inflation, or changes in the parameters
describing future investment opportunities: interest rates, volatility, market-risk premiums,
and betas. For example, returns on an asset whose return increases when inflation increases
can be used to hedge uncertainty in the future inflation rate. Its risk premium may fall as a
result of investors’ extra demand for this asset.
Risk premiums of individual securities should reflect their sensitivities to changes in
extra-market risk factors just as their betas on the market index determine their risk premiums in the simple CAPM. When securities can be used to hedge these factors, the resulting
hedging demands will make the SML multifactor, with each risk source that can be hedged
adding an additional factor to the SML. Risk factors can be represented either by returns
on these hedge portfolios (just as the index portfolio represents the market factor), or more
directly by changes in the risk factors themselves, for example, changes in interest rates
or inflation.
Factor Models of Security Returns
We begin with a familiar single-factor model like the one introduced in Chapter 8. Uncertainty in asset returns has two sources: a common or macroeconomic factor and firmspecific events. The common factor is constructed to have zero expected value, because
we use it to measure new information concerning the macroeconomy, which, by definition,
has zero expected value.
If we call F the deviation of the common factor from its expected value, bi the sensitivity of firm i to that factor, and ei the firm-specific disturbance, the factor model states
that the actual excess return on firm i will equal its initially expected value plus a (zero
expected value) random amount attributable to unanticipated economywide events, plus
another (zero expected value) random amount attributable to firm-specific events.
Formally, the single-factor model of excess returns is described by Equation 10.1:
Ri 5 E (Ri ) 1 bi F 1 ei
(10.1)
where E(Ri ) is the expected excess return on stock i. Notice that if the macro factor has a
value of 0 in any particular period (i.e., no macro surprises), the excess return on the security will equal its previously expected value, E(Ri ), plus the effect of firm-specific events
only. The nonsystematic components of returns, the eis, are assumed to be uncorrelated
across stocks and with the factor F.
Example 10.1
Factor Models
To make the factor model more concrete, consider an example. Suppose that the macro
factor, F, is taken to be news about the state of the business cycle, measured by the
unexpected percentage change in gross domestic product (GDP), and that the consensus is that GDP will increase by 4% this year. Suppose also that a stock’s b value is 1.2.
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PART III
Equilibrium in Capital Markets
If GDP increases by only 3%, then the value of F would be 21%, representing a 1%
disappointment in actual growth versus expected growth. Given the stock’s beta value,
this disappointment would translate into a return on the stock that is 1.2% lower than
previously expected. This macro surprise, together with the firm-specific disturbance, ei,
determines the total departure of the stock’s return from its originally expected value.
CONCEPT CHECK
10.1
Suppose you currently expect the stock in Example 10.1 to earn a 10% rate of return. Then some
macroeconomic news suggests that GDP growth will come in at 5% instead of 4%. How will you revise
your estimate of the stock’s expected rate of return?
The factor model’s decomposition of returns into systematic and firm-specific
components is compelling, but confining systematic risk to a single factor is not. Indeed,
when we motivated systematic risk as the source of risk premiums in Chapter 9, we noted
that extra market sources of risk may arise from a number of sources such as uncertainty
about interest rates, inflation, and so on. The market return reflects macro factors as well as
the average sensitivity of firms to those factors.
It stands to reason that a more explicit representation of systematic risk, allowing for
different stocks to exhibit different sensitivities to its various components, would constitute
a useful refinement of the single-factor model. It is easy to see that models that allow for
several factors—multifactor models—can provide better descriptions of security returns.
Apart from their use in building models of equilibrium security pricing, multifactor
models are useful in risk management applications. These models give us a simple way
to measure investor exposure to various macroeconomic risks and construct portfolios to
hedge those risks.
Let’s start with a two-factor model. Suppose the two most important macroeconomic
sources of risk are uncertainties surrounding the state of the business cycle, news of which
we will again measure by unanticipated growth in GDP and changes in interest rates. We
will denote by IR any unexpected change in interest rates. The return on any stock will
respond both to sources of macro risk and to its own firm-specific influences. We can write
a two-factor model describing the excess return on stock i in some time period as follows:
Ri 5 E (Ri ) 1 biGDPGDP 1 biIRIR 1 ei
(10.2)
The two macro factors on the right-hand side of the equation comprise the systematic factors in the economy. As in the single-factor model, both of these macro factors
have zero expectation: They represent changes in these variables that have not already been
anticipated. The coefficients of each factor in Equation 10.2 measure the sensitivity of share
returns to that factor. For this reason the coefficients are sometimes called factor loadings
or, equivalently, factor betas. An increase in interest rates is bad news for most firms, so
we would expect interest rate betas generally to be negative. As before, ei reflects firmspecific influences.
To illustrate the advantages of multifactor models, consider two firms, one a regulated
electric-power utility in a mostly residential area, the other an airline. Because residential
demand for electricity is not very sensitive to the business cycle, the utility has a low beta
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CHAPTER 10
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327
on GDP. But the utility’s stock price may have a relatively high sensitivity to interest rates.
Because the cash flow generated by the utility is relatively stable, its present value behaves
much like that of a bond, varying inversely with interest rates. Conversely, the performance
of the airline is very sensitive to economic activity but is less sensitive to interest rates. It
will have a high GDP beta and a lower interest rate beta. Suppose that on a particular day, a
news item suggests that the economy will expand. GDP is expected to increase, but so are
interest rates. Is the “macro news” on this day good or bad? For the utility, this is bad news:
Its dominant sensitivity is to rates. But for the airline, which responds more to GDP, this
is good news. Clearly a one-factor or single-index model cannot capture such differential
responses to varying sources of macroeconomic uncertainty.
Example 10.2
Risk Assessment Using Multifactor Models
Suppose we estimate the two-factor model in Equation 10.2 for Northeast Airlines and
find the following result:
R 5 .133 1 1.2(GDP) 2 .3(IR) 1 e
This tells us that, based on currently available information, the expected excess rate
of return for Northeast is 13.3%, but that for every percentage point increase in GDP
beyond current expectations, the return on Northeast shares increases on average by
1.2%, while for every unanticipated percentage point that interest rates increases,
Northeast’s shares fall on average by .3%.
Factor betas can provide a framework for a hedging strategy. The idea for an investor
who wishes to hedge a source of risk is to establish an opposite factor exposure to offset
that particular source of risk. Often, futures contracts can be used to hedge particular factor
exposures. We explore this application in Chapter 22.
As it stands, however, the multifactor model is no more than a description of the factors that affect security returns. There is no “theory” in the equation. The obvious question
left unanswered by a factor model like Equation 10.2 is where E(R ) comes from, in other
words, what determines a security’s expected excess rate of return. This is where we need a
theoretical model of equilibrium security returns. We therefore now turn to arbitrage pricing theory to help determine the expected value, E(R), in Equations 10.1 and 10.2.
10.2 Arbitrage Pricing Theory
Stephen Ross developed the arbitrage pricing theory (APT) in 1976.1 Like the CAPM,
the APT predicts a security market line linking expected returns to risk, but the path it takes
to the SML is quite different. Ross’s APT relies on three key propositions: (1) security
returns can be described by a factor model; (2) there are sufficient securities to diversify
away idiosyncratic risk; and (3) well-functioning security markets do not allow for the persistence of arbitrage opportunities. We begin with a simple version of Ross’s model, which
assumes that only one systematic factor affects security returns.
1
Stephen A. Ross, “Return, Risk and Arbitrage,” in I. Friend and J. Bicksler, eds., Risk and Return in Finance
(Cambridge, MA: Ballinger, 1976).
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PART III
Equilibrium in Capital Markets
Arbitrage, Risk Arbitrage, and Equilibrium
An arbitrage opportunity arises when an investor can earn riskless profits without making
a net investment. A trivial example of an arbitrage opportunity would arise if shares of a
stock sold for different prices on two different exchanges. For example, suppose IBM sold
for $195 on the NYSE but only $193 on NASDAQ. Then you could buy the shares on
NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per
share without tying up any of your own capital. The Law of One Price states that if two
assets are equivalent in all economically relevant respects, then they should have the same
market price. The Law of One Price is enforced by arbitrageurs: If they observe a violation
of the law, they will engage in arbitrage activity—simultaneously buying the asset where it
is cheap and selling where it is expensive. In the process, they will bid up the price where
it is low and force it down where it is high until the arbitrage opportunity is eliminated.
The idea that market prices will move to rule out arbitrage opportunities is perhaps the
most fundamental concept in capital market theory. Violation of this restriction would indicate the grossest form of market irrationality.
The critical property of a risk-free arbitrage portfolio is that any investor, regardless of
risk aversion or wealth, will want to take an infinite position in it. Because those large positions will quickly force prices up or down until the opportunity vanishes, security prices
should satisfy a “no-arbitrage condition,” that is, a condition that rules out the existence of
arbitrage opportunities.
There is an important difference between arbitrage and risk–return dominance arguments in support of equilibrium price relationships. A dominance argument holds that
when an equilibrium price relationship is violated, many investors will make limited portfolio changes, depending on their degree of risk aversion. Aggregation of these limited
portfolio changes is required to create a large volume of buying and selling, which in turn
restores equilibrium prices. By contrast, when arbitrage opportunities exist, each investor wants to take as large a position as possible; hence it will not take many investors to
bring about the price pressures necessary to restore equilibrium. Therefore, implications
for prices derived from no-arbitrage arguments are stronger than implications derived from
a risk–return dominance argument.
The CAPM is an example of a dominance argument, implying that all investors hold
mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their
portfolios toward the underpriced and away from the overpriced securities. Pressure on
equilibrium prices results from many investors shifting their portfolios, each by a relatively
small dollar amount. The assumption that a large number of investors are mean-variance
optimizers is critical. In contrast, the implication of a no-arbitrage condition is that a few
investors who identify an arbitrage opportunity will mobilize large dollar amounts and
quickly restore equilibrium.
Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict
definition. Arbitrageur often refers to a professional searching for mispriced securities in
specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free)
arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it
from pure arbitrage.
To leap ahead, in Part Four we will discuss “derivative” securities such as futures and
options, whose market values are determined by prices of other securities. For example,
the value of a call option on a stock is determined by the price of the stock. For such securities, strict arbitrage is a practical possibility, and the condition of no-arbitrage leads to
exact pricing. In the case of stocks and other “primitive” securities whose values are not
determined strictly by another bundle of assets, no-arbitrage conditions must be obtained
by appealing to diversification arguments.
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Well-Diversified Portfolios
Consider the risk of a portfolio of stocks in a single-factor market. We first show that if a
portfolio is well diversified, its firm-specific or nonfactor risk becomes negligible, so that
only factor (or systematic) risk remains. The excess return on an n-stock portfolio with
weights wi, Swi 5 1, is
RP 5 E (RP) 1 bP F 1 eP
where
(10.3)
bP 5 g w i b i ; E ( RP) 5 g wi E (Ri )
are the weighted averages of the bi and risk premiums of the n securities. The portfolio
nonsystematic component (which is uncorrelated with F ) is eP 5 Swi ei , which similarly is
a weighted average of the ei of the n securities.
We can divide the variance of this portfolio into systematic and nonsystematic sources:
s 2P 5 b 2P s 2F 1 s2 (eP )
where s2F is the variance of the factor F and s2(eP) is the nonsystematic risk of the portfolio, which is given by
s2 (eP ) 5 Variance ( g wi ei ) 5 g w2i s2 (ei )
Note that in deriving the nonsystematic variance of the portfolio, we depend on the fact
that the firm-specific eis are uncorrelated and hence that the variance of the “portfolio” of
nonsystematic eis is the weighted sum of the individual nonsystematic variances with the
square of the investment proportions as weights.
If the portfolio were equally weighted, wi 5 1/n, then the nonsystematic variance would be
2
2
1
1
1 s (ei )
s2(eP) 5 Variance ( g wiei ) 5 a a n b s2(ei ) 5 n a n 5 n s2 (ei )
where the last term is the average value of nonsystematic variance across securities. In
words, the nonsystematic variance of the portfolio equals the average nonsystematic variance divided by n. Therefore, when n is large, nonsystematic variance approaches zero.
This is the effect of diversification.
This property is true of portfolios other than the equally weighted one. Any portfolio for
which each wi becomes consistently smaller as n gets large (more precisely, for which each
w2i approaches zero as n increases) will satisfy the condition that the portfolio nonsystematic risk will approach zero. This property motivates us to define a well-diversified portfolio
as one with each weight, wi , small enough that for practical purposes the nonsystematic
variance, s2(eP), is negligible.
CONCEPT CHECK
10.2
a. A portfolio is invested in a very large number of shares (n is large). However, one-half of the portfolio
is invested in stock 1, and the rest of the portfolio is equally divided among the other n 2 1 shares. Is
this portfolio well diversified?
b. Another portfolio also is invested in the same n shares, where n is very large. Instead of equally
weighting with portfolio weights of 1/n in each stock, the weights in half the securities are 1.5/n
while the weights in the other shares are .5/n. Is this portfolio well diversified?
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PART III
Equilibrium in Capital Markets
Because the expected value of eP for any well-diversified portfolio is zero, and its
variance also is effectively zero, we can conclude that any realized value of eP will be
virtually zero. Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio,
for all practical purposes
RP 5 E (RP ) 1 bP F
The solid line in Figure 10.1, panel A plots the excess return of a well-diversified portfolio A with E (RA) 5 10% and bA 5 1 for various realizations of the systematic factor. The
expected return of portfolio A is 10%; this is where the solid line crosses the vertical axis.
At this point the systematic factor is zero, implying no macro surprises. If the macro factor
is positive, the portfolio’s return exceeds its expected value; if it is negative, the portfolio’s
return falls short of its mean. The excess return on the portfolio is therefore
E (RA ) 1 bA F 5 10% 1 1.0 3 F
Compare panel A in Figure 10.1 with panel B, which is a similar graph for a single stock (S)
with bs 5 1. The undiversified stock is subject to nonsystematic risk, which is seen in
a scatter of points around the line. The well-diversified portfolio’s return, in contrast, is
determined completely by the systematic factor.
In a single-factor world, all pairs of well-diversified portfolios are perfectly correlated:
Their risk is fully determined by the same systematic factor. Consider a second welldiversified portfolio, Portfolio Q, with RQ 5 E(RQ) 1 bQF. We can compute the standard
deviations of P and Q, as well as the covariance and correlation between them:
sP 5 bP sF; sQ 5 bQ sF
Cov ( RP , RQ ) 5 Cov (bP F, bQ F ) 5 bP bQ s2F
r PQ 5
Cov (RP, RQ )
51
sP sQ
Excess Return (%)
Excess Return (%)
A
S
10
10
F
0
A
F
0
B
Figure 10.1 Excess returns as a function of the systematic factor. Panel A,
Well-diversified portfolio A. Panel B, Single stock (S).
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Perfect correlation means that in a plot of expected return versus standard deviation (such
as Figure 7.5), any two well-diversified portfolios lie on a straight line. We will see later
that this common line is the CML.
Diversification and Residual Risk in Practice
What is the effect of diversification on portfolio residual SD in practice, where portfolio size is not unlimited? In reality, we may find (annualized) residual SDs as high as
50% for large stocks and even 100% for small stocks. To illustrate the impact of diversification, we examine portfolios of two configurations. One portfolio is equally weighted;
this achieves the highest benefits of diversification with equal-SD stocks. For comparison, we form the other portfolio using far-from-equal weights. We select stocks in groups
of four, with relative weights in each group of 70%, 15%, 10%, and 5%. The highest
weight is 14 times greater than the lowest, which will severely reduce potential benefits of
diversification. However, extended diversification in which we add to the portfolio more
and more groups of four stocks with the same relative weights will overcome this problem because the highest portfolio weight still falls with additional diversification. In an
equally weighted 1,000-stock portfolio, each weight is 0.1%; in the unequally weighted
portfolio, with 1,000/4 5 250 groups of four stocks, the highest and lowest weights are
70%/250 5 0.28% and 5%/250 5 0.02%, respectively.
What is a large portfolio? Many widely held ETFs each include hundreds of stocks, and
some funds such as the Wilshire 5000 hold thousands. These portfolios are accessible to
the public since the annual expense ratios of investment companies that offer such funds
are of the order of only 10 basis points. Thus a portfolio of 1,000 stocks is not unheard of,
but a portfolio of 10,000 stocks is.
Table 10.1 shows portfolio residual SD as a function of the number of stocks. Equally
weighted, 1,000-stock portfolios achieve small but not negligible standard deviations of
1.58% when residual risk is 50% and 3.16% when residual risk is 100%. The SDs for the
unbalanced portfolios are about double these values. For 10,000-stock portfolios, the SDs
are negligible, verifying that diversification can eliminate risk even in very unbalanced
portfolios, at least in principle, if the investment universe is large enough.
Residual SD of Each Stock 5 50%
Residual SD of Each Stock 5 100%
N
N
SD(eP)
Equal weights: wi 5 1/N
4
25.00
4
60
6.45
60
200
3.54
200
1,000
1.58
1,000
10,000
0.50
10,000
Sets of four relative weights: w1 5 0.65, w2 5 0.2, w3 5 0.1, w4 5 0.05
4
36.23
4
60
9.35
60
200
5.12
200
1,000
2.29
1,000
10,000
0.72
10,000
SD(eP)
50.00
12.91
7.07
3.16
1.00
72.46
18.71
10.25
4.58
1.45
Table 10.1
Residual variance with even and uneven portfolio weights
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PART III
Equilibrium in Capital Markets
Executing Arbitrage
Imagine a single-factor market where the well-diversified portfolio, M, represents
the market factor, F, of Equation 10.1. The excess return on any security is given by
Ri 5 ai 1 bi RM 1 ei , and that of a well-diversified (therefore zero residual) portfolio, P, is
RP 5 aP 1 bP RM
(10.4)
E (RP) 5 aP 1 bP E (RM)
(10.5)
Now suppose that security analysis reveals that portfolio P has a positive alpha.2 We also
estimate the risk premium of the index portfolio, M, from macro analysis.
Since neither M nor portfolio P have residual risk, the only risk to the returns of the two
portfolios is systematic, derived from their betas on the common factor (the beta of the
index is 1.0). Therefore, you can eliminate the risk of P altogether: Construct a zero-beta
portfolio, called Z, from P and M by appropriately selecting weights wP and wM 5 1 2 wP
on each portfolio:
bZ 5 wPbP 1 (1 2 wP)bM 5 0
bM 5 1
wP 5
(10.6)
2bP
1
; w 5 1 2 wP 5
1 2 bP M
1 2 bP
Therefore, portfolio Z is riskless, and its alpha is
aZ 5 wP aP 1 (1 2 wP) aM 5 wP aP
(10.7)
The risk premium on Z must be zero because the risk of Z is zero. If its risk premium
were not zero, you could earn arbitrage profits. Here is how:
Since the beta of Z is zero, Equation 10.5 implies that its risk premium is just its alpha.
Using Equation 10.7, its alpha is wP aP, so
E (RZ ) 5 wP aP 5
1
a
1 2 bP P
(10.8)
You now form a zero-net-investment arbitrage portfolio: If bP , 1 and the risk premium
of Z is positive (implying that Z returns more than the risk-free rate), borrow and invest
the proceeds in Z. For every borrowed dollar invested in Z, you get a net return (i.e., net of
1
paying the interest on your loan) of
aP. This is a money machine, which you would
1 2 bP
work as hard as you can.3 Similarly if bP . 1, Equation 10.8 tells us that the risk premium is negative; therefore, sell Z short and invest the proceeds at the risk-free rate. Once
again, a money machine has been created. Neither situation can persist, as the large volume
of trades from arbitrageurs pursuing these strategies will push prices until the arbitrage
opportunity disappears (i.e., until the risk premium of portfolio Z equals zero).
2
If the portfolio alpha is negative, we can still pursue the following strategy. We would simply switch to a short
position in P, which would have a positive alpha of the same absolute value as P’s, and a beta that is the negative
of P’s.
3
The function in Equation 10.8 becomes unstable at bP 5 1. For values of bP near 1, it becomes infinitely large
with the sign of aP. This isn’t an economic absurdity, since in that case, the sizes of your long position in P and
short position in M will be almost identical, and the arbitrage profit you earn per dollar invested will be nearly
infinite.
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The No-Arbitrage Equation of the APT
We’ve seen that arbitrage activity will quickly pin the risk premium of any zero-beta welldiversified portfolio to zero.4 Setting the expression in Equation 10.8 to zero implies that
the alpha of any well-diversified portfolio must also be zero. From Equation 10.5, this
means that for any well-diversified P,
E (RP) 5 bP E (RM)
(10.9)
In other words, the risk premium (expected excess return) on portfolio P is the product of
its beta and the market-index risk premium. Equation 10.9 thus establishes that the SML
of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage”
requirement of the APT.
Another demonstration that the APT results in the same SML as the CAPM is more
graphical in nature. First we show why all well-diversified portfolios with the same beta must
have the same expected return. Figure 10.2 plots the returns on two such portfolios, A and
B, both with betas of 1, but with differing expected returns: E (rA) 5 10% and E (rB) 5 8%.
Could portfolios A and B coexist with the return pattern depicted? Clearly not: No matter
what the systematic factor turns out to be, portfolio A outperforms portfolio B, leading to
an arbitrage opportunity.
If you sell short $1 million of B and buy $1 million of A, a zero-net-investment strategy,
you would have a riskless payoff of $20,000, as follows:
(.10 1 1.0 3 F ) 3 $1 million
2(.08 1 1.0 3 F ) 3 $1 million
.02 3 $1 million 5 $20,000
from long position in A
from short position in B
net proceeds
Your profit is risk-free because the factor risk cancels out across the long and short
positions. Moreover, the strategy requires zero-net-investment. You should pursue it on an
Return (%)
A
B
10
8
0
F (Realization of
Macro Factor)
Figure 10.2 Returns as a function of the systematic factor: an arbitrage
opportunity
4
As an exercise, show that when aP , 0 you reverse the position of P in Z, and the arbitrage portfolio will still
earn a riskless excess return.
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Equilibrium in Capital Markets
Expected Return (%)
A
10
rf = 4
Risk
Premium
D
7
6
C
F
.5
1
β (With Respect
to Macro Factor)
Figure 10.3 An arbitrage opportunity
infinitely large scale until the return discrepancy between the two portfolios disappears.
Well-diversified portfolios with equal betas must have equal expected returns in market
equilibrium, or arbitrage opportunities exist.
What about portfolios with different betas? Their risk premiums must be proportional to beta. To see why, consider Figure 10.3. Suppose that the risk-free rate is 4%
and that a well-diversified portfolio, C, with a beta of .5, has an expected return of 6%.
Portfolio C plots below the line from the risk-free asset to portfolio A. Consider, therefore, a new portfolio, D, composed of half of portfolio A and half of the risk-free asset.
Portfolio D’s beta will be (.5 3 0 1 .5 3 1.0) 5 .5, and its expected return will be
(.5 3 4 1 .5 3 10) 5 7%. Now portfolio D has an equal beta but a greater expected return
than portfolio C. From our analysis in the previous paragraph we know that this constitutes an arbitrage opportunity. We conclude that, to preclude arbitrage opportunities, the
expected return on all well-diversified portfolios must lie on the straight line from the
risk-free asset in Figure 10.3.
Notice in Figure 10.3 that risk premiums are indeed proportional to portfolio betas. The
risk premium is depicted by the vertical arrow, which measures the distance between the
risk-free rate and the expected return on the portfolio. As in the simple CAPM, the risk
premium is zero for b 5 0 and rises in direct proportion to b.
10.3 The APT, the CAPM, and the Index Model
Equation 10.9 raises three questions:
1. Does the APT also apply to less-than-well-diversified portfolios?
2. Is the APT as a model of risk and return superior or inferior to the CAPM? Do we
need both models?
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3. Suppose a security analyst identifies a positive-alpha portfolio with some remaining
residual risk. Don’t we already have a prescription for this case from the TreynorBlack (T-B) procedure applied to the index model (Chapter 8)? Is this framework
preferred to the APT?
The APT and the CAPM
The APT is built on the foundation of well-diversified portfolios. However, we’ve seen,
for example in Table 10.1, that even large portfolios may have non-negligible residual risk.
Some indexed portfolios may have hundreds or thousands of stocks, but active portfolios
generally cannot, as there is a limit to how many stocks can be actively analyzed in search
of alpha. How does the APT stand up to these limitations?
Suppose we order all portfolios in the universe by residual risk. Think of Level 0 portfolios as having zero residual risk; in other words, they are the theoretically well-diversified
portfolios of the APT. Level 1 portfolios have very small residual risk, say up to 0.5%.
Level 2 portfolios have yet greater residual SD, say up to 1%, and so on.
If the SML described by Equation 10.9 applies to all well-diversified Level 0 portfolios,
it must at least approximate the risk premiums of Level 1 portfolios. Even more important,
while Level 1 risk premiums may deviate slightly from Equation 10.9, such deviations
should be unbiased, with alphas equally likely to be positive or negative. Deviations should
be uncorrelated with beta or residual SD and should average to zero.
We can apply the same logic to portfolios of slightly higher Level 2 residual risk. Since
all Level 1 portfolios are still well approximated by Equation 10.9, so must be risk premiums of Level 2 portfolios, albeit with slightly less accuracy. Here too, we may take
comfort in the lack of bias and zero average deviations from the risk premiums predicted by Equation 10.9. But still, the precision of predictions of risk premiums from
Equation 10.9 consistently deteriorates with increasing residual risk. (One might ask why
we don’t transform Level 2 portfolios into Level 1 or even Level 0 portfolios by further
diversifying, but as we’ve pointed out, this may not be feasible in practice for assets with
considerable residual risk when active portfolio size or the size of the investment universe
is limited.) If residual risk is sufficiently high and the impediments to complete diversification are too onerous, we cannot have full confidence in the APT and the arbitrage activities
that underpin it.
Despite this shortcoming, the APT is valuable. First, recall that the CAPM requires that
almost all investors be mean-variance optimizers. We may well suspect that they are not.
The APT frees us of this assumption. It is sufficient that a small number of sophisticated
arbitrageurs scour the market for arbitrage opportunities. This alone produces an SML,
Equation 10.9, that is a good and unbiased approximation for all assets but those with
significant residual risk.
Perhaps even more important is the fact that the APT is anchored by observable portfolios such as the market index. The CAPM is not even testable because it relies on an
unobserved, all-inclusive portfolio. The reason that the APT is not fully superior to the
CAPM is that at the level of individual assets and high residual risk, pure arbitrage may
be insufficient to enforce Equation 10.9. Therefore, we need to turn to the CAPM as a
complementary theoretical construct behind equilibrium risk premiums.
It should be noted, however, that when we replace the unobserved market portfolio of
the CAPM with an observed, broad index portfolio that may not be efficient, we no longer
can be sure that the CAPM predicts risk premiums of all assets with no bias. Neither model
therefore is free of limitations. Comparing the APT arbitrage strategy to maximization of
the Sharpe ratio in the context of an index model may well be the more useful framework
for analysis.
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The APT and Portfolio Optimization in a Single-Index Market
The APT is couched in a single-factor market5 and applies with perfect accuracy to welldiversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk premium of a well-diversified portfolio deviates from Equation 10.9. The trades executed by
these arbitrageurs are the enforcers of the accuracy of this equation.
In effect, the APT shows how to take advantage of security mispricing when diversification opportunities are abundant. When you lock in and scale up an arbitrage opportunity
you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio,
but only if the arbitrage portfolio is truly risk-free! However, if the arbitrage position is not
perfectly well diversified, an increase in its scale (borrowing cash, or borrowing shares to
go short) will increase the risk of the arbitrage position, potentially without bound. The
APT ignores this complication.
Now consider an investor who confronts the same single factor market, and whose
security analysis reveals an underpriced asset (or portfolio), that is, one whose risk
premium implies a positive alpha. This investor can follow the advice weaved throughout
Chapters 6–8 to construct an optimal risky portfolio. The optimization process will
consider both the potential profit from a position in the mispriced asset, as well as the risk
of the overall portfolio and efficient diversification. As we saw in Chapter 8, the TreynorBlack (T-B) procedure can be summarized as follows.6
1. Estimate the risk premium and standard deviation of the benchmark (index) portfolio, RPM and sM.
2. Place all the assets that are mispriced into an active portfolio. Call the alpha of the
active portfolio aA, its systematic-risk coefficient bA, and its residual risk s(eA).
Your optimal risky portfolio will allocate to the active portfolio a weight, w*A:
A
w0A
2
(eA ) *
; wA
E ( RM )
2
M
w0A
1 1 w0A (1 2 bA)
The allocation to the passive portfolio is then, w*M 5 1 2 w*A. With this allocation, the increase in the Sharpe ratio of the optimal portfolio, SP, over that of the
passive portfolio, SM, depends on the size of the information ratio of the active
portfolio, IR A 5 aA/s(eA). The optimized portfolio can attain a Sharpe ratio of
SP 5 "S 2M 1 IR 2A.
3. To maximize the Sharpe ratio of the risky portfolio, you maximize the IR of the
active portfolio. This is achieved by allocating to each asset in the active portfolio
a portfolio weight proportional to: wAi 5 ai/s2(ei). When this is done, the square
of the information ratio of the active portfolio will be the sum of the squared individual information ratios: IR 2A 5 a IR 2i .
Now see what happens in the T-B model when the residual risk of the active portfolio
is zero. This is essentially the assumption of the APT, that a well-diversified portfolio
(with zero residual risk) can be formed. When the residual risk of the active portfolio goes
to zero, the position in it goes to infinity. This is precisely the same implication as the
APT: When portfolios are well-diversified, you will scale up an arbitrage position without
5
The APT is easily extended to a multifactor market as we show later.
The tediousness of some of the expressions involved in the T-B method should not deter anyone. The calculations are pretty straightforward, especially in a spreadsheet. The estimation of the risk parameters also is a relatively straightforward statistical task. The real difficulty is to uncover security alphas and the macro-factor risk
premium, RPM.
6
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bound. Similarly, when the residual risk of an asset in the active T-B portfolio is zero, it
will displace all other assets from that portfolio, and thus the residual risk of the active
portfolio will be zero and elicit the same extreme portfolio response.
When residual risks are nonzero, the T-B procedure produces the optimal risky portfolio, which is a compromise between seeking alpha and shunning potentially diversifiable risk. The APT ignores residual risk altogether, assuming it has been diversified away.
Obviously, we have no use for the APT in this context. When residual risk can be made
small through diversification, the T-B model prescribes very aggressive (large) positions
in mispriced securities that exert great pressure on equilibrium risk premiums to eliminate
nonzero alpha values. The T-B model does what the APT is meant to do but with more
flexibility in terms of accommodating the practical limits to diversification. In this sense,
Treynor and Black anticipated the development of the APT.
Example 10.3
Exploiting Alpha
Table 10.2 summarizes a rudimentary experiment that compares the prescriptions and
predictions of the APT and T-B model in the presence of realistic values of residual risk.
We use relatively small alpha values (1 and 3%), three levels of residual risk consistent
with values in Table 10.1 (2, 3, and 4%), and two levels of beta (0.5 and 2) to span the
likely range of reasonable parameters.
The first set of columns in Table 10.2, titled Active Portfolio, show the parameter
values in each example. The second set of columns, titled Zero-Net-Investment, Arbitrage
(Zero-Beta), shows the weight in the active portfolio and resultant information ratio of
the active portfolio. This would be the Sharpe ratio if the arbitrage position (the positivealpha, zero-beta portfolio) made up the entire risky portfolio (as would be prescribed
Index Risk Premium 5 7
Index SD 5 20
Index Sharpe Ratio 5 0.35
Active Portfolio
Zero-Net-Investment,
Arbitrage (Zero-Beta)
Portfolio
Treynor-Black Procedure
Alpha (%)
Residual SD
Beta
w in Active
Info Ratio
w(beta 5 0)
w(beta)
1
1
1
1
1
1
3
3
3
3
3
3
4
4
3
3
2
2
4
4
3
3
2
2
0.5
2
0.5
2
0.5
2
0.5
2
0.5
2
0.5
2
2
1
2
1
2
1
2
1
2
1
2
1
0.25
0.25
0.33
0.33
0.50
0.50
0.75
0.75
1.00
1.00
1.50
1.50
3.57
3.57
6.35
6.35
14.29
14.29
10.71
10.71
19.05
19.05
42.86
42.86
1.28
1.00
1.52
1.00
1.75
1.00
1.69
1.00
1.81
1.00
1.91
1.00
Sharpe Incremental
Ratio Sharpe Ratio
0.43
0.43
0.48
0.48
0.61
0.61
0.83
0.83
1.06
1.06
1.54
1.54
0.18
0.18
0.15
0.15
0.11
0.11
0.08
0.08
0.06
0.06
0.04
0.04
Table 10.2
Performance of APT vs. Index Model when diversification of residual SD is incomplete
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by the APT). The last set of columns shows the T-B position in the active portfolio that
maximizes the Sharpe ratio of the overall risky portfolio. The final column shows the
increment to the Sharpe ratio of the T-B portfolio relative to the APT portfolio.
Keep in mind that even when the two models call for a similar weight in the active
portfolio (compare w in Active for the APT model to w(beta) for the T-B model), they
nevertheless prescribe a different overall risky portfolio. The APT assumes zero investment beyond what is necessary to hedge out the market risk of the active portfolio. In
contrast, the T-B procedure chooses a mix of active and index portfolios to maximize the
Sharpe ratio. With identical investment in the active portfolio, the T-B portfolio can still
include additional investment in the index portfolio.
To obtain the Sharpe ratio of the risky portfolio, we need the Sharpe ratio of the
index portfolio. As an estimate, we use the average return and standard deviation of
the broad market index (NYSE 1 AMEX 1 NASDAQ) over the period 1926–2012. The
top row (over the column titles) of Table 10.2 shows an annual Sharpe ratio of 0.35. The
rows of the table are ordered by the information ratio of the active portfolio.
Table 10.2 shows that the T-B procedure noticeably improves the Sharpe ratio beyond
the information ratio of the APT (for which the IR is also the Sharpe ratio). However, as
the information ratio of the active portfolio increases, the difference in the T-B and APT
active portfolio positions declines, as does the difference between their Sharpe ratios.
Put differently, the higher the information ratio, the closer we are to a risk-free arbitrage
opportunity, and the closer are the prescriptions of the APT and T-B models.
10.4 A Multifactor APT
We have assumed so far that only one systematic factor affects stock returns. This simplifying assumption is in fact too simplistic. We’ve noted that it is easy to think of several
factors driven by the business cycle that might affect stock returns: interest rate fluctuations, inflation rates, and so on. Presumably, exposure to any of these factors will affect a
stock’s risk and hence its expected return. We can derive a multifactor version of the APT
to accommodate these multiple sources of risk.
Suppose that we generalize the single-factor model expressed in Equation 10.1 to a
two-factor model:
Ri 5 E (Ri ) 1 bi1 F1 1 bi2 F2 1 ei
(10.10)
In Example 10.2, factor 1 was the departure of GDP growth from expectations, and factor 2
was the unanticipated change in interest rates. Each factor has zero expected value because
each measures the surprise in the systematic variable rather than the level of the variable.
Similarly, the firm-specific component of unexpected return, ei , also has zero expected
value. Extending such a two-factor model to any number of factors is straightforward.
We can now generalize the simple APT to a more general multifactor version. But first
we must introduce the concept of a factor portfolio, which is a well-diversified portfolio
constructed to have a beta of 1 on one of the factors and a beta of zero on any other factor.
We can think of a factor portfolio as a tracking portfolio. That is, the returns on such a portfolio track the evolution of particular sources of macroeconomic risk but are uncorrelated
with other sources of risk. It is possible to form such factor portfolios because we have a
large number of securities to choose from, and a relatively small number of factors. Factor
portfolios will serve as the benchmark portfolios for a multifactor security market line.
The multidimensional SML predicts that exposure to each risk factor contributes to the
security’s total risk premium by an amount equal to the factor beta times the risk premium
of the factor portfolio tracking that source of risk. We illustrate with an example.
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Example 10.4
Arbitrage Pricing Theory and Multifactor Models of Risk and Return
339
Multifactor SML
Suppose that the two factor portfolios, portfolios 1 and 2, have expected returns
E (r1) 5 10% and E (r2) 5 12%. Suppose further that the risk-free rate is 4%. The risk
premium on the first factor portfolio is 10% 2 4% 5 6%, whereas that on the second
factor portfolio is 12% 2 4% 5 8%.
Now consider a well-diversified portfolio, portfolio A, with beta on the first factor,
bA1 5 .5, and beta on the second factor, bA2 5 .75. The multifactor APT states that the
overall risk premium on this portfolio must equal the sum of the risk premiums required
as compensation for each source of systematic risk. The risk premium attributable to risk
factor 1 should be the portfolio’s exposure to factor 1, bA1, multiplied by the risk premium earned on the first factor portfolio, E (r1) 2 rf. Therefore, the portion of portfolio
A’s risk premium that is compensation for its exposure to the first factor is bA1 [E (r1) 2 rf] 5
.5(10% 2 4%) 5 3%, whereas the risk premium attributable to risk factor 2 is
bA2[E (r2) 2 rf] 5 .75(12% 2 4%) 5 6%. The total risk premium on the portfolio should
be 3% 1 6% 5 9% and the total return on the portfolio should be 4% 1 9% 5 13%.
To generalize the argument in Example 10.4, note that the factor exposures of any portfolio, P, are given by its betas, bP1 and bP2. A competing portfolio, Q, can be formed by
investing in factor portfolios with the following weights: bP1 in the first factor portfolio,
bP2 in the second factor portfolio, and 1 2 bP1 2 bP2 in T-bills. By construction, portfolio
Q will have betas equal to those of portfolio P and expected return of
E (rQ ) 5 bP1E (r1) 1 bP2 E (r2) 1 (1 2 bP1 2 bP2) rf
5 rf 1 bP1 3 E (r1) 2 rf 4 1 bP2 3 E (r2) 2 rf 4
(10.11)
Using the numbers in Example 10.4:
E (rQ ) 5 4 1 .5 3 (10 2 4) 1 .75 3 (12 2 4) 5 13%
Example 10.5
Mispricing and Arbitrage
Suppose that the expected return on portfolio A from Example 10.4 were 12% rather
than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from
the factor portfolios with the same betas as portfolio A. This requires weights of .5 on
the first factor portfolio, .75 on the second factor portfolio, and 2.25 on the risk-free
asset. This portfolio has exactly the same factor betas as portfolio A: It has a beta of .5 on
the first factor because of its .5 weight on the first factor portfolio, and a beta of .75
on the second factor. (The weight of 2.25 on risk-free T-bills does not affect the sensitivity to either factor.)
Now invest $1 in portfolio Q and sell (short) $1 in portfolio A. Your net investment is
zero, but your expected dollar profit is positive and equal to
$1 3 E(rQ) 2 $1 3 E(rA) 5 $1 3 .13 2 $1 3 .12 5 $.01
Moreover, your net position is riskless. Your exposure to each risk factor cancels out
because you are long $1 in portfolio Q and short $1 in portfolio A, and both of these
well-diversified portfolios have exactly the same factor betas. Thus, if portfolio A’s
expected return differs from that of portfolio Q’s, you can earn positive risk-free profits
on a zero-net-investment position. This is an arbitrage opportunity.
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Because portfolio Q in Example 10.5 has precisely the same exposures as portfolio A to
the two sources of risk, their expected returns also ought to be equal. So portfolio A also ought
to have an expected return of 13%. If it does not, then there will be an arbitrage opportunity.7
We conclude that any well-diversified portfolio with betas bP1 and bP2 must
have the return given in Equation 10.11 if arbitrage opportunities are to be precluded.
Equation 10.11 simply generalizes the one-factor SML.
Finally, the extension of the multifactor SML of Equation 10.11 to individual assets
is precisely the same as for the one-factor APT. Equation 10.11 cannot be satisfied by
every well-diversified portfolio unless it is satisfied approximately by individual securities.
Equation 10.11 thus represents the multifactor SML for an economy with multiple sources
of risk.
We pointed out earlier that one application of the CAPM is to provide “fair” rates
of return for regulated utilities. The multifactor APT can be used to the same ends. The
nearby box summarizes a study in which the
APT was applied to find the cost of capital
for regulated electric companies. Notice that
CONCEPT CHECK 10.3
empirical estimates for interest rate and inflation risk premiums in the box are negative, as
Using the factor portfolios of Example 10.4, find the equilibwe argued was reasonable in our discussion
rium rate of return on a portfolio with b1 5 .2 and b2 5 1.4.
of Example 10.2.
10.5 The Fama-French (FF) Three-Factor Model
The currently dominant approach to specifying factors as candidates for relevant sources
of systematic risk uses firm characteristics that seem on empirical grounds to proxy for
exposure to systematic risk. The factors chosen are variables that on past evidence seem to
predict average returns well and therefore may be capturing risk premiums. One example
of this approach is the Fama and French three-factor model and its variants, which have
come to dominate empirical research and industry applications:8
Rit 5 a i 1 biM RMt 1 biSMB SMBt 1 biHMLHMLt 1 eit
(10.12)
where
SMB 5 Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the
return on a portfolio of large stocks.
HML 5 High Minus Low, i.e., the return of a portfolio of stocks with a high
book-to-market ratio in excess of the return on a portfolio of stocks with a
low book-to-market ratio.
Note that in this model the market index does play a role and is expected to capture systematic risk originating from macroeconomic factors.
These two firm-characteristic variables are chosen because of long-standing observations that corporate capitalization (firm size) and book-to-market ratio predict deviations
7
The risk premium on portfolio A is 9% (more than the historical risk premium of the S&P 500) despite the fact
that its betas, which are both below 1, might seem defensive. This highlights another distinction between multifactor and single-factor models. Whereas a beta greater than 1 in a single-factor market is aggressive, we cannot
say in advance what would be aggressive or defensive in a multifactor economy where risk premiums depend on
the sum of the contributions of several factors.
8
Eugene F. Fama and Kenneth R. French, “Multifactor Explanations of Asset Pricing Anomalies,” Journal of
Finance 51 (1996), pp. 55–84.
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Elton, Gruber, and Mei* use the APT to derive the cost of
capital for electric utilities. They assume that the relevant
risk factors are unanticipated developments in the term
structure of interest rates, the level of interest rates, inflation rates, the business cycle (measured by GDP), foreign
exchange rates, and a summary measure they devise to
measure other macro factors.
Their first step is to estimate the risk premium associated with exposure to each risk source. They accomplish
this in a two-step strategy (which we will describe in considerable detail in Chapter 13):
1. Estimate “factor loadings” (i.e., betas) of a large sample
of firms. Regress returns of 100 randomly selected
stocks against the systematic risk factors. They use a
time-series regression for each stock (e.g., 60 months
of data), therefore estimating 100 regressions, one for
each stock.
2. Estimate the reward earned per unit of exposure to
each risk factor. For each month, regress the return
of each stock against the five betas estimated. The
coefficient on each beta is the extra average return
earned as beta increases, i.e., it is an estimate of the
risk premium for that risk factor from that month’s
data. These estimates are of course subject to sampling error. Therefore, average the risk premium estimates across the 12 months in each year. The average
response of return to risk is less subject to sampling
error.
The risk premiums are in the middle column of the table at
the top of the next column.
Notice that some risk premiums are negative. The
interpretation of this result is that risk premium should be
positive for risk factors you don’t want exposure to, but
negative for factors you do want exposure to. For example,
you should desire securities that have higher returns when
inflation increases and be willing to accept lower expected
returns on such securities; this shows up as a negative risk
premium.
Factor
Term structure
Interest rates
Exchange rates
Business cycle
Inflation
Other macro factors
Factor Risk
Premium
Factor Betas for
Niagara Mohawk
.425
2.051
2.049
.041
2.069
.530
1.0615
22.4167
1.3235
.1292
2.5220
.3046
Therefore, the expected return on any security should
be related to its factor betas as follows:
rf 1 .425 bterm struc 2 .051 bint rate
2.049 bex rate 1 .041 bbus cycle 2 .069 binflation 1 .530 bother
Finally, to obtain the cost of capital for a particular firm,
the authors estimate the firm’s betas against each source of
risk, multiply each factor beta by the “cost of factor risk”
from the table above, sum over all risk sources to obtain
the total risk premium, and add the risk-free rate.
For example, the beta estimates for Niagara Mohawk
appear in the last column of the table above. Therefore, its
cost of capital is
WORDS FROM THE STREET
Using the APT to Find Cost of Capital
Cost of capital 5 rf 1 .425 3 1.0615 2 .051(22.4167)
2.049(1.3235) 1 .041(.1292)
2.069(2.5220) 1 .530(.3046)
5 rf 1 .72
In other words, the monthly cost of capital for Niagara
Mohawk is .72% above the monthly risk-free rate. Its annualized risk premium is therefore .72% 3 12 5 8.64%.
*Edwin J. Elton, Martin J. Gruber, and Jianping Mei, “Cost of
Capital Using Arbitrage Pricing Theory: A Case Study of Nine New
York Utilities,” Financial Markets, Institutions, and Instruments
3 (August 1994), pp. 46–68.
of average stock returns from levels consistent with the CAPM. Fama and French justify this model on empirical grounds: While SMB and HML are not themselves obvious
candidates for relevant risk factors, the argument is that these variables may proxy for
yet-unknown more-fundamental variables. For example, Fama and French point out that
firms with high ratios of book-to-market value are more likely to be in financial distress
and that small stocks may be more sensitive to changes in business conditions. Thus, these
variables may capture sensitivity to risk factors in the macroeconomy. More evidence on
the Fama-French model appears in Chapter 13.
The problem with empirical approaches such as the Fama-French model, which use proxies for extramarket sources of risk, is that none of the factors in the proposed models can
be clearly identified as hedging a significant source of uncertainty. Black9 points out that
9
Fischer Black, “Beta and Return,” Journal of Portfolio Management 20 (1993), pp. 8–18.
341
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when researchers scan and rescan the database of security returns in search of explanatory
factors (an activity often called data-snooping), they may eventually uncover past “patterns”
that are due purely to chance. Black observes that return premiums to factors such as firm
size have proven to be inconsistent since first discovered. However, Fama and French have
shown that size and book-to-market ratios have predicted average returns in various time
periods and in markets all over the world, thus mitigating potential effects of data-snooping.
The firm-characteristic basis of the Fama-French factors raises the question of whether
they reflect a multi-index ICAPM based on extra-market hedging demands or just represent yet-unexplained anomalies, where firm characteristics are correlated with alpha values. This is an important distinction for the debate over the proper interpretation of the
model, because the validity of FF-style models may signify either a deviation from rational
equilibrium (as there is no rational reason to prefer one or another of these firm characteristics per se), or that firm characteristics identified as empirically associated with average
returns are correlated with other (yet unknown) risk factors.
The issue is still unresolved and is discussed in Chapter 13.
SUMMARY
1. Multifactor models seek to improve the explanatory power of single-factor models by explicitly
accounting for the various systematic components of security risk. These models use indicators
intended to capture a wide range of macroeconomic risk factors.
2. Once we allow for multiple risk factors, we conclude that the security market line also ought to be
multidimensional, with exposure to each risk factor contributing to the total risk premium of the
security.
3. A (risk-free) arbitrage opportunity arises when two or more security prices enable investors to
construct a zero-net-investment portfolio that will yield a sure profit. The presence of arbitrage
opportunities will generate a large volume of trades that puts pressure on security prices. This
pressure will continue until prices reach levels that preclude such arbitrage.
4. When securities are priced so that there are no risk-free arbitrage opportunities, we say that they
satisfy the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are
important because we expect them to hold in real-world markets.
5. Portfolios are called “well-diversified” if they include a large number of securities and the investment proportion in each is sufficiently small. The proportion of a security in a well-diversified
portfolio is small enough so that for all practical purposes a reasonable change in that security’s
rate of return will have a negligible effect on the portfolio’s rate of return.
6. In a single-factor security market, all well-diversified portfolios have to satisfy the expected
return–beta relationship of the CAPM to satisfy the no-arbitrage condition. If all well-diversified
portfolios satisfy the expected return–beta relationship, then individual securities also must satisfy this relationship, at least approximately.
7. The APT does not require the restrictive assumptions of the CAPM and its (unobservable) market
portfolio. The price of this generality is that the APT does not guarantee this relationship for all
securities at all times.
8. A multifactor APT generalizes the single-factor model to accommodate several sources of systematic risk. The multidimensional security market line predicts that exposure to each risk factor
contributes to the security’s total risk premium by an amount equal to the factor beta times the
risk premium of the factor portfolio that tracks that source of risk.
Related Web sites
for this chapter are
available at www.
mhhe.com/bkm
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9. A multifactor extension of the single-factor CAPM, the ICAPM, is a model of the risk–return
trade-off that predicts the same multidimensional security market line as the APT. The ICAPM
suggests that priced risk factors will be those sources of risk that lead to significant hedging
demand by a substantial fraction of investors.
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CHAPTER 10
Arbitrage Pricing Theory and Multifactor Models of Risk and Return
arbitrage pricing theory
arbitrage
Law of One Price
risk arbitrage
single-factor model
multifactor model
factor loading
factor beta
well-diversified portfolio
factor portfolio
343
KEY TERMS
KEY EQUATIONS
Single factor model: Ri 5 E(Ri) 1 b1F 1 ei
Multifactor model (here, 2 factors, F1 and F2): Ri 5 E(Ri) 1 b1F1 1 b2F2 1 ei
Single-index model: Ri 5 ai 1 bi RM 1 ei
Multifactor SML (here, 2 factors, labeled 1 and 2)
E(ri) 5 rf 1 b1 3 E(r1) 2 rf 4 1 b2 3 E(r2 ) 2 rf 4
5 rf 1 b1E(R1) 1 b2 E(R2)
1. Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial
production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a
beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12%. If industrial
production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised
estimate of the expected rate of return on the stock?
PROBLEM SETS
2. The APT itself does not provide guidance concerning the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Why, for example,
is industrial production a reasonable factor to test for a risk premium?
Basic
3. If the APT is to be a useful theory, the number of systematic factors in the economy must be
small. Why?
4. Suppose that there are two independent economic factors, F1 and F2. The risk-free rate is 6%,
and all stocks have independent firm-specific components with a standard deviation of 45%. The
following are well-diversified portfolios:
Portfolio
Beta on F1
Beta on F2
Expected Return
1.5
2.2
2.0
20.2
31%
27%
A
B
Intermediate
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where the risk premiums on the two factor portfolios are E(R1) and E(R2 )
What is the expected return–beta relationship in this economy?
5. Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio
E(r)
Beta
A
F
12%
6%
1.2
0.0
Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected
return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
6. Assume that both portfolios A and B are well diversified, that E(rA) 5 12%, and E(rB) 5 9%.
If the economy has only one factor, and bA 5 1.2, whereas bB 5 .8, what must be the riskfree rate?
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PART III
Equilibrium in Capital Markets
7. Assume that stock market returns have the market index as a common factor, and that all stocks
in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard
deviation of 30%.
Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 12%, and
the other half an alpha of 22%. Suppose the analyst buys $1 million of an equally weighted
portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of
the negative alpha stocks.
a. What is the expected profit (in dollars) and standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100
stocks?
8. Assume that security returns are generated by the single-index model,
Ri 5 a i 1 bi RM 1 ei
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where Ri is the excess return for security i and RM is the market’s excess return. The risk-free rate
is 2%. Suppose also that there are three securities A, B, and C, characterized by the following
data:
Security
bi
E (Ri )
s(ei )
A
B
C
0.8
1.0
1.2
10%
12
14
25%
10
20
a. If sM 5 20%, calculate the variance of returns of securities A, B, and C.
b. Now assume that there are an infinite number of assets with return characteristics identical to
those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities,
what will be the mean and variance of the portfolio’s excess returns? What about portfolios
composed only of type B or C stocks?
c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically.
9. The SML relationship states that the expected risk premium on a security in a one-factor model
must be directly proportional to the security’s beta. Suppose that this were not the case. For example, suppose that expected return rises more than proportionately with beta as in the figure below.
E(r)
B
C
A
β
a. How could you construct an arbitrage portfolio? (Hint: Consider combinations of portfolios A
and B, and compare the resultant portfolio to C.)
b. Some researchers have examined the relationship between average returns on diversified portfolios and the b and b2 of those portfolios. What should they have discovered about the effect
of b2 on portfolio return?
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CHAPTER 10
Arbitrage Pricing Theory and Multifactor Models of Risk and Return
345
10. Consider the following multifactor (APT) model of security returns for a particular stock.
Factor
Factor Beta
Inflation
Industrial production
Oil prices
Factor Risk Premium
1.2
0.5
0.3
6%
8
3
a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the market views the stock as fairly priced.
b. Suppose that the market expected the values for the three macro factors given in column 1
below, but that the actual values turn out as given in column 2. Calculate the revised expectations for the rate of return on the stock once the “surprises” become known.
Expected Rate of Change
Inflation
Industrial production
Oil prices
Actual Rate of Change
5%
3
2
4%
6
0
11. Suppose that the market can be described by the following three sources of systematic risk with
associated risk premiums.
Factor
Risk Premium
Industrial production (I)
Interest rates (R)
Consumer confidence (C)
6%
2
4
The return on a particular stock is generated according to the following equation:
r 5 15% 1 1.0 I 1 .5R 1 .75C 1 e
Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6%. Is the stock
over- or underpriced? Explain.
12. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh
insights concerning the firm’s cost of capital. She decides that this would be a good opportunity
to try out the new material on the APT that she learned last semester. She decides that three
promising factors would be (i) the return on a broad-based index such as the S&P 500; (ii) the
level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and
(iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta
of Pork Products against each of these factors by using a multiple regression and to estimate the
risk premium associated with each exposure factor. Comment on Jennifer’s choice of factors.
Which are most promising with respect to the likely impact on her firm’s cost of capital? Can
you suggest improvements to her specification?
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Factor
Use the following information to Answer Problems 13–16:
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The
only model that Orb’s senior management has promoted in the past is the capital asset pricing model
(CAPM). Now Orb’s management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk
premium for real GDP is 8% while the factor risk premium for inflation is 2%. He estimates for
Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively.
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PART III
Equilibrium in Capital Markets
Using his APT results, he computes the expected return of the fund. For comparison purposes, he
then uses fundamental analysis to also compute the expected return of Orb’s High Growth Fund.
McCracken finds that the two estimates of the Orb High Growth Fund’s expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of
Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the
expected return is 8.5% above the risk-free rate. McCracken then applies the APT model to the Large
Cap Fund. He finds that the sensitivities to real GDP and inflation are .75 and 1.25, respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to compose a portfolio that has a unit
sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT
estimates for the High Growth Fund and the Large Cap Fund. He then computes the sensitivities for
a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken
will use his APT results for these three funds to accomplish the task of creating a portfolio with a
unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles
says such a GDP Fund would be good for clients who are retirees who live off the steady income of
their investments. McCracken says that the fund would be a good choice if upcoming supply side
macroeconomic policies of the government are successful.
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13. According to the APT, if the risk-free rate is 4%, what should be McCracken’s estimate of the
expected return of Orb’s High Growth Fund?
14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information
Kwon provides, is an arbitrage opportunity available?
15. The GDP Fund composed from the other three funds would have a weight in Utility Fund equal
to (a) 22.2; (b) 23.2; or (c) .3.
16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund
would be appropriate:
a. McCracken was correct and Stiles was wrong.
b. Both were correct.
c. Stiles was correct and McCracken was wrong.
Challenge
17. Assume a universe of n (large) securities for which the largest residual variance is not larger
than ns2M. Construct as many different weighting schemes as you can that generate welldiversified portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT
security market line:
a. For a single-factor market.
b. For a multifactor market.
19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor.
a. Explain why.
b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit
of the merged company. Would you expect the stock market behavior of the merged firm to
differ from that of a portfolio of the two previously independent firms? How does the merger
affect market capitalization? What is the prediction of the Fama-French model for the risk
premium on the combined firm? Do we see here a flaw in the FF model?
1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced
securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing
CAPM and APT, the consultant made the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio.
b. Neither the CAPM nor APT assumes normally distributed security returns.
c. The CAPM assumes that one specific factor explains security returns but APT does not.
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Arbitrage Pricing Theory and Multifactor Models of Risk and Return
347
State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each incorrect argument, why the argument is incorrect.
2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%.
Portfolio
X
Y
Expected Return
16%
12
Beta
1.00
0.25
In this situation you would conclude that portfolios X and Y:
a.
b.
c.
d.
Are in equilibrium.
Offer an arbitrage opportunity.
Are both underpriced.
Are both fairly priced.
a.
b.
c.
d.
The expected return of the portfolio equals zero.
The capital market line is tangent to the opportunity set.
The Law of One Price remains unviolated.
A risk-free arbitrage opportunity exists.
4. According to the theory of arbitrage:
a.
b.
c.
d.
High-beta stocks are consistently overpriced.
Low-beta stocks are consistently overpriced.
Positive alpha investment opportunities will quickly disappear.
Rational investors will pursue arbitrage consistent with their risk tolerance.
5. The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing
model (CAPM) because the APT:
a.
b.
c.
d.
Places more emphasis on market risk.
Minimizes the importance of diversification.
Recognizes multiple unsystematic risk factors.
Recognizes multiple systematic risk factors.
6. An investor takes as large a position as possible when an equilibrium price relationship is violated. This is an example of:
a.
b.
c.
d.
A dominance argument.
The mean-variance efficient frontier.
Arbitrage activity.
The capital asset pricing model.
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3. A zero-investment portfolio with a positive alpha could arise if:
7. The feature of the general version of the arbitrage pricing theory (APT) that offers the greatest
potential advantage over the simple CAPM is the:
a. Identification of anticipated changes in production, inflation, and term structure of interest
rates as key factors explaining the risk–return relationship.
b. Superior measurement of the risk-free rate of return over historical time periods.
c. Variability of coefficients of sensitivity to the APT factors for a given asset over time.
d. Use of several factors instead of a single market index to explain the risk–return relationship.
8. In contrast to the capital asset pricing model, arbitrage pricing theory:
a.
b.
c.
d.
Requires that markets be in equilibrium.
Uses risk premiums based on micro variables.
Specifies the number and identifies specific factors that determine expected returns.
Does not require the restrictive assumptions concerning the market portfolio.
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348
PART III
Equilibrium in Capital Markets
E-INVESTMENTS EXERCISES
One of the factors in the APT model specified by Chen, Roll, and Ross is the percent change
in unanticipated inflation. Who gains and who loses when inflation changes? Go to
http://hussmanfunds.com/rsi/infsurprises.htm to see a graph of the Inflation Surprise
Index and Economists’ Inflation Forecasts.
SOLUTIONS TO CONCEPT CHECKS
1. The GDP beta is 1.2 and GDP growth is 1% better than previously expected. So you will increase
your forecast for the stock return by 1.2 3 1% 5 1.2%. The revised forecast is for an 11.2%
return.
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2. a. This portfolio is not well diversified. The weight on the first security does not decline as n
increases. Regardless of how much diversification there is in the rest of the portfolio, you will
not shed the firm-specific risk of this security.
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b. This portfolio is well diversified. Even though some stocks have three times the weight of
other stocks (1.5/n versus .5/n), the weight on all stocks approaches zero as n increases. The
impact of any individual stock’s firm-specific risk will approach zero as n becomes ever larger.
3. The equilibrium return is E (r) 5 rf 1 bP1[E (r1) 2 rf ] 1 bP2 [E (r2) 2 rf ]. Using the data in
Example 10.4:
E(r) 5 4 1 .2 3 (10 2 4) 1 1.4 3 (12 2 4) 5 16.4%
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CHAPTER FOURTEEN
Bond Prices and Yields
bod61671_ch14_445-486.indd 445
for our analysis of the universe of potential
investment vehicles.
The bond is the basic debt security, and
this chapter starts with an overview of the
universe of bond markets, including Treasury,
corporate, and international bonds. We turn
next to bond pricing, showing how bond
prices are set in accordance with market
interest rates and why bond prices change
with those rates. Given this background, we
can compare the myriad measures of bond
returns such as yield to maturity, yield to call,
holding-period return, and realized compound rate of return. We show how bond
prices evolve over time, discuss certain tax
rules that apply to debt securities, and show
how to calculate after-tax returns. Finally, we
consider the impact of default or credit risk
on bond pricing and look at the determinants
of credit risk and the default premium built
into bond yields. Credit risk is central to both
collateralized debt obligations and credit
default swaps, so we examine these instruments as well.
PART IV
IN THE PREVIOUS chapters on risk and return
relationships, we treated securities at a high
level of abstraction. We assumed implicitly
that a prior, detailed analysis of each security
already had been performed, and that its risk
and return features had been assessed.
We turn now to specific analyses of particular security markets. We examine valuation
principles, determinants of risk and return,
and portfolio strategies commonly used
within and across the various markets.
We begin by analyzing debt securities. A
debt security is a claim on a specified periodic
stream of income. Debt securities are often
called fixed-income securities because they
promise either a fixed stream of income or
one that is determined according to a specified formula. These securities have the advantage of being relatively easy to understand
because the payment formulas are specified
in advance. Uncertainty about their cash flows
is minimal as long as the issuer of the security is sufficiently creditworthy. That makes
these securities a convenient starting point
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PART IV
Fixed-Income Securities
14.1 Bond Characteristics
A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues (i.e., sells) a bond to the lender for some amount of cash; the bond is the
“IOU” of the borrower. The arrangement obligates the issuer to make specified payments
to the bondholder on specified dates. A typical coupon bond obligates the issuer to make
semiannual payments of interest to the bondholder for the life of the bond. These are called
coupon payments because in precomputer days, most bonds had coupons that investors
would clip off and present to claim the interest payment. When the bond matures, the issuer
repays the debt by paying the bond’s par value (equivalently, its face value). The coupon
rate of the bond determines the interest payment: The annual payment is the coupon rate
times the bond’s par value. The coupon rate, maturity date, and par value of the bond are
part of the bond indenture, which is the contract between the issuer and the bondholder.
To illustrate, a bond with par value of $1,000 and coupon rate of 8% might be sold to a
buyer for $1,000. The bondholder is then entitled to a payment of 8% of $1,000, or $80 per
year, for the stated life of the bond, say, 30 years. The $80 payment typically comes in two
semiannual installments of $40 each. At the end of the 30-year life of the bond, the issuer
also pays the $1,000 par value to the bondholder.
Bonds usually are issued with coupon rates set just high enough to induce investors to
pay par value to buy the bond. Sometimes, however, zero-coupon bonds are issued that
make no coupon payments. In this case, investors receive par value at the maturity date but
receive no interest payments until then: The bond has a coupon rate of zero. These bonds
are issued at prices considerably below par value, and the investor’s return comes solely
from the difference between issue price and
the payment of par value at maturity. We will
return to these bonds later.
U.S. Treasury Quotes
ASKED
CHANGE YIELD (%)
MATURITY
COUPON
BID
ASKED
Jun 15 13
1.125
100.8203
100.8281
0.0078
0.177
Jan 15 15
0.250
100.0391
100.0469
0.0547
0.231
Jun 30 16
1.500
104.1563
104.1875
0.1016
0.421
Jul 31 18
Nov 15 18
2.250
108.4922
108.5391
0.0938
0.790
9.000
150.3750
150.4219
0.1719
0.770
Feb 15 21
7.875
154.2344
154.3281
0.1875
1.172
Feb 15 26
6.000
149.1484
149.2266
0.1484
1.869
May 15 30
6.250
160.2190
161.0000
0.0781
2.117
Feb 15 36
4.500
138.4063
138.4844
0.1641
2.362
May 15 42
3.000
108.8047
108.8672
0.1328
2.572
Figure 14.1 Prices and yields of U.S. Treasury bonds
Source: The Wall Street Journal Online, July 31, 2012. Reprinted by
permission of Dow Jones & Company, Inc. © 2012 Dow Jones & Company.
All Rights Reserved Worldwide.
Treasury Bonds and Notes
Figure 14.1 is an excerpt from the listing of
Treasury issues. Treasury notes are issued
with original maturities ranging between 1
and 10 years, while Treasury bonds are issued
with maturities ranging from 10 to 30 years.
Both bonds and notes may be purchased
directly from the Treasury in denominations
of only $100, but denominations of $1,000
are far more common. Both make semiannual
coupon payments.
The highlighted bond in Figure 14.1
matures on July 31, 2018. Its coupon rate is
2.25%. Par value typically is $1,000; thus the
bond pays interest of $22.50 per year in two
semiannual payments of $11.25. Payments
are made in January and July of each year.
Although bonds usually are sold in denominations of $1,000, the bid and ask prices
are quoted as a percentage of par value.1
1
Recall that the bid price is the price at which you can sell the bond to a dealer. The ask price, which is slightly
higher, is the price at which you can buy the bond from a dealer.
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CHAPTER 14
Bond Prices and Yields
447
Therefore, the ask price is 108.5391% of par, or $1,085.391. The minimum price
increment, or tick size, in The Wall Street Journal listing is 1/128, so this bond may also be
viewed as selling for 108 69⁄128 percent of par value.2
The last column, labeled “Ask yield,” is the yield to maturity on the bond based on the
ask price. The yield to maturity is a measure of the average rate of return to an investor
who purchases the bond for the ask price and holds it until its maturity date. We will have
much to say about yield to maturity below.
Accrued Interest and Quoted Bond Prices The bond prices that you see
quoted in the financial pages are not actually the prices that investors pay for the bond.
This is because the quoted price does not include the interest that accrues between coupon
payment dates.
If a bond is purchased between coupon payments, the buyer must pay the seller for
accrued interest, the prorated share of the upcoming semiannual coupon. For example, if
30 days have passed since the last coupon payment, and there are 182 days in the semiannual coupon period, the seller is entitled to a payment of accrued interest of 30/182 of the
semiannual coupon. The sale, or invoice, price of the bond would equal the stated price
(sometimes called the flat price) plus the accrued interest.
In general, the formula for the amount of accrued interest between two dates is
Accrued interest 5
Example 14.1
Annual coupon payment
Days since last coupon payment
3
2
Days separating coupon payments
Accrued Interest
Suppose that the coupon rate is 8%. Then the annual coupon is $80 and the semiannual
coupon payment is $40. Because 30 days have passed since the last coupon payment,
the accrued interest on the bond is $40 3 (30/182) 5 $6.59. If the quoted price of the
bond is $990, then the invoice price will be $990 1 $6.59 5 $996.59.
The practice of quoting bond prices net of accrued interest explains why the price of
a maturing bond is listed at $1,000 rather than $1,000 plus one coupon payment. A purchaser of an 8% coupon bond 1 day before the bond’s maturity would receive $1,040 (par
value plus semiannual interest) on the following day and so should be willing to pay a total
price of $1,040 for the bond. The bond price is quoted net of accrued interest in the financial pages and thus appears as $1,000.3
Corporate Bonds
Like the government, corporations borrow money by issuing bonds. Figure 14.2 is a
sample of listings for a few actively traded corporate bonds. Although some bonds trade
2
Bonds traded on formal exchanges are subject to minimum tick sizes set by the exchange. For example, the
minimum price increment on the 2-year Treasury bond futures contract (traded on the Chicago Board of Trade) is
1/128, although longer-term T-bonds have larger tick sizes. Private traders can negotiate their own tick size. For
example, one can find price quotes on Bloomberg screens with tick sizes as low as 1/256.
3
In contrast to bonds, stocks do not trade at flat prices with adjustments for “accrued dividends.” Whoever owns
the stock when it goes “ex-dividend” receives the entire dividend payment, and the stock price reflects the value
of the upcoming dividend. The price therefore typically falls by about the amount of the dividend on the “ex-day.”
There is no need to differentiate between reported and invoice prices for stocks.
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PART IV
Fixed-Income Securities
MOODY'S/S&P/
FITCH
HIGH
LOW
LAST
May 2013
A2 /A+ /AA–
103.7060
103.4240
103.6590
1.3090
0.5662
5.750%
Jan 2022
A3 /A– /A
111.8040
100.3290
109.4040
0.2034
4.5187
BUD3876843
2.500%
Jul 2022
A3 /A /A
104.1200
101.8410
102.1360
0.4110
2.2590
JPMORGAN CHASE
& CO
JPM.KPG
4.750%
May 2013
A2 /A /A+
103.1790
102.6250
103.1790
0.2720
0.4663
HOUSEHOLD FIN
CORP
HBC.IGQ
4.750%
Jul 2013
Baa1 /A /AA–
105.3220
103.0760
103.3780
0.2280
1.1635
ANHEUSER BUSCH
INBEV WORLDWIDE
BUD3876840
1.375%
Jul 2017
A3 /A /A
101.3000
100.9150
101.0460
–0.0820
1.1569
ISSUER NAME
SYMBOL
COUPON MATURITY
WACHOVIA CORP
GLOBAL MTN
WFC.PO
5.500%
GOLDMAN SACHS
GROUP INC
GD.AEH
ANHEUSER BUSCH
INBEV WORLDWIDE
CHANGE YIELD %
Figure 14.2 Listing of corporate bonds
Source: FINRA (Financial Industry Regulatory Authority), August 1, 2012.
electronically on the NYSE Bonds platform, most bonds are traded over-the-counter in
a network of bond dealers linked by a computer quotation system. In practice, the bond
market can be quite “thin,” with few investors interested in trading a particular issue at any
particular time.
The bond listings in Figure 14.2 include the coupon, maturity, price, and yield to maturity of each bond. The “rating” column is the estimation of bond safety given by the three
major bond-rating agencies—Moody’s, Standard & Poor’s, and Fitch. Bonds with gradations of A ratings are safer than those with B ratings or below. As a general rule, safer
bonds with higher ratings promise lower yields to maturity than other bonds with similar
maturities. We will return to this topic toward the end of the chapter.
Call Provisions on Corporate Bonds Some corporate bonds are issued with call
provisions allowing the issuer to repurchase the bond at a specified call price before the
maturity date. For example, if a company issues a bond with a high coupon rate when market
interest rates are high, and interest rates later fall, the firm might like to retire the high-coupon
debt and issue new bonds at a lower coupon rate to reduce interest payments. This is called
refunding. Callable bonds typically come with a period of call protection, an initial time during which the bonds are not callable. Such bonds are referred to as deferred callable bonds.
The option to call the bond is valuable to the firm,
allowing it to buy back the bonds and refinance at lower
interest rates when market rates fall. Of course, the firm’s
CONCEPT CHECK 14.1
benefit is the bondholder’s burden. Holders of called
bonds must forfeit their bonds for the call price, thereby
Suppose that Verizon issues two bonds with
giving up the attractive coupon rate on their original
identical coupon rates and maturity dates. One
investment. To compensate investors for this risk, callbond is callable, however, whereas the other is
not. Which bond will sell at a higher price?
able bonds are issued with higher coupons and promised
yields to maturity than noncallable bonds.
Convertible Bonds Convertible bonds give bondholders an option to exchange
each bond for a specified number of shares of common stock of the firm. The conversion
ratio is the number of shares for which each bond may be exchanged. Suppose a convertible bond is issued at par value of $1,000 and is convertible into 40 shares of a firm’s
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stock. The current stock price is $20 per share, so the option to convert is not profitable
now. Should the stock price later rise to $30, however, each bond may be converted profitably into $1,200 worth of stock. The market conversion value is the current value of
the shares for which the bonds may be exchanged. At the $20 stock price, for example,
the bond’s conversion value is $800. The conversion premium is the excess of the bond
value over its conversion value. If the bond were selling currently for $950, its premium
would be $150.
Convertible bondholders benefit from price appreciation of the company’s stock.
Again, this benefit comes at a price: Convertible bonds offer lower coupon rates and stated
or promised yields to maturity than do nonconvertible bonds. However, the actual return
on the convertible bond may exceed the stated yield to maturity if the option to convert
becomes profitable.
We discuss convertible and callable bonds further in Chapter 20.
Puttable Bonds While the callable bond gives the issuer the option to extend or retire
the bond at the call date, the extendable or put bond gives this option to the bondholder.
If the bond’s coupon rate exceeds current market yields, for instance, the bondholder will
choose to extend the bond’s life. If the bond’s coupon rate is too low, it will be optimal not
to extend; the bondholder instead reclaims principal, which can be invested at current yields.
Floating-Rate Bonds Floating-rate bonds make interest payments that are tied to
some measure of current market rates. For example, the rate might be adjusted annually
to the current T-bill rate plus 2%. If the 1-year T-bill rate at the adjustment date is 4%, the
bond’s coupon rate over the next year would then be 6%. This arrangement means that the
bond always pays approximately current market rates.
The major risk involved in floaters has to do with changes in the firm’s financial
strength. The yield spread is fixed over the life of the security, which may be many years. If
the financial health of the firm deteriorates, then investors will demand a greater yield premium than is offered by the security. In this case, the price of the bond will fall. Although
the coupon rate on floaters adjusts to changes in the general level of market interest rates,
it does not adjust to changes in the financial condition of the firm.
Preferred Stock
Although preferred stock strictly speaking is considered to be equity, it often is included
in the fixed-income universe. This is because, like bonds, preferred stock promises to pay
a specified stream of dividends. However, unlike bonds, the failure to pay the promised
dividend does not result in corporate bankruptcy. Instead, the dividends owed simply
cumulate, and the common stockholders may not receive any dividends until the preferred
stockholders have been paid in full. In the event of bankruptcy, preferred stockholders’
claims to the firm’s assets have lower priority than those of bondholders but higher priority
than those of common stockholders.
Preferred stock commonly pays a fixed dividend. Therefore, it is in effect a perpetuity,
providing a level cash flow indefinitely. In contrast, floating-rate preferred stock is much
like floating-rate bonds. The dividend rate is linked to a measure of current market interest
rates and is adjusted at regular intervals.
Unlike interest payments on bonds, dividends on preferred stock are not considered
tax-deductible expenses to the firm. This reduces their attractiveness as a source of capital
to issuing firms. On the other hand, there is an offsetting tax advantage to preferred stock.
When one corporation buys the preferred stock of another corporation, it pays taxes on
only 30% of the dividends received. For example, if the firm’s tax bracket is 35%, and
it receives $10,000 in preferred-dividend payments, it will pay taxes on only $3,000 of
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that income: Total taxes owed on the income will be .35 3 $3,000 5 $1,050. The firm’s
effective tax rate on preferred dividends is therefore only .30 3 35% 5 10.5%. Given this
tax rule, it is not surprising that most preferred stock is held by corporations.
Preferred stock rarely gives its holders full voting privileges in the firm. However, if
the preferred dividend is skipped, the preferred stockholders may then be provided some
voting power.
Other Domestic Issuers
There are, of course, several issuers of bonds in addition to the Treasury and private corporations. For example, state and local governments issue municipal bonds. The outstanding
feature of these is that interest payments are tax-free. We examined municipal bonds, the
value of the tax exemption, and the equivalent taxable yield of these bonds in Chapter 2.
Government agencies such as the Federal Home Loan Bank Board, the Farm Credit
agencies, and the mortgage pass-through agencies Ginnie Mae, Fannie Mae, and Freddie
Mac also issue considerable amounts of bonds. These too were reviewed in Chapter 2.
International Bonds
International bonds are commonly divided into two categories, foreign bonds and
Eurobonds. Foreign bonds are issued by a borrower from a country other than the one in
which the bond is sold. The bond is denominated in the currency of the country in which
it is marketed. For example, if a German firm sells a dollar-denominated bond in the
United States, the bond is considered a foreign bond. These bonds are given colorful names
based on the countries in which they are marketed. For example, foreign bonds sold in the
United States are called Yankee bonds. Like other bonds sold in the United States, they are
registered with the Securities and Exchange Commission. Yen-denominated bonds sold
in Japan by non-Japanese issuers are called Samurai bonds. British pound-denominated
foreign bonds sold in the United Kingdom are called bulldog bonds.
In contrast to foreign bonds, Eurobonds are denominated in one currency, usually that
of the issuer, but sold in other national markets. For example, the Eurodollar market refers
to dollar-denominated bonds sold outside the United States (not just in Europe), although
London is the largest market for Eurodollar bonds. Because the Eurodollar market falls
outside U.S. jurisdiction, these bonds are not regulated by U.S. federal agencies. Similarly,
Euroyen bonds are yen-denominated bonds selling outside Japan, Eurosterling bonds are
pound-denominated Eurobonds selling outside the United Kingdom, and so on.
Innovation in the Bond Market
Issuers constantly develop innovative bonds with unusual features; these issues illustrate
that bond design can be extremely flexible. Here are examples of some novel bonds. They
should give you a sense of the potential variety in security design.
Inverse Floaters These are similar to the floating-rate bonds we described earlier,
except that the coupon rate on these bonds falls when the general level of interest rates
rises. Investors in these bonds suffer doubly when rates rise. Not only does the present
value of each dollar of cash flow from the bond fall as the discount rate rises, but the level
of those cash flows falls as well. Of course, investors in these bonds benefit doubly when
rates fall.
Asset-Backed Bonds Miramax once issued bonds with coupon rates tied to the financial performance of Pulp Fiction and other films. Domino’s Pizza has issued bonds with
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payments backed by revenues from its pizza franchises. These are examples of asset-backed
securities. The income from a specified group of assets is used to service the debt. More
conventional asset-backed securities are mortgage-backed securities or securities backed by
auto or credit card loans, as we discussed in Chapter 2.
Catastrophe Bonds Oriental Land Company, which manages Tokyo Disneyland,
issued a bond in 1999 with a final payment that depended on whether there had been an
earthquake near the park. More recently, FIFA (the Fédération Internationale de Football Association) issued catastrophe bonds with payments that would be halted if terrorism forced the cancellation of the 2006 World Cup. These bonds are a way to transfer
“catastrophe risk” from the firm to the capital markets. Investors in these bonds receive
compensation for taking on the risk in the form of higher coupon rates. But in the event
of a catastrophe, the bondholders will give up all or part of their investments. “Disaster”
can be defined by total insured losses or by criteria such as wind speed in a hurricane or
Richter level in an earthquake. Issuance of catastrophe bonds has grown in recent years
as insurers have sought ways to spread their risks across a wider spectrum of the capital
market.
Indexed Bonds Indexed bonds make payments that are tied to a general price index
or the price of a particular commodity. For example, Mexico has issued bonds with payments that depend on the price of oil. Some bonds are indexed to the general price level.
The United States Treasury started issuing such inflation-indexed bonds in January 1997.
They are called Treasury Inflation Protected Securities (TIPS). By tying the par value of
the bond to the general level of prices, coupon payments as well as the final repayment of
par value on these bonds increase in direct proportion to the Consumer Price Index. Therefore, the interest rate on these bonds is a risk-free real rate.
To illustrate how TIPS work, consider a newly issued bond with a 3-year maturity, par
value of $1,000, and a coupon rate of 4%. For simplicity, we will assume the bond makes
annual coupon payments. Assume that inflation turns out to be 2%, 3%, and 1% in the
next 3 years. Table 14.1 shows how the bond cash flows will be calculated. The first payment comes at the end of the first year, at t 5 1. Because inflation over the year was 2%,
the par value of the bond increases from $1,000 to $1,020; because the coupon rate is 4%,
the coupon payment is 4% of this amount, or $40.80. Notice that par value increases by the
inflation rate, and because the coupon payments are 4% of par, they too increase in proportion to the general price level. Therefore, the cash flows paid by the bond are fixed in real
terms. When the bond matures, the investor receives a final coupon payment of $42.44 plus
the (price-level-indexed) repayment of principal, $1,061.11.4
Inflation in Year
Coupon
Principal
Time
Just Ended
Par Value Payment 1 Repayment 5 Total Payment
0
1
2
3
2%
3
1
$1,000.00
1,020.00
1,050.60
1,061.11
$40.80
42.02
42.44
$
0
0
1,061.11
$
40.80
42.02
1,103.55
Table 14.1
Principal and interest
payments for a Treasury
Inflation Protected
Security
4
By the way, total nominal income (i.e., coupon plus that year’s increase in principal) is treated as taxable income
in each year.
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The nominal rate of return on the bond in the first year is
Nominal return 5
Interest 1 Price appreciation
40.80 1 20
5
5 6.08%
Initial price
1,000
The real rate of return is precisely the 4% real yield on the bond:
Real return 5
1 1 Nominal return
1.0608
215
2 1 5 .04, or 4%
1 1 Inflation
1.02
One can show in a similar manner (see Problem 18 in the end-of-chapter problems) that
the rate of return in each of the 3 years is 4% as long as the real yield on the bond remains
constant. If real yields do change, then there will be capital gains or losses on the bond. In
mid-2013, the real yield on long-term TIPS bonds was less than 0.5%.
14.2 Bond Pricing
Because a bond’s coupon and principal repayments all occur months or years in the future,
the price an investor would be willing to pay for a claim to those payments depends on the
value of dollars to be received in the future compared to dollars in hand today. This “present value” calculation depends in turn on market interest rates. As we saw in Chapter 5,
the nominal risk-free interest rate equals the sum of (1) a real risk-free rate of return and (2)
a premium above the real rate to compensate for expected inflation. In addition, because
most bonds are not riskless, the discount rate will embody an additional premium that
reflects bond-specific characteristics such as default risk, liquidity, tax attributes, call risk,
and so on.
We simplify for now by assuming there is one interest rate that is appropriate for
discounting cash flows of any maturity, but we can relax this assumption easily. In practice, there may be different discount rates for cash flows accruing in different periods. For
the time being, however, we ignore this refinement.
To value a security, we discount its expected cash flows by the appropriate discount
rate. The cash flows from a bond consist of coupon payments until the maturity date plus
the final payment of par value. Therefore,
Bond value 5 Present value of coupons 1 Present value of par value
If we call the maturity date T and call the interest rate r, the bond value can be written as
T Coupon
Par value
Bond value 5 a
t 1
(
)
(1 1 r)T
1
1
r
t51
(14.1)
The summation sign in Equation 14.1 directs us to add the present value of each coupon
payment; each coupon is discounted based on the time until it will be paid. The first term
on the right-hand side of Equation 14.1 is the present value of an annuity. The second term
is the present value of a single amount, the final payment of the bond’s par value.
You may recall from an introductory finance class that the present value of a $1 annuity
1
1
R. We call this
that lasts for T periods when the interest rate equals r is B1 2
r
(1 1 r)T
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CHAPTER 14
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453
1
(1 1 r)T
the PV factor, that is, the present value of a single payment of $1 to be received in T periods. Therefore, we can write the price of the bond as
expression the T-period annuity factor for an interest rate of r.5 Similarly, we call
Price 5 Coupon 3
1
1
1
B1 2
R 1 Par value 3
r
(1 1 r)T
(1 1 r ) T
(14.2)
5 Coupon 3 Annuity factor(r, T) 1 Par value 3 PV factor(r, T)
Example 14.2
Bond Pricing
We discussed earlier an 8% coupon, 30-year maturity bond with par value of $1,000
paying 60 semiannual coupon payments of $40 each. Suppose that the interest rate is 8%
annually, or r 5 4% per 6-month period. Then the value of the bond can be written as
60
$1,000
$40
Price 5 a
1
t
(14.3)
(1.04)60
t51 (1.04)
5 $40 3 Annuity factor(4%, 60) 1 $1,000 3 PV factor(4%, 60)
It is easy to confirm that the present value of the bond’s 60 semiannual coupon payments of $40 each is $904.94 and that the $1,000 final payment of par value has a
present value of $95.06, for a total bond value of $1,000. You can calculate the value
directly from Equation 14.2, perform these calculations on any financial calculator (see
Example 14.3 below), use a spreadsheet program (see the Excel Applications box), or use
a set of present value tables.
In this example, the coupon rate equals the market interest rate, and the bond price
equals par value. If the interest rate were not equal to the bond’s coupon rate, the bond
would not sell at par value. For example, if the interest rate were to rise to 10% (5% per
6 months), the bond’s price would fall by $189.29 to $810.71, as follows:
$40 3 Annuity factor(5%, 60) 1 $1,000 3 PV factor(5%, 60)
5 $757.17 1 $53.54 5 $810.71
At a higher interest rate, the present value of the payments to be received by the
bondholder is lower. Therefore, bond prices fall as market interest rates rise. This illustrates a crucial general rule in bond valuation.6
5
Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting T periods can be
viewed as equivalent to a perpetuity whose first payment comes at the end of the current period less another perpetuity whose first payment comes at the end of the (T 1 1)st period. The immediate perpetuity net of the delayed
perpetuity provides exactly T payments. We know that the value of a $1 per period perpetuity is $1/r. Therefore,
1
1
the present value of the delayed perpetuity is $1/r discounted for T additional periods, or 3
.
r
(1 1 r)T
The present value of the annuity is the present value of the first perpetuity minus the present value of the delayed
1
1
perpetuity, or B 1 2
R.
r
(1 1 r)T
6
Here is a trap to avoid. You should not confuse the bond’s coupon rate, which determines the interest paid to the
bondholder, with the market interest rate. Once a bond is issued, its coupon rate is fixed. When the market interest
rate increases, investors discount any fixed payments at a higher discount rate, which implies that present values
and bond prices fall.
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PART IV
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Bond prices are tedious to calculate without a spreadsheet or a financial calculator,
but they are easy to calculate with either. Financial calculators designed with present and
future value formulas already programmed can greatly simplify calculations of the sort we
just encountered in Example 14.2. The basic financial calculator uses five keys that correspond to the inputs for time-value-of-money problems such as bond pricing:
1. n is the number of time periods. In the case of a bond, n equals the number of
periods until the bond matures. If the bond makes semiannual payments, n is the
number of half-year periods or, equivalently, the number of semiannual coupon
payments. For example, if the bond has 10 years until maturity, you would enter
20 for n, since each payment period is one-half year.
2. i is the interest rate per period, expressed as a percentage (not as a decimal). For
example, if the interest rate is 6%, you would enter 6, not .06.
3. PV is the present value. Many calculators require that PV be entered as a negative
number, in recognition of the fact that purchase of the bond is a cash outflow, while
the receipt of coupon payments and face value are cash inflows.
4. FV is the future value or face value of the bond. In general, FV is interpreted as
a one-time future payment of a cash flow, which, for bonds, is the face (i.e., par)
value.
5. PMT is the amount of any recurring payment. For coupon bonds, PMT is the coupon payment; for zero-coupon bonds, PMT will be zero.
Given any four of these inputs, the calculator will solve for the fifth. We can illustrate with
the bond in Example 14.2.
Example 14.3
Bond Pricing on a Financial Calculator
To find the bond’s price when the annual market interest rate is 8%, you would enter
these inputs (in any order):
n
i
FV
PMT
60
4
1,000
40
The bond has a maturity of 30 years, so it makes 60 semiannual payments.
The semiannual market interest rate is 4%.
The bond will provide a one-time cash flow of $1,000 when
it matures.
Each semiannual coupon payment is $40.
On most calculators, you now punch the “compute” key (labeled COMP or CPT) and
then enter PV to obtain the bond price, that is the present value today of the bond’s
cash flows. If you do this, you should find a value of 21,000. The negative sign signifies
that while the investor receives cash flows from the bond, the price paid to buy the bond
is a cash outflow, or a negative cash flow.
If you want to find the value of the bond when the interest rate is 10% (the second
part of Example 14.2), just enter 5% for the semiannual interest rate (type “5” and
then “i”), and when you compute PV, you will find that it is 2810.71.
Figure 14.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates,
including 8%, at which the bond sells at par, and 10%, at which it sells for $810.71. The
negative slope illustrates the inverse relationship between prices and yields. The shape
of the curve in Figure 14.3 implies that an increase in the interest rate results in a price
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455
decline that is smaller than the price gain resulting from a decrease of equal magnitude
in the interest rate. This property of bond prices is called convexity because of the convex
shape of the bond price curve. This curvature reflects the fact that progressive increases in
the interest rate result in progressively smaller reductions in the bond price.7 Therefore, the
price curve becomes flatter at higher interest rates. We return to convexity in Chapter 16.
CONCEPT CHECK
14.2
Calculate the price of the 30-year, 8% coupon bond for a market interest rate of 3% per half-year.
Compare the capital gains for the interest rate decline to the losses incurred when the rate increases to 5%.
Bond Price ($)
Corporate bonds typically are issued at
par value. This means that the underwriters
4,000
of the bond issue (the firms that market the
3,500
bonds to the public for the issuing corpora3,000
tion) must choose a coupon rate that very
2,500
closely approximates market yields. In a
primary issue, the underwriters attempt to
2,000
sell the newly issued bonds directly to their
1,500
customers. If the coupon rate is inadequate,
1,000
investors will not pay par value for the
810.71
bonds.
500
After the bonds are issued, bondholders
0
may buy or sell bonds in secondary mar0
5
8
10
15
20
kets. In these markets, bond prices fluctuate
Interest Rate (%)
inversely with the market interest rate.
The inverse relationship between price
and yield is a central feature of fixedFigure 14.3 The inverse relationship between bond prices
income securities. Interest rate fluctuations
and yields. Price of an 8% coupon bond with 30-year
represent the main source of risk in the
maturity making semiannual payments
fixed-income market, and we devote considerable attention in Chapter 16 to assessing the sensitivity of bond prices to market yields. For now, however, we simply highlight
one key factor that determines that sensitivity, namely, the maturity of the bond.
As a general rule, keeping all other factors the same, the longer the maturity of the
bond, the greater the sensitivity of price to fluctuations in the interest rate. For example,
consider Table 14.2, which presents the price of an 8% coupon bond at different market
yields and times to maturity. For any departure of the interest rate from 8% (the rate
at which the bond sells at par value), the change in the bond price is greater for longer
times to maturity.
This makes sense. If you buy the bond at par with an 8% coupon rate, and market
rates subsequently rise, then you suffer a loss: You have tied up your money earning 8%
when alternative investments offer higher returns. This is reflected in a capital loss on
7
The progressively smaller impact of interest increases results largely from the fact that at higher rates the bond
is worth less. Therefore, an additional increase in rates operates on a smaller initial base, resulting in a smaller
price decline.
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PART IV
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Table 14.2
Bond Price at Given Market Interest Rate
Bond prices at different
interest rates (8% coupon bond, coupons paid
semiannually)
Time to
Maturity
2%
4%
6%
8%
10%
1 year
1,059.11
1,038.83
1,019.13
1,000.00
981.41
10 years
20 years
30 years
1,541.37
1,985.04
2,348.65
1,327.03
1,547.11
1,695.22
1,148.77
1,231.15
1,276.76
1,000.00
1,000.00
1,000.00
875.35
828.41
810.71
the bond—a fall in its market price. The longer the period for which your money is tied
up, the greater the loss, and correspondingly the greater the drop in the bond price. In
Table 14.2, the row for 1-year maturity bonds shows little price sensitivity—that is, with
only 1 year’s earnings at stake, changes in interest rates are not too threatening. But for
30-year maturity bonds, interest rate swings have a large impact on bond prices. The
force of discounting is greatest for the longest-term bonds.
This is why short-term Treasury securities such as T-bills are considered to be the safest.
They are free not only of default risk but also largely of price risk attributable to interest
rate volatility.
Bond Pricing between Coupon Dates
Equation 14.2 for bond prices assumes that the next coupon payment is in precisely one
payment period, either a year for an annual payment bond or 6 months for a semiannual
payment bond. But you probably want to be able to price bonds all 365 days of the year,
not just on the one or two dates each year that it makes a coupon payment!
In principle, the fact that the bond is between coupon dates does not affect the pricing problem. The procedure is always the same: Compute the present value of each
remaining payment and sum up. But if you are between coupon dates, there will be
fractional periods remaining until each payment, and this does complicate the arithmetic
computations.
Fortunately, bond pricing functions are included in most spreadsheet programs such
as Excel. The spreadsheet allows you to enter today’s date as well as the maturity date
of the bond, and so can provide prices for bonds at any date. The nearby box shows
you how.
As we pointed out earlier, bond prices are typically quoted net of accrued interest. These
prices, which appear in the financial press, are called flat prices. The actual invoice price
that a buyer pays for the bond includes accrued interest. Thus,
Invoice price 5 Flat price 1 Accrued interest
When a bond pays its coupon, flat price equals invoice price, because at that moment
accrued interest reverts to zero. However, this will be the exceptional case, not the rule.
Excel pricing functions provide the flat price of the bond. To find the invoice price, we
need to add accrued interest. Fortunately, Excel also provides functions that count the days
since the last coupon payment date and thus can be used to compute accrued interest. The
nearby box also illustrates how to use these functions. The box provides examples using
bonds that have just paid a coupon and so have zero accrued interest, as well as a bond that
is between coupon dates.
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eXcel APPLICATIONS: Bond Pricing
E
xcel and most other spreadsheet programs provide
built-in functions to compute bond prices and yields.
They typically ask you to input both the date you buy the
bond (called the settlement date) and the maturity date of
the bond. The Excel function for bond price is
5 PRICE(settlement date, maturity date, annual coupon
rate, yield to maturity, redemption value as percent of
par value, number of coupon payments per year)
For the 2.25% coupon July 2018 maturity bond highlighted in Figure 14.1, we would enter the values in
Spreadsheet 14.1. (Notice that in spreadsheets, we must
enter interest rates as decimals, not percentages). Alternatively, we could simply enter the following function in
Excel:
5 PRICE(DATE(2012,7,31), DATE(2018,7,31), .0225, .0079,
100, 2)
The DATE function in Excel, which we use for both
the settlement and maturity date, uses the format
DATE(year,month,day). The first date is July 31, 2012, when
the bond is purchased, and the second is July 31, 2018,
when it matures. Most bonds pay coupons either on the
15th or the last business day of the month.
Notice that the coupon rate and yield to maturity are
expressed as decimals, not percentages. In most cases,
redemption value is 100 (i.e., 100% of par value), and the
resulting price similarly is expressed as a percent of par
value. Occasionally, however, you may encounter bonds
that pay off at a premium or discount to par value. One
example would be callable bonds, discussed shortly.
The value of the bond returned by the pricing function is 108.5392 (cell B12), which nearly matches the
price reported in Table 14.1. (The yield to maturity is
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
2.25% coupon bond,
maturing July 31, 2018
Settlement date
Maturity date
Annual coupon rate
Yield to maturity
Redemption value (% of face value)
Coupon payments per year
7/31/2012
7/31/2018
0.0225
0.0079
100
2
Flat price (% of par)
Days since last coupon
Days in coupon period
Accrued interest
Invoice price
108.5392
0
184
0
108.5392
Spreadsheet 14.1
Bond Pricing in Excel
reported to only three decimal places, which results in a
little rounding error.) This bond has just paid a coupon. In
other words, the settlement date is precisely at the beginning of the coupon period, so no adjustment for accrued
interest is necessary.
To illustrate the procedure for bonds between coupon
payments, consider the 6.25% coupon May 2030 bond, also
appearing in Figure 14.1. Using the entries in column D of
the spreadsheet, we find in cell D12 that the (flat) price of
the bond is 161.002, which matches the price given in the
figure except for a few cents’ rounding error.
What about the bond’s invoice price? Rows 13
through 16 make the necessary adjustments. The function
described in cell C13 counts the days since the last coupon.
This day count is based on the bond’s settlement date,
maturity date, coupon period (1 5 annual; 2 5 semiannual), and day count convention (choice 1 uses actual
days). The function described in cell C14 counts the total
days in each coupon payment period. Therefore, the
entries for accrued interest in row 15 are the semiannual
coupon multiplied by the fraction of a coupon period that
has elapsed since the last payment. Finally, the invoice
prices in row 16 are the sum of flat price plus accrued
interest.
As a final example, suppose you wish to find the price
of the bond in Example 14.2. It is a 30-year maturity bond
with a coupon rate of 8% (paid semiannually). The market
interest rate given in the latter part of the example is 10%.
However, you are not given a specific settlement or maturity date. You can still use the PRICE function to value
the bond. Simply choose an arbitrary settlement date
(January 1, 2000, is convenient) and let the maturity date be
30 years hence. The appropriate inputs appear in column F
of the spreadsheet, with the resulting price, 81.0707% of
face value, appearing in cell F16.
C
Formula in column B
= DATE (2012, 7, 31)
= DATE (2018, 7, 31)
=PRICE(B4,B5,B6,B7,B8,B9)
=COUPDAYBS(B4,B5,2,1)
=COUPDAYS(B4,B5,2,1)
=(B13/B14)*B6*100/2
=B12 +B15
D
E
6.25% coupon bond,
maturing May 2030
F
G
8% coupon bond,
30-year maturity
7/31/2012
5/15/2030
0.0625
0.02117
100
2
1/1/2000
1/1/2030
0.08
0.1
100
2
161.0020
77
184
1.308
162.3097
81.0707
0
182
0
81.0707
eXcel
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PART IV
Fixed-Income Securities
14.3 Bond Yields
Most bonds do not sell for par value. But ultimately, barring default, they will mature to
par value. Therefore, we would like a measure of rate of return that accounts for both current income and the price increase or decrease over the bond’s life. The yield to maturity is
the standard measure of the total rate of return. However, it is far from perfect, and we will
explore several variations of this measure.
Yield to Maturity
In practice, an investor considering the purchase of a bond is not quoted a promised rate
of return. Instead, the investor must use the bond price, maturity date, and coupon payments to infer the return offered by the bond over its life. The yield to maturity (YTM) is
defined as the interest rate that makes the present value of a bond’s payments equal to its
price. This interest rate is often interpreted as a measure of the average rate of return that
will be earned on a bond if it is bought now and held until maturity. To calculate the yield
to maturity, we solve the bond price equation for the interest rate given the bond’s price.
Example 14.4
Yield to Maturity
Suppose an 8% coupon, 30-year bond is selling at $1,276.76. What average rate of
return would be earned by an investor purchasing the bond at this price? We find the
interest rate at which the present value of the remaining 60 semiannual payments equals
the bond price. This is the rate consistent with the observed price of the bond. Therefore, we solve for r in the following equation:
60
$1,000
$40
$1,276.76 5 a
1
t
(
)
(
1
1
r
1
1 r)60
t51
or, equivalently,
1,276.76 5 40 3 Annuity factor(r, 60) 1 1,000 3 PV factor(r, 60)
These equations have only one unknown variable, the interest rate, r. You can use a
financial calculator or spreadsheet to confirm that the solution is r 5 .03, or 3%, per
half-year.8 This is the bond’s yield to maturity.
The financial press reports yields on an annualized basis, and annualizes the bond’s
semiannual yield using simple interest techniques, resulting in an annual percentage
rate, or APR. Yields annualized using simple interest are also called “bond equivalent
yields.” Therefore, the semiannual yield would be doubled and reported in the newspaper as a bond equivalent yield of 6%. The effective annual yield of the bond, however,
accounts for compound interest. If one earns 3% interest every 6 months, then after
1 year, each dollar invested grows with interest to $1 3 (1.03)2 5 $1.0609, and the
effective annual interest rate on the bond is 6.09%.
8
On your financial calculator, you would enter the following inputs: n 5 60 periods; PV 5 21276.76; FV 5 1000;
PMT 5 40; then you would compute the interest rate (COMP i or CPT i). Notice that we enter the present value,
or PV, of the bond as minus $1,276.76. Again, this is because most calculators treat the initial purchase price
of the bond as a cash outflow. Spreadsheet 14.2 shows how to find yield to maturity using Excel. Without a
financial calculator or spreadsheet, you still could solve the equation, but you would need to use a trial-and-error
approach.
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CHAPTER 14
Bond Prices and Yields
459
Excel also provides a function for yield to maturity that is especially useful in-between
coupon dates. It is
5 YIELD(settlement date, maturity date, annual coupon rate, bond price, redemption
value as percent of par value, number of coupon payments per year)
The bond price used in the function should be the reported flat price, without accrued interest. For example, to find the yield to maturity of the bond in Example 14.4, we would use
column B of Spreadsheet 14.2. If the coupons were paid only annually, we would change
the entry for payments per year to 1 (see cell D8), and the yield would fall slightly to
5.99%.
The bond’s yield to maturity is the internal rate of return on an investment in the bond.
The yield to maturity can be interpreted as the compound rate of return over the life of the
bond under the assumption that all bond coupons can be reinvested at that yield.9 Yield to
maturity is widely accepted as a proxy for average return.
Yield to maturity differs from the current yield of a bond, which is the bond’s annual
coupon payment divided by the bond price. For example, for the 8%, 30-year bond currently selling at $1,276.76, the current yield would be $80/$1,276.76 5 .0627, or 6.27%,
per year. In contrast, recall that the effective annual yield to maturity is 6.09%. For this
bond, which is selling at a premium over par value ($1,276 rather than $1,000), the coupon
rate (8%) exceeds the current yield (6.27%), which exceeds the yield to maturity (6.09%).
The coupon rate exceeds current yield because the coupon rate divides the coupon payments by par value ($1,000) rather than by the bond price ($1,276). In turn, the current
yield exceeds yield to maturity because the yield to maturity accounts for the built-in capital loss on the bond; the bond bought today for $1,276 will eventually fall in value to
$1,000 at maturity.
Example 14.4 illustrates a general rule: For premium bonds (bonds selling above
par value), coupon rate is greater than current yield, which in turn is greater than yield
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
Semiannual coupons
Settlement date
Maturity date
Annual coupon rate
Bond price (flat)
Redemption value (% of face value)
Coupon payments per year
Yield to maturity (decimal)
D
E
Annual coupons
1/1/2000
1/1/2030
0.08
127.676
100
2
1/1/2000
1/1/2030
0.08
127.676
100
1
0.0600
0.0599
The formula entered here is: =YIELD(B3,B4,B5,B6,B7,B8)
Spreadsheet 14.2
Finding yield to maturity in Excel
eXcel
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9
If the reinvestment rate does not equal the bond’s yield to maturity, the compound rate of return will differ from
YTM. This is demonstrated below in Examples 14.6 and 14.7.
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PART IV
Fixed-Income Securities
to maturity. For discount bonds (bonds selling below par value), these relationships are
reversed (see Concept Check 3).
It is common to hear people talking loosely about the yield on a bond. In these cases,
they almost always are referring to the yield to maturity.
CONCEPT CHECK
14.3
What will be the relationship among coupon rate, current yield, and yield to maturity for bonds selling at
discounts from par? Illustrate using the 8% (semiannual payment) coupon bond, assuming it is selling at a
yield to maturity of 10%.
Yield to Call
Prices ($)
Yield to maturity is calculated on the assumption that the bond will be held until maturity.
What if the bond is callable, however, and may be retired prior to the maturity date? How
should we measure average rate of return for bonds subject to a call provision?
Figure 14.4 illustrates the risk of call to the bondholder. The top line is the value of a
“straight” (i.e., noncallable) bond with par value $1,000, an 8% coupon rate, and a 30-year
time to maturity as a function of the market interest rate. If interest rates fall, the bond
price, which equals the present value of the promised payments, can rise substantially.
Now consider a bond that has the same coupon rate and maturity date but is callable
at 110% of par value, or $1,100. When interest rates fall, the present value of the bond’s
scheduled payments rises, but the call provision allows the issuer to repurchase the bond at
the call price. If the call price is less than the present value of the scheduled payments, the
issuer may call the bond back from the bondholder.
The bottom line in Figure 14.4 is the value of the callable bond. At high interest rates,
the risk of call is negligible because the present value of scheduled payments is less than
the call price; therefore the values of the
straight and callable bonds converge. At
lower rates, however, the values of the bonds
2,000
begin to diverge, with the difference reflect1,800
ing the value of the firm’s option to reclaim
1,600
the callable bond at the call price. At very
Straight Bond
1,400
low rates, the present value of scheduled pay1,200
1,100
ments exceeds the call price, so the bond is
1,000
Callable
called. Its value at this point is simply the call
800
Bond
price, $1,100.
600
This analysis suggests that bond market
400
analysts
might be more interested in a bond’s
200
yield
to
call rather than yield to maturity,
0
especially if the bond is likely to be called.
3
4
5
6
7
8
9
10 11 12 13
The yield to call is calculated just like the
Interest Rate (%)
yield to maturity except that the time until
call replaces time until maturity, and the call
price replaces the par value. This computaFigure 14.4 Bond prices: Callable and straight debt.
tion is sometimes called “yield to first call,”
Coupon 5 8%; maturity 5 30 years; semiannual
as it assumes the issuer will call the bond as
payments.
soon as it may do so.
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CHAPTER 14
Example 14.5
Bond Prices and Yields
461
Yield to Call
Suppose the 8% coupon, 30-year maturity bond sells for $1,150 and is callable in
10 years at a call price of $1,100. Its yield to maturity and yield to call would be calculated using the following inputs:
Coupon payment
Number of semiannual periods
Final payment
Price
Yield to Call
Yield to Maturity
$40
20 periods
$1,100
$1,150
$40
60 periods
$1,000
$1,150
Yield to call is then 6.64%. [To confirm this on a calculator, input n 5 20; PV 5
(2)1150; FV 5 1100; PMT 5 40; compute i as 3.32%, or 6.64% bond equivalent yield.]
Yield to maturity is 6.82%. [To confirm, input n 5 60; PV 5 (2)1150; FV 5 1000;
PMT 5 40; compute i as 3.41% or 6.82% bond equivalent yield.] In Excel, you can calculate yield to call as 5 YIELD(DATE(2000,01,01), DATE(2010,01,01), .08, 115, 110, 2).
Notice that redemption value is input as 110, i.e., 110% of par value.
We have noted that most callable bonds are issued with an initial period of call protection. In addition, an implicit form of call protection operates for bonds selling at deep discounts from their call prices. Even if interest rates fall a bit, deep-discount bonds still will
sell below the call price and thus will not be subject to a call.
Premium bonds that might be selling near their call prices, however, are especially apt
to be called if rates fall further. If interest rates fall, a callable premium bond is likely to
provide a lower return than could be earned on a discount bond whose potential price
appreciation is not limited by the likelihood of a call. Investors in premium bonds therefore
may be more interested in the bond’s yield to call than its yield to maturity because it may
appear to them that the bond will be retired at the call date.
CONCEPT CHECK
14.4
a. The yield to maturity on two 10-year maturity bonds currently is 7%. Each bond has a call price of
$1,100. One bond has a coupon rate of 6%, the other 8%. Assume for simplicity that bonds are called
as soon as the present value of their remaining payments exceeds their call price. What will be the
capital gain on each bond if the market interest rate suddenly falls to 6%?
b. A 20-year maturity 9% coupon bond paying coupons semiannually is callable in 5 years at a call price
of $1,050. The bond currently sells at a yield to maturity of 8%. What is the yield to call?
Realized Compound Return versus Yield to Maturity
We have noted that yield to maturity will equal the rate of return realized over the life of
the bond if all coupons are reinvested at an interest rate equal to the bond’s yield to maturity. Consider, for example, a 2-year bond selling at par value paying a 10% coupon once a
year. The yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest
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PART IV
Fixed-Income Securities
A: Reinvestment Rate 5 10%
$1,100
Cash Flow:
Time: 0
$100
1
2
$1,100
Future
Value:
= $1,100
100 × 1.10 = $ 110
$1,210
B: Reinvestment Rate 5 8%
rate of 10%, the $1,000 investment in the bond
will grow after 2 years to $1,210, as illustrated
in Figure 14.5, panel A. The coupon paid in the
first year is reinvested and grows with interest to
a second-year value of $110, which together with
the second coupon payment and payment of par
value in the second year results in a total value of
$1,210.
To summarize, the initial value of the investment is V0 5 $1,000. The final value in 2 years is
V2 5 $1,210. The compound rate of return, therefore, is calculated as follows:
V0 (1 1 r)2 5 V2
$1,000 (1 1 r)2 5 $1,210
$1,100
r 5 .10 5 10%
Cash Flow:
Time: 0
Future
Value:
$100
1
2
$1,100
= $1,100
100 × 1.08 = $ 108
$1,208
Figure 14.5 Growth of invested funds
Example 14.6
With a reinvestment rate equal to the 10% yield to
maturity, the realized compound return equals
yield to maturity.
But what if the reinvestment rate is not 10%?
If the coupon can be invested at more than 10%,
funds will grow to more than $1,210, and the realized compound return will exceed 10%. If the
reinvestment rate is less than 10%, so will be the
realized compound return. Consider the following
example.
Realized Compound Return
If the interest rate earned on the first coupon is less than 10%, the final value of the
investment will be less than $1,210, and the realized compound return will be less than
10%. To illustrate, suppose the interest rate at which the coupon can be invested is only
8%. The following calculations are illustrated in Figure 14.5, panel B.
Future value of first coupon payment with interest earnings 5 $100 3 1.08 5 $ 108
1 Cash payment in second year (final coupon plus par value)
$1,100
5 Total value of investment with reinvested coupons
$1,208
The realized compound return is the compound rate of growth of invested funds, assuming that all coupon payments are reinvested. The investor purchased the bond for par at
$1,000, and this investment grew to $1,208.
V0(1 1 r)2 5 V2
$1,000(1 1 r)2 5 $1,208
r 5 .0991 5 9.91%
Example 14.6 highlights the problem with conventional yield to maturity when reinvestment rates can change over time. Conventional yield to maturity will not equal realized
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CHAPTER 14
Bond Prices and Yields
463
compound return. However, in an economy with future interest rate uncertainty, the rates at
which interim coupons will be reinvested are not yet known. Therefore, although realized
compound return can be computed after the investment period ends, it cannot be computed
in advance without a forecast of future reinvestment rates. This reduces much of the attraction of the realized return measure.
Forecasting the realized compound yield over various holding periods or investment
horizons is called horizon analysis. The forecast of total return depends on your forecasts
of both the price of the bond when you sell it at the end of your horizon and the rate at
which you are able to reinvest coupon income. The sales price depends in turn on the yield
to maturity at the horizon date. With a longer investment horizon, however, reinvested coupons will be a larger component of your final proceeds.
Example 14.7
Horizon Analysis
Suppose you buy a 30-year, 7.5% (annual payment) coupon bond for $980 (when its
yield to maturity is 7.67%) and plan to hold it for 20 years. Your forecast is that the
bond’s yield to maturity will be 8% when it is sold and that the reinvestment rate on
the coupons will be 6%. At the end of your investment horizon, the bond will have
10 years remaining until expiration, so the forecast sales price (using a yield to maturity
of 8%) will be $966.45. The 20 coupon payments will grow with compound interest to
$2,758.92. (This is the future value of a 20-year $75 annuity with an interest rate of 6%.)
On the basis of these forecasts, your $980 investment will grow in 20 years to $966.45
1 $2,758.92 5 $3,725.37. This corresponds to an annualized compound return of
6.90%:
V0(1 1 r)20 5 V20
$980(1 1 r)20 5 $3,725.37
r 5 .0690 5 6.90%
Examples 14.6 and 14.7 demonstrate that as interest rates change, bond investors are
actually subject to two sources of offsetting risk. On the one hand, when rates rise, bond
prices fall, which reduces the value of the portfolio. On the other hand, reinvested coupon
income will compound more rapidly at those higher rates. This reinvestment rate risk
will offset the impact of price risk. In Chapter 16, we will explore this trade-off in more
detail and will discover that by carefully tailoring their bond portfolios, investors can precisely balance these two effects for any given investment horizon.
14.4 Bond Prices over Time
As we noted earlier, a bond will sell at par value when its coupon rate equals the market
interest rate. In these circumstances, the investor receives fair compensation for the time
value of money in the form of the recurring coupon payments. No further capital gain is
necessary to provide fair compensation.
When the coupon rate is lower than the market interest rate, the coupon payments alone
will not provide investors as high a return as they could earn elsewhere in the market. To
receive a competitive return on such an investment, investors also need some price appreciation on their bonds. The bonds, therefore, must sell below par value to provide a “builtin” capital gain on the investment.
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PART IV
Fixed-Income Securities
Example 14.8
Fair Holding-Period Return
To illustrate built-in capital gains or losses, suppose a bond was issued several years ago
when the interest rate was 7%. The bond’s annual coupon rate was thus set at 7%. (We
will suppose for simplicity that the bond pays its coupon annually.) Now, with 3 years left
in the bond’s life, the interest rate is 8% per year. The bond’s market price is the present
value of the remaining annual coupons plus payment of par value. That present value is10
$70 3 Annuity factor(8%, 3) 1 $1,000 3 PV factor(8%, 3) 5 $974.23
which is less than par value.
In another year, after the next coupon is paid and remaining maturity falls to two
years, the bond would sell at
$70 3 Annuity factor(8%, 2) 1 $1,000 3 PV factor(8%, 2) 5 $982.17
thereby yielding a capital gain over the year of $7.94. If an investor had purchased the
bond at $974.23, the total return over the year would equal the coupon payment plus
capital gain, or $70 1 $7.94 5 $77.94. This represents a rate of return of $77.94/$974.23,
or 8%, exactly the rate of return currently available elsewhere in the market.
When bond prices are set according to the present
value formula, any discount from par value provides an
anticipated capital gain that will augment a below-market
At what price will the bond in Example 14.8 sell
coupon rate by just enough to provide a fair total rate of
in yet another year, when only 1 year remains
return. Conversely, if the coupon rate exceeds the market
until maturity? What is the rate of return to an
interest rate, the interest income by itself is greater than
investor who purchases the bond when its price
that available elsewhere in the market. Investors will bid
is $982.17 and sells it 1 year hence?
up the price of these bonds above their par values. As the
bonds approach maturity, they will fall in value because
fewer of these above-market coupon payments remain. The resulting capital losses offset
the large coupon payments so that the bondholder again receives only a competitive rate
of return.
Problem 14 at the end of the chapter asks you to work through the case of the highcoupon bond. Figure 14.6 traces out the price paths of high- and low-coupon bonds (net of
accrued interest) as time to maturity approaches, at least for the case in which the market
interest rate is constant. The low-coupon bond enjoys capital gains, whereas the highcoupon bond suffers capital losses.11
We use these examples to show that each bond offers investors the same total rate of
return. Although the capital gains versus income components differ, the price of each
bond is set to provide competitive rates, as we should expect in well-functioning capital
markets. Security returns all should be comparable on an after-tax risk-adjusted basis. If
they are not, investors will try to sell low-return securities, thereby driving down their
prices until the total return at the now-lower price is competitive with other securities.
CONCEPT CHECK
14.5
10
Using a calculator, enter n 5 3, i 5 8, PMT 5 70, FV 5 1000, and compute PV.
If interest rates are volatile, the price path will be “jumpy,” vibrating around the price path in Figure 14.6 and
reflecting capital gains or losses as interest rates fluctuate. Ultimately, however, the price must reach par value at
the maturity date, so the price of the premium bond will fall over time while that of the discount bond will rise.
11
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CHAPTER 14
Bond Prices and Yields
465
Price (% of Par Value)
Prices should continue to adjust until
all securities are fairly priced in that
160
expected returns are comparable, given
140
appropriate risk and tax adjustments.
We see evidence of this price
120
adjustment in Figure 14.1. Compare
100
the highlighted bond with the one
just below it. The July 2018 bond
80
has a coupon rate of 2.25%, while
60
the November 2018 bond has a
much higher coupon rate, 9%. But
40
Coupon = 12%
the higher coupon rate on that bond
Coupon = 4%
20
does not mean that it offers a higher
return; instead, it sells at a much
0
higher price. The yields to maturity on
0
5
10
15
20
25
30
the two bonds are nearly identical, a
Time (years)
shade below .8%. This makes sense,
since investors should care about their
total return, including both coupon
Figure 14.6 Price path of two 30-year maturity bonds, each
income as well as price change. In the
selling at a yield to maturity of 8%. Bond price approaches par
end, prices of similar-maturity bonds
value as maturity date approaches.
adjust until yields are pretty much
equalized.
Of course, the yields across bonds in Figure 14.1 are not all equal. Clearly, longer term
bonds at this time offered higher promised yields, a common pattern, and one that reflects
the relative risks of the bonds. We will explore the relation between yield and time to maturity in the next chapter.
Yield to Maturity versus Holding-Period Return
In Example 14.8, the holding-period return and the yield to maturity were equal. The bond
yield started and ended the year at 8%, and the bond’s holding-period return also equaled
8%. This turns out to be a general result. When the yield to maturity is unchanged over the
period, the rate of return on the bond will equal that yield. As we noted, this should not be
surprising: The bond must offer a rate of return competitive with those available on other
securities.
However, when yields fluctuate, so will a bond’s rate of return. Unanticipated changes
in market rates will result in unanticipated changes in bond returns and, after the fact, a
bond’s holding-period return can be better or worse than the yield at which it initially sells.
An increase in the bond’s yield acts to reduce its price, which reduces the holding period
return. In this event, the holding period return is likely to be less than the initial yield to
maturity.12 Conversely, a decline in yield will result in a holding-period return greater than
the initial yield.
12
We have to be a bit careful here. When yields increase, coupon income can be reinvested at higher rates, which
offsets the impact of the initial price decline. If your holding period is sufficiently long, the positive impact of the
higher reinvestment rate can more than offset the initial price decline. But common performance evaluation periods for portfolio managers are no more than 1 year, and over these shorter horizons the price impact will almost
always dominate the impact of the reinvestment rate. We discuss the trade-off between price risk and reinvestment
rate risk more fully in Chapter 16.
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PART IV
Fixed-Income Securities
Example 14.9
Yield to Maturity versus Holding-Period Return
Consider a 30-year bond paying an annual coupon of $80 and selling at par value of
$1,000. The bond’s initial yield to maturity is 8%. If the yield remains at 8% over the
year, the bond price will remain at par, so the holding-period return also will be 8%. But
if the yield falls below 8%, the bond price will increase. Suppose the yield falls and the
price increases to $1,050. Then the holding-period return is greater than 8%:
Holding-period return 5
CONCEPT CHECK
$80 1 ($1,050 2 $1,000)
5 .13, or 13%
$1,000
14.6
Show that if yield to maturity increases, then holding-period return is less than initial yield. For example,
suppose in Example 14.9 that by the end of the first year, the bond’s yield to maturity is 8.5%. Find the
1-year holding-period return and compare it to the bond’s initial 8% yield to maturity.
Here is another way to think about the difference between yield to maturity and
holding-period return. Yield to maturity depends only on the bond’s coupon, current
price, and par value at maturity. All of these values are observable today, so yield to
maturity can be easily calculated. Yield to maturity can be interpreted as a measure of
the average rate of return if the investment in the bond is held until the bond matures. In
contrast, holding-period return is the rate of return over a particular investment period
and depends on the market price of the bond at the end of that holding period; of course
this price is not known today. Because bond prices over the holding period will respond
to unanticipated changes in interest rates, holding-period return can at most be forecast.
Zero-Coupon Bonds and Treasury Strips
Original-issue discount bonds are less common than coupon bonds issued at par. These are
bonds that are issued intentionally with low coupon rates that cause the bond to sell at a discount from par value. The most common example of this type of bond is the zero-coupon
bond, which carries no coupons and provides all its return in the form of price appreciation.
Zeros provide only one cash flow to their owners, on the maturity date of the bond.
U.S. Treasury bills are examples of short-term zero-coupon instruments. If the bill has
face value of $10,000, the Treasury issues or sells it for some amount less than $10,000,
agreeing to repay $10,000 at maturity. All of the investor’s return comes in the form of
price appreciation.
Longer-term zero-coupon bonds are commonly created from coupon-bearing notes and
bonds. A bond dealer who purchases a Treasury coupon bond may ask the Treasury to break
down the cash flows to be paid by the bond into a series of independent securities, where
each security is a claim to one of the payments of the original bond. For example, a 10-year
coupon bond would be “stripped” of its 20 semiannual coupons, and each coupon payment
would be treated as a stand-alone zero-coupon bond. The maturities of these bonds would
thus range from 6 months to 10 years. The final payment of principal would be treated as
another stand-alone zero-coupon security. Each of the payments is now treated as an independent security and is assigned its own CUSIP number (by the Committee on Uniform
Securities Identification Procedures), the security identifier that allows for electronic trading over the Fedwire system, a network that connects all Federal Reserve banks and their
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Bond Prices and Yields
30
27
24
21
18
15
12
9
6
3
0
Price ($)
branches. The payments are still considered obligations of the U.S. Treasury. The Treasury pro1,000
900
gram under which coupon stripping is performed
800
is called STRIPS (Separate Trading of Registered
700
Interest and Principal of Securities), and these
600
zero-coupon securities are called Treasury strips.
500
400
What should happen to prices of zeros as time
300
passes? On their maturity dates, zeros must sell
200
for par value. Before maturity, however, they
100
should sell at discounts from par, because of the
0
time value of money. As time passes, price should
approach par value. In fact, if the interest rate is
Time (years)
constant, a zero’s price will increase at exactly
Today
Maturity
Date
the rate of interest.
To illustrate, consider a zero with 30 years
until maturity, and suppose the market interest
rate is 10% per year. The price of the bond today
Figure 14.7 The price of a 30-year zero-coupon bond
is $1,000/(1.10)30 5 $57.31. Next year, with only
over time at a yield to maturity of 10%. Price equals
29 years until maturity, if the yield is still 10%,
1,000/(1.10)T, where T is time until maturity.
29
the price will be $1,000/(1.10) 5 $63.04, a 10%
increase over its previous-year value. Because the
par value of the bond is now discounted for one less year, its price has increased by the
1-year discount factor.
Figure 14.7 presents the price path of a 30-year zero-coupon bond for an annual market
interest rate of 10%. The bond prices rise exponentially, not linearly, until its maturity.
After-Tax Returns
The tax authorities recognize that the “built-in” price appreciation on original-issue discount (OID) bonds such as zero-coupon bonds represents an implicit interest payment to
the holder of the security. The IRS, therefore, calculates a price appreciation schedule to
impute taxable interest income for the built-in appreciation during a tax year, even if the
asset is not sold or does not mature until a future year. Any additional gains or losses that
arise from changes in market interest rates are treated as capital gains or losses if the OID
bond is sold during the tax year.
Example 14.10
Taxation of Original-Issue Discount Bonds
If the interest rate originally is 10%, the 30-year zero would be issued at a price of
$1,000/(1.10)30 5 $57.31. The following year, the IRS calculates what the bond price
would be if the yield were still 10%. This is $1,000/(1.10)29 5 $63.04. Therefore, the
IRS imputes interest income of $63.04 2 $57.31 5 $5.73. This amount is subject to tax.
Notice that the imputed interest income is based on a “constant yield method” that
ignores any changes in market interest rates.
If interest rates actually fall, let’s say to 9.9%, the bond price will be $1,000/(1.099)29 5
$64.72. If the bond is sold, then the difference between $64.72 and $63.04 is treated
as capital gains income and taxed at the capital gains tax rate. If the bond is not sold,
then the price difference is an unrealized capital gain and does not result in taxes in that
year. In either case, the investor must pay taxes on the $5.73 of imputed interest at the
rate on ordinary income.
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The procedure illustrated in Example 14.10 applies as well to the taxation of other
original-issue discount bonds, even if they are not zero-coupon bonds. Consider, as an
example, a 30-year maturity bond that is issued with a coupon rate of 4% and a yield to
maturity of 8%. For simplicity, we will assume that the bond pays coupons once annually.
Because of the low coupon rate, the bond will be issued at a price far below par value,
specifically at $549.69. If the bond’s yield to maturity is still 8%, then its price in
1 year will rise to $553.66. (Confirm this for yourself.) This would provide a pretax
holding-period return (HPR) of exactly 8%:
HPR 5
$40 1 ($553.66 2 $549.69)
5 .08
$549.69
The increase in the bond price based on a constant yield, however, is treated as interest
income, so the investor is required to pay taxes on the explicit coupon income, $40, as well
as the imputed interest income of $553.66 2 $549.69 5 $3.97. If the bond’s yield actually
changes during the year, the difference between the bond’s price and the constant-yield
value of $553.66 would be treated as capital gains income if the bond is sold.
CONCEPT CHECK
14.7
Suppose that the yield to maturity of the 4% coupon, 30-year maturity bond falls to 7% by the end of the
first year and that the investor sells the bond after the first year. If the investor’s federal plus state tax rate
on interest income is 38% and the combined tax rate on capital gains is 20%, what is the investor’s aftertax rate of return?
14.5 Default Risk and Bond Pricing
Although bonds generally promise a fixed flow of income, that income stream is not riskless unless the investor can be sure the issuer will not default on the obligation. While U.S.
government bonds may be treated as free of default risk, this is not true of corporate bonds.
Therefore, the actual payments on these bonds are uncertain, for they depend to some
degree on the ultimate financial status of the firm.
Bond default risk, usually called credit risk, is measured by Moody’s Investor Services,
Standard & Poor’s Corporation, and Fitch Investors Service, all of which provide financial information on firms as well as quality ratings of large corporate and municipal bond
issues. International sovereign bonds, which also entail default risk, especially in emerging markets, also are commonly rated for default risk. Each rating firm assigns letter
grades to the bonds of corporations and municipalities to reflect their assessment of the
safety of the bond issue. The top rating is AAA or Aaa, a designation awarded to only
about a dozen firms. Moody’s modifies each rating class with a 1, 2, or 3 suffix (e.g.,
Aaa1, Aaa2, Aaa3) to provide a finer gradation of ratings. The other agencies use a 1 or 2
modification.
Those rated BBB or above (S&P, Fitch) or Baa and above (Moody’s) are considered
investment-grade bonds, whereas lower-rated bonds are classified as speculativegrade or junk bonds. Defaults on low-grade issues are not uncommon. For example,
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469
almost half of the bonds rated CCC by Standard & Poor’s at issue have defaulted within
10 years. Highly rated bonds rarely default, but even these bonds are not free of credit risk.
For example, in 2001 WorldCom sold $11.8 billion of bonds with an investment-grade
rating. Only a year later, the firm filed for bankruptcy and its bondholders lost more than
80% of their investment. Certain regulated institutional investors such as insurance companies have not always been allowed to invest in speculative-grade bonds.
Figure 14.8 provides the definitions of each bond rating classification.
Bond Ratings
Very High
Quality
High Quality
Speculative
Very Poor
Standard & Poor’s
AAA AA
A BBB
BB B
CCC D
Moody’s
Aaa Aa
A Baa
Ba B
Caa C
At times both Moody’s and Standard & Poor’s have used adjustments to these ratings:
S&P uses plus and minus signs: A+ is the strongest A rating and A− the weakest.
Moody’s uses a 1, 2, or 3 designation, with 1 indicating the strongest.
Moody’s
S&P
Aaa
AAA
Aa
AA
A
A
Baa
BBB
Ba
B
Caa
Ca
BB
B
CCC
CC
Debt rated in these categories is regarded, on balance, as predominantly
speculative with respect to capacity to pay interest and repay principal in
accordance with the terms of the obligation. BB and Ba indicate the lowest
degree of speculation, and CC and Ca the highest degree of speculation.
Although such debt will likely have some quality and protective
characteristics, these are outweighed by large uncertainties or major risk
exposures to adverse conditions. Some issues may be in default.
C
D
C
D
This rating is reserved for income bonds on which no interest is being paid.
Debt rated D is in default, and payment of interest and/or repayment of
principal is in arrears.
Debt rated Aaa and AAA has the highest rating. Capacity to pay interest
and principal is extremely strong.
Debt rated Aa and AA has a very strong capacity to pay interest and repay
principal. Together with the highest rating, this group comprises the highgrade bond class.
Debt rated A has a strong capacity to pay interest and repay principal,
although it is somewhat more susceptible to the adverse effects of
changes in circumstances and economic conditions than debt in
higher-rated categories.
Debt rated Baa and BBB is regarded as having an adequate capacity to
pay interest and repay principal. Whereas it normally exhibits adequate
protection parameters, adverse economic conditions or changing
circumstances are more likely to lead to a weakened capacity to pay
interest and repay principal for debt in this category than in higher-rated
categories. These bonds are medium-grade obligations.
Figure 14.8 Definitions of each bond rating class
Source: Stephen A. Ross and Randolph W. Westerfield, Corporate Finance, Copyright 1988 (St. Louis: Times Mirror/
Mosby College Publishing, reproduced with permission from the McGraw-Hill Companies, Inc.). Data from various editions of Standard & Poor’s Bond Guide and Moody’s Bond Guide.
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Junk Bonds
Junk bonds, also known as high-yield bonds, are nothing more than speculative-grade
(low-rated or unrated) bonds. Before 1977, almost all junk bonds were “fallen angels,” that
is, bonds issued by firms that originally had investment-grade ratings but that had since
been downgraded. In 1977, however, firms began to issue “original-issue junk.”
Much of the credit for this innovation is given to Drexel Burnham Lambert, and especially its trader Michael Milken. Drexel had long enjoyed a niche as a junk bond trader and
had established a network of potential investors in junk bonds. Firms not able to muster an
investment-grade rating were happy to have Drexel (and other investment bankers) market
their bonds directly to the public, as this opened up a new source of financing. Junk issues
were a lower-cost financing alternative than borrowing from banks.
High-yield bonds gained considerable notoriety in the 1980s when they were used as
financing vehicles in leveraged buyouts and hostile takeover attempts. Shortly thereafter, however, the junk bond market suffered. The legal difficulties of Drexel and Michael
Milken in connection with Wall Street’s insider trading scandals of the late 1980s tainted
the junk bond market.
At the height of Drexel’s difficulties, the high-yield bond market nearly dried up. Since
then, the market has rebounded dramatically. However, it is worth noting that the average
credit quality of newly issued high-yield debt issued today is higher than the average quality in the boom years of the 1980s. Of course, junk bonds are more vulnerable to economic
distress than investment-grade bonds. During the financial crisis of 2008–2009, prices on
these bonds fell dramatically, and their yields to maturity rose equally dramatically. The
spread between yields on B-rated bonds and Treasuries widened from around 3% in early
2007 to an astonishing 19% by the beginning of 2009.
Determinants of Bond Safety
Bond rating agencies base their quality ratings largely on an analysis of the level and trend
of some of the issuer’s financial ratios. The key ratios used to evaluate safety are
1. Coverage ratios—Ratios of company earnings to fixed costs. For example, the
times-interest-earned ratio is the ratio of earnings before interest payments and
taxes to interest obligations. The fixed-charge coverage ratio includes lease payments and sinking fund payments with interest obligations to arrive at the ratio of
earnings to all fixed cash obligations (sinking funds are described below). Low or
falling coverage ratios signal possible cash flow difficulties.
2. Leverage ratio, debt-to-equity ratio—A too-high leverage ratio indicates excessive
indebtedness, signaling the possibility the firm will be unable to earn enough to
satisfy the obligations on its bonds.
3. Liquidity ratios—The two most common liquidity ratios are the current ratio
(current assets/current liabilities) and the quick ratio (current assets excluding
inventories/current liabilities). These ratios measure the firm’s ability to pay bills
coming due with its most liquid assets.
4. Profitability ratios—Measures of rates of return on assets or equity. Profitability ratios are indicators of a firm’s overall financial health. The return on assets
(earnings before interest and taxes divided by total assets) or return on equity (net
income/equity) are the most popular of these measures. Firms with higher returns
on assets or equity should be better able to raise money in security markets because
they offer prospects for better returns on the firm’s investments.
5. Cash flow-to-debt ratio—This is the ratio of total cash flow to outstanding debt.
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Bond Prices and Yields
3-year medians
EBIT interest coverage multiple
EBITDA interest coverage multiple
Funds from operations/total debt (%)
Free operating cash flow/total debt (%)
Total debt/EBITDA multiple
Return on capital (%)
Total debt/total debt 1 equity (%)
AAA
AA
A
BBB
BB
B
CCC
23.8
25.5
203.3
127.6
0.4
27.6
12.4
19.5
24.6
79.9
44.5
0.9
27.0
28.3
8.0
10.2
48.0
25.0
1.6
17.5
37.5
4.7
6.5
35.9
17.3
2.2
13.4
42.5
2.5
3.5
22.4
8.3
3.5
11.3
53.7
1.2
1.9
11.5
2.8
5.3
8.7
75.9
0.4
0.9
5.0
(2.1)
7.9
3.2
113.5
Table 14.3
Financial ratios by rating class, long-term debt
Note: EBITDA is earnings before interest, taxes, depreciation, and amortization
Source: Corporate Rating Criteria, Standard & Poor’s, 2006.
ROE
Standard & Poor’s periodically computes median values of selected ratios for firms in
several rating classes, which we present in Table 14.3. Of course, ratios must be evaluated in the context of industry standards, and analysts differ in the weights they place on
particular ratios. Nevertheless, Table 14.3 demonstrates the tendency of ratios to improve
along with the firm’s rating class. And default rates vary dramatically with bond rating.
Historically, only about 1% of industrial bonds originally rated AA or better at issuance
had defaulted after 15 years. That ratio is around 7.5% for BBB-rated bonds, and 40% for
B-rated bonds. Credit risk clearly varies dramatically across rating classes.
Many studies have tested whether financial ratios can in fact be used to predict default
risk. One of the best-known series of tests was conducted by Edward Altman, who used
discriminant analysis to predict bankruptcy. With this technique a firm is assigned a score
based on its financial characteristics. If its score exceeds a cut-off value, the firm is deemed
creditworthy. A score below the cut-off value indicates significant bankruptcy risk in the
near future.
To illustrate the technique, suppose that we were to collect data on the return on
equity (ROE) and coverage ratios of a sample of firms, and then keep records of any
corporate bankruptcies. In Figure 14.9 we plot the
ROE and coverage ratios for each firm, using X
for firms that eventually went bankrupt and O for
those that remained solvent. Clearly, the X and O
firms show different patterns of data, with the solvent firms typically showing higher values for the
two ratios.
The discriminant analysis determines the equation of the line that best separates the X and O
observations. Suppose that the equation of the line
is .75 5 .9 3 ROE 1 .4 3 Coverage. Then, based
Coverage Ratio
on its own financial ratios, each firm is assigned
a “Z-score” equal to .9 3 ROE 1 .4 3 Coverage.
If its Z-score exceeds .75, the firm plots above the
line and is considered a safe bet; Z-scores below
Figure 14.9 Discriminant analysis
.75 foretell financial difficulty.
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Altman found the following equation to best separate failing and nonfailing firms:
Z 5 3.1
Shareholders’ equity
EBIT
Sales
1 1.0
1 .42
Total assets
Assets
Total liabilities
1 .85
Retained earnings
Working capital
1 .72
Total assets
Total assets
where EBIT 5 earnings before interest and taxes.13 Z-scores below 1.23 indicate vulnerability to bankruptcy, scores between 1.23 and 2.90 are a gray area, and scores above 2.90
are considered safe.
CONCEPT CHECK
14.8
Suppose we add a new variable equal to current liabilities/current assets to Altman’s
equation. Would you expect this variable to receive a positive or negative coefficient?
Bond Indentures
A bond is issued with an indenture, which is the contract between the issuer and the bondholder. Part of the indenture is a set of restrictions that protect the rights of the bondholders.
Such restrictions include provisions relating to collateral, sinking funds, dividend policy,
and further borrowing. The issuing firm agrees to these protective covenants in order to
market its bonds to investors concerned about the safety of the bond issue.
Sinking Funds Bonds call for the payment of par value at the end of the bond’s life.
This payment constitutes a large cash commitment for the issuer. To help ensure the commitment does not create a cash flow crisis, the firm agrees to establish a sinking fund to
spread the payment burden over several years. The fund may operate in one of two ways:
1. The firm may repurchase a fraction of the outstanding bonds in the open market
each year.
2. The firm may purchase a fraction of the outstanding bonds at a special call price
associated with the sinking fund provision. The firm has an option to purchase the
bonds at either the market price or the sinking fund price, whichever is lower. To
allocate the burden of the sinking fund call fairly among bondholders, the bonds
chosen for the call are selected at random based on serial number.14
The sinking fund call differs from a conventional bond call in two important ways.
First, the firm can repurchase only a limited fraction of the bond issue at the sinking fund
call price. At best, some indentures allow firms to use a doubling option, which allows
repurchase of double the required number of bonds at the sinking fund call price. Second,
while callable bonds generally have call prices above par value, the sinking fund call price
usually is set at the bond’s par value.
13
Altman’s original work was published in Edward I. Altman, “Financial Ratios, Discriminant Analysis, and the
Prediction of Corporate Bankruptcy,” Journal of Finance 23 (September 1968). This equation is from his updated
study, Corporate Financial Distress and Bankruptcy, 2nd ed. (New York: Wiley, 1993), p. 29. Altman’s analysis
is updated and extended in W. H. Beaver, M. F. McNichols, and J-W. Rhie, “Have Financial Statements become
Less Informative? Evidence from the Ability of Financial Ratios to Predict Bankruptcy,” Review of Accounting
Studies 10 (2005), pp. 93–122.
14
Although it is less common, the sinking fund provision also may call for periodic payments to a trustee, with the
payments invested so that the accumulated sum can be used for retirement of the entire issue at maturity.
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473
Although sinking funds ostensibly protect bondholders by making principal repayment
more likely, they can hurt the investor. The firm will choose to buy back discount bonds
(selling below par) at market price, while exercising its option to buy back premium bonds
(selling above par) at par. Therefore, if interest rates fall and bond prices rise, firms will
benefit from the sinking fund provision that enables them to repurchase their bonds at
below-market prices. In these circumstances, the firm’s gain is the bondholder’s loss.
One bond issue that does not require a sinking fund is a serial bond issue, in which the
firm sells bonds with staggered maturity dates. As bonds mature sequentially, the principal
repayment burden for the firm is spread over time, just as it is with a sinking fund. One
advantage of serial bonds over sinking fund issues is that there is no uncertainty introduced
by the possibility that a particular bond will be called for the sinking fund. The disadvantage of serial bonds, however, is that bonds of different maturity dates are not interchangeable, which reduces the liquidity of the issue.
Subordination of Further Debt One of the factors determining bond safety is
total outstanding debt of the issuer. If you bought a bond today, you would be understandably distressed to see the firm tripling its outstanding debt tomorrow. Your bond would be
riskier than it appeared when you bought it. To prevent
firms from harming bondholders in this manner, subordination clauses restrict the amount of additional borrow& Mobil Corp. debenture 8s, due 2032:
Rating — Aa2
ing. Additional debt might be required to be subordinated
AUTH----$250,000,000.
in priority to existing debt; that is, in the event of bankOUTSTG----Dec. 31, 1993, $250,000,000.
DATED----Oct. 30, 1991.
ruptcy, subordinated or junior debtholders will not be
INTEREST----F&A 12.
TRUSTEE----Chemical Bank.
paid unless and until the prior senior debt is fully paid off.
DENOMINATION----Fully registered, $1,000 and
Dividend Restrictions Covenants also limit the
dividends firms may pay. These limitations protect the
bondholders because they force the firm to retain assets
rather than paying them out to stockholders. A typical
restriction disallows payments of dividends if cumulative
dividends paid since the firm’s inception exceed cumulative retained earnings plus proceeds from sales of stock.
Collateral Some bonds are issued with specific collateral behind them. Collateral is a particular asset that the
bondholders receive if the firm defaults on the bond. If the
collateral is property, the bond is called a mortgage bond. If
the collateral takes the form of other securities held by the
firm, the bond is a collateral trust bond. In the case of equipment, the bond is known as an equipment obligation bond.
This last form of collateral is used most commonly by firms
such as railroads, where the equipment is fairly standard and
can be easily sold to another firm should the firm default.
Collateralized bonds generally are considered safer
than general debenture bonds, which are unsecured,
meaning they do not provide for specific collateral. Credit
risk of unsecured bonds depends on the general earning
power of the firm. If the firm defaults, debenture owners
become general creditors of the firm. Because they are
safer, collateralized bonds generally offer lower yields
than general debentures.
Figure 14.10 shows the terms of a bond issued by
Mobil as described in Moody’s Industrial Manual. The
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integral multiplies thereof. Transferable and
exchangeable without service charge.
CALLABLE----As a whole or in part, at any time,
on or after Aug. 12, 2002, at the option of Co. on
at least 30 but not more than the 60 days' notice to
each Aug. 11 as follows:
2003..........105.007
2006..........104.256
2009..........103.505
2012..........102.754
2015..........102.003
2018..........101.252
2021..........100.501
2004..........104.756
2007..........104.005
2010..........103.254
2013..........102.503
2016..........101.752
2019..........101.001
2022..........100.250
2005..........104.506
2008..........103.755
2011..........103.004
2014..........102.253
2017..........101.502
2020..........100.751
and thereafter at 100 plus accrued interest.
SECURITY----Not secured. Ranks equally with all
other
unsecured
and
unsubordinated
indebtedness
of Co. Co. nor any Affiliate will not incur any
indebtedness; provided that Co. will not create as
security for any indebtedness for borrowed money,
any mortgage, pledge, security interest or lien on
any stock or indebtedness is directly owned by
Co. without effectively providing that the debt
securities shall be secured equally and ratably with
such indebtedness. so long as such indebtedness
shall be so secured.
INDENTURE
MODIFICATION----Indenture
may be modified, except as provided with, consent
of 66 2/3% of debs. outstg.
RIGHTS ON DEFAULT----Trustee, or 25% of
debs. outstg., may declare principal due and payable (30 days' grace for payment of interest).
LISTED----On New York Stock Exchange.
PURPOSE----Proceeds used for general corporate
purposes.
OFFERED----($250,000,000) at 99.51 plus accrued
interest (proceeds to Co., 99.11) on Aug. 5, 1992
thru Merrill Lynch & Co., Donaldson, Lufkin &
Jenerette Securities Corp., PaineWebber Inc., Prudential
Securities
Inc.,
Smith
Barney,
Harris
Upham & Co. Inc. and associates.
Figure 14.10 Callable bond issued by Mobil
Source: Mergent’s Industrial Manual, Mergent’s Investor
Services, 1994. Reprinted with permission. All rights reserved.
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bond is registered and listed on the NYSE. It was issued in 1991 but was not callable until
2002. Although the call price started at 105.007% of par value, it declines gradually until
reaching par after 2020. Most of the terms of the bond are typical and illustrate many of the
indenture provisions we have mentioned. However, in recent years there has been a marked
trend away from the use of call provisions.
Yield to Maturity and Default Risk
Because corporate bonds are subject to default risk, we must distinguish between the
bond’s promised yield to maturity and its expected yield. The promised or stated yield will
be realized only if the firm meets the obligations of the bond issue. Therefore, the stated
yield is the maximum possible yield to maturity of the bond. The expected yield to maturity
must take into account the possibility of a default.
For example, at the height of the financial crisis in October 2008, as Ford Motor
Company struggled, its bonds due in 2028 were rated CCC and were selling at about 33%
of par value, resulting in a yield to maturity of about 20%. Investors did not really believe
the expected rate of return on these bonds was 20%. They recognized that there was a
decent chance that bondholders would not receive all the payments promised in the bond
contract and that the yield based on expected cash flows was far less than the yield based
on promised cash flows. As it turned out, of course, Ford weathered the storm, and investors who purchased its bonds made a very nice profit: The bonds were selling in mid-2012
for about 110% of par value, more than triple their value in 2008.
Example 14.11
Expected vs. Promised Yield to Maturity
Suppose a firm issued a 9% coupon bond 20 years ago. The bond now has 10 years
left until its maturity date, but the firm is having financial difficulties. Investors believe
that the firm will be able to make good on the remaining interest payments, but at the
maturity date, the firm will be forced into bankruptcy, and bondholders will receive only
70% of par value. The bond is selling at $750.
Yield to maturity (YTM) would then be calculated using the following inputs:
Coupon payment
Number of semiannual periods
Final payment
Price
Expected YTM
Stated YTM
$45
20 periods
$700
$750
$45
20 periods
$1,000
$750
The yield to maturity based on promised payments is 13.7%. Based on the expected
payment of $700 at maturity, however, the yield to maturity would be only 11.6%. The
stated yield to maturity is greater than the yield investors actually expect to receive.
Example 14.11 suggests that when a bond becomes more subject to default risk,
its price will fall, and therefore its promised yield to maturity will rise. Similarly, the
default premium, the spread between the stated yield to maturity and that on otherwisecomparable Treasury bonds, will rise. However, its expected yield to maturity, which
ultimately is tied to the systematic risk of the bond, will be far less affected. Let’s
continue Example 14.11.
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CHAPTER 14
Example 14.12
475
Bond Prices and Yields
Default Risk and the Default Premium
Suppose that the condition of the firm in Example 14.11 deteriorates further, and investors now believe that the bond will pay off only 55% of face value at maturity. Investors
now demand an expected yield to maturity of 12% (i.e., 6% semiannually), which is
.4% higher than in Example 14.11. But the price of the bond will fall from $750 to $688
[n 5 20; i 5 6; FV 5 550; PMT 5 $45]. At this price, the stated yield to maturity based
on promised cash flows is 15.2%. While the expected yield to maturity has increased by
.4%, the drop in price has caused the promised yield to maturity to rise by 1.5%.
To compensate for the possibility of default, corporate
CONCEPT CHECK 14.9
bonds must offer a default premium. The default premium is the difference between the promised yield on a
What is the expected yield to maturity in Example
corporate bond and the yield of an otherwise-identical
14.12 if the firm is in even worse condition?
government bond that is riskless in terms of default. If the
Investors expect a final payment of only $500,
firm remains solvent and actually pays the investor all of
and the bond price has fallen to $650.
the promised cash flows, the investor will realize a higher
yield to maturity than would be realized from the government bond. If, however, the firm goes bankrupt, the corporate bond is likely to provide a
lower return than the government bond. The corporate bond has the potential for both better and worse performance than the default-free Treasury bond. In other words, it is riskier.
The pattern of default premiums offered on risky bonds is sometimes called the risk
structure of interest rates. The greater the default risk, the higher the default premium.
Figure 14.11 shows spreads between yields to maturity of bonds of different risk classes.
You can see here clear evidence of credit-risk premiums on promised yields. Note, for
example, the incredible run-up of
credit spreads during the financial
20
crisis of 2008–2009.
12
8
4
2012
2009
2006
2003
2000
1997
1994
1991
1988
1985
1982
1979
1976
–4
1973
0
1970
A credit default swap (CDS)
is in effect an insurance policy
on the default risk of a bond or
loan. To illustrate, the annual premium in July 2012 on a 5-year
German government CDS was
about 0.75%, meaning that the
CDS buyer would pay the seller
an annual premium of $.75 for
each $100 of bond principal. The
seller collects these annual payments for the term of the contract
but must compensate the buyer for
loss of bond value in the event of a
default.15
Yield Spread (%)
Credit Default Swaps
High Yield
Baa-Rated
Aaa-Rated
16
Figure 14.11 Yield spreads between corporate and 10-year Treasury
bonds
Source: Federal Reserve Bank of St. Louis.
15
Actually, credit default swaps may pay off even short of an actual default. The contract specifies the particular
“credit events” that will trigger a payment. For example, restructuring (rewriting the terms of a firm’s outstanding
debt as an alternative to formal bankruptcy proceedings) may be defined as a triggering credit event.
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PART IV
Fixed-Income Securities
As originally envisioned, credit default swaps were designed to allow lenders to buy protection against default risk. The natural buyers of CDSs would then be large bondholders
or banks that wished to enhance the creditworthiness of their outstanding loans. Even if the
borrower had a shaky credit standing, the “insured” debt would be as safe as the issuer of the
CDS. An investor holding a bond with a BB rating could, in principle, raise the effective
quality of the debt to AAA by buying a CDS on the issuer.
This insight suggests how CDS contracts should be priced. If a BB-rated corporate
bond bundled with insurance via a CDS is effectively equivalent to a AAA-rated bond,
then the premium on the swap ought to approximate the yield spread between AAA-rated
and BB-rated bonds.16 The risk structure of interest rates and CDS prices ought to be
tightly aligned.
Figure 14.12, panel A, shows the premiums on 5-year CDSs on German government
debt between 2008 and 2012. Even as the strongest economy in the eurozone, German
CDS prices nevertheless reflect financial strain, first in the deep recession of 2009 and
then again in 2011 as the prospects of defaults (and German-led bailouts) of Greece and
A: Premiums on German Sovereign Debt CDS Contracts
Premium (bp/year)
200
150
100
50
May 12
Jan 12
Sep 11
May 11
Jan 11
Sep 10
May 10
Jan 10
Sep 09
May 09
Jan 09
Sep 08
May 08
Jan 08
0
B: Premiums on Spanish Sovereign Debt CDS Contracts
Premium (bp/year)
900
750
500
250
May 12
Jan 12
Sep 11
May 11
Jan 11
Sep 10
May 10
Jan 10
Sep 09
May 09
Jan 09
Sep 08
May 08
Jan 08
0
Figure 14.12 Pricing of 5-year credit default swaps
Source: Bloomberg, August 1, 2012, http://www.bloomberg.com/quote/CDBR1U5:IND/chart.
16
We say approximately because there are some differences between highly rated bonds and bonds synthetically
enhanced with credit default swaps. For example, the term of the swap may not match the maturity of the bond.
Tax treatment of coupon payments versus swap payments may differ, as may the liquidity of the bonds. Finally,
some CDSs may entail one-time up-front payments as well as annual premiums.
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The credit crisis of 2008–2009, when lending among banks
and other financial institutions effectively seized up, was in
large measure a crisis of transparency. The biggest problem
was a widespread lack of confidence in the financial standing of counterparties to a trade. If one institution could not
be confident that another would remain solvent, it would
understandably be reluctant to offer it a loan. When doubt
about the credit exposure of customers and trading partners spiked to levels not seen since the Great Depression,
the market for loans dried up.
Credit default swaps were particularly cited for fostering doubts about counterparty reliability. By August 2008,
$63 trillion of such swaps were reportedly outstanding.
(By comparison, U.S. gross domestic product in 2008 was
about $14 trillion.) As the subprime mortgage market collapsed and the economy entered a deep recession, the
potential obligations on these contracts ballooned to
levels previously considered unimaginable and the ability
of CDS sellers to honor their commitments appeared in
doubt. For example, the huge insurance firm AIG alone
had sold more than $400 billion of CDS contracts on subprime mortgages and other loans and was days from
insolvency. But AIG’s insolvency could have triggered the
insolvency of other firms that had relied on its promise of
protection against loan defaults. These in turn might have
triggered further defaults. In the end, the government
felt compelled to rescue AIG to prevent a chain reaction
of insolvencies.
Counterparty risk and lax reporting requirements made
it effectively impossible to tease out firms’ exposures to
credit risk. One problem was that CDS positions do not have
to be accounted for on balance sheets. And the possibility
of one default setting off a sequence of further defaults
means that lenders may be exposed to the default of an
institution with which they do not even directly trade. Such
knock-on effects create systemic risk, in which the entire
financial system can freeze up. With the ripple effects of
bad debt extending in ever-widening circles, lending to
anyone can seem imprudent.
In the aftermath of the credit crisis, the Dodd-Frank Act
called for new regulation and reforms. One proposal is for
a central clearinghouse for credit derivatives such as CDS
contracts. Such a system would foster transparency of positions, would allow the clearinghouse to replace traders’ offsetting long and short positions with a single net position,
and would require daily recognition of gains or losses on
positions through a margin or collateral account. If losses
were to mount, positions would have to be unwound
before growing to unsustainable levels. Allowing traders
to accurately assess counterparty risk, and limiting such risk
through margin accounts and the extra back-up of the clearinghouse, would go a long way in limiting systemic risk.
WORDS FROM THE STREET
Credit Default Swaps, Systemic Risk, and the Financial
Crisis of 2008–2009
other eurozone countries worsened. As its perceived credit risk increased, so did the cost
of insuring its debt.
Panel B of Figure 14.12 shows the premiums on 5-year CDS contracts on Spanish
government debt. Spain’s economy was far shakier than Germany’s, and its CDS prices
reflected this fact. By the summer of 2012, its 5-year CDS premiums were around 600
basis points, about eight times the price to insure German debt.
While CDSs were conceived as a form of bond insurance, it wasn’t long before investors realized that they could be used to speculate on the financial health of particular issuers. As Figure 14.12 makes clear, someone in early 2011 wishing to bet against Spain
might have purchased CDS contracts on its sovereign debt and would have profited as
CDS prices spiked over the next 18 months. The nearby box discusses the role of credit
default swaps in the financial crisis of 2008–2009.
Credit Risk and Collateralized Debt Obligations
Collateralized debt obligations, or CDOs, emerged in the last decade as a major mechanism to reallocate credit risk in the fixed-income markets. To create a CDO, a financial
institution, commonly a bank, first would establish a legally distinct entity to buy and later
resell a portfolio of bonds or other loans. A common vehicle for this purpose was the socalled Structured Investment Vehicle (SIV).17 An SIV raises funds, often by issuing shortterm commercial paper, and uses the proceeds to buy corporate bonds or other forms of debt
17
The legal separation of the bank from the SIV allows the ownership of the loans to be conducted off the bank’s
balance sheet, and thus avoids capital requirements the bank would otherwise encounter.
477
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PART IV
Fixed-Income Securities
Senior-Subordinated
Tranche Structure
Senior tranche
Bank
Typical Terms
70–90% of notional
principal, coupon similar to
Aa-Aaa rated bonds
Mezzanine 1
5–15% of principal,
investment-grade rating
Mezzanine 2
5–15% of principal, higherquality junk rating
<2%, unrated, coupon rate
with 20% credit spread
Structured
investment
vehicle, SIV
Equity/first loss/
residual tranche
Figure 14.13 Collateralized debt obligations
such as mortgage loans or credit card debt. These loans are first pooled together and then
split into a series of classes known as tranches. (Tranche is the French word for “slice.”)
Each tranche is given a different level of seniority in terms of its claims on the underlying
loan pool, and each can be sold as a stand-alone security. As the loans in the underlying pool
make their interest payments, the proceeds are distributed to pay interest to each tranche in
order of seniority. This priority structure implies that each tranche has a different exposure
to credit risk.
Figure 14.13 illustrates a typical setup. The senior tranche is on top. Its investors may
account for perhaps 80% of the principal of the entire pool. But it has first claim on all the
debt service. Using our numbers, even if 20% of the debt pool defaults, the senior tranche
can be paid in full. Once the highest seniority tranche is paid off, the next-lower class (e.g.,
the mezzanine 1 tranche in Figure 14.13) receives the proceeds from the pool of loans until
its claims also are satisfied. Using junior tranches to insulate senior tranches from credit
risk in this manner, one can create Aaa-rated bonds even from a junk-bond portfolio.
Of course, shielding senior tranches from default risk means that the risk is concentrated on the lower tranches. The bottom tranche—called alternatively the equity, first-loss,
or residual tranche—has last call on payments from the pool of loans, or, put differently, is
at the head of the line in terms of absorbing default or delinquency risk.
Not surprisingly, investors in tranches with the greatest exposure to credit risk demand
the highest coupon rates. Therefore, while the lower mezzanine and equity tranches bear
the most risk, they will provide the highest returns if credit experience turns out favorably.
Mortgage-backed CDOs were an investment disaster in 2007–2009. These were CDOs
formed by pooling subprime mortgage loans made to individuals whose credit standing did
not allow them to qualify for conventional mortgages. When home prices stalled in 2007
and interest rates on these typically adjustable-rate loans reset to market levels, mortgage
delinquencies and home foreclosures soared, and investors in these securities lost billions
of dollars. Even investors in highly rated tranches experienced large losses.
Not surprisingly, the rating agencies that had certified these tranches as investmentgrade came under considerable fire. Questions were raised concerning conflicts of interest: Because the rating agencies are paid by bond issuers, the agencies were accused of
responding to pressure to ease their standards.
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CHAPTER 14
Bond Prices and Yields
1. Fixed-income securities are distinguished by their promise to pay a fixed or specified stream of
income to their holders. The coupon bond is a typical fixed-income security.
479
SUMMARY
2. Treasury notes and bonds have original maturities greater than 1 year. They are issued at or near
par value, with their prices quoted net of accrued interest.
3. Callable bonds should offer higher promised yields to maturity to compensate investors for the fact
that they will not realize full capital gains should the interest rate fall and the bonds be called away
from them at the stipulated call price. Bonds often are issued with a period of call protection. In
addition, discount bonds selling significantly below their call price offer implicit call protection.
4. Put bonds give the bondholder rather than the issuer the option to terminate or extend the life of
the bond.
5. Convertible bonds may be exchanged, at the bondholder’s discretion, for a specified number of
shares of stock. Convertible bondholders “pay” for this option by accepting a lower coupon rate
on the security.
6. Floating-rate bonds pay a coupon rate at a fixed premium over a reference short-term interest
rate. Risk is limited because the rate is tied to current market conditions.
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7. The yield to maturity is the single interest rate that equates the present value of a security’s cash
flows to its price. Bond prices and yields are inversely related. For premium bonds, the coupon
rate is greater than the current yield, which is greater than the yield to maturity. The order of
these inequalities is reversed for discount bonds.
8. The yield to maturity is often interpreted as an estimate of the average rate of return to an investor who purchases a bond and holds it until maturity. This interpretation is subject to error,
however. Related measures are yield to call, realized compound yield, and expected (versus
promised) yield to maturity.
9. Prices of zero-coupon bonds rise exponentially over time, providing a rate of appreciation equal
to the interest rate. The IRS treats this built-in price appreciation as imputed taxable interest
income to the investor.
10. When bonds are subject to potential default, the stated yield to maturity is the maximum possible yield to maturity that can be realized by the bondholder. In the event of default, however,
that promised yield will not be realized. To compensate bond investors for default risk, bonds
must offer default premiums, that is, promised yields in excess of those offered by default-free
government securities. If the firm remains healthy, its bonds will provide higher returns than
government bonds. Otherwise the returns may be lower.
11. Bond safety is often measured using financial ratio analysis. Bond indentures are another safeguard to protect the claims of bondholders. Common indentures specify sinking fund requirements, collateralization of the loan, dividend restrictions, and subordination of future debt.
12. Credit default swaps provide insurance against the default of a bond or loan. The swap buyer
pays an annual premium to the swap seller, but collects a payment equal to lost value if the loan
later goes into default.
13. Collateralized debt obligations are used to reallocate the credit risk of a pool of loans. The pool
is sliced into tranches, with each tranche assigned a different level of seniority in terms of its
claims on the cash flows from the underlying loans. High seniority tranches are usually quite
safe, with credit risk concentrated on the lower level tranches. Each tranche can be sold as a
stand-alone security.
debt securities
bond
par value
face value
bod61671_ch14_445-486.indd 479
coupon rate
bond indenture
zero-coupon bonds
callable bonds
convertible bonds
put bond
floating-rate bonds
yield to maturity
Related Web sites
for this chapter are
available at www.
mhhe.com/bkm
KEY TERMS
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PART IV
Fixed-Income Securities
current yield
premium bonds
discount bonds
realized compound return
horizon analysis
reinvestment rate risk
KEY EQUATIONS
credit risk
investment-grade bonds
speculative-grade or junk
bonds
sinking fund
subordination clauses
collateral
debenture
default premium
credit default swap (CDS)
collateralized debt obligations
(CDOs)
Price of a coupon bond:
Price 5 Coupon 3
1
1
1
B1 2
R 1 Par value 3
r
(1 1 r ) T
(1 1 r ) T
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5 Coupon 3 Annuity factor(r, T) 1 Par value 3 PV factor(r, T)
PROBLEM SETS
1. Define the following types of bonds:
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
Basic
Catastrophe bond.
Eurobond.
Zero-coupon bond.
Samurai bond.
Junk bond.
Convertible bond.
Serial bond.
Equipment obligation bond.
Original issue discount bond.
Indexed bond.
Callable bond.
Puttable bond.
2. Two bonds have identical times to maturity and coupon rates. One is callable at 105, the other at
110. Which should have the higher yield to maturity? Why?
3. The stated yield to maturity and realized compound yield to maturity of a (default-free) zerocoupon bond will always be equal. Why?
Intermediate
4. Why do bond prices go down when interest rates go up? Don’t lenders like high interest rates?
5. A bond with an annual coupon rate of 4.8% sells for $970. What is the bond’s current yield?
6. Which security has a higher effective annual interest rate?
a. A 3-month T-bill selling at $97,645 with par value $100,000.
b. A coupon bond selling at par and paying a 10% coupon semiannually.
7. Treasury bonds paying an 8% coupon rate with semiannual payments currently sell at par value.
What coupon rate would they have to pay in order to sell at par if they paid their coupons annually? (Hint: What is the effective annual yield on the bond?)
8. Consider a bond with a 10% coupon and with yield to maturity 5 8%. If the bond’s yield to
maturity remains constant, then in 1 year, will the bond price be higher, lower, or unchanged?
Why?
9. Consider an 8% coupon bond selling for $953.10 with 3 years until maturity making annual
coupon payments. The interest rates in the next 3 years will be, with certainty, r1 5 8%,
r2 5 10%, and r3 5 12%. Calculate the yield to maturity and realized compound yield of
the bond.
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CHAPTER 14
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481
10. Assume you have a 1-year investment horizon and are trying to choose among three bonds. All
have the same degree of default risk and mature in 10 years. The first is a zero-coupon bond that
pays $1,000 at maturity. The second has an 8% coupon rate and pays the $80 coupon once per
year. The third has a 10% coupon rate and pays the $100 coupon once per year.
a. If all three bonds are now priced to yield 8% to maturity, what are their prices?
b. If you expect their yields to maturity to be 8% at the beginning of next year, what will their
prices be then? What is your before-tax holding-period return on each bond? If your tax
bracket is 30% on ordinary income and 20% on capital gains income, what will your aftertax rate of return be on each?
c. Recalculate your answer to (b) under the assumption that you expect the yields to maturity
on each bond to be 7% at the beginning of next year.
11. A 20-year maturity bond with par value of $1,000 makes semiannual coupon payments at a
coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond
if the bond price is:
12. Repeat Problem 11 using the same data, but assuming that the bond makes its coupon payments
annually. Why are the yields you compute lower in this case?
13. Fill in the table below for the following zero-coupon bonds, all of which have par values of $1,000.
Price
Maturity (years)
Bond-Equivalent
Yield to Maturity
$400
$500
$500
—
—
$400
20
20
10
10
10
—
—
—
—
10%
8%
8%
14. Consider a bond paying a coupon rate of 10% per year semiannually when the market interest
rate is only 4% per half-year. The bond has 3 years until maturity.
a. Find the bond’s price today and 6 months from now after the next coupon is paid.
b. What is the total (6-month) rate of return on the bond?
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a. $950.
b. $1,000.
c. $1,050.
15. A bond with a coupon rate of 7% makes semiannual coupon payments on January 15 and
July 15 of each year. The Wall Street Journal reports the ask price for the bond on January 30
at 100.125. What is the invoice price of the bond? The coupon period has 182 days.
16. A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond selling above or
below par value? Explain.
17. Is the coupon rate of the bond in Problem 16 more or less than 9%?
18. Return to Table 14.1 and calculate both the real and nominal rates of return on the TIPS bond in
the second and third years.
19. A newly issued 20-year maturity, zero-coupon bond is issued with a yield to maturity of 8%
and face value $1,000. Find the imputed interest income in the first, second, and last year of the
bond’s life.
20. A newly issued 10-year maturity, 4% coupon bond making annual coupon payments is sold to
the public at a price of $800. What will be an investor’s taxable income from the bond over the
coming year? The bond will not be sold at the end of the year. The bond is treated as an originalissue discount bond.
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PART IV
Fixed-Income Securities
21. A 30-year maturity, 8% coupon bond paying coupons semiannually is callable in 5 years
at a call price of $1,100. The bond currently sells at a yield to maturity of 7% (3.5% per
half-year).
a. What is the yield to call?
b. What is the yield to call if the call price is only $1,050?
c. What is the yield to call if the call price is $1,100, but the bond can be called in 2 years
instead of 5 years?
22. A 10-year bond of a firm in severe financial distress has a coupon rate of 14% and sells for
$900. The firm is currently renegotiating the debt, and it appears that the lenders will allow the
firm to reduce coupon payments on the bond to one-half the originally contracted amount.
The firm can handle these lower payments. What is the stated and expected yield to maturity
of the bonds? The bond makes its coupon payments annually.
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23. A 2-year bond with par value $1,000 making annual coupon payments of $100
is priced at $1,000. What is the yield to maturity of the bond? What will be the realized compound yield to maturity if the 1-year interest rate next year turns out to be (a) 8%, (b) 10%,
(c) 12%?
24. Suppose that today’s date is April 15. A bond with a 10% coupon paid semiannually every
January 15 and July 15 is listed in The Wall Street Journal as selling at an ask price of 101.25.
If you buy the bond from a dealer today, what price will you pay for it?
25. Assume that two firms issue bonds with the following characteristics. Both bonds are issued
at par.
Issue size
Maturity
Coupon
Collateral
Callable
Call price
Sinking fund
ABC Bonds
XYZ Bonds
$1.2 billion
10 years*
9%
First mortgage
Not callable
None
None
$150 million
20 years
10%
General debenture
In 10 years
110
Starting in 5 years
*Bond is extendible at the discretion of the bondholder for an additional 10 years.
Ignoring credit quality, identify four features of these issues that might account for the lower
coupon on the ABC debt. Explain.
26. An investor believes that a bond may temporarily increase in credit risk. Which of the following
would be the most liquid method of exploiting this?
a. The purchase of a credit default swap.
b. The sale of a credit default swap.
c. The short sale of the bond.
27. Which of the following most accurately describes the behavior of credit default swaps?
a. When credit risk increases, swap premiums increase.
b. When credit and interest rate risk increases, swap premiums increase.
c. When credit risk increases, swap premiums increase, but when interest rate risk increases,
swap premiums decrease.
28. What would be the likely effect on the yield to maturity of a bond resulting from:
a. An increase in the issuing firm’s times-interest-earned ratio.
b. An increase in the issuing firm’s debt-to-equity ratio.
c. An increase in the issuing firm’s quick ratio.
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CHAPTER 14
Bond Prices and Yields
483
Issue size
Original maturity
Current price (% of par)
Current coupon
Coupon adjusts
Coupon reset rule
Callable
Call price
Sinking fund
Yield to maturity
Price range since issued
9% Coupon Notes
Floating-Rate Note
$250 million
20 years
93
9%
Fixed coupon
—
10 years after issue
106
None
9.9%
$85–$112
$280 million
10 years
98
8%
Every year
1-year T-bill rate 1 2%
10 years after issue
102.50
None
—
$97–$102
Why is the price range greater for the 9% coupon bond than the floating-rate note?
What factors could explain why the floating-rate note is not always sold at par value?
Why is the call price for the floating-rate note not of great importance to investors?
Is the probability of a call for the fixed-rate note high or low?
If the firm were to issue a fixed-rate note with a 15-year maturity, what coupon rate would it
need to offer to issue the bond at par value?
f. Why is an entry for yield to maturity for the floating-rate note not appropriate?
a.
b.
c.
d.
e.
30. Masters Corp. issues two bonds with 20-year maturities. Both bonds are callable at $1,050. The
first bond is issued at a deep discount with a coupon rate of 4% and a price of $580 to yield
8.4%. The second bond is issued at par value with a coupon rate of 8¾%.
a. What is the yield to maturity of the par bond? Why is it higher than the yield of the discount
bond?
b. If you expect rates to fall substantially in the next 2 years, which bond would you prefer to hold?
c. In what sense does the discount bond offer “implicit call protection”?
31. A newly issued bond pays its coupons once annually. Its coupon rate is 5%, its maturity is
20 years, and its yield to maturity is 8%.
a. Find the holding-period return for a 1-year investment period if the bond is selling at a yield
to maturity of 7% by the end of the year.
b. If you sell the bond after 1 year, what taxes will you owe if the tax rate on interest income is
40% and the tax rate on capital gains income is 30%? The bond is subject to original-issue
discount tax treatment.
c. What is the after-tax holding-period return on the bond?
d. Find the realized compound yield before taxes for a 2-year holding period, assuming that
(1) you sell the bond after 2 years, (2) the bond yield is 7% at the end of the second year, and
(3) the coupon can be reinvested for 1 year at a 3% interest rate.
e. Use the tax rates in (b) above to compute the after-tax 2-year realized compound yield.
Remember to take account of OID tax rules.
Challenge
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29. A large corporation issued both fixed- and floating-rate notes 5 years ago, with terms given in
the following table:
1. Leaf Products may issue a 10-year maturity fixed-income security, which might include a sinking
fund provision and either refunding or call protection.
a. Describe a sinking fund provision.
b. Explain the impact of a sinking fund provision on:
i. The expected average life of the proposed security.
ii. Total principal and interest payments over the life of the proposed security.
c. From the investor’s point of view, explain the rationale for demanding a sinking fund provision.
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PART IV
Fixed-Income Securities
2. Bonds of Zello Corporation with a par value of $1,000 sell for $960, mature in 5 years, and have
a 7% annual coupon rate paid semiannually.
a. Calculate the:
i. Current yield.
ii. Yield to maturity (to the nearest whole percent, i.e., 3%, 4%, 5%, etc.).
iii. Realized compound yield for an investor with a 3-year holding period and a reinvestment
rate of 6% over the period. At the end of 3 years the 7% coupon bonds with 2 years remaining will sell to yield 7%.
b. Cite one major shortcoming for each of the following fixed-income yield measures:
i. Current yield.
ii. Yield to maturity.
iii. Realized compound yield.
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3. On May 30, 2012, Janice Kerr is considering one of the newly issued 10-year AAA corporate
bonds shown in the following exhibit.
Description
Coupon
Price
Callable
Call Price
Sentinal, due May 30, 2022
Colina, due May 30, 2022
6.00%
6.20%
100
100
Noncallable
Currently callable
NA
102
a. Suppose that market interest rates decline by 100 basis points (i.e., 1%). Contrast the effect of
this decline on the price of each bond.
b. Should Kerr prefer the Colina over the Sentinal bond when rates are expected to rise or to fall?
c. What would be the effect, if any, of an increase in the volatility of interest rates on the prices
of each bond?
4. A convertible bond has the following features:
Coupon
Maturity
Market price of bond
Market price of underlying common stock
Annual dividend
Conversion ratio
5.25%
June 15, 2030
$77.50
$28.00
$1.20
20.83 shares
Calculate the conversion premium for this bond.
5. a. Explain the impact on the offering yield of adding a call feature to a proposed bond issue.
b. Explain the impact on the bond’s expected life of adding a call feature to a proposed bond
issue.
c. Describe one advantage and one disadvantage of including callable bonds in a portfolio.
6. a. An investment in a coupon bond will provide the investor with a return equal to the bond’s
yield to maturity at the time of purchase if:
i. The bond is not called for redemption at a price that exceeds its par value.
ii. All sinking fund payments are made in a prompt and timely fashion over the life of the issue.
iii. The reinvestment rate is the same as the bond’s yield to maturity and the bond is held until
maturity.
iv. All of the above.
b. A bond with a call feature:
i. Is attractive because the immediate receipt of principal plus premium produces a high return.
ii. Is more apt to be called when interest rates are high because the interest savings will be
greater.
iii. Will usually have a higher yield to maturity than a similar noncallable bond.
iv. None of the above.
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Bond Prices and Yields
485
c. In which one of the following cases is the bond selling at a discount?
i. Coupon rate is greater than current yield, which is greater than yield to maturity.
ii. Coupon rate, current yield, and yield to maturity are all the same.
iii. Coupon rate is less than current yield, which is less than yield to maturity.
iv. Coupon rate is less than current yield, which is greater than yield to maturity.
d. Consider a 5-year bond with a 10% coupon that has a present yield to maturity of 8%. If interest rates remain constant, 1 year from now the price of this bond will be:
i. Higher.
ii. Lower.
iii. The same.
iv. Par.
1. Go to the Web site of Standard & Poor’s at www.standardandpoors.com. Look for
Rating Services (Find a Rating). Find the ratings on bonds of at least 10 companies.
Try to choose a sample with a wide range of ratings. Then go to a Web site such as
money.msn.com or finance.yahoo.com and obtain, for each firm, as many of the
financial ratios tabulated in Table 14.3 as you can find. Which ratios seem to best
explain credit ratings?
2. At www.bondsonline.com review the Industrial Spreads for various ratings (click the
links on the left-side menus to follow the links to Today’s Markets, Corporate Bond
Spreads). These are spreads above U.S. Treasuries of comparable maturities. What factors tend to explain the yield differences? How might these yield spreads differ during
an economic boom versus a recession?
SOLUTIONS TO CONCEPT CHECKS
1. The callable bond will sell at the lower price. Investors will not be willing to pay as much if they
know that the firm retains a valuable option to reclaim the bond for the call price if interest rates
fall.
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E-INVESTMENTS EXERCISES
2. At a semiannual interest rate of 3%, the bond is worth $40 3 Annuity factor (3%, 60) 1 $1,000 3
PV factor(3%, 60) 5 $1,276.76, which results in a capital gain of $276.76. This exceeds the
capital loss of $189.29 (i.e., $1,000 2 $810.71) when the semiannual interest rate increased to 5%.
3. Yield to maturity exceeds current yield, which exceeds coupon rate. Take as an example the 8%
coupon bond with a yield to maturity of 10% per year (5% per half-year). Its price is $810.71, and
therefore its current yield is 80/810.71 5 .0987, or 9.87%, which is higher than the coupon rate
but lower than the yield to maturity.
4. a. The bond with the 6% coupon rate currently sells for 30 3 Annuity factor(3.5%, 20) 1 1,000
3 PV factor(3.5%, 20) 5 $928.94. If the interest rate immediately drops to 6% (3% per
half-year), the bond price will rise to $1,000, for a capital gain of $71.06, or 7.65%. The 8%
coupon bond currently sells for $1,071.06. If the interest rate falls to 6%, the present value of
the scheduled payments increases to $1,148.77. However, the bond will be called at $1,100,
for a capital gain of only $28.94, or 2.70%.
b. The current price of the bond can be derived from its yield to maturity. Using your calculator,
set: n 5 40 (semiannual periods); payment 5 $45 per period; future value 5 $1,000; interest
rate 5 4% per semiannual period. Calculate present value as $1,098.96. Now we can calculate
yield to call. The time to call is 5 years, or 10 semiannual periods. The price at which the
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bond will be called is $1,050. To find yield to call, we set: n 5 10 (semiannual periods);
payment 5 $45 per period; future value 5 $1,050; present value 5 $1,098.96. Calculate yield
to call as 3.72%.
5. Price 5 $70 3 Annuity factor(8%, 1) 1 $1,000 3 PV factor(8%, 1) 5 $990.74
Rate of return to investor 5
$70 1 ($990.74 2 $982.17)
5 .080 5 8%
$982.17
6. By year-end, remaining maturity is 29 years. If the yield to maturity were still 8%, the bond
would still sell at par and the holding-period return would be 8%. At a higher yield, price
and return will be lower. Suppose, for example, that the yield to maturity rises to 8.5%. With
annual payments of $80 and a face value of $1,000, the price of the bond will be $946.70
[n 5 29; i 5 8.5%; PMT 5 $80; FV 5 $1,000]. The bond initially sold at $1,000 when issued at
the start of the year. The holding-period return is
HPR 5
80 1 (946.70 2 1,000)
5 .0267 5 2.67%
1,000
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which is less than the initial yield to maturity of 8%.
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7. At the lower yield, the bond price will be $631.67 [n 5 29, i 5 7%, FV 5 $1,000, PMT 5 $40].
Therefore, total after-tax income is
Coupon
$40 3 (1 2 .38)
Imputed interest
($553.66 2 $549.69) 3 (1 2 .38)
Capital gains
($631.67 2 $553.66) 3 (1 2 .20)
Total income after taxes
Rate of return 5 89.67/549.69 5 .163 5 16.3%.
5 $24.80
5 2.46
5 62.41
$89.67
8. It should receive a negative coefficient. A high ratio of liabilities to assets is a bad omen for a
firm, and that should lower its credit rating.
9. The coupon payment is $45. There are 20 semiannual periods. The final payment is assumed
to be $500. The present value of expected cash flows is $650. The expected yield to maturity is
6.317% semiannual or annualized, 12.63%, bond equivalent yield.
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CHAPTER FIFTEEN
The Term Structure
of Interest Rates
IN CHAPTER 14 we assumed for the sake
of simplicity that the same constant interest rate is used to discount cash flows of any
maturity. In the real world this is rarely the
case. We have seen, for example, that in
2012 short-term Treasury bonds and notes
carried yields to maturity less than 1% while
the longest-term bonds offered yields of
about 2.5%. At the time that these bond
prices were quoted, anyway, the longer-term
securities had higher yields. This, in fact, is
a typical pattern, but as we shall see below,
the relationship between time to maturity
and yield to maturity can vary dramatically
from one period to another. In this chapter
we explore the pattern of interest rates for
different-term assets. We attempt to identify
the factors that account for that pattern and
determine what information may be derived
from an analysis of the so-called term structure of interest rates, the structure of interest
rates for discounting cash flows of different
maturities.
We demonstrate how the prices of Treasury
bonds may be derived from prices and yields
of stripped zero-coupon Treasury securities.
We also examine the extent to which the term
structure reveals market-consensus forecasts
of future interest rates and how the presence
of interest rate risk may affect those inferences. Finally, we show how traders can use
the term structure to compute forward rates
that represent interest rates on “forward,” or
deferred, loans, and consider the relationship
between forward rates and future interest
rates.
15.1 The Yield Curve
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Figure 14.1 demonstrated that bonds of different maturities typically sell at different yields
to maturity. When these bond prices and yields were compiled, long-term bonds sold at
higher yields than short-term bonds. Practitioners commonly summarize the relationship between yield and maturity graphically in a yield curve, which is a plot of yield to
maturity as a function of time to maturity. The yield curve is one of the key concerns of
fixed-income investors. It is central to bond valuation and, as well, allows investors to
gauge their expectations for future interest rates against those of the market. Such a comparison is often the starting point in the formulation of a fixed-income portfolio strategy.
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Treasury Yield Curve
Treasury Yield Curve
Yields as of 4:30 P.M. Eastern time Yields as of 4:30 P.M. Eastern time
Treasury Yield Curve
Yields as of 4:30 P.M. Eastern time
Treasury Yield Curve
Yields as of 4:30 P.M. Eastern time
Percent
6.0
Percent
6.0
Percent
6.50
Percent
8.60
5.0
5.0
6.25
8.40
4.0
4.0
8.20
6.00
8.00
3.0
3.0
5.75
2.0
2.0
1.0
1.0
0.0
1 3
6 2
5 10
30
Maturities
Months Year
A. (January 2006)
Flat Yield Curve
7.80
5.50
7.60
7.40
5.25
1
3
6 1 2 56 710
30
Months Year
Maturities
B. (December 2012)
Rising Yield Curve
3
6
1 2 3 5 10
30
Months Year
Maturities
C. (September 11, 2000)
Inverted Yield Curve
3
6
1 2 3 45 710
30
Months Year
Maturities
D. (October 4, 1989)
Hump-Shaped Yield Curve
Figure 15.1 Treasury yield curves
Source: Various editions of The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1989, 2000, 2006, and 2012
Dow Jones & Company, Inc. All Rights Reserved Worldwide.
In 2012, the yield curve was rising, with long-term bonds offering yields higher than
those of short-term bonds. But the relationship between yield and maturity can vary
widely. Figure 15.1 illustrates yield curves of several different shapes. Panel A is the
almost-flat curve of early 2006. Panel B is a more typical upward-sloping curve from
2012. Panel C is a downward-sloping or “inverted” curve, and panel D is hump-shaped,
first rising and then falling.
Bond Pricing
If yields on different-maturity bonds are not all equal, how should we value coupon bonds
that make payments at many different times? For example, suppose that yields on zerocoupon Treasury bonds of different maturities are as given in Table 15.1. The table tells us
that zero-coupon bonds with 1-year maturity sell at a yield to maturity of y1 5 5%, 2-year
zeros sell at yields of y2 5 6%, and 3-year zeros sell at yields of y3 5 7%. Which of these
rates should we use to discount bond cash flows? The answer: all of them. The trick is to
consider each bond cash flow—either coupon or principal payment—as at least potentially
sold off separately as a stand-alone zero-coupon bond.
Recall the Treasury STRIPS program we introduced in the last chapter (Section 14.4).
Stripped Treasuries are zero-coupon bonds created by selling each coupon or principal
payment from a whole Treasury bond as a separate cash flow. For example, a 1-year
Table 15.1
Prices and yields to maturity
on zero-coupon bonds
($1,000 face value)
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Maturity (years)
Yield to Maturity (%)
Price
1
2
3
4
5%
6
7
8
$952.38 5 $1,000/1.05
$890.00 5 $1,000/1.062
$816.30 5 $1,000/1.073
$735.03 5 $1,000/1.084
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489
maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero
(by selling the first coupon payment as a stand-alone security) and a 12-month zero
(corresponding to payment of final coupon and principal). Treasury stripping suggests
exactly how to value a coupon bond. If each cash flow can be (and in practice often is) sold
off as a separate security, then the value of the whole bond should be the same as the value
of its cash flows bought piece by piece in the STRIPS market.
What if it weren’t? Then there would be easy profits to be made. For example, if
investment bankers ever noticed a bond selling for less than the amount at which the
sum of its parts could be sold, they would buy the bond, strip it into stand-alone zerocoupon securities, sell off the stripped cash flows, and profit by the price difference. If
the bond were selling for more than the sum of the values of its individual cash flows,
they would run the process in reverse: buy the individual zero-coupon securities in the
STRIPS market, reconstitute (i.e., reassemble) the cash flows into a coupon bond, and
sell the whole bond for more than the cost of the pieces. Both bond stripping and bond
reconstitution offer opportunities for arbitrage—the exploitation of mispricing among
two or more securities to clear a riskless economic profit. Any violation of the Law of
One Price, that identical cash flow bundles must sell for identical prices, gives rise to
arbitrage opportunities.
Now, we know how to value each stripped cash flow. We simply look up its appropriate
discount rate in The Wall Street Journal. Because each coupon payment matures at a
different time, we discount by using the yield appropriate to its particular maturity—this
is the yield on a Treasury strip maturing at the time of that cash flow. We can illustrate
with an example.
Example 15.1
Valuing Coupon Bonds
Suppose the yields on stripped Treasuries are as given in Table 15.1, and we wish to
value a 10% coupon bond with a maturity of 3 years. For simplicity, assume the bond
makes its payments annually. Then the first cash flow, the $100 coupon paid at the end
of the first year, is discounted at 5%; the second cash flow, the $100 coupon at the
end of the second year, is discounted at 6%; and the final cash flow consisting of the final
coupon plus par value, or $1,100, is discounted at 7%. The value of the coupon bond
is therefore
1,100
100
100
1
1
5 95.238 1 89.000 1 897.928 5 $1,082.17
2
1.05
1.06
1.073
Calculate the yield to maturity of the coupon bond in Example 15.1, and you may be
surprised. Its yield to maturity is 6.88%; so while its maturity matches that of the 3-year
zero in Table 15.1, its yield is a bit lower.1 This reflects the fact that the 3-year coupon
bond may usefully be thought of as a portfolio of three implicit zero-coupon bonds, one
corresponding to each cash flow. The yield on the coupon bond is then an amalgam of the
yields on each of the three components of the “portfolio.” Think about what this means: If
their coupon rates differ, bonds of the same maturity generally will not have the same yield
to maturity.
What then do we mean by “the” yield curve? In fact, in practice, traders refer to
several yield curves. The pure yield curve refers to the curve for stripped, or zero-coupon,
1
Remember that the yield to maturity of a coupon bond is the single interest rate at which the present value
of cash flows equals market price. To calculate the bond’s yield to maturity on your calculator or spreadsheet,
set n 5 3; price 5 21,082.17; future value 5 1,000; payment 5 100. Then compute the interest rate.
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Treasuries. In contrast, the on-the-run yield curve refers to the plot of yield as a function
of maturity for recently issued coupon bonds selling at or near par value. As we’ve just seen,
there may be significant differences in these two curves. The yield curves published in the
financial press, for example, in Figure 15.1, are typically on-the-run curves. On-the-run
Treasuries have the greatest liquidity, so traders have keen interest in their yield curve.
CONCEPT CHECK
15.1
Calculate the price and yield to maturity of a 3-year bond with a coupon rate of 4% making annual
coupon payments. Does its yield match that of either the 3-year zero or the 10% coupon bond considered
in Example 15.1? Why is the yield spread between the 4% bond and the zero smaller than the yield spread
between the 10% bond and the zero?
15.2 The Yield Curve and Future Interest Rates
We’ve told you what the yield curve is, but we haven’t yet had much to say about where
it comes from. For example, why is the curve sometimes upward-sloping and other times
downward-sloping? How do expectations for the evolution of interest rates affect the shape
of today’s yield curve?
These questions do not have simple answers, so we will begin with an admittedly idealized framework, and then extend the discussion to more realistic settings. To start, consider
a world with no uncertainty, specifically, one in which all investors already know the path
of future interest rates.
The Yield Curve under Certainty
If interest rates are certain, what should we make of the fact that the yield on the 2-year
zero coupon bond in Table 15.1 is greater than that on the 1-year zero? It can’t be that one
bond is expected to provide a higher rate of return than the other. This would not be possible in a certain world—with no risk, all bonds (in fact, all securities!) must offer identical
returns, or investors will bid up the price of the high-return bond until its rate of return is
no longer superior to that of other bonds.
Instead, the upward-sloping yield curve is evidence that short-term rates are going to
be higher next year than they are now. To see why, consider two 2-year bond strategies.
The first strategy entails buying the 2-year zero offering a 2-year yield to maturity of
y2 5 6%, and holding it until maturity. The zero with face value $1,000 is purchased today
for $1,000/1.062 5 $890 and matures in 2 years to $1,000. The total 2-year growth factor
for the investment is therefore $1,000/$890 5 1.062 5 1.1236.
Now consider an alternative 2-year strategy. Invest the same $890 in a 1-year zerocoupon bond with a yield to maturity of 5%. When that bond matures, reinvest the proceeds in another 1-year bond. Figure 15.2 illustrates these two strategies. The interest rate
that 1-year bonds will offer next year is denoted as r2.
Remember, both strategies must provide equal returns—neither entails any risk.
Therefore, the proceeds after 2 years to either strategy must be equal:
Buy and hold 2-year zero 5 Roll over 1-year bonds
$890 3 1.062 5 $890 3 1.05 3 (1 1 r2 )
We find next year’s interest rate by solving 1 1 r2 5 1.062/1.05 5 1.0701, or r2 5 7.01%. So
while the 1-year bond offers a lower yield to maturity than the 2-year bond (5% versus 6%),
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CHAPTER 15
0
1
2
The Term Structure of Interest Rates
491
Time Line
Alternative 1: Buy and
hold 2-year zero
2-Year Investment
$890 ×1.062 = $1000
$890
Alternative 2: Buy a 1-year
zero, and reinvest proceeds in
another 1-year zero
1-Year Investment
$890
1-Year Investment
$890 × 1.05
= $934.50
$934.50(1 + r2)
Figure 15.2 Two 2-year investment programs
we see that it has a compensating advantage: It allows you to roll over your funds into
another short-term bond next year when rates will be higher. Next year’s interest rate is
higher than today’s by just enough to make rolling over 1-year bonds equally attractive as
investing in the 2-year bond.
To distinguish between yields on long-term bonds versus short-term rates that will be
available in the future, practitioners use the following terminology. They call the yield to
maturity on zero-coupon bonds the spot rate, meaning the rate that prevails today for a
time period corresponding to the zero’s maturity. In contrast, the short rate for a given time
interval (e.g., 1 year) refers to the interest rate for that interval available at different points in
time. In our example, the short rate today is 5%, and the short rate next year will be 7.01%.
Not surprisingly, the 2-year spot rate is an average of today’s short rate and next year’s
short rate. But because of compounding, that average is a geometric one.2 We see this by
again equating the total return on the two competing 2-year strategies:
(1 1 y2 )2 5 (1 1 r1 ) 3 (1 1 r2 )
1 1 y2 5 3(1 1 r1 ) 3 (1 1 r2 )4 1/2
(15.1)
Equation 15.1 begins to tell us why the yield curve might take on different shapes at
different times. When next year’s short rate, r2, is greater than this year’s short rate, r1, the
average of the two rates is higher than today’s rate, so y2 . r1 and the yield curve slopes
upward. If next year’s short rate were less than r1, the yield curve would slope downward.
2
In an arithmetic average, we add n numbers and divide by n. In a geometric average, we multiply n numbers and
take the nth root.
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Thus, at least in part, the yield curve reflects the market’s assessments of coming interest rates.
The following example uses a similar analysis to find the short rate that will prevail in year 3.
Example 15.2
Finding a Future Short Rate
Now we compare two 3-year strategies. One is to buy a 3-year zero, with a yield to
maturity from Table 15.1 of 7%, and hold it until maturity. The other is to buy a 2-year
zero yielding 6%, and roll the proceeds into a 1-year bond in year 3, at the short rate r3.
The growth factor for the invested funds under each policy will be:
Buy and hold 3-year zero 5 Buy 2-year zero; roll proceeds into 1-year bond
(1 1 y3)3 5 (1 1 y2)2 3 (1 1 r3)
1.073 5 1.062 3 (1 1 r3)
which implies that r3 5 1.073/1.062 2 1 5 .09025 5 9.025%. Again, notice that the yield
on the 3-year bond reflects a geometric average of the discount factors for the next 3 years:
1 1 y3 5 [(1 1 r1) 3 (1 1 r2) 3 (1 1 r3)]1/3
1.07 5 [1.05 3 1.0701 3 1.09025]1/3
We conclude that the yield or spot rate on a long-term bond reflects the path of short
rates anticipated by the market over the life of the bond.
CONCEPT CHECK
15.2
Use Table 15.1 to find the short rate that will prevail in the fourth year. Confirm that the discount factor
on the 4-year zero is a geometric average of 11 the short rates in the next 4 years.
Figure 15.3 summarizes the results of our analysis and emphasizes the difference
between short rates and spot rates. The top line presents the short rates for each year. The
lower lines present spot rates—or, equivalently, yields to maturity on zero-coupon bonds
for different holding periods—extending from the present to each relevant maturity date.
Holding-Period Returns
We’ve argued that the multiyear cumulative returns on all of our competing bonds ought
to be equal. What about holding-period returns over shorter periods such as a year? You
might think that bonds selling at higher yields to maturity will offer higher 1-year returns,
but this is not the case. In fact, once you stop to think about it, it’s clear that this cannot be
true. In a world of certainty, all bonds must offer identical returns, or investors will flock
to the higher-return securities, bidding up their prices, and reducing their returns. We can
illustrate by using the bonds in Table 15.1.
Example 15.3
Holding-Period Returns on Zero-Coupon Bonds
The 1-year bond in Table 15.1 can be bought today for $1,000/1.05 5 $952.38 and
will mature to its par value in 1 year. It pays no coupons, so total investment income is
just its price appreciation, and its rate of return is ($1,000 2 $952.38)/$952.38 5 .05.
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493
The 2-year bond can be bought for $1,000/1.062 5 $890.00. Next year, the bond will
have a remaining maturity of 1 year and the 1-year interest rate will be 7.01%. Therefore,
its price next year will be $1,000/1.0701 5 $934.49, and its 1-year holding-period rate
of return will be ($934.49 2 $890.00)/$890.00 5 .05, for an identical 5% rate of return.
CONCEPT CHECK
15.3
Show that the rate of return on the 3-year zero in Table 15.1 also will be 5%. Hint: Next year, the bond will
have a maturity of 2 years. Use the short rates derived in Figure 15.3 to compute the 2-year spot rate that
will prevail a year from now.
Forward Rates
The following equation generalizes our approach to inferring a future short rate from the
yield curve of zero-coupon bonds. It equates the total return on two n-year investment
strategies: buying and holding an n-year zero-coupon bond versus buying an (n 2 1)-year
zero and rolling over the proceeds into a 1-year bond.
(1 1 yn )n 5 (1 1 yn21 )n21 3 (1 1 rn )
(15.2)
where n denotes the period in question, and yn is the yield to maturity of a zero-coupon
bond with an n-period maturity. Given the observed yield curve, we can solve Equation 15.2
for the short rate in the last period:
(1 1 rn ) 5
(1 1 yn )n
(1 1 yn21 )n21
1
2
3
4
r1 = 5%
r2 = 7.01%
r3 = 9.025%
r4 = 11.06%
(15.3)
Year
Short Rate in Each Year
Current Spot Rates
(Yields to Maturity)
for Various Maturities
y1 = 5%
1-Year Investment
y2 = 6%
2-Year Investment
y3 = 7%
3-Year Investment
y4 = 8%
4-Year Investment
Figure 15.3 Short rates versus spot rates
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Equation 15.3 has a simple interpretation. The numerator on the right-hand side is the
total growth factor of an investment in an n-year zero held until maturity. Similarly,
the denominator is the growth factor of an investment in an (n 2 1)-year zero. Because the
former investment lasts for one more year than the latter, the difference in these growth
factors must be the rate of return available in year n when the (n 2 1)-year zero can be
rolled over into a 1-year investment.
Of course, when future interest rates are uncertain, as they are in reality, there is no
meaning to inferring “the” future short rate. No one knows today what the future interest
rate will be. At best, we can speculate as to its expected value and associated uncertainty.
Nevertheless, it still is common to use Equation 15.3 to investigate the implications of the
yield curve for future interest rates. Recognizing that future interest rates are uncertain,
we call the interest rate that we infer in this matter the forward interest rate rather than
the future short rate, because it need not be the interest rate that actually will prevail at the
future date.
If the forward rate for period n is denoted fn, we then define fn by the equation
(1 1 fn) 5
(1 1 yn)n
(1 1 yn21)n21
(15.4)
Equivalently, we may rewrite Equation 15.4 as
(1 1 yn)n 5 (1 1 yn21)n21(1 1 fn)
(15.5)
In this formulation, the forward rate is defined as the “break-even” interest rate that equates
the return on an n-period zero-coupon bond to that of an (n 2 1)-period zero-coupon bond
rolled over into a 1-year bond in year n. The actual total returns on the two n-year strategies will be equal if the short interest rate in year n turns out to equal fn.
Example 15.4
Forward Rates
Suppose a bond trader uses the data presented in Table 15.1. The forward rate for year 4
would be computed as
1 1 f4 5
(1 1 y4)4
(1 1 y3)3
5
1.084
5 1.1106
1.073
Therefore, the forward rate is f4 5 .1106, or 11.06%.
We emphasize again that the interest rate that actually will prevail in the future need
not equal the forward rate, which is calculated from today’s data. Indeed, it is not even
necessarily the case that the forward rate
equals the expected value of the future
CONCEPT CHECK 15.4
short interest rate. This is an issue that we
address in the next section. For now, howYou’ve been exposed to many “rates” in the last few pages.
ever, we note that forward rates equal future
Explain the differences between spot rates, short rates, and
forward rates.
short rates in the special case of interest rate
certainty.
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eXcel APPLICATIONS: Spot and Forward Yields
T
he spreadsheet below (available at www.mhhe.com/
bkm) can be used to estimate prices and yields of coupon
bonds and to calculate the forward rates for both single-year
and multiyear periods. Spot yields are derived for the yield
curve of bonds that are selling at their par value, also referred
to as the current coupon or “on-the-run” bond yield curve.
The spot rates for each maturity date are used to calculate the present value of each period’s cash flow. The sum
of these cash flows is the price of the bond. Given its price,
the bond’s yield to maturity can then be computed. If you
were to err and use the yield to maturity of the on-the-run
A
B
C
D
Forward Rate Calculations
56
57
Spot Rate
1-yr for.
2-yr for.
58
59 Period
60
1 8.0000%
7.9792%
7.6770%
61
2 7.9896%
7.3757%
6.9205%
62
3 7.7846%
6.4673%
6.2695%
63
4 7.4537%
6.0720%
6.3065%
5 7.1760%
6.5414%
6.2920%
64
6 7.0699%
6.0432%
6.4299%
65
7 6.9227%
6.8181%
5.8904%
66
8 6.9096%
4.9707%
5.3993%
67
bond to discount each of the bond’s coupon payments, you
could find a significantly different price. That difference is
calculated in the worksheet.
Excel Questions
1. Change the spot rate in the spreadsheet to 8% for all maturities.
The forward rates will all be 8%. Why is this not surprising?
2. The spot rates in column B decrease for longer maturities,
and the forward rates decrease even more rapidly with
maturity. What happens to the pattern of forward rates if
you input spot rates that increase with maturity? Why?
E
F
G
H
3-yr for.
4-yr for.
5-yr for.
6-yr for.
7.2723%
6.6369%
6.3600%
6.2186%
6.4671%
5.9413%
5.8701%
5.2521%
6.9709%
6.6131%
6.2807%
6.3682%
6.0910%
5.9134%
5.6414%
5.2209%
6.8849%
6.4988%
6.3880%
6.0872%
6.0387%
5.7217%
5.5384%
5.1149%
6.7441%
6.5520%
6.1505%
6.0442%
5.8579%
5.6224%
5.3969%
5.1988%
15.3 Interest Rate Uncertainty and Forward Rates
Let us turn now to the more difficult analysis of the term structure when future interest
rates are uncertain. We have argued so far that, in a certain world, different investment
strategies with common terminal dates must provide equal rates of return. For example,
two consecutive 1-year investments in zeros would need to offer the same total return as an
equal-sized investment in a 2-year zero. Therefore, under certainty,
(1 1 r1)(1 1 r2 ) 5 (1 1 y2 )2
(15.6)
What can we say when r2 is not known today?
For example, suppose that today’s rate is r1 5 5% and that the expected short rate for
the following year is E(r2) 5 6%. If investors cared only about the expected value of the
interest rate, then the yield to maturity on a 2-year zero would be determined by using
the expected short rate in Equation 15.6:
(1 1 y2 )2 5 (1 1 r1 ) 3 3 1 1 E(r2 ) 4 5 1.05 3 1.06
The price of a 2-year zero would be $1,000/(1 1 y2)2 5 $1,000/(1.05 3 1.06) 5 $898.47.
But now consider a short-term investor who wishes to invest only for 1 year. She can
purchase the 1-year zero for $1,000/1.05 5 $952.38, and lock in a riskless 5% return
because she knows that at the end of the year, the bond will be worth its maturity value of
$1,000. She also can purchase the 2-year zero. Its expected rate of return also is 5%: Next
year, the bond will have 1 year to maturity, and we expect that the 1-year interest rate will
be 6%, implying a price of $943.40 and a holding-period return of 5%.
495
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PART IV
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But the rate of return on the 2-year bond is risky. If next year’s interest rate turns out to
be above expectations, that is, greater than 6%, the bond price will be below $943.40; conversely if r2 turns out to be less than 6%, the bond price will exceed $943.40. Why should
this short-term investor buy the risky 2-year bond when its expected return is 5%, no better
than that of the risk-free 1-year bond? Clearly, she would not hold the 2-year bond unless
it offered a higher expected rate of return. This requires that the 2-year bond sell at a price
lower than the $898.47 value we derived when we ignored risk.
Example 15.5
Bond Prices and Forward Rates with Interest Rate Risk
Suppose that most investors have short-term horizons and therefore are willing to hold
the 2-year bond only if its price falls to $881.83. At this price, the expected holdingperiod return on the 2-year bond is 7% (because 943.40/881.83 5 1.07). The risk premium of the 2-year bond, therefore, is 2%; it offers an expected rate of return of 7%
versus the 5% risk-free return on the 1-year bond. At this risk premium, investors are
willing to bear the price risk associated with interest rate uncertainty.
When bond prices reflect a risk premium, however, the forward rate, f2, no longer
equals the expected short rate, E(r2). Although we have assumed that E(r2) 5 6%, it
is easy to confirm that f2 5 8%. The yield to maturity on the 2-year zeros selling at
$881.83 is 6.49%, and
1 1 f2 5
(1 1 y2)2
1 1 y1
5
1.06492
5 1.08
1.05
The result in Example 15.5—that the forward rate exceeds the expected short rate—
should not surprise us. We defined the forward rate as the interest rate that would need to
prevail in the second year to make the long- and short-term investments equally attractive,
ignoring risk. When we account for risk, it is clear that short-term investors will shy away
from the long-term bond unless it offers an expected return greater than that of the 1-year
bond. Another way of putting this is to say that investors will require a risk premium to hold
the longer-term bond. The risk-averse investor would be willing to hold the long-term bond
only if the expected value of the short rate is less than the break-even value, f2, because the
lower the expectation of r2, the greater the anticipated return on the long-term bond.
Therefore, if most individuals are short-term investors, bonds must have prices that
make f2 greater than E(r2). The forward rate will embody a premium compared with the
expected future short-interest rate. This liquidity premium compensates short-term investors for the uncertainty about the price at which they will be able to sell their long-term
bonds at the end of the year.3
CONCEPT CHECK
15.5
Suppose that the required liquidity premium for the short-term investor is 1%.
What must E(r2) be if f2 is 7%?
Perhaps surprisingly, we also can imagine scenarios in which long-term bonds can be
perceived by investors to be safer than short-term bonds. To see how, we now consider a
“long-term” investor, who wishes to invest for a full 2-year period. Suppose that the investor
3
Liquidity refers to the ability to sell an asset easily at a predictable price. Because long-term bonds have greater
price risk, they are considered less liquid in this context and thus must offer a premium.
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CHAPTER 15
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497
can purchase a $1,000 par value 2-year zero-coupon bond for $890 and lock in a guaranteed
yield to maturity of y2 5 6%. Alternatively, the investor can roll over two 1-year investments. In this case an investment of $890 would grow in 2 years to 890 3 1.05 3 (1 1 r2),
which is an uncertain amount today because r2 is not yet known. The break-even year-2
interest rate is, once again, the forward rate, 7.01%, because the forward rate is defined as
the rate that equates the terminal value of the two investment strategies.
The expected value of the payoff of the rollover strategy is 890 3 1.05 3 [1 1 E(r2)].
If E(r2) equals the forward rate, f2, then the expected value of the payoff from the rollover
strategy will equal the known payoff from the 2-year-maturity bond strategy.
Is this a reasonable presumption? Once again, it is only if the investor does not care
about the uncertainty surrounding the final value of the rollover strategy. Whenever that
risk is important, however, the long-term investor will not be willing to engage in the
rollover strategy unless its expected return exceeds that of the 2-year bond. In this case the
investor would require that
(1.05)3 1 1 E(r2)4 . (1.06)2 5 (1.05)(1 1 f2)
which implies that E(r2) exceeds f2. The investor would require that the expected value of
next year’s short rate exceed the forward rate.
Therefore, if all investors were long-term investors, no one would be willing to hold
short-term bonds unless those bonds offered a reward for bearing interest rate risk. In this
situation bond prices would be set at levels such that rolling over short bonds resulted in
greater expected return than holding long bonds. This would cause the forward rate to be
less than the expected future spot rate.
For example, suppose that in fact E(r2) 5 8%. The liquidity premium therefore is negative: f2 2 E(r2) 5 7.01% 2 8% 5 2.99%. This is exactly opposite from the conclusion that
we drew in the first case of the short-term investor. Clearly, whether forward rates will equal
expected future short rates depends on investors’ readiness to bear interest rate risk, as well
as their willingness to hold bonds that do not correspond to their investment horizons.
15.4 Theories of the Term Structure
The Expectations Hypothesis
The simplest theory of the term structure is the expectations hypothesis. A common version of this hypothesis states that the forward rate equals the market consensus expectation of the future short interest rate; that is, f2 5 E(r2), and liquidity premiums are zero.
If f2 5 E(r2), we may relate yields on long-term bonds to expectations of future interest
rates. In addition, we can use the forward rates derived from the yield curve to infer market
expectations of future short rates. For example, with (1 1 y2)2 5 (1 1 r1) 3 (1 1 f2 ) from
Equation 15.5, if the expectations hypothesis is correct we may also write that (1 1 y2)2 5
(1 1 r1) 3 [1 1 E(r2 )]. The yield to maturity would thus be determined solely by current
and expected future one-period interest rates. An upward-sloping yield curve would be
clear evidence that investors anticipate increases in interest rates.
CONCEPT CHECK
15.6
If the expectations hypothesis is valid, what can we conclude about the premiums
necessary to induce investors to hold bonds of different maturities from their
investment horizons?
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The Expectations Hypothesis and Forward Inflation Rates
Forward rates derived from conventional bonds are nominal interest rates. But using price-level-indexed bonds such
as TIPS, we can also calculate forward real interest rates.
Recall that the difference between the real rate and the
nominal rate is approximately the expected inflation rate.
Therefore, comparing real and nominal forward rates
might give us a glimpse of the market’s expectation of
future inflation rates. The real versus nominal spread is a
sort of forward inflation rate.
As part of its monetary policy, the Federal Reserve
Board periodically reduces its target federal funds rate
in an attempt to stimulate the economy. The following
page capture from a Bloomberg screen shows the minuteby-minute spread between the 5-year forward nominal
interest rate and forward real rate on one day the Fed
announced such a policy change. The spread immediately
widened at the announcement, signifying that the market
expected the more expansionary monetary policy to eventually result in a higher inflation rate. The increase in the
inflation rate implied by the graph is fairly mild, about
.05%, from about 2.53% to 2.58%, but the impact of the
announcement is very clear, and the speed of adjustment
to the announcement was impressive.
By the way, nothing limits us to nominal bonds when using the expectations hypothesis.
The nearby box points out that we can apply the theory to the term structure of real
interest rates as well, and thereby learn something about market expectations of coming
inflation rates.
Liquidity Preference
We noted earlier that short-term investors will be unwilling to hold long-term bonds unless
the forward rate exceeds the expected short interest rate, f2 . E(r2), whereas long-term
investors will be unwilling to hold short bonds unless
E(r2) . f2. In other words, both groups of investors
CONCEPT CHECK 15.7
require a premium to hold bonds with maturities different from their investment horizons. Advocates of the
The liquidity premium hypothesis also holds
liquidity preference theory of the term structure believe
that issuers of bonds prefer to issue long-term
that short-term investors dominate the market so that the
bonds to lock in borrowing costs. How would
forward rate will generally exceed the expected short
this preference contribute to a positive liquidity
rate. The excess of f2 over E(r2), the liquidity premium, is
premium?
predicted to be positive.
To illustrate the differing implications of these theories
for the term structure of interest rates, suppose the short interest rate is expected to be
constant indefinitely. Suppose that r1 5 5% and that E(r2) 5 5%, E(r3) 5 5%, and so on.
Under the expectations hypothesis the 2-year yield to maturity could be derived from the
following:
(1 1 y2)2 5 (1 1 r1) 3 1 1 E(r2) 4
5 (1.05)(1.05)
so that y2 equals 5%. Similarly, yields on bonds of all maturities would equal 5%.
498
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CHAPTER 15
The Term Structure of Interest Rates
499
In contrast, under the liquidity preference theory, f2 would exceed E(r2). To illustrate,
suppose the liquidity premium is 1%, so f2 is 6%. Then, for 2-year bonds:
(1 1 y2)2 5 (1 1 r1)(1 1 f2)
5 1.05 3 1.06 5 1.113
implying that 1 1 y2 5 1.055. Similarly, if f3 also equals 6%, then the yield on 3-year
bonds would be determined by
(1 1 y3)3 5 (1 1 r1)(1 1 f2)(1 1 f3)
5 1.05 3 1.06 3 1.06 5 1.17978
7.0
YTM
E(r)
Forward Rate
A
Interest Rate (%)
6.5
6.0
Constant Liquidity
Premium
5.5
5.0
4.5
4.0
0
1
2
7.0
3
4
YTM
5
6
Maturity
E(r)
7
8
9
10
Forward Rate
B
Interest Rate (%)
6.5
6.0
5.5
Liquidity Premium
Increases with
Maturity
5.0
4.5
4.0
0
1
2
3
4
5
6
Maturity
7
8
9
10
Figure 15.4 Yield curves. Panel A, Constant expected short rate. Liquidity
premium of 1%. Result is a rising yield curve. Panel B, Declining expected short
rates. Increasing liquidity premiums. Result is a rising yield curve despite falling
expected interest rates. (continued on next page)
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PART IV
Fixed-Income Securities
implying that 1 1 y3 5 1.0567. The plot of the yield curve in this situation would be given
as in Figure 15.4, panel A. Such an upward-sloping yield curve is commonly observed in
practice.
If interest rates are expected to change over time, then the liquidity premium may be
overlaid on the path of expected spot rates to determine the forward interest rate. Then
the yield to maturity for each date will be an average of the single-period forward rates.
Several such possibilities for increasing and declining interest rates appear in Figure 15.4,
panels B to D.
7.0
YTM
E(r)
Forward Rate
C
Interest Rate (%)
6.5
6.0
5.5
5.0
Constant Liquidity
Premium
4.5
4.0
0
1
2
7.0
3
4
YTM
5
6
Maturity
E(r)
7
8
9
10
8
9
10
Forward Rate
D
Interest Rate (%)
6.5
Liquidity Premium
Increases with
Maturity
6.0
5.5
5.0
4.5
4.0
0
1
2
3
4
5
6
Maturity
7
Figure 15.4 (Concluded) Panel C, Declining expected short rates. Constant
liquidity premiums. Result is a hump-shaped yield curve. Panel D, Increasing
expected short rates. Increasing liquidity premiums. Result is a sharply rising
yield curve.
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CHAPTER 15
The Term Structure of Interest Rates
501
15.5 Interpreting the Term Structure
If the yield curve reflects expectations of future short rates, then it offers a potentially
powerful tool for fixed-income investors. If we can use the term structure to infer the
expectations of other investors in the economy, we can use those expectations as benchmarks for our own analysis. For example, if we are relatively more optimistic than other
investors that interest rates will fall, we will be more willing to extend our portfolios into
longer-term bonds. Therefore, in this section, we will take a careful look at what information can be gleaned from a careful analysis of the term structure. Unfortunately, while the
yield curve does reflect expectations of future interest rates, it also reflects other factors
such as liquidity premiums. Moreover, forecasts of interest rate changes may have different
investment implications depending on whether those changes are driven by changes in
the expected inflation rate or the real rate, and this adds another layer of complexity to
the proper interpretation of the term structure.
We have seen that under certainty, 1 plus the yield to maturity on a zero-coupon bond
is simply the geometric average of 1 plus the future short rates that will prevail over the
life of the bond. This is the meaning of Equation 15.1, which we give in general form
here:
1 1 yn 5 3(1 1 r1)(1 1 r2)c(1 1 rn) 4 1/n
When future rates are uncertain, we modify Equation 15.1 by replacing future short rates
with forward rates:
1 1 yn 5 3(1 1 r1)(1 1 f2)(1 1 f3)c(1 1 fn) 4 1/n
(15.7)
Thus there is a direct relationship between yields on various maturity bonds and forward
interest rates.
First, we ask what factors can account for a rising yield curve. Mathematically, if the
yield curve is rising, fn11 must exceed yn. In words, the yield curve is upward-sloping at
any maturity date, n, for which the forward rate for the coming period is greater than the
yield at that maturity. This rule follows from the notion of the yield to maturity as an average (albeit a geometric average) of forward rates.
If the yield curve is to rise as one moves to longer maturities, it must be the case that
extension to a longer maturity results in the inclusion of a “new” forward rate that is higher
than the average of the previously observed rates. This is analogous to the observation
that if a new student’s test score is to increase the class average, that student’s score must
exceed the class’s average without her score. To increase the yield to maturity, an aboveaverage forward rate must be added to the other rates used in the averaging computation.
Example 15.6
Forward Rates and the Slopes of the Yield Curve
If the yield to maturity on 3-year zero-coupon bonds is 7%, then the yield on 4-year
bonds will satisfy the following equation:
(1 1 y4 )4 5 (1.07)3(1 1 f4 )
If f4 5 .07, then y4 also will equal .07. (Confirm this!) If f4 is greater than 7%, y4 will
exceed 7%, and the yield curve will slope upward. For example, if f4 5 .08, then
(1 1 y4)4 5 (1.07)3(1.08) 5 1.3230, and y4 5 .0725.
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PART IV
CONCEPT CHECK
Fixed-Income Securities
15.8
Look back at Table 15.1. Show that y4 will exceed y3 if and only if the forward interest rate for period 4 is
greater than 7%, which is the yield to maturity on the 3-year bond, y3.
Given that an upward-sloping yield curve is always associated with a forward rate higher
than the spot, or current, yield to maturity, we ask next what can account for that higher forward rate. Unfortunately, there always are two possible answers to this question. Recall that
the forward rate can be related to the expected future short rate according to this equation:
fn 5 E(rn) 1 Liquidity premium
(15.8)
where the liquidity premium might be necessary to induce investors to hold bonds of
maturities that do not correspond to their preferred investment horizons.
By the way, the liquidity premium need not be positive, although that is the position
generally taken by advocates of the liquidity premium hypothesis. We showed previously that
if most investors have long-term horizons, the liquidity premium in principle could be negative.
In any case, Equation 15.8 shows that there are two reasons that the forward rate could be
high. Either investors expect rising interest rates, meaning that E(rn) is high, or they require a
large premium for holding longer-term bonds. Although it is tempting to infer from a rising
yield curve that investors believe that interest rates will eventually increase, this is not a valid
inference. Indeed, panel A in Figure 15.4 provides a simple counter-example to this line of
reasoning. There, the short rate is expected to stay at 5% forever. Yet there is a constant 1%
liquidity premium so that all forward rates are 6%. The result is that the yield curve continually rises, starting at a level of 5% for 1-year bonds, but eventually approaching 6% for longterm bonds as more and more forward rates at 6% are averaged into the yields to maturity.
Therefore, although it is true that expectations of increases in future interest rates can
result in a rising yield curve, the converse is not true: A rising yield curve does not in and
of itself imply expectations of higher future interest rates. The effects of possible liquidity
premiums confound any simple attempt to extract expectations from the term structure.
But estimating the market’s expectations is crucial because only by comparing your own
expectations to those reflected in market prices can you determine whether you are relatively bullish or bearish on interest rates.
One very rough approach to deriving expected future spot rates is to assume that liquidity
premiums are constant. An estimate of that premium can be subtracted from the forward rate
to obtain the market’s expected interest rate. For example, again making use of the example
plotted in panel A of Figure 15.4, the researcher would estimate from historical data that a
typical liquidity premium in this economy is 1%. After calculating the forward rate from the
yield curve to be 6%, the expectation of the future spot rate would be determined to be 5%.
This approach has little to recommend it for two reasons. First, it is next to impossible
to obtain precise estimates of a liquidity premium. The general approach to doing so would
be to compare forward rates and eventually realized future short rates and to calculate
the average difference between the two. However, the deviations between the two values can be quite large and unpredictable because of unanticipated economic events that
affect the realized short rate. The data are too noisy to calculate a reliable estimate of the
expected premium. Second, there is no reason to believe that the liquidity premium should
be constant. Figure 15.5 shows the rate of return variability of prices of long-term Treasury
bonds since 1971. Interest rate risk fluctuated dramatically during the period. So we should
expect risk premiums on various maturity bonds to fluctuate, and empirical evidence
suggests that liquidity premiums do in fact fluctuate over time.
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CHAPTER 15
503
The Term Structure of Interest Rates
2013
2011
2009
2007
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
Standard Deviation of Monthly Returns (%)
Still, very steep yield
curves are interpreted by
6.0
many market professionals as warning signs of
5.0
impending rate increases.
In fact, the yield curve is
4.0
a good predictor of the
business cycle as a whole,
because long-term rates
3.0
tend to rise in anticipation
of an expansion in eco2.0
nomic activity.
The usually observed
upward slope of the yield
1.0
curve, especially for short
maturities, is the empiri0.0
cal basis for the liquidity premium doctrine that
long-term bonds offer a
positive liquidity premium.
Because the yield curve
Figure 15.5 Price volatility of long-term Treasury bonds
normally has an upward
slope due to risk premiums,
a downward-sloping yield curve is taken as a strong indication that yields are more likely
than not to fall. The prediction of declining interest rates is in turn often interpreted as a
signal of a coming recession. Short-term rates exceeded long-term ones in each of the seven
recessions since 1970. For this reason, it is not surprising that the slope of the yield curve is
one of the key components of the index of leading economic indicators.
Figure 15.6 presents a history of yields on 90-day Treasury bills and 10-year Treasury
bonds. Yields on the longer-term bonds generally exceed those on the bills, meaning that the yield curve generally slopes upward. Moreover, the exceptions to this rule
do seem to precede episodes of falling short rates, which, if anticipated, would induce a
downward-sloping yield curve. For example, the figure shows that 1980–1981 were years
in which 90-day yields exceeded long-term yields. These years preceded both a drastic
drop in the general level of rates and a steep recession.
Why might interest rates fall? There are two factors to consider: the real rate and the
inflation premium. Recall that the nominal interest rate is composed of the real rate plus a
factor to compensate for the effect of inflation:
1 1 Nominal rate 5 (1 1 Real rate)(1 1 Inflation rate)
or, approximately,
Nominal rate < Real rate 1 Inflation rate
Therefore, an expected change in interest rates can be due to changes in either expected
real rates or expected inflation rates. Usually, it is important to distinguish between these
two possibilities because the economic environments associated with them may vary substantially. High real rates may indicate a rapidly expanding economy, high government
budget deficits, and tight monetary policy. Although high inflation rates can arise out of
a rapidly expanding economy, inflation also may be caused by rapid expansion of the
money supply or supply-side shocks to the economy such as interruptions in oil supplies.
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PART IV
Fixed-Income Securities
16
10-Year Treasury
90-Day T-Bills
Difference
Interest Rate (%)
12
8
4
2014
2012
2010
2008
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
−4
1970
0
Figure 15.6 Term spread: Yields on 10-year versus 90-day Treasury securities
These factors have very different implications for investments. Even if we conclude from
an analysis of the yield curve that rates will fall, we need to analyze the macroeconomic
factors that might cause such a decline.
15.6 Forward Rates as Forward Contracts
We have seen that forward rates may be derived from the yield curve, using Equation 15.5.
In general, forward rates will not equal the eventually realized short rate, or even today’s
expectation of what that short rate will be. But there is still an important sense in which
the forward rate is a market interest rate. Suppose that you wanted to arrange now to make
a loan at some future date. You would agree today on the interest rate that will be charged,
but the loan would not commence until some time in the future. How would the interest
rate on such a “forward loan” be determined? Perhaps not surprisingly, it would be the
forward rate of interest for the period of the loan. Let’s use an example to see how this
might work.
Example 15.7
Forward Interest Rate Contract
Suppose the price of 1-year maturity zero-coupon bonds with face value $1,000 is
$952.38 and the price of 2-year zeros with $1,000 face value is $890. The yield to maturity
on the 1-year bond is therefore 5%, while that on the 2-year bond is 6%. The forward
rate for the second year is thus
f2 5
bod61671_ch15_487-514.indd 504
(1 1 y2)2
(1 1 y1)
215
1.062
2 1 5 .0701, or 7.01%
1.05
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CHAPTER 15
The Term Structure of Interest Rates
505
Now consider the strategy laid out in the following table. In the first column we
present data for this example, and in the last column we generalize. We denote by B0(T )
today’s price of a zero-coupon bond with face value $1,000 maturing at time T.
Initial Cash Flow
Buy a 1-year zero-coupon bond
Sell 1.0701 2-year zeros
2952.38
1890 3 1.0701 5 952.38
0
In General
2B0(1)
1B0(2) 3 (1 1 f2)
0
The initial cash flow (at time 0) is zero. You pay $952.38, or in general B0(1), for a zero
maturing in 1 year, and you receive $890, or in general B0(2), for each zero you sell maturing in 2 years. By selling 1.0701 of these bonds, you set your initial cash flow to zero.4
At time 1, the 1-year bond matures and you receive $1,000. At time 2, the 2-year
maturity zero-coupon bonds that you sold mature, and you have to pay 1.0701 3
$1,000 5 $1,070.10. Your cash flow stream is shown in Figure 15.7, panel A. Notice
that you have created a “synthetic” forward loan: You effectively will borrow $1,000 a
year from now, and repay $1,070.10 a year later. The rate on this forward loan is therefore 7.01%, precisely equal to the forward rate for the second year.
In general, to construct the synthetic forward loan, you sell (1 1 f2) 2-year zeros for
every 1-year zero that you buy. This makes your initial cash flow zero because the prices of
the 1- and 2-year zeros differ by the factor (1 1 f2); notice that
B0(1) 5
$1,000
(1 1 y1)
while B0(2) 5
A: Forward Rate = 7.01%
$1,000
$1,000
5
2
(1 1 y1)(1 1 f2)
(1 1 y2)
$1,000
0
1
2
−$1,070.10
B:
For a General Forward Rate. The short rates in the two periods are r1 (which
is observable today) and r2 (which is not). The rate that can be locked in for a
one-period-ahead loan is f2.
$1,000
0
r1
1
r2
2
−$1,000(1 + f2)
Figure 15.7 Engineering a synthetic forward loan
4
Of course, in reality one cannot sell a fraction of a bond, but you can think of this part of the transaction as
follows. If you sold one of these bonds, you would effectively be borrowing $890 for a 2-year period. Selling
1.0701 of these bonds simply means that you are borrowing $890 3 1.0701 5 $952.38.
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PART IV
Fixed-Income Securities
Therefore, when you sell (1 1 f2) 2-year zeros you generate just enough cash to buy one
1-year zero. Both zeros mature to a face value of $1,000, so the difference between the
cash inflow at time 1 and the cash outflow at time 2 is the same factor, 1 1 f2, as illustrated
in Figure 15.7, panel B. As a result, f2 is the rate on the forward loan.
Obviously, you can construct a synthetic forward loan for periods beyond the second
year, and you can construct such loans for multiple periods. Challenge Problems 18 and 19
at the end of the chapter lead you through some of these variants.
CONCEPT CHECK
15.9
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Suppose that the price of 3-year zero-coupon bonds is $816.30. What is the forward rate for the third year?
How would you construct a synthetic 1-year forward loan that commences at t 5 2 and matures at t 5 3?
SUMMARY
1. The term structure of interest rates refers to the interest rates for various terms to maturity embodied in the prices of default-free zero-coupon bonds.
2. In a world of certainty all investments must provide equal total returns for any investment
period. Short-term holding-period returns on all bonds would be equal in a risk-free economy,
and all equal to the rate available on short-term bonds. Similarly, total returns from rolling over
short-term bonds over longer periods would equal the total return available from long-maturity
bonds.
3. The forward rate of interest is the break-even future interest rate that would equate the total
return from a rollover strategy to that of a longer-term zero-coupon bond. It is defined by the
equation
(1 1 yn)n(1 1 fn11) 5 (1 1 yn11)n11
where n is a given number of periods from today. This equation can be used to show that yields to
maturity and forward rates are related by the equation
(1 1 yn)n 5 (1 1 r1)(1 1 f2)(1 1 f3) c(1 1 fn)
4. A common version of the expectations hypothesis holds that forward interest rates are unbiased
estimates of expected future interest rates. However, there are good reasons to believe that forward rates differ from expected short rates because of a risk premium known as a liquidity premium. A positive liquidity premium can cause the yield curve to slope upward even if no increase
in short rates is anticipated.
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for this chapter are
available at www.
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KEY TERMS
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5. The existence of liquidity premiums makes it extremely difficult to infer expected future interest rates from the yield curve. Such an inference would be made easier if we could assume the
liquidity premium remained reasonably stable over time. However, both empirical and theoretical
considerations cast doubt on the constancy of that premium.
6. Forward rates are market interest rates in the important sense that commitments to forward (i.e.,
deferred) borrowing or lending arrangements can be made at these rates.
term structure of interest rates
yield curve
bond stripping
bond reconstitution
pure yield curve
on-the-run yield curve
spot rate
short rate
forward interest rate
liquidity premium
expectations hypothesis
liquidity preference theory
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CHAPTER 15
Forward rate of interest: 1 1 fn 5
The Term Structure of Interest Rates
(1 1 yn)n
507
KEY EQUATIONS
(1 1 yn21)n21
Yield to maturity given sequence of forward rates: 1 1 yn 5 [(1 1 r1) (1 1 f2) (1 1 f3) ? ? ? (1 1 fn)]1/n
Liquidity premium 5 Forward rate 2 Expected short rate
1. What is the relationship between forward rates and the market’s expectation of future short rates?
Explain in the context of both the expectations and liquidity preference theories of the term structure of interest rates.
PROBLEM SETS
2. Under the expectations hypothesis, if the yield curve is upward-sloping, the market must expect
an increase in short-term interest rates. True/false/uncertain? Why?
Basic
3. Under the liquidity preference theory, if inflation is expected to be falling over the next few years,
long-term interest rates will be higher than short-term rates. True/false/uncertain? Why?
a. Upward sloping.
b. Downward sloping.
c. Flat.
5. Which of the following is true according to the pure expectations theory? Forward rates:
a. Exclusively represent expected future short rates.
b. Are biased estimates of market expectations.
c. Always overestimate future short rates.
6. Assuming the pure expectations theory is correct, an upward-sloping yield curve implies:
a. Interest rates are expected to increase in the future.
b. Longer-term bonds are riskier than short-term bonds.
c. Interest rates are expected to decline in the future.
7. The following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields
to maturity of each bond and the implied sequence of forward rates.
Maturity (Years)
Price of Bond
1
2
3
4
$943.40
898.47
847.62
792.16
Intermediate
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4. If the liquidity preference hypothesis is true, what shape should the term structure curve have in a
period where interest rates are expected to be constant?
8. Assuming that the expectations hypothesis is valid, compute the expected price path of the 4-year
bond in the previous problem as time passes. What is the rate of return of the bond in each year?
Show that the expected return equals the forward rate for each year.
9. Consider the following $1,000 par value zero-coupon bonds:
Bond
Years to Maturity
A
B
C
D
1
2
3
4
YTM(%)
5%
6
6.5
7
According to the expectations hypothesis, what is the expected 1-year interest rate 3 years from
now?
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PART IV
Fixed-Income Securities
10. The term structure for zero-coupon bonds is currently:
Maturity (Years)
YTM (%)
1
4%
2
5
3
6
Next year at this time, you expect it to be:
Maturity (Years)
YTM (%)
1
5%
2
6
3
7
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a. What do you expect the rate of return to be over the coming year on a 3-year zero-coupon bond?
b. Under the expectations theory, what yields to maturity does the market expect to observe on
1- and 2-year zeros at the end of the year? Is the market’s expectation of the return on the
3-year bond greater or less than yours?
11. The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2-year zeros is
8%. The Treasury plans to issue a 2-year maturity coupon bond, paying coupons once per year
with a coupon rate of 9%. The face value of the bond is $100.
a. At what price will the bond sell?
b. What will the yield to maturity on the bond be?
c. If the expectations theory of the yield curve is correct, what is the market expectation of the
price that the bond will sell for next year?
d. Recalculate your answer to (c) if you believe in the liquidity preference theory and you
believe that the liquidity premium is 1%.
12. Below is a list of prices for zero-coupon bonds of various maturities.
Maturity (Years)
Price of $1,000 Par Bond
(Zero-Coupon)
1
$943.40
2
873.52
3
816.37
a. An 8.5% coupon $1,000 par bond pays an annual coupon and will mature in 3 years. What
should the yield to maturity on the bond be?
b. If at the end of the first year the yield curve flattens out at 8%, what will be the 1-year
holding-period return on the coupon bond?
13. Prices of zero-coupon bonds reveal the following pattern of forward rates:
Year
1
2
3
Forward Rate
5%
7
8
In addition to the zero-coupon bond, investors also may purchase a 3-year bond making annual
payments of $60 with par value $1,000.
a. What is the price of the coupon bond?
b. What is the yield to maturity of the coupon bond?
c. Under the expectations hypothesis, what is the expected realized compound yield of the
coupon bond?
d. If you forecast that the yield curve in 1 year will be flat at 7%, what is your forecast for the
expected rate of return on the coupon bond for the 1-year holding period?
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CHAPTER 15
The Term Structure of Interest Rates
509
14. You observe the following term structure:
Effective Annual YTM
1-year zero-coupon bond
2-year zero-coupon bond
3-year zero-coupon bond
4-year zero-coupon bond
6.1%
6.2
6.3
6.4
a. If you believe that the term structure next year will be the same as today’s, will the 1-year or
the 4-year zeros provide a greater expected 1-year return?
b. What if you believe in the expectations hypothesis?
16. Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34,
while a 2-year zero sells at $84.99. You are considering the purchase of a 2-year-maturity bond
making annual coupon payments. The face value of the bond is $100, and the coupon rate is
12% per year.
a. What is the yield to maturity of the 2-year zero? The 2-year coupon bond?
b. What is the forward rate for the second year?
c. If the expectations hypothesis is accepted, what are (1) the expected price of the coupon bond
at the end of the first year and (2) the expected holding-period return on the coupon
bond over the first year?
d. Will the expected rate of return be higher or lower if you accept the liquidity preference
hypothesis?
17. The current yield curve for default-free zero-coupon bonds is as follows:
Maturity (Years)
YTM (%)
1
2
3
10%
11
12
a. What are the implied 1-year forward rates?
b. Assume that the pure expectations hypothesis of the term structure is correct. If market
expectations are accurate, what will be the pure yield curve (that is, the yields to maturity on
1- and 2-year zero coupon bonds) next year?
c. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over
the next year? What if you purchase a 3-year zero-coupon bond? (Hint: Compute the current
and expected future prices.) Ignore taxes.
d. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid
annually? If you purchased it at that price, what would your total expected rate of return be
over the next year (coupon plus price change)? Ignore taxes.
Challenge
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15. The yield to maturity (YTM) on 1-year zero-coupon bonds is 5% and the YTM on 2-year zeros
is 6%. The yield to maturity on 2-year-maturity coupon bonds with coupon rates of 12% (paid
annually) is 5.8%. What arbitrage opportunity is available for an investment banking firm?
What is the profit on the activity?
18. Suppose that the prices of zero-coupon bonds with various maturities are given in the following
table. The face value of each bond is $1,000.
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Maturity (Years)
Price
1
2
3
4
5
$925.93
853.39
782.92
715.00
650.00
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PART IV
Fixed-Income Securities
a. Calculate the forward rate of interest for each year.
b. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on
that loan equals the forward rate.
c. Repeat (b) for a 1-year forward loan beginning in year 4.
19. Continue to use the data in the preceding problem. Suppose that you want to construct a 2-year
maturity forward loan commencing in 3 years.
a. Suppose that you buy today one 3-year maturity zero-coupon bond. How many 5-year maturity zeros would you have to sell to make your initial cash flow equal to zero?
b. What are the cash flows on this strategy in each year?
c. What is the effective 2-year interest rate on the effective 3-year-ahead forward loan?
d. Confirm that the effective 2-year interest rate equals (1 1 f4) 3 (1 1 f5) 2 1. You therefore
can interpret the 2-year loan rate as a 2-year forward rate for the last 2 years. Alternatively,
show that the effective 2-year forward rate equals
(1 1 y5)5
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(1 1 y3)3
21
1. Briefly explain why bonds of different maturities have different yields in terms of the expectations and liquidity preference hypotheses. Briefly describe the implications of each hypothesis
when the yield curve is (1) upward-sloping and (2) downward-sloping.
2. Which one of the following statements about the term structure of interest rates is true?
a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates
exceed current short-term rates.
b. The expectations hypothesis contends that the long-term rate is equal to the anticipated shortterm rate.
c. The liquidity premium theory indicates that, all else being equal, longer maturities will have
lower yields.
d. The liquidity preference theory contends that lenders prefer to buy securities at the short end
of the yield curve.
3. The following table shows yields to maturity of zero-coupon Treasury securities.
Term to Maturity (Years)
1
2
3
4
5
10
Yield to Maturity (%)
3.50%
4.50
5.00
5.50
6.00
6.60
a. Calculate the forward 1-year rate of interest for year 3.
b. Describe the conditions under which the calculated forward rate would be an unbiased estimate of the 1-year spot rate of interest for that year.
c. Assume that a few months earlier, the forward 1-year rate of interest for that year had been
significantly higher than it is now. What factors could account for the decline in the forward rate?
4. The 6-month Treasury bill spot rate is 4%, and the 1-year Treasury bill spot rate is 5%. What is
the implied 6-month forward rate for 6 months from now?
5. The tables below show, respectively, the characteristics of two annual-pay bonds from the same
issuer with the same priority in the event of default, and spot interest rates. Neither bond’s price
is consistent with the spot rates. Using the information in these tables, recommend either bond A
or bond B for purchase.
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The Term Structure of Interest Rates
511
Bond Characteristics
Coupons
Maturity
Coupon rate
Yield to maturity
Price
Bond A
Bond B
Annual
3 years
10%
10.65%
98.40
Annual
3 years
6%
10.75%
88.34
Spot Interest Rates
Spot Rates (Zero-Coupon)
1
5%
2
8
3
11
6. Sandra Kapple is a fixed-income portfolio manager who works with large institutional clients.
Kapple is meeting with Maria VanHusen, consultant to the Star Hospital Pension Plan, to discuss
management of the fund’s approximately $100 million Treasury bond portfolio. The current U.S.
Treasury yield curve is given in the following table. VanHusen states, “Given the large differential
between 2- and 10-year yields, the portfolio would be expected to experience a higher return over
a 10-year horizon by buying 10-year Treasuries, rather than buying 2-year Treasuries and reinvesting the proceeds into 2-year T-bonds at each maturity date.”
Maturity
Yield
Maturity
Yield
1 year
2
3
4
5
2.00%
2.90
3.50
3.80
4.00
6 years
7
8
9
10
4.15%
4.30
4.45
4.60
4.70
a. Indicate whether VanHusen’s conclusion is correct, based on the pure expectations hypothesis.
b. VanHusen discusses with Kapple alternative theories of the term structure of interest rates and
gives her the following information about the U.S. Treasury market:
Maturity (years)
Liquidity premium (%)
2
.55
3
.55
4
.65
5
.75
6
.90
7
1.10
8
1.20
9
1.50
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Term (Years)
10
1.60
Use this additional information and the liquidity preference theory to determine what the slope of
the yield curve implies about the direction of future expected short-term interest rates.
7. A portfolio manager at Superior Trust Company is structuring a fixed-income portfolio to meet
the objectives of a client. The portfolio manager compares coupon U.S. Treasuries with zerocoupon stripped U.S. Treasuries and observes a significant yield advantage for the stripped bonds:
Term
Coupon
U.S. Treasuries
Zero-Coupon Stripped
U.S. Treasuries
3 years
5.50%
5.80%
7
6.75
7.25
10
7.25
7.60
30
7.75
8.20
Briefly discuss why zero-coupon stripped U.S. Treasuries could yield more than coupon U.S.
Treasuries with the same final maturity.
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PART IV
Fixed-Income Securities
8. The shape of the U.S. Treasury yield curve appears to reflect two expected Federal Reserve
reductions in the federal funds rate. The current short-term interest rate is 5%. The first reduction of approximately 50 basis points (bp) is expected 6 months from now and the second
reduction of approximately 50 bp is expected 1 year from now. The current U.S. Treasury term
premiums are 10 bp per year for each of the next 3 years (out through the 3-year benchmark).
However, the market also believes that the Federal Reserve reductions will be reversed in a
single 100-bp increase in the federal funds rate 2½ years from now. You expect liquidity premiums to remain 10 bp per year for each of the next 3 years (out through the 3-year benchmark).
Describe or draw the shape of the Treasury yield curve out through the 3-year benchmark. Which term structure theory supports the shape of the U.S. Treasury yield curve you’ve
described?
9. U.S. Treasuries represent a significant holding in many pension portfolios. You decide to analyze the yield curve for U.S. Treasury notes.
a. Using the data in the table below, calculate the 5-year spot and forward rates assuming
annual compounding. Show your calculations.
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U.S. Treasury Note Yield Curve Data
Years to Maturity
Par Coupon
Yield to Maturity
Calculated
Spot Rates
Calculated
Forward Rates
1
2
3
4
5
5.00
5.20
6.00
7.00
7.00
5.00
5.21
6.05
7.16
?
5.00
5.42
7.75
10.56
?
b. Define and describe each of the following three concepts:
i. Short rate
ii. Spot rate
iii. Forward rate
Explain how these concepts are related.
c. You are considering the purchase of a zero-coupon U.S. Treasury note with 4 years to maturity. On the basis of the above yield-curve analysis, calculate both the expected yield to
maturity and the price for the security. Show your calculations.
10. The spot rates of interest for five U.S. Treasury securities are shown in the following exhibit.
Assume all securities pay interest annually.
Spot Rates of Interest
Term to Maturity
Spot Rate of Interest
1 year
2
3
4
5
13.00%
12.00
11.00
10.00
9.00
a. Compute the 2-year implied forward rate for a deferred loan beginning in 3 years.
b. Compute the price of a 5-year annual-pay Treasury security with a coupon rate of 9% by
using the information in the exhibit.
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CHAPTER 15
The Term Structure of Interest Rates
513
E-INVESTMENTS EXERCISES
Go to www.smartmoney.com. Access the Living Yield Curve (look for the Economy and
Bonds tab), a moving picture of the yield curve. Is the yield curve usually upward- or
downward-sloping? What about today’s yield curve? How much does the slope of the
curve vary? Which varies more: short-term or long-term rates? Can you explain why this
might be the case?
SOLUTIONS TO CONCEPT CHECKS
1. The price of the 3-year bond paying a $40 coupon is
At this price, the yield to maturity is 6.945% [n 5 3; PV 5 (2)922.65; FV 5 1,000; PMT 5 40].
This bond’s yield to maturity is closer to that of the 3-year zero-coupon bond than is the yield to
maturity of the 10% coupon bond in Example 15.1. This makes sense: This bond’s coupon rate is
lower than that of the bond in Example 15.1. A greater fraction of its value is tied up in the final
payment in the third year, and so it is not surprising that its yield is closer to that of a pure 3-year
zero-coupon security.
2. We compare two investment strategies in a manner similar to Example 15.2:
Buy and hold 4-year zero 5 Buy 3-year zero; roll proceeds into 1-year bond
(1 1 y4 )4 5 (1 1 y3 )3 3 (1 1 r4 )
1.084 5 1.073 3 (1 1 r4 )
which implies that r4 5 1.084/1.073 2 1 5 .11056 5 11.056%. Now we confirm that the yield on
the 4-year zero is a geometric average of the discount factors for the next 3 years:
1 1 y4 5 3(1 1 r1 ) 3 (1 1 r2 ) 3 (1 1 r3 ) 3 (1 1 r4 ) 4 1/4
1.08 5 3 1.05 3 1.0701 3 1.09025 3 1.11056 4 1/4
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40
40
1040
1
1
5 38.095 1 35.600 1 848.950 5 $922.65
2
1.05
1.06
1.073
3. The 3-year bond can be bought today for $1,000/1.073 5 $816.30. Next year, it will have a
remaining maturity of 2 years. The short rate in year 2 will be 7.01% and the short rate in year 3
will be 9.025%. Therefore, the bond’s yield to maturity next year will be related to these short
rates according to
(1 1 y2 )2 5 1.0701 3 1.09025 5 1.1667
and its price next year will be $1,000/(1 1 y2)2 5 $1,000/1.1667 5 $857.12. The 1-year holdingperiod rate of return is therefore ($857.12 2 $816.30)/$816.30 5 .05, or 5%.
4. The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods.
The short rate for period n is the one-period interest rate that will prevail in period n. Finally, the
forward rate for period n is the short rate that would satisfy a “break-even condition” equating
the total returns on two n-period investment strategies. The first strategy is an investment in an
n-period zero-coupon bond; the second is an investment in an n 2 1 period zero-coupon bond
“rolled over” into an investment in a one-period zero. Spot rates and forward rates are observable
today, but because interest rates evolve with uncertainty, future short rates are not. In the special
case in which there is no uncertainty in future interest rates, the forward rate calculated from the
yield curve would equal the short rate that will prevail in that period.
5. 7% 2 1% 5 6%.
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PART IV
Fixed-Income Securities
6. The risk premium will be zero.
7. If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest
costs on long bonds over short bonds. This willingness combines with investors’ demands for
higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.
8. In general, from Equation 15.5, (1 1 yn)n 5 (1 1 yn21)n21 3 (1 1 fn). In this case,
(1 1 y4 )4 5 (1.07)3 3 (1 1 f4 ). If f4 5 .07, then (1 1 y4 )4 5 (1.07)4 and y4 5 .07. If f4 is greater
than .07, then y4 also will be greater, and conversely if f4 is less than .07, then y4 will be as well.
1,000 1/3
b 2 1 5 .07 5 7.0%
816.30
The forward rate for the third year is therefore
9. The 3-year yield to maturity is a
f3 5
(1 1 y3 )3
(1 1 y2
)2
215
1.073
2 1 5 .0903 5 9.03%
1.062
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(Alternatively, note that the ratio of the price of the 2-year zero to the price of the 3-year zero is
1 1 f3 5 1.0903.) To construct the synthetic loan, buy one 2-year maturity zero, and sell 1.0903
3-year maturity zeros. Your initial cash flow is zero, your cash flow at time 2 is 1$1,000, and
your cash flow at time 3 is 2$1,090.30, which corresponds to the cash flows on a 1-year forward
loan commencing at time 2 with an interest rate of 9.03%.
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CHAPTER EIGHTEEN
Equity Valuation Models
AS OUR DISCUSSION of market efficiency
indicated, finding undervalued securities
is hardly easy. At the same time, there are
enough chinks in the armor of the efficient
market hypothesis that the search for such
securities should not be dismissed out of
hand. Moreover, it is the ongoing search for
mispriced securities that maintains a nearly
efficient market. Even minor mispricing
would allow a stock market analyst to earn
his salary.
This chapter describes the valuation models that stock market analysts use to uncover
mispriced securities. The models presented
are those used by fundamental analysts,
those analysts who use information concerning the current and prospective profitability
of a company to assess its fair market value.
We start with a discussion of alternative measures of the value of a company. From there,
we progress to quantitative tools called dividend discount models, which security analysts commonly use to measure the value of
a firm as an ongoing concern. Next we turn
to price–earnings, or P/E, ratios, explaining
why they are of such interest to analysts but
also highlighting some of their shortcomings.
We explain how P/E ratios are tied to dividend
valuation models and, more generally, to the
growth prospects of the firm.
We close the chapter with a discussion and
extended example of free cash flow models
used by analysts to value firms based on forecasts of the cash flows that will be generated
from the firms’ business endeavors. Finally,
we apply the several valuation tools covered
in the chapter to a real firm and find some
disparity in their conclusions—a conundrum
that will confront any security analyst—and
consider reasons for these discrepancies.
The purpose of fundamental analysis is to identify stocks that are mispriced relative to some
measure of “true” value that can be derived from observable financial data. There are many convenient sources of relevant data. For U.S. companies, the Securities and Exchange Commission
provides information at its EDGAR Web site, www.sec.gov/edgar.shtml. The SEC requires all
public companies (except foreign companies and companies with less than $10 million in assets
and 500 shareholders) to file registration statements, periodic reports, and other forms electronically through EDGAR. Anyone can access and download this information. Many Web sites such
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as finance.yahoo.com, money.msn.com, or finance.google.com also provide analysis
and data derived from the EDGAR reports.
Table 18.1 shows some financial highlights for Microsoft as well as some comparable
data for other firms in the software applications industry. The price per share of Microsoft’s
common stock is $30.63, and the total market value or capitalization of those shares (called
market cap for short) is $258 billion. Under the heading “Valuation,” Table 18.1 shows the
ratio of Microsoft’s stock price to five benchmarks. Its share price is 15.4 times its (per share)
earnings in the most recent 12 months, 3.9 times its recent book value, 3.5 times its sales, and
10.9 times its cash flow. The last valuation ratio, PEG, is the P/E ratio divided by the growth
rate of earnings. We would expect more rapidly growing firms to sell at higher multiples of
current earnings (more on this below), so PEG normalizes the P/E ratio by the growth rate.
These valuation ratios are commonly used to assess the valuation of one firm compared
to others in the same industry, and we will consider all of them. The column to the right
gives comparable ratios for other firms in the software applications industry. For example,
an analyst might note that Microsoft’s price/earnings ratio and price/CF ratio are both
below the industry average. Microsoft’s ratio of market value to book value, the net worth
of the company as reported on the balance sheet, is also below industry norms, 3.9 versus
10.5. These ratios might indicate that its stock is underpriced. However, Microsoft is a
more mature firm than many in the industry, and perhaps this discrepancy reflects a lower
expected future growth rate. In fact, its PEG ratio is nearly identical to the industry average. Clearly, rigorous valuation models will be necessary to sort through these sometimes
conflicting signals of value.
Limitations of Book Value
Shareholders in a firm are sometimes called “residual claimants,” which means that the
value of their stake is what is left over when the liabilities of the firm are subtracted from
Table 18.1
Financial highlights
for Microsoft
Corporation,
September 12, 2012
Price per share
Common shares outstanding (billion)
Market capitalization ($ billion)
Latest 12 Months
Sales ($ billion)
EBITDA ($ billion)
Net income ($ billion)
Earnings per share
Valuation
Price/Earnings
Price/Book
Price/Sales
Price/Cash flow
PEG
Profitability
ROE (%)
ROA (%)
Operating profit margin (%)
Net profit margin (%)
$ 30.63
8.38
$258
$ 73.72
$ 30.71
$ 16.98
$ 2.00
Microsoft
15.4
3.9
3.5
10.9
1.1
27.5
15.0
37.9
23.0
Industry Avg
17.5
10.5
20.5
1.2
24.9
23.2
Source: Compiled from data available at finance.yahoo.com, September 12, 2012.
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its assets. Shareholders’ equity is this net worth. However, the values of both assets and
liabilities recognized in financial statements are based on historical—not current—values.
For example, the book value of an asset equals the original cost of acquisition less some
adjustment for depreciation, even if the market price of that asset has changed over time.
Moreover, depreciation allowances are used to allocate the original cost of the asset over
several years, but do not reflect loss of actual value.
Whereas book values are based on original cost, market values measure current values
of assets and liabilities. The market value of the shareholders’ equity investment equals the
difference between the current values of all assets and liabilities. We’ve emphasized that
current values generally will not match historical ones. Equally or even more important,
many assets, such as the value of a good brand name or specialized expertise developed
over many years, may not even be included on the financial statements. Market prices
therefore reflect the value of the firm as a going concern. It would be unusual if the market
price of a stock were exactly equal to its book value.
Can book value represent a “floor” for the stock’s price, below which level the market
price can never fall? Although Microsoft’s book value per share in 2012 was less than
its market price, other evidence disproves this notion. While it is not common, there are
always some firms selling at a market price below book value. In 2012, for example, such
troubled firms included Sprint/Nextel, Bank of America, Mitsubishi, and AOL.
A better measure of a floor for the stock price is the firm’s liquidation value per share.
This represents the amount of money that could be realized by breaking up the firm, selling
its assets, repaying its debt, and distributing the remainder to the shareholders. If the market price of equity drops below liquidation value, the firm becomes attractive as a takeover
target. A corporate raider would find it profitable to buy enough shares to gain control and
then actually to liquidate.
Another measure of firm value is the replacement cost of assets less liabilities. Some
analysts believe the market value of the firm cannot remain for long too far above its
replacement cost because if it did, competitors would enter the market. The resulting competitive pressure would drive down the market value of all firms until they fell to replacement cost.
This idea is popular among economists, and the ratio of market price to replacement cost
is known as Tobin’s q, after the Nobel Prize–winning economist James Tobin. In the long
run, according to this view, the ratio of market price to replacement cost will tend toward
1, but the evidence is that this ratio can differ significantly from 1 for very long periods.
Although focusing on the balance sheet can give some useful information about a firm’s
liquidation value or its replacement cost, the analyst must usually turn to expected future
cash flows for a better estimate of the firm’s value as a going concern. We therefore turn
to the quantitative models that analysts use to value common stock based on forecasts of
future earnings and dividends.
18.2 Intrinsic Value versus Market Price
The most popular model for assessing the value of a firm as a going concern starts from
the observation that an investor in stock expects a return consisting of cash dividends and
capital gains or losses. We begin by assuming a 1-year holding period and supposing that
ABC stock has an expected dividend per share, E(D1), of $4; the current price of a share,
P0, is $48; and the expected price at the end of a year, E(P1), is $52. For now, don’t worry
about how you derive your forecast of next year’s price. At this point we ask only whether
the stock seems attractively priced today given your forecast of next year’s price.
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The expected holding-period return is E(D1) plus the expected price appreciation,
E(P1) 2 P0, all divided by the current price, P0:
E(D1) 1 3 E(P1) 2 P0 4
P0
4 1 (52 2 48)
5
5 .167, or 16.7%
48
Expected HPR 5 E(r) 5
Thus, the stock’s expected holding-period return is the sum of the expected dividend
yield, E(D1)/P0, and the expected rate of price appreciation, the capital gains yield,
[E(P1) 2 P0]/P0.
But what is the required rate of return for ABC stock? The CAPM states that when
stock market prices are at equilibrium levels, the rate of return that investors can expect
to earn on a security is rf 1 b[E(rM) 2 rf ]. Thus, the CAPM may be viewed as providing
an estimate of the rate of return an investor can reasonably expect to earn on a security
given its risk as measured by beta. This is the return that investors will require of any other
investment with equivalent risk. We will denote this required rate of return as k. If a stock
is priced “correctly,” it will offer investors a “fair” return, that is, its expected return will
equal its required return. Of course, the goal of a security analyst is to find stocks that are
mispriced. For example, an underpriced stock will provide an expected return greater than
the required return.
Suppose that rf 5 6%, E(rM) 2 rf 5 5%, and the beta of ABC is 1.2. Then the value
of k is
k 5 6% 1 1.2 3 5% 5 12%
The expected holding-period return, 16.7%, therefore exceeds the required rate of return
based on ABC’s risk by a margin of 4.7%. Naturally, the investor will want to include more
of ABC stock in the portfolio than a passive strategy would indicate.
Another way to see this is to compare the intrinsic value of a share of stock to its market
price. The intrinsic value, denoted V0, is defined as the present value of all cash payments
to the investor in the stock, including dividends as well as the proceeds from the ultimate
sale of the stock, discounted at the appropriate risk-adjusted interest rate, k. If the intrinsic
value, or the investor’s own estimate of what the stock is really worth, exceeds the market
price, the stock is considered undervalued and a good investment. For ABC, using a 1-year
investment horizon and a forecast that the stock can be sold at the end of the year at price
P1 5 $52, the intrinsic value is
V0 5
E (D1) 1 E (P1)
$4 1 $52
5
5 $50
11k
1.12
Equivalently, at a price of $50, the investor would derive a 12% rate of return—just
equal to the required rate of return—on an investment in the stock. However, at the current
price of $48, the stock is underpriced compared to intrinsic value. At this price, it provides
better than a fair rate of return relative to its risk. Using the terminology of the CAPM, it is
a positive-alpha stock, and investors will want to buy more of it than they would following
a passive strategy.
If the intrinsic value turns out to be lower than the current market price, investors should
buy less of it than under the passive strategy. It might even pay to go short on ABC stock,
as we discussed in Chapter 3.
In market equilibrium, the current market price will reflect the intrinsic value estimates
of all market participants. This means the individual investor whose V0 estimate differs
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from the market price, P0, in effect must disagree with some or all of the market consensus
estimates of E(D1), E(P1), or k. A common term for the market consensus value of the
required rate of return, k, is the market capitalization rate, which we use often throughout this chapter.
CONCEPT CHECK
18.1
You expect the price of IBX stock to be $59.77 per share a year from now. Its current market price is $50,
and you expect it to pay a dividend 1 year from now of $2.15 per share.
a. What are the stock’s expected dividend yield, rate of price appreciation, and holding-period return?
b. If the stock has a beta of 1.15, the risk-free rate is 6% per year, and the expected rate of return on the
market portfolio is 14% per year, what is the required rate of return on IBX stock?
c. What is the intrinsic value of IBX stock, and how does it compare to the current market price?
18.3 Dividend Discount Models
Consider an investor who buys a share of Steady State Electronics stock, planning to hold
it for 1 year. The intrinsic value of the share is the present value of the dividend to be
received at the end of the first year, D1, and the expected sales price, P1. We will henceforth
use the simpler notation P1 instead of E(P1) to avoid clutter. Keep in mind, though, that
future prices and dividends are unknown, and we are dealing with expected values, not
certain values. We’ve already established
V0 5
D1 1 P1
11k
(18.1)
Although this year’s dividends are fairly predictable given a company’s history, you
might ask how we can estimate P1, the year-end price. According to Equation 18.1, V1 (the
year-end intrinsic value) will be
V1 5
D2 1 P2
11k
If we assume the stock will be selling for its intrinsic value next year, then V1 5 P1, and we
can substitute this value for P1 into Equation 18.1 to find
V0 5
D1
D2 1 P2
1
(1 1 k)2
11k
This equation may be interpreted as the present value of dividends plus sales price
for a 2-year holding period. Of course, now we need to come up with a forecast of P2.
Continuing in the same way, we can replace P2 by (D3 1 P3)/(1 1 k), which relates P0 to
the value of dividends plus the expected sales price for a 3-year holding period.
More generally, for a holding period of H years, we can write the stock value as the
present value of dividends over the H years, plus the ultimate sale price, PH:
V0 5
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D1
D2
DH 1 PH
1
1 c1
2
(1 1 k)H
(1 1 k)
11k
(18.2)
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Note the similarity between this formula and the bond valuation formula developed
in Chapter 14. Each relates price to the present value of a stream of payments (coupons
in the case of bonds, dividends in the case of stocks) and a final payment (the face value
of the bond, or the sales price of the stock). The key differences in the case of stocks are
the uncertainty of dividends, the lack of a fixed maturity date, and the unknown sales
price at the horizon date. Indeed, one can continue to substitute for price indefinitely, to
conclude
V0 5
D3
D1
D2
1
1
1c
2
(
)
(
11k
11k
1 1 k)3
(18.3)
Equation 18.3 states that the stock price should equal the present value of all expected
future dividends into perpetuity. This formula is called the dividend discount model
(DDM) of stock prices.
It is tempting, but incorrect, to conclude from Equation 18.3 that the DDM focuses
exclusively on dividends and ignores capital gains as a motive for investing in stock. Indeed,
we assume explicitly in Equation 18.1 that capital gains (as reflected in the expected sales
price, P1) are part of the stock’s value. Our point is that the price at which you can sell a
stock in the future depends on dividend forecasts at that time.
The reason only dividends appear in Equation 18.3 is not that investors ignore capital
gains. It is instead that those capital gains will reflect dividend forecasts at the time the
stock is sold. That is why in Equation 18.2 we can write the stock price as the present value
of dividends plus sales price for any horizon date. PH is the present value at time H of all
dividends expected to be paid after the horizon date. That value is then discounted back
to today, time 0. The DDM asserts that stock prices are determined ultimately by the cash
flows accruing to stockholders, and those are dividends.1
The Constant-Growth DDM
Equation 18.3 as it stands is still not very useful in valuing a stock because it requires
dividend forecasts for every year into the indefinite future. To make the DDM practical,
we need to introduce some simplifying assumptions. A useful and common first pass at the
problem is to assume that dividends are trending upward at a stable growth rate that we
will call g. For example, if g 5 .05, and the most recently paid dividend was D0 5 3.81,
expected future dividends are
D1 5 D0(1 1 g) 5 3.81 3 1.05 5 4.00
D2 5 D0(1 1 g)2 5 3.81 3 (1.05)2 5 4.20
D3 5 D0(1 1 g)3 5 3.81 3 (1.05)3 5 4.41
and so on. Using these dividend forecasts in Equation 18.3, we solve for intrinsic
value as
V0 5
D0 (1 1 g)2
D0 (1 1 g)
D0 (1 1 g)3 c
1
1
1
2
(1 1 k)
(1 1 k)3
11k
1
If investors never expected a dividend to be paid, then this model implies that the stock would have no value. To
reconcile the DDM with the fact that non-dividend-paying stocks do have a market value, one must assume that
investors expect that some day it may pay out some cash, even if only a liquidating dividend.
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This equation can be simplified to2
V0 5
D0(1 1 g)
D1
5
k2g
k2g
(18.4)
Note in Equation 18.4 that we divide D1 (not D0) by k 2 g to calculate intrinsic value. If
the market capitalization rate for Steady State is 12%, we can use Equation 18.4 to show
that the intrinsic value of a share of Steady State stock is
$3.81(1 1 .05)
$4.00
5
5 $57.14
.12 2 .05
.12 2 .05
Equation 18.4 is called the constant-growth DDM, or the Gordon model, after Myron
J. Gordon, who popularized the model. It should remind you of the formula for the present value of a perpetuity. If dividends were expected not to grow, then the dividend stream
would be a simple perpetuity, and the valuation formula would be3 V0 5 D1/k. Equation
18.4 is a generalization of the perpetuity formula to cover the case of a growing perpetuity.
As g increases (for a given value of D1), the stock price also rises.
Example 18.1
Preferred Stock and the DDM
Preferred stock that pays a fixed dividend can be valued using the constant-growth dividend discount model. The constant-growth rate of dividends is simply zero. For example,
to value a preferred stock paying a fixed dividend of $2 per share when the discount rate
is 8%, we compute
V0 5
$2
5 $25
.08 2 0
D1
We prove that the intrinsic value, V0, of a stream of cash dividends growing at a constant rate g is equal to
k2g
as follows. By definition,
2
V0 5
D1(1 1 g)
D1(1 1 g)2
D1
1
1
1c
2
11k
(1 1 k )
(1 1 k )3
(a)
Multiplying through by (1 1 k)/(1 1 g), we obtain
D1(1 1 g)
D1
D1
(1 1 k )
V 5
1c
1
1
(1 1 g ) 0 (1 1 g )
(1 1 k )
(1 1 k )2
(b)
Subtracting equation (a) from equation (b), we find that
D1
11k
V 2 V0 5
(1 1 g )
11g 0
which implies
(k 2 g)V0
D1
5
(1 1 g )
(1 1 g )
D1
V0 5
k2g
3
Recall from introductory finance that the present value of a $1 per year perpetuity is 1/k. For example, if
k 5 10%, the value of the perpetuity is $1/.10 5 $10. Notice that if g 5 0 in Equation 18.4, the constant-growth
DDM formula is the same as the perpetuity formula.
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Example 18.2
The Constant-Growth DDM
High Flyer Industries has just paid its annual dividend of $3 per share. The dividend is
expected to grow at a constant rate of 8% indefinitely. The beta of High Flyer stock is
1.0, the risk-free rate is 6%, and the market risk premium is 8%. What is the intrinsic
value of the stock? What would be your estimate of intrinsic value if you believed that
the stock was riskier, with a beta of 1.25?
Because a $3 dividend has just been paid and the growth rate of dividends is 8%,
the forecast for the year-end dividend is $3 3 1.08 5 $3.24. The market capitalization
rate (using the CAPM) is 6% 1 1.0 3 8% 5 14%. Therefore, the value of the stock is
V0 5
D1
$3.24
5
5 $54
k2g
.14 2 .08
If the stock is perceived to be riskier, its value must be lower. At the higher beta, the
market capitalization rate is 6% 1 1.25 3 8% 5 16%, and the stock is worth only
$3.24
5 $40.50
.16 2 .08
The constant-growth DDM is valid only when g is less than k. If dividends were
expected to grow forever at a rate faster than k, the value of the stock would be infinite. If
an analyst derives an estimate of g greater than k, that growth rate must be unsustainable in
the long run. The appropriate valuation model to use in this case is a multistage DDM such
as those discussed below.
The constant-growth DDM is so widely used by stock market analysts that it is worth
exploring some of its implications and limitations. The constant-growth rate DDM implies
that a stock’s value will be greater:
1. The larger its expected dividend per share.
2. The lower the market capitalization rate, k.
3. The higher the expected growth rate of dividends.
Another implication of the constant-growth model is that the stock price is expected to
grow at the same rate as dividends. To see this, suppose Steady State stock is selling at its
intrinsic value of $57.14, so that V0 5 P0. Then
D1
P0 5
k2g
Note that price is proportional to dividends. Therefore, next year, when the dividends
paid to Steady State stockholders are expected to be higher by g 5 5%, price also should
increase by 5%. To confirm this, note
D2 5 $4(1.05) 5 $4.20
D2
$4.20
P1 5
5
5 $60.00
k2g
.12 2 .05
which is 5% higher than the current price of $57.14. To generalize,
P1 5
D1(1 1 g)
D2
D1
(1 1 g) 5 P0(1 1 g)
5
5
k2g
k2g
k2g
Therefore, the DDM implies that in the case of constant growth of dividends, the
rate of price appreciation in any year will equal that constant-growth rate, g. For a stock
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whose market price equals its intrinsic value (V0 5 P0), the expected holding-period
return will be
E(r) 5 Dividend yield 1 Capital gains yield
P1 2 P0
D1
D1
5
1
5
1g
(18.5)
P0
P0
P0
This formula offers a means to infer the market capitalization rate of a stock, for if the
stock is selling at its intrinsic value, then E(r) 5 k, implying that k 5 D1/P0 1 g. By
observing the dividend yield, D1/P0, and estimating the growth rate of dividends, we can
compute k. This equation is also known as the discounted cash flow (DCF) formula.
This approach is often used in rate hearings for regulated public utilities. The regulatory
agency responsible for approving utility pricing decisions is mandated to allow the firms to
charge just enough to cover costs plus a “fair” profit, that is, one that allows a competitive
return on the investment the firm has made in its productive capacity. In turn, that return is
taken to be the expected return investors require on the stock of the firm. The D1/P0 1 g
formula provides a means to infer that required return.
Example 18.3
The Constant-Growth Model
Suppose that Steady State Electronics wins a major contract for its new computer chip.
The very profitable contract will enable it to increase the growth rate of dividends from
5% to 6% without reducing the current dividend from the projected value of $4.00 per
share. What will happen to the stock price? What will happen to future expected rates
of return on the stock?
The stock price ought to increase in response to the good news about the contract,
and indeed it does. The stock price jumps from its original value of $57.14 to a postannouncement price of
D1
$4.00
5
5 $66.67
k2g
.12 2 .06
Investors who are holding the stock when the good news about the contract is
announced will receive a substantial windfall.
On the other hand, at the new price the expected rate of return on the stock is 12%,
just as it was before the new contract was announced.
E(r) 5
D1
$4.00
1g5
1 .06 5 .12, or 12%
P0
$66.67
This result makes sense. Once the news about the contract is reflected in the stock price,
the expected rate of return will be consistent with the risk of the stock. Because the risk
of the stock has not changed, neither should the expected rate of return.
CONCEPT CHECK
18.2
a. IBX’s stock dividend at the end of this year is expected to be $2.15, and it is expected to grow at 11.2%
per year forever. If the required rate of return on IBX stock is 15.2% per year, what is its intrinsic value?
b. If IBX’s current market price is equal to this intrinsic value, what is next year’s expected price?
c. If an investor were to buy IBX stock now and sell it after receiving the $2.15 dividend a year from now,
what is the expected capital gain (i.e., price appreciation) in percentage terms? What are the dividend
yield and the holding-period return?
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Convergence of Price to Intrinsic Value
Now suppose that the current market price of ABC stock is only $48 per share and,
therefore, that the stock is undervalued by $2 per share. In this case the expected rate of
price appreciation depends on an additional assumption about whether the discrepancy
between the intrinsic value and the market price will disappear, and if so, when.
One fairly common assumption is that the discrepancy will never disappear and that the
market price will trend upward at rate g forever. This implies that the discrepancy between
intrinsic value and market price also will grow at the same rate. In our example:
Now
Next Year
V0 5 $50
P0 5 $48
V0 2 P0 5 $2
V1 5 $50 3 1.04 5 $52
P1 5 $48 3 1.04 5 $49.92
V1 2 P1 5 $2 3 1.04 5 $2.08
Under this assumption the expected HPR will exceed the required rate, because the
dividend yield is higher than it would be if P0 were equal to V0. In our example the dividend yield would be 8.33% instead of 8%, so that the expected HPR would be 12.33%
rather than 12%:
D1
$4
E(r) 5
1g5
1 .04 5 .0833 1 .04 5 .1233
P0
$48
An investor who identifies this undervalued stock can get an expected dividend that
exceeds the required yield by 33 basis points. This excess return is earned each year, and
the market price never catches up to intrinsic value.
An alternative assumption is that the gap between market price and intrinsic value will
disappear by the end of the year. In that case we would have P1 5 V1 5 $52, and
P1 2 P0
D1
4
52 2 48
E(r) 5
1
5
1
5 .0833 1 .0833 5 .1667
P0
P0
48
48
The assumption of complete catch-up to intrinsic value produces a much larger 1-year
HPR. In future years, however, the stock is expected to generate only fair rates of return.
Many stock analysts assume that a stock’s price will approach its intrinsic value gradually over time—for example, over a 5-year period. This puts their expected 1-year HPR
somewhere between the bounds of 12.33% and 16.67%.
Stock Prices and Investment Opportunities
Consider two companies, Cash Cow, Inc., and Growth Prospects, each with expected earnings in the coming year of $5 per share. Both companies could in principle pay out all
of these earnings as dividends, maintaining a perpetual dividend flow of $5 per share. If
the market capitalization rate were k 5 12.5%, both companies would then be valued at
D1/k 5 $5/.125 5 $40 per share. Neither firm would grow in value, because with all earnings paid out as dividends, and no earnings reinvested in the firm, both companies’ capital
stock and earnings capacity would remain unchanged over time; earnings4 and dividends
would not grow.
4
Actually, we are referring here to earnings net of the funds necessary to maintain the productivity of the firm’s
capital, that is, earnings net of “economic depreciation.” In other words, the earnings figure should be interpreted
as the maximum amount of money the firm could pay out each year in perpetuity without depleting its productive
capacity. For this reason, the net earnings number may be quite different from the accounting earnings figure that
the firm reports in its financial statements. We explore this further in the next chapter.
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Dividends per Share ($)
Now suppose one of the firms, Growth Prospects,
12
engages in projects that generate a return on investment of 15%, which is greater than the required rate
10
of return, k 5 12.5%. It would be foolish for such a
High Reinvestment
8
company to pay out all of its earnings as dividends.
6
If Growth Prospects retains or plows back some of
its earnings into its profitable projects, it can earn a
Low Reinvestment
4
15% rate of return for its shareholders, whereas if
2
it pays out all earnings as dividends, it forgoes the
0
projects, leaving shareholders to invest the dividends
0
10
20
30
in other opportunities at a fair market rate of only
Year
12.5%. Suppose, therefore, that Growth Prospects
chooses a lower dividend payout ratio (the fraction
Figure 18.1 Dividend growth for two earnings
of earnings paid out as dividends), reducing payout
reinvestment policies
from 100% to 40%, maintaining a plowback ratio
(the fraction of earnings reinvested in the firm) at
60%. The plowback ratio is also referred to as the
earnings retention ratio.
The dividend of the company, therefore, will be $2 (40% of $5 earnings) instead of $5.
Will share price fall? No—it will rise! Although dividends initially fall under the earnings reinvestment policy, subsequent growth in the assets of the firm because of reinvested profits will generate growth in future dividends, which will be reflected in today’s
share price.
Figure 18.1 illustrates the dividend streams generated by Growth Prospects under two
dividend policies. A low-reinvestment-rate plan allows the firm to pay higher initial dividends, but results in a lower dividend growth rate. Eventually, a high-reinvestment-rate
plan will provide higher dividends. If the dividend growth generated by the reinvested
earnings is high enough, the stock will be worth more under the high-reinvestment
strategy.
How much growth will be generated? Suppose Growth Prospects starts with plant and
equipment of $100 million and is all equity financed. With a return on investment or equity
(ROE) of 15%, total earnings are ROE 3 $100 million 5 .15 3 $100 million 5 $15 million.
There are 3 million shares of stock outstanding, so earnings per share are $5, as posited
above. If 60% of the $15 million in this year’s earnings is reinvested, then the value of the
firm’s assets will increase by .60 3 $15 million 5 $9 million, or by 9%. The percentage
increase in assets is the rate at which income was generated (ROE) times the plowback
ratio (the fraction of earnings reinvested in the firm), which we will denote as b.
Now endowed with 9% more assets, the company earns 9% more income, and pays out
9% higher dividends. The growth rate of the dividends, therefore, is5
g 5 ROE 3 b 5 .15 3 .60 5 .09
If the stock price equals its intrinsic value, it should sell at
P0 5
D1
$2
5
5 $57.14
k2g
.125 2 .09
5
We can derive this relationship more generally by noting that with a fixed ROE, earnings (which equal
ROE 3 book value) will grow at the same rate as the book value of the firm. Abstracting from issuance of new
shares of stock, the growth rate of book value equals reinvested earnings/book value. Therefore,
g5
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Reinvested earnings
Book value
5
Reinvested earnings
Total earnings
3
Total earnings
Book value
5 b 3 ROE
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When Growth Prospects pursued a no-growth policy and paid out all earnings as
dividends, the stock price was only $40. Therefore, you can think of $40 as the value per
share of the assets the company already has in place.
When Growth Prospects decided to reduce current dividends and reinvest some of its
earnings in new investments, its stock price increased. The increase in the stock price
reflects the fact that the planned investments provide an expected rate of return greater than
the required rate. In other words, the investment opportunities have positive net present
value. The value of the firm rises by the NPV of these investment opportunities. This net
present value is also called the present value of growth opportunities, or PVGO.
Therefore, we can think of the value of the firm as the sum of the value of assets already
in place, or the no-growth value of the firm, plus the net present value of the future investments the firm will make, which is the PVGO. For Growth Prospects, PVGO 5 $17.14 per
share:
Price 5 No-growth value per share 1 PVGO
E1
P0 5
1 PVGO
k
57.14 5 40 1 17.14
(18.6)
We know that in reality, dividend cuts almost always are accompanied by steep drops
in stock prices. Does this contradict our analysis? Not necessarily: Dividend cuts are usually taken as bad news about the future prospects of the firm, and it is the new information
about the firm—not the reduced dividend yield per se—that is responsible for the stock
price decline.
For example, when J.P. Morgan cut its quarterly dividend from 38 cents to 5 cents a
share in 2009, its stock price actually increased by about 5%. The company was able to
convince investors that the cut would conserve cash and prepare the firm to weather a
severe recession. When investors were convinced that the dividend cut made sense, the
stock price actually increased. Similarly, when BP announced in the wake of the massive
2010 Gulf oil spill that it would suspend dividends for the rest of the year, its stock price
did not budge. The cut already had been widely anticipated, so it was not new information.
These examples show that stock price declines in response to dividend cuts are really a
response to the information conveyed by the cut.
It is important to recognize that growth per se is not what investors desire. Growth
enhances company value only if it is achieved by investment in projects with attractive
profit opportunities (i.e., with ROE . k). To see why, let’s now consider Growth Prospects’s
unfortunate sister company, Cash Cow, Inc. Cash Cow’s ROE is only 12.5%, just equal to
the required rate of return, k. The net present value of its investment opportunities is zero.
We’ve seen that following a zero-growth strategy with b 5 0 and g 5 0, the value of Cash
Cow will be E1 / k 5 $5/.125 5 $40 per share. Now suppose Cash Cow chooses a plowback
ratio of b 5 .60, the same as Growth Prospects’s plowback. Then g would increase to
g 5 ROE 3 b 5 .125 3 .60 5 .075
but the stock price is still $40:
P0 5
D1
$2
5
5 $40
k2g
.125 2 .075
which is no different from the no-growth strategy.
In the case of Cash Cow, the dividend reduction used to free funds for reinvestment in the firm generates only enough growth to maintain the stock price at the current level. This is as it should be: If the firm’s projects yield only what investors can
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earn on their own, shareholders cannot be made better off by a high-reinvestment-rate
policy. This demonstrates that “growth” is not the same as growth opportunities. To justify reinvestment, the firm must engage in projects with better prospective returns than
those shareholders can find elsewhere. Notice also that the PVGO of Cash Cow is zero:
PVGO 5 P0 2 E1/k 5 40 2 40 5 0. With ROE 5 k, there is no advantage to plowing
funds back into the firm; this shows up as PVGO of zero. In fact, this is why firms with
considerable cash flow but limited investment prospects are called “cash cows.” The cash
these firms generate is best taken out of, or “milked from,” the firm.
Example 18.4
Growth Opportunities
Takeover Target is run by entrenched management that insists on reinvesting 60% of its
earnings in projects that provide an ROE of 10%, despite the fact that the firm’s capitalization rate is k 5 15%. The firm’s year-end dividend will be $2 per share, paid out
of earnings of $5 per share. At what price will the stock sell? What is the present value
of growth opportunities? Why would such a firm be a takeover target for another firm?
Given current management’s investment policy, the dividend growth rate will be
g 5 ROE 3 b 5 10% 3 .60 5 6%
and the stock price should be
P0 5
$2
5 $22.22
.15 2 .06
The present value of growth opportunities is
PVGO 5 Price per share 2 No-growth value per share
5 $22.22 2 E1 /k 5 $22.22 2 $5/.15 5 2$11.11
PVGO is negative. This is because the net present value of the firm’s projects is negative:
The rate of return on those assets is less than the opportunity cost of capital.
Such a firm would be subject to takeover, because another firm could buy the
firm for the market price of $22.22 per share and increase the value of the firm by
changing its investment policy. For example, if the new management simply paid out
all earnings as dividends, the value of the firm would increase to its no-growth value,
E1/k 5 $5/.15 5 $33.33.
CONCEPT CHECK
18.3
a. Calculate the price of a firm with a plowback ratio of .60 if its ROE is 20%. Current earnings, E1, will be
$5 per share, and k 5 12.5%.
b. What if ROE is 10%, which is less than the market capitalization rate? Compare the firm’s price in this
instance to that of a firm with the same ROE and E1, but a plowback ratio of b 5 0.
Life Cycles and Multistage Growth Models
As useful as the constant-growth DDM formula is, you need to remember that it is based
on a simplifying assumption, namely, that the dividend growth rate will be constant forever. In fact, firms typically pass through life cycles with very different dividend profiles in
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different phases. In early years, there are ample opportunities for profitable reinvestment in
the company. Payout ratios are low, and growth is correspondingly rapid. In later years, the
firm matures, production capacity is sufficient to meet market demand, competitors enter
the market, and attractive opportunities for reinvestment may become harder to find. In this
mature phase, the firm may choose to increase the dividend payout ratio, rather than retain
earnings. The dividend level increases, but thereafter it grows at a slower rate because the
company has fewer growth opportunities.
Table 18.2 illustrates this pattern. It gives Value Line’s forecasts of return on capital,
dividend payout ratio, and 3-year projected growth rate in earnings per share for a sample
of the firms included in the computer software industry versus those of East Coast electric
utilities. (We compare return on capital rather than return on equity because the latter is
affected by leverage, which tends to be far greater in the electric utility industry than in the
software industry. Return on capital measures operating income per dollar of total longterm financing, regardless of whether the source of the capital supplied is debt or equity.
We will return to this issue in the next chapter.)
By and large, the software firms have attractive investment opportunities. The median
return on capital of these firms is forecast to be 16.5%, and the firms have responded with
high plowback ratios. Most of these firms pay no dividends at all. The high return on capital and high plowback result in rapid growth. The median projected growth rate of earnings
per share in this group is 13.2%.
Table 18.2
Financial ratios in
two industries
Ticker
Computer Software
Adobe Systems
Cognizant
Compuware
Intuit
Microsoft
Oracle
Red Hat
Parametric Tech
SAP
Median
Electric Utilities (East Coast)
Central Hudson G&E
Consolidated Edison
Duke Energy
Northeast Utilities
Pennsylvania Power
Public Service Enterprise
South Carolina E & G
Southern Company
Tampa Electric
United Illuminating
Median
ADBE
CTSH
CPWR
INTU
MSFT
ORCL
RHT
PMTC
SAP
CHG
ED
DUK
NU
PPL
PEG
SCG
SO
TE
UIL
Return on
Capital (%)
Payout
Ratio (%)
Growth Rate
2014–2016
12.0%
18.5
13.5
20.0
31.5
20.5
13.0
15.0
16.5
16.5%
0.0%
0.0
0.0
22.0
34.0
12.0
0.0
0.0
28.0
0.0%
13.2%
20.5
16.6
10.9
11.7
7.0
18.2
16.0
9.1
13.2%
6.0%
6.5
5.5
6.0
7.0
7.5
6.0
7.0
7.5
6.0
6.3%
66.0%
58.0
66.0
53.0
58.0
53.0
57.0
69.0
59.0
71.0
58.5%
2.0%
2.9
4.0
7.7
7.7
6.3
3.8
5.1
8.3
2.1
4.5%
Source: Value Line Investment Survey, July and August, 2012. Reprinted with permission of Value Line
Investment Survey. © 2012 Value Line Publishing, Inc. All rights reserved.
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605
In contrast, the electric utilities are more representative of mature firms. Their median
return on capital is lower, 6.3%; dividend payout is higher, 58.5%; and median growth is
lower, 4.5%. We conclude that the higher payouts of the electric utilities reflect their more
limited opportunities to reinvest earnings at attractive rates of return.
To value companies with temporarily high growth, analysts use a multistage version of
the dividend discount model. Dividends in the early high-growth period are forecast and
their combined present value is calculated. Then, once the firm is projected to settle down
to a steady-growth phase, the constant-growth DDM is applied to value the remaining
stream of dividends.
We can illustrate this with a real-life example. Figure 18.2 is a Value Line Investment
Survey report on Honda Motor Co. Some of the relevant information for 2012 is highlighted.
Honda’s beta appears at the circled A, its recent stock price at the B, the per-share
dividend payments at the C, the ROE (referred to as return on shareholder equity) at the
D, and the dividend payout ratio (referred to as all dividends to net profits) at the E.6 The
rows ending at C, D, and E are historical time series. The boldfaced, italicized entries
under 2013 are estimates for that year. Similarly, the entries in the far right column (labeled
15–17) are forecasts for some time between 2015–2017, which we will take to be 2016.
Value Line projects fairly rapid growth in the near term, with dividends rising from $.78
in 2013 to $1.00 in 2016. This growth rate cannot be sustained indefinitely. We can obtain
dividend inputs for this initial period by using the explicit forecasts for 2013 and 2016 and
linear interpolation for the years between:
2013
2014
$.78
$.85
2015
2016
$ .92
$1.00
Now let us assume the dividend growth rate levels off in 2016. What is a good guess for
that steady-state growth rate? Value Line forecasts a dividend payout ratio of .25 and an
ROE of 10%, implying long-term growth will be
g 5 ROE 3 b 5 10.0% 3 (1 2 .25) 5 7.5%
Our estimate of Honda’s intrinsic value using an investment horizon of 2016 is therefore
obtained from Equation 18.2, which we restate here:
V2012 5
5
D2013
D2014
D2015
D2016 1 P2016
1
1
1
(1 1 k)2
(1 1 k)3
(1 1 k)4
11k
1.00 1 P2016
.78
.92
.85
1
1
1
2
3
(1 1 k)
(1 1 k)
(1 1 k)4
11k
Here, P2016 represents the forecast price at which we can sell our shares at the end of 2016,
when dividends are assumed to enter their constant-growth phase. That price, according to
the constant-growth DDM, should be
P2016 5
D2017
D2016(1 1 g)
1.00 3 1.075
5
5
k2g
k2g
k 2 .075
The only variable remaining to be determined to calculate intrinsic value is the market
capitalization rate, k.
6
Because Honda is a Japanese firm, Americans would hold its shares via ADRs, or American Depository
Receipts. ADRs are not shares of the firm, but are claims to shares of the underlying foreign stock that are then
traded in U.S. security markets. Value Line notes that each Honda ADR is a claim on one common share, but in
other cases, each ADR may represent a claim to multiple shares or even fractional shares.
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B
A
C
D
E
Figure 18.2 Value Line Investment Survey report on Honda Motor Co.
Source: Jason A. Smith, Value Line Investment Survey, May 25, 2012. Reprinted with permission of Value Line Investment Survey
© 2012 Value Line Publishing, Inc. All rights reserved.
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607
One way to obtain k is from the CAPM. Observe from the Value Line report that
Honda’s beta is .95. The risk-free rate on long-term Treasury bonds in 2012 was about
2.0%.7 Suppose that the market risk premium were forecast at 8%, roughly in line with its
historical average. This would imply that the forecast for the market return was
Risk-free rate 1 Market risk premium 5 2% 1 8% 5 10%
Therefore, we can solve for the market capitalization rate as
k 5 rf 1 b 3 E(rM) 2 rf 4 5 2% 1 .95(10% 2 2%) 5 9.6%
Our forecast for the stock price in 2016 is thus
P2016 5
$1.00 3 1.075
5 $51.19
.096 2 .075
and today’s estimate of intrinsic value is
V2012 5
.78
.85
.92
1.00 1 51.19
1
1
1
5 $38.29
(1.096)2
(1.096)3
(1.096)4
1.096
We know from the Value Line report that Honda’s actual price was $32.88 (at the circled B). Our intrinsic value analysis indicates that the stock was underpriced. Should we
increase our holdings?
Perhaps. But before betting the farm, stop to consider how firm our estimate is. We’ve
had to guess at dividends in the near future, the ultimate growth rate of those dividends,
and the appropriate discount rate. Moreover, we’ve assumed Honda will follow a relatively
simple two-stage growth process. In practice, the growth of dividends can follow more
complicated patterns. Even small errors in these approximations could upset a conclusion.
For example, suppose that we have overestimated Honda’s growth prospects and that
the actual ROE in the post-2016 period will be 9% rather than 10%. Using the lower return
on equity in the dividend discount model would result in an intrinsic value in 2012 of only
$28.77, which is less than the stock price. Our conclusion regarding intrinsic value versus
price is reversed.
The exercise also highlights the importance of performing sensitivity analysis when
you attempt to value stocks. Your estimates of stock values are no better than your assumptions. Sensitivity analysis will highlight the inputs that need to be most carefully examined. For example, even modest changes in the estimated ROE for the post-2016 period
can result in big changes in intrinsic value. Similarly, small changes in the assumed capitalization rate would change intrinsic value substantially. On the other hand, reasonable
changes in the dividends forecast between 2013 and 2016 would have a small impact on
intrinsic value.
CONCEPT CHECK
18.4
Confirm that the intrinsic value of Honda using ROE 5 9% is $28.77. (Hint: First calculate the stock price
in 2016. Then calculate the present value of all interim dividends plus the present value of the 2016 sales
price.)
7
When valuing long-term assets such as stocks, it is common to treat the long-term Treasury bond, rather than
short-term T-bills, as the risk-free asset.
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Multistage Growth Models
The two-stage growth model that we just considered for Honda is a good start toward
realism, but clearly we could do even better if our valuation model allowed for more
flexible patterns of growth. Multistage growth models allow dividends per share to grow
at several different rates as the firm matures. Many analysts use three-stage growth models. They may assume an initial period of high dividend growth (or instead make yearby-year forecasts of dividends for the short term), a final period of sustainable growth,
and a transition period between, during which dividend growth rates taper off from
the initial rapid rate to the ultimate sustainable rate. These models are conceptually no
harder to work with than a two-stage model, but they require many more calculations
and can be tedious to do by hand. It is easy, however, to build an Excel spreadsheet for
such a model.
Spreadsheet 18.1 is an example of such a model. Column B contains the inputs we have
used so far for Honda. Column E contains dividend forecasts. In cells E2 through E5 we
present the Value Line estimates for the next 4 years. Dividend growth in this period is
about 8.6% annually. Rather than assume a sudden transition to constant dividend growth
starting in 2016, we assume instead that the dividend growth rate in 2016 will be 8.6% and
that it will decline steadily through 2027, finally reaching the constant terminal growth rate
of 7.5% (see column F). Each dividend in the transition period is the previous year’s dividend times that year’s growth rate. Terminal value once the firm enters a constant-growth
stage (cell G17) is computed from the constant-growth DDM. Finally, investor cash flow
in each period (column H) equals dividends in each year plus the terminal value in 2027.
The present value of these cash flows is computed in cell H19 as $40.29, about 5% higher
than the value we found from the two-stage model. We obtain a greater intrinsic value in
this case because we assume that dividend growth only gradually declines to its steadystate value.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A
Inputs
beta
mkt_prem
rf
k_equity
plowback
roe
term_gwth
B
C
D
Year
0.95
0.08
0.02
0.0960
0.75
0.1
0.075
Value line
forecasts of
annual dividends
Transitional period
with slowing dividend
growth
Beginning of constant
growth period
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
E
F
G
Dividend Div growth Term value
0.78
0.85
0.92
1.00
1.09
0.0863
1.18
0.0852
1.28
0.0841
1.38
0.0829
1.50
0.0818
1.62
0.0807
1.75
0.0795
1.88
0.0784
2.03
0.0773
2.18
0.0761
2.35
0.0750
2.52
0.0750
129.18
40.29
I
= PV of CF
E17*(1+F17)/(B5 − F17)
Spreadsheet 18.1
A three-stage growth model for Honda Motor Co.
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H
Investor CF
0.78
0.85
0.92
1.00
1.09
1.18
1.28
1.38
1.50
1.62
1.75
1.88
2.03
2.18
2.35
131.71
NPV(B5,H2:H17)
eXcel
Please visit us at
www.mhhe.com/bkm
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18.4 Price–Earnings Ratio
The Price–Earnings Ratio and Growth Opportunities
Much of the real-world discussion of stock market valuation concentrates on the firm’s
price–earnings multiple, the ratio of price per share to earnings per share, commonly
called the P/E ratio. Our discussion of growth opportunities shows why stock market analysts focus on the P/E ratio. Both companies considered, Cash Cow and Growth Prospects,
had earnings per share (EPS) of $5, but Growth Prospects reinvested 60% of earnings in
prospects with an ROE of 15%, whereas Cash Cow paid out all earnings as dividends.
Cash Cow had a price of $40, giving it a P/E multiple of 40/5 5 8.0, whereas Growth Prospects sold for $57.14, giving it a multiple of 57.14/5 5 11.4. This observation suggests the
P/E ratio might serve as a useful indicator of expectations of growth opportunities.
We can see how growth opportunities are reflected in P/E ratios by rearranging
Equation 18.6 to
P0
1
PVGO
5 ¢1 1
≤
E1
k
E/k
(18.7)
When PVGO 5 0, Equation 18.7 shows that P0 5 E1/k. The stock is valued like a nongrowing perpetuity of E1, and the P/E ratio is just 1/k. However, as PVGO becomes an
increasingly dominant contributor to price, the P/E ratio can rise dramatically.
The ratio of PVGO to E/k has a straightforward interpretation. It is the ratio of the
component of firm value due to growth opportunities to the component of value due to
assets already in place (i.e., the no-growth value of the firm, E/k). When future growth
opportunities dominate the estimate of total value, the firm will command a high price
relative to current earnings. Thus a high P/E multiple indicates that a firm enjoys ample
growth opportunities.
P/E multiples do vary with growth prospects. Between 1996 and 2012, for example,
FedEx’s P/E ratio averaged about 17.4 while Consolidated Edison’s (an electric utility)
average P/E was only 15.7. These numbers do not necessarily imply that FedEx was overpriced compared to Con Ed. If investors believed FedEx would grow faster than Con Ed,
the higher price per dollar of earnings would be justified. That is, an investor might well
pay a higher price per dollar of current earnings if he or she expects that earnings stream
to grow more rapidly. In fact, FedEx’s growth rate has been consistent with its higher P/E
multiple. Over this period, its earnings per share grew at 10.2% per year while Con Ed’s
earnings growth rate was only 1.6%. Figure 18.4 on page 615 shows the EPS history of the
two companies.
We conclude that the P/E ratio reflects the market’s optimism concerning a firm’s
growth prospects. Analysts must decide whether they are more or less optimistic than the
belief implied by the market multiple. If they are more optimistic, they will recommend
buying the stock.
There is a way to make these insights more precise. Look again at the constant-growth
DDM formula, P0 5 D1/(k 2 g). Now recall that dividends equal the earnings that are not
reinvested in the firm: D1 5 E1(1 2 b). Recall also that g 5 ROE 3 b. Hence, substituting
for D1 and g, we find that
P0 5
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E1(1 2 b)
k 2 ROE 3 b
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implying the P/E ratio is
P0
12b
5
E1
k 2 ROE 3 b
(18.8)
It is easy to verify that the P/E ratio increases with ROE. This makes sense, because highROE projects give the firm good opportunities for growth.8 We also can verify that the P/E
ratio increases for higher plowback, b, as long as ROE exceeds k. This too makes sense.
When a firm has good investment opportunities, the market will reward it with a higher
P/E multiple if it exploits those opportunities more aggressively by plowing back more
earnings into those opportunities.
Remember, however, that growth is not desirable for its own sake. Examine Table 18.3
where we use Equation 18.8 to compute both growth rates and P/E ratios for different combinations of ROE and b. Although growth always increases with the plowback rate (move
across the rows in panel A), the P/E ratio does not (move across the rows in panel B). In
the top row of panel B, the P/E falls as the plowback rate increases. In the middle row, it is
unaffected by plowback. In the third row, it increases.
This pattern has a simple interpretation. When the expected ROE is less than the
required return, k, investors prefer that the firm pay out earnings as dividends rather than
reinvest earnings in the firm at an inadequate rate of return. That is, for ROE lower than
k, the value of the firm falls as plowback increases. Conversely, when ROE exceeds k, the
firm offers attractive investment opportunities, so the value of the firm is enhanced as those
opportunities are more fully exploited by increasing the plowback rate.
Finally, where ROE just equals k, the firm offers “break-even” investment opportunities with a fair rate of return. In this case, investors are indifferent between reinvestment
of earnings in the firm or elsewhere at the market capitalization rate, because the rate of
return in either case is 12%. Therefore, the stock price is unaffected by the plowback rate.
We conclude that the higher the plowback rate, the higher the growth rate, but a higher
plowback rate does not necessarily mean a higher P/E ratio. Higher plowback increases
P/E only if investments undertaken by the firm offer an expected rate of return greater than
the market capitalization rate. Otherwise, increasing plowback hurts investors because
more money is sunk into projects with inadequate rates of return.
Table 18.3
Effect of ROE
and plowback on
growth and the
P/E ratio
Plowback Rate (b)
0
ROE
10%
12
14
0
0
0
10%
12
14
8.33
8.33
8.33
.25
.50
A. Growth rate, g
2.5%
5.0%
3.0
6.0
3.5
7.0
B. P/E ratio
7.89
7.14
8.33
8.33
8.82
10.00
.75
7.5%
9.0
10.5
5.56
8.33
16.67
Assumption: k 5 12% per year.
8
Note that Equation 18.8 is a simple rearrangement of the DDM formula, with ROE 3 b 5 g. Because that
formula requires that g , k, Equation 18.8 is valid only when ROE 3 b , k.
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CHAPTER 18
Equity Valuation Models
611
Notwithstanding these fine points, P/E ratios frequently are taken as proxies for the
expected growth in dividends or earnings. In fact, a common Wall Street rule of thumb is
that the growth rate ought to be roughly equal to the P/E ratio. In other words, the ratio of
P/E to g, often called the PEG ratio, should be about 1.0. Peter Lynch, the famous portfolio
manager, puts it this way in his book One Up on Wall Street:
The P/E ratio of any company that’s fairly priced will equal its growth rate. I’m talking
here about growth rate of earnings here. . . . If the P/E ratio of Coca Cola is 15, you’d
expect the company to be growing at about 15% per year, etc. But if the P/E ratio is less
than the growth rate, you may have found yourself a bargain.
Example 18.5
P/E Ratio versus Growth Rate
Let’s try Lynch’s rule of thumb. Assume that
rf 5 8%
rM 2 rf 5 8%
b 5 .4
(roughly the value when Peter Lynch was writing)
(about the historical average market risk premium)
(a typical value for the plowback ratio in the United States)
Therefore, rM 5 rf 1 market risk premium 5 8% 1 8% 5 16%, and k 5 16% for an
average (b 5 1) company. If we also accept as reasonable that ROE 5 16% (the same
value as the expected return on the stock), we conclude that
g 5 ROE 3 b 5 16% 3 .4 5 6.4%
and
P
1 2 .4
5
5 6.26
E
.16 2 .064
Thus, the P/E ratio and g are about equal using these assumptions, consistent with the
rule of thumb.
However, note that this rule of thumb, like almost all others, will not work in all
circumstances. For example, the yield on long-term Treasury bonds today is more like
2%, so a comparable forecast of rM today would be
rf 1 Market risk premium 5 2% 1 8% 5 10%
If we continue to focus on a firm with b 5 1, and if ROE still is about the same as k, then
g 5 10% 3 .4 5 4.0%
while
P
1 2 .4
5
5 10
E
.10 2 .04
The P/E ratio and g now diverge and the PEG ratio is now 2.5. Nevertheless, lowerthan-average PEG ratios are still widely seen as signaling potential underpricing.
The importance of growth opportunities is most evident in the valuation of start-up
firms. For example, in the dot-com boom of the late 1990s, many companies that had yet to
turn a profit were valued by the market at billions of dollars. The perceived value of these
companies was exclusively as growth opportunities. For example, the online auction firm
eBay had 1998 profits of $2.4 million, far less than the $45 million profit earned by the
traditional auctioneer Sotheby’s; yet eBay’s market value was more than 10 times greater:
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$22 billion versus $1.9 billion. (As it turns out, the market was quite right to value eBay
so much more aggressively than Sotheby’s. Its net income in 2011 was $1.8 billion, more
than 10 times that of Sotheby’s.
Of course, when company valuation is determined primarily by growth opportunities,
those values can be very sensitive to reassessments of such prospects. When the market
became more skeptical of the business prospects of most Internet retailers at the close of
the 1990s, that is, as it revised the estimates of growth opportunities downward, their stock
prices plummeted.
As perceptions of future prospects wax and wane, share price can swing wildly. Growth
prospects are intrinsically difficult to tie down; ultimately, however, those prospects drive
the value of the most dynamic firms in the economy.
The nearby box contains a simple valuation analysis. As Facebook headed toward its
highly anticipated IPO in 2012, there was widespread speculation about the price at which
it would eventually trade in the stock market. Notice that the discussion in the article
focused on two key questions. First, what was a reasonable projection of the growth rate
of Facebook’s profits? Second, what multiple of earnings was appropriate to translate an
earnings forecast into a price forecast? These are precisely the questions addressed by our
stock valuation models.
CONCEPT CHECK
18.5
ABC stock has an expected ROE of 12% per year, expected earnings per share of $2, and expected dividends of $1.50 per share. Its market capitalization rate is 10% per year.
a. What are its expected growth rate, its price, and its P/E ratio?
b. If the plowback ratio were .4, what would be the expected dividend per share, the growth rate, price,
and the P/E ratio?
P/E Ratios and Stock Risk
One important implication of any stock-valuation model is that (holding all else equal)
riskier stocks will have lower P/E multiples. We can see this quite easily in the context of
the constant-growth model by examining the formula for the P/E ratio (Equation 18.8):
P
12b
5
E
k2g
Riskier firms will have higher required rates of return, that is, higher values of k. Therefore, the P/E multiple will be lower. This is true even outside the context of the constantgrowth model. For any expected earnings and dividend stream, the present value of those
cash flows will be lower when the stream is perceived to be riskier. Hence the stock price
and the ratio of price to earnings will be lower.
Of course, you can find many small, risky, start-up companies with very high P/E multiples. This does not contradict our claim that P/E multiples should fall with risk; instead it
is evidence of the market’s expectations of high growth rates for those companies. This is
why we said that high-risk firms will have lower P/E ratios holding all else equal. Given a
growth projection, the P/E multiple will be lower when risk is perceived to be higher.
Pitfalls in P/E Analysis
No description of P/E analysis is complete without mentioning some of its pitfalls. First,
consider that the denominator in the P/E ratio is accounting earnings, which are influenced
by somewhat arbitrary accounting rules such as the use of historical cost in depreciation
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WHAT IS FACEBOOK WORTH?
As investors dug into the company’s freshly released
financials Wednesday, analysts and investors began
circulating a range of values—from as little as $50 billion to
as much as $125 billion—for the social-networking Web site.
It will be months before the market sets a final price,
but already the valuation question has become a tug of
war over two essential questions: Just how fast can the
company continue to grow? And can it extract value from
advertising in the way it plans?
Facebook’s revenue grew 88% in 2011, and net income
grew 65%. Facebook’s growth has already decelerated from
154% from 2009 to 2010 to the 88% it experienced last year.
Francis Gaskins, president of IPOdesktop.com, which
analyzes IPOs for investors, says he doesn’t believe Facebook is worth more than $50 billion—50 times its reported
profits for 2011 of $1 billion, or more than triple the market’s average price-to-earnings ratio. Google Inc.’s profits
are 10 times that of Facebook, but its stock-market value is
$190 billion, he notes.
A $100 billion valuation “would have us believe that
Facebook is worth 53% of Google, even though Google’s
sales and profits are 10 times that of Facebook,” he said.
Martin Pyykkonen, an analyst at Denver banking boutique Wedge Partners, is more bullish, saying the value
could top $100 billion. He says Facebook could trade at 15
to 18 times next year’s expected earnings before interest,
taxes, and certain noncash charges, a cash-flow measure
known as EBITDA. By comparison, he says, mature companies trade at 8 to 10 times EBITDA. Microsoft Corp. trades
at 7 times, and Google about 10 times.
While that math only justifies an $81 billion valuation,
he says Facebook may be able to unlock faster growth in
ad spending and reach $5.5 billion in EBITDA, which could
justify a higher multiple of 20 times, implying a $110 billion
valuation.
Source: Randall Smith, “Facebook’s $100 Billion Question,” The
Wall Street Journal, February 3, 2012. Reprinted by permission of
The Wall Street Journal, Copyright © 2012 Dow Jones & Company,
Inc. All Rights Reserved Worldwide.
WORDS FROM THE STREET
Facebook’s $100 Billion Question
and inventory valuation. In times of high inflation, historic cost depreciation and inventory
costs will tend to underrepresent true economic values, because the replacement cost of
both goods and capital equipment will rise with the general level of prices. As Figure 18.3
demonstrates, P/E ratios generally have been inversely related to the inflation rate. In part,
this reflects the market’s assessment that earnings in high inflation periods are of “lower
quality,” artificially distorted by inflation, and warranting lower P/E ratios.
Earnings management is the practice of using flexibility in accounting rules to
improve the apparent profitability of the firm. We will have much to say on this topic in the
next chapter on interpreting financial statements. A version of earnings management that
became common in the 1990s was the reporting of “pro forma earnings” measures.
60
50
40
30
P/E Ratio
20
10
0
1955
Inflation Rate
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Figure 18.3 P/E ratios of the S&P 500 Index and inflation
613
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Pro forma earnings are calculated ignoring certain expenses, for example, restructuring charges, stock-option expenses, or write-downs of assets from continuing operations.
Firms argue that ignoring these expenses gives a clearer picture of the underlying profitability of the firm. Comparisons with earlier periods probably would make more sense if
those costs were excluded.
But when there is too much leeway for choosing what to exclude, it becomes hard for
investors or analysts to interpret the numbers or to compare them across firms. The lack of
standards gives firms considerable leeway to manage earnings.
Even GAAP allows firms considerable discretion to manage earnings. For example,
in the late 1990s, Kellogg took restructuring charges, which are supposed to be one-time
events, nine quarters in a row. Were these really one-time events, or were they more appropriately treated as ordinary expenses? Given the available leeway in managing earnings,
the justified P/E multiple becomes difficult to gauge.
Another confounding factor in the use of P/E ratios is related to the business cycle. We
were careful in deriving the DDM to define earnings as being net of economic depreciation, that is, the maximum flow of income that the firm could pay out without depleting
its productive capacity. But reported earnings are computed in accordance with generally
accepted accounting principles and need not correspond to economic earnings. Beyond
this, however, notions of a normal or justified P/E ratio, as in Equation 18.7 or 18.8, assume
implicitly that earnings rise at a constant rate, or, put another way, on a smooth trend line.
In contrast, reported earnings can fluctuate dramatically around a trend line over the course
of the business cycle.
Another way to make this point is to note that the “normal” P/E ratio predicted by
Equation 18.8 is the ratio of today’s price to the trend value of future earnings, E1. The
P/E ratio reported in the financial pages of the newspaper, by contrast, is the ratio of price
to the most recent past accounting earnings. Current accounting earnings can differ considerably from future economic earnings. Because ownership of stock conveys the right to
future as well as current earnings, the ratio of price to most recent earnings can vary substantially over the business cycle, as accounting earnings and the trend value of economic
earnings diverge by greater and lesser amounts.
As an example, Figure 18.4 graphs the earnings per share of FedEx and Con Ed since
1996. Note that FedEx’s EPS is far more variable. Because the market values the entire
stream of future dividends generated by the company, when earnings are temporarily depressed, the P/E ratio should tend to be high—that is, the denominator of the ratio
responds more sensitively to the business cycle than the numerator. This pattern is borne
out well.
Figure 18.5 graphs the P/E ratios of the two firms. FedEx has greater earnings volatility
and more variability in its P/E ratio. Its clearly higher average growth rate shows up in its
generally higher P/E ratio. The only period in which Con Ed’s ratio exceeded FedEx’s was
in 2012, a year when FedEx’s earnings rose at a far faster rate than its underlying trend.
The market seems to have decided that this earnings performance was not likely to be sustainable, and FedEx’s price rose less dramatically than its annual earnings. Consequently,
its P/E ratio declined.
This example shows why analysts must be careful in using P/E ratios. There is no way
to say P/E ratio is overly high or low without referring to the company’s long-run growth
prospects, as well as to current earnings per share relative to the long-run trend line.
Nevertheless, Figures 18.4 and 18.5 demonstrate a clear relation between P/E ratios and
growth. Despite considerable short-run fluctuations, FedEx’s EPS clearly trended upward
over the period. Con Ed’s earnings were essentially flat. FedEx’s growth prospects are
reflected in its consistently higher P/E multiple.
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CHAPTER 18
Combining P/E
Analysis and the
DDM
Earnings per Share (1992 = 1.0)
5
FedEx
4
3
2
Con Ed
1
0
1996
1998
2000
2002
2004
2006
2008
2010
2012
2010
2012
Figure 18.4 Earnings growth for two companies
25
20
P/E FedEx
15
10
P/E Con Ed
Some analysts use P/E
5
1996
1998
2000
2002
2004
2006
2008
ratios in conjunction with
earnings forecasts to estimate the price of a stock
at an investor’s horizon
Figure 18.5 Price–earnings ratios
date. The Honda analysis in
Figure 18.2 shows that
Value Line forecast a P/E ratio for 2016 of 14. EPS for 2016 was forecast at $4, implying
a price in 2016 of 14 3 $4 5 $56. Given an estimate of $56 for the 2016 sales price, we
would compute intrinsic value in 2012 as
V2012 5
615
6
P/E Ratio
This analysis suggests
that P/E ratios should vary
across industries, and in
fact they do. Figure 18.6
shows P/E ratios in 2012
for a sample of industries.
Notice that the industries
with the highest multiples—
such as business software
or biotech—have attractive
investment opportunities and
relatively high growth rates,
whereas the industries
with the lowest ratios—for
example, aerospace or computer manufacturers—are
in more mature or less profitable industries with limited growth opportunities.
The relationship between
P/E and growth is not perfect, which is not surprising in light of the pitfalls
discussed in this section,
but as a general rule, the
P/E multiple does appear to
track growth opportunities.
Equity Valuation Models
.78
.92
1.00 1 56
.85
1
1
5 $41.62
1
(1.096)2
(1.096)3
(1.096)4
1.096
Other Comparative Valuation Ratios
The price–earnings ratio is an example of a comparative valuation ratio. Such ratios are
used to assess the valuation of one firm versus another based on a fundamental indicator
such as earnings. For example, an analyst might compare the P/E ratios of two firms in the
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PART V
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Biotech
Business Software
Heavy Construction
Trucking
Auto Manufacturers
Restaurants
Food Products
Application Software
Asset Management
Chemical Products
Pharmaceuticals
Integrated Oil & Gas
Home Improvement
Electric Utilities
Industrial Metals
Iron Steel
Telecom Services
Computer Systems
Health Care Plans
Money Center Banks
Aerospace/Defense
57.8
34.7
32.4
28.0
25.3
21.4
21.1
17.5
17.5
17.4
17.2
16.8
16.5
15.6
14.9
14.8
14.7
13.1
11.8
11.3
8.5
0
10
20
30
P/E Ratio
40
50
60
Figure 18.6 P/E ratios for different industries
Source: Yahoo! Finance, finance.yahoo.com, September 12, 2012.
same industry to test whether the market is valuing one firm “more aggressively” than the
other. Other such comparative ratios are commonly used:
Price-to-Book Ratio This is the ratio of price per share divided by book value per
share. As we noted earlier in this chapter, some analysts view book value as a useful measure of value and therefore treat the ratio of price to book value as an indicator of how
aggressively the market values the firm.
Price-to-Cash-Flow Ratio Earnings as reported on the income statement can be
affected by the company’s choice of accounting practices, and thus are commonly viewed
as subject to some imprecision and even manipulation. In contrast, cash flow—which
tracks cash actually flowing into or out of the firm—is less affected by accounting decisions. As a result, some analysts prefer to use the ratio of price to cash flow per share
rather than price to earnings per share. Some analysts use operating cash flow when calculating this ratio; others prefer “free cash flow,” that is, operating cash flow net of new
investment.
Price-to-Sales Ratio Many start-up firms have no earnings. As a result, the price–
earnings ratio for these firms is meaningless. The price-to-sales ratio (the ratio of stock
price to the annual sales per share) has recently become a popular valuation benchmark for
these firms. Of course, price-to-sales ratios can vary markedly across industries, because
profit margins vary widely.
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CHAPTER 18
Equity Valuation Models
617
40
35
Price
Earnings
30
25
20
15
Price
Cash Flow
10
Price
Sales
5
0
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure 18.7 Market valuation statistics
Be Creative Sometimes a standard valuation ratio will simply not be available, and you
will have to devise your own. In the 1990s, some analysts valued retail Internet firms based
on the number of hits their Web sites received. As it turns out, they valued these firms
using too-generous “price-to-hits” ratios. Nevertheless, in a new investment environment,
these analysts used the information available to them to devise the best valuation tools they
could.
Figure 18.7 presents the behavior of several valuation measures. While the levels of
these ratios differ considerably, for the most part, they track each other fairly closely, with
upturns and downturns at the same times.
18.5 Free Cash Flow Valuation Approaches
An alternative approach to the dividend discount model values the firm using free cash
flow, that is, cash flow available to the firm or its equityholders net of capital expenditures. This approach is particularly useful for firms that pay no dividends, for which the
dividend discount model would be difficult to implement. But free cash flow models
may be applied to any firm and can provide useful insights about firm value beyond
the DDM.
One approach is to discount the free cash flow for the firm (FCFF) at the weightedaverage cost of capital to obtain the value of the firm, and subtract the then-existing value
of debt to find the value of equity. Another is to focus from the start on the free cash flow to
equityholders (FCFE), discounting those directly at the cost of equity to obtain the market
value of equity.
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The free cash flow to the firm is the after-tax cash flow generated by the firm’s
operations, net of investments in capital, and net working capital. It includes cash flows
available to both debt- and equityholders.9 It equals:
FCFF 5 EBIT (1 2 tc ) 1 Depreciation 2 Capital expenditures 2 Increase in NWC
(18.9)
where
EBIT 5 earnings before interest and taxes
tc 5 the corporate tax rate
NWC 5 net working capital
Alternatively, we can focus on cash flow available to equityholders. This will differ
from free cash flow to the firm by after-tax interest expenditures, as well as by cash flow
associated with net issuance or repurchase of debt (i.e., principal repayments minus proceeds from issuance of new debt).
FCFE 5 FCFF 2 Interest expense 3 (1 2 tc ) 1 Increases in net debt (18.10)
A free cash flow to the firm valuation model discounts year-by-year cash flows plus
some estimate of terminal value, VT. In Equation 18.11, we use the constant-growth model
to estimate terminal value and discount at the weighted-average cost of capital.
T
FCFFt
VT
FCFFT11
Firm value 5 a
t 1
T , where VT 5
(
)
(
)
WACC
2g
1
1
WACC
1
1
WACC
t51
(18.11)
To find equity value, we subtract the existing market value of debt from the derived value
of the firm.
Alternatively, we can discount free cash flows to equity (FCFE) at the cost of equity, kE.
T
FCFEt
VT
FCFET11
Intrinsic value of equity 5 a
t 1
T , where VT 5
(
)
(
)
kE 2 g
1
1
k
1 1 kE
t51
E
(18.12)
As in the dividend discount model, free cash flow models use a terminal value to avoid
adding the present values of an infinite sum of cash flows. That terminal value may simply
be the present value of a constant-growth perpetuity (as in the formulas above) or it may
be based on a multiple of EBIT, book value, earnings, or free cash flow. As a general rule,
estimates of intrinsic value depend critically on terminal value.
Spreadsheet 18.2 presents a free cash flow valuation of Honda using the data supplied
by Value Line in Figure 18.2. We start with the free cash flow to the firm approach given in
Equation 18.9. Panel A of the spreadsheet lays out values supplied by Value Line. Entries
for middle years are interpolated from beginning and final values. Panel B calculates free
cash flow. The sum of after-tax profits in row 11 (from Value Line) plus after-tax interest
payments in row 12 [i.e., interest expense 3 (1 2 tc )] equals EBIT(1 2 tc ). In row 13 we
subtract the change in net working capital, in row 14 we add back depreciation, and in row
15 we subtract capital expenditures. The result in row 17 is the free cash flow to the firm,
FCFF, for each year between 2013 and 2016.
To find the present value of these cash flows, we will discount at WACC, which is
calculated in panel C. WACC is the weighted average of the after-tax cost of debt and the
9
This is firm cash flow assuming all-equity financing. Any tax advantage to debt financing is recognized by using
an after-tax cost of debt in the computation of weighted-average cost of capital. This issue is discussed in any
introductory corporate finance text.
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CHAPTER 18
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
B
A. Input data
P/E
Cap spending/shr
LT Debt
Shares
EPS
Working Capital
B. Cash flow calculations
Profits (after tax)
Interest (after tax)
Chg Working Cap
Depreciation
Cap Spending
I
Equity Valuation Models
C
D
E
F
G
2012
2013
2014
2015
2016
14.35
2.65
30000
1800
3.10
36825
14.25
2.70
28500
1800
3.75
37750
14.17
2.82
27333
1798
3.83
41173
14.08
2.93
26167
1797
3.92
44597
14.00
3.05
25000
1795
4.00
48020
5700.0
702.0
6850.0
666.9
925.0
7250.0
4860.0
6970.0
639.6
3423.3
7166.67
5064.9
7090.0
612.3
3423.3
7083
5269.8
7210.0
585.0
3423.3
7000.0
5474.8
8981.9
6815.0
6288.0
4481.8
6092.5
4313.5
Terminal value
5896.9 104098.2
4145.3 83203.2
assumes fixed debt ratio after 2016
7000.0
FCFF
FCFE
C. Discount rate calculations
Current beta
0.95
Unlevered beta
0.767
terminal growth
0.02
tax_rate
0.35
r_debt
0.036
risk-free rate
0.02
market risk prem
0.08
MV equity
81795
0.27
Debt/Value
0.950
Levered beta
0.096
k_equity
0.077
WACC
PV factor for FCFF
1.000
PV factor for FCFE
1.000
0.25
0.934
0.095
0.077
0.929
0.913
0.23
0.919
0.094
0.077
0.862
0.835
0.22
0.904
0.092
0.077
0.800
0.765
100940
0.20
0.891
0.091
0.078
0.742
0.701
D. Present values
PV(FCFF)
PV(FCFE)
8341
6225
5421
3744
4875
3299
4378
2905
H
J
K
L
619
M
= (1-tax_rate) × r_debt × LT Debt
from Value Line
current beta/[1 + (1-tax)*debt/equity]
from Value Line
YTM in 2012 on A+ rated LT debt
Spreadsheet 18.2
Free cash flow valuation of Honda Motor Co.
*2012 P/E ratio is from Yahoo! Finance. Other inputs are from Value Line.
0.091
0.078
0.742
0.701
77283
58307
Row 3 × Row 11
linear trend from initial to final value
unlevered beta × [1 + (1-tax)*debt/equity]
from CAPM and levered beta
(1-t)*r_debt*D/V + k_equity*(1-D/V)
Discount each year at WACC
Discount each year at k_equity
Intrinsic val Equity val Intrin/share
100298
70298
39.05
74479
74479
41.38
eXcel
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cost of equity in each year. When computing WACC, we must account for the change in
leverage forecast by Value Line. To compute the cost of equity, we will use the CAPM as
in our earlier (dividend discount model) valuation exercise, but accounting for the fact that
equity beta will decline each year as the firm reduces leverage.10
10
Call bL the firm’s equity beta at the initial level of leverage as provided by Value Line. Equity betas reflect both
business risk and financial risk. When a firm changes its capital structure (debt/equity mix), it changes financial risk,
and therefore equity beta changes. How should we recognize the change in financial risk? As you may remember
from an introductory corporate finance class, you must first unleverage beta. This leaves us with business risk. We
use the following formula to find unleveraged beta, bU (where D/E is the firm’s current debt-equity ratio):
bL
bU 5
1 1 (D/E)(1 2 tc )
Then, we re-leverage beta in any particular year using the forecast capital structure for that year (which reintroduces the financial risk associated with that year’s capital structure):
bL 5 bU 3 1 1 (D/E)(1 2 tc ) 4
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To find Honda’s cost of debt, we note that its long-term bonds were rated A1 in 2012
and that yields to maturity on this quality debt at the time were about 3.6%. Honda’s debtto-value ratio (assuming its debt is selling near par value) is computed in row 29. In 2012,
the ratio was .27. Based on Value Line forecasts, it will fall to .20 by 2016. We interpolate
the debt-to-value ratio for the intermediate years. WACC is computed in row 32. WACC
increases slightly over time as the debt-to-value ratio declines between 2012 and 2016.
The present value factor for cash flows accruing in each year is the previous year’s factor
divided by (1 1 WACC) for that year. The present value of each cash flow (row 37) is the
free cash flow times the cumulative discount factor.
The terminal value of the firm (cell H17) is computed from the constant-growth model
as FCFF2016 3 (1 1 g)/(WACC2016 2 g), where g (cell B23) is the assumed value for the
steady growth rate. We assume in the spreadsheet that g 5 .02, roughly in line with the
long-run growth rate of the broad economy.11 Terminal value is also discounted back to
2012 (cell H37), and the intrinsic value of the firm is thus found as the sum of discounted
free cash flows between 2013 and 2016 plus the discounted terminal value. Finally, the
value of debt in 2012 is subtracted from firm value to arrive at the intrinsic value of equity
in 2012 (cell K37), and value per share is calculated in cell L37 as equity value divided by
number of shares in 2012.
The free cash flow to equity approach yields a similar intrinsic value for the stock.12
FCFE (row 18) is obtained from FCFF by subtracting after-tax interest expense and net
debt repurchases. The cash flows are then discounted at the equity rate. Like WACC, the
cost of equity changes each period as leverage changes. The present value factor for equity
cash flows is presented in row 34. Equity value is reported in cell J38, which is put on a per
share basis in cell L38.
Spreadsheet 18.2 is available at the Online Learning Center for this text, www.mhhe.
com/bkm.
Comparing the Valuation Models
In principle, the free cash flow approach is fully consistent with the dividend discount
model and should provide the same estimate of intrinsic value if one can extrapolate to
a period in which the firm begins to pay dividends growing at a constant rate. This was
demonstrated in two famous papers by Modigliani and Miller.13 However, in practice, you
will find that values from these models may differ, sometimes substantially. This is due
to the fact that in practice, analysts are always forced to make simplifying assumptions.
For example, how long will it take the firm to enter a constant-growth stage? How should
depreciation best be treated? What is the best estimate of ROE? Answers to questions
like these can have a big impact on value, and it is not always easy to maintain consistent
assumptions across the models.
11
In the long run a firm can’t grow forever at a rate higher than the aggregate economy. So by the time we assert
that growth is in a stable stage, it seems reasonable that the growth rate should not be significantly greater than
that of the overall economy (although it can be less if the firm is in a declining industry).
12
Over the 2013–2016 period, Value Line predicts that Honda will retire a considerable fraction of its outstanding debt. The implied debt repurchases are a use of cash and reduce the cash flow available to equity. Such
repurchases cannot be sustained indefinitely, however, for debt outstanding would soon be run down to zero.
Therefore, in our estimate of the terminal value of equity, we compute the final cash flow assuming that starting
in 2016 Honda will begin issuing enough debt to maintain its debt-to-value ratio. This approach is consistent with
the assumption of constant growth and constant discount rates after 2016.
13
Franco Modigliani and M. Miller, “The Cost of Capital, Corporation Finance, and the Theory of Investment,”
American Economic Review, June 1958, and “Dividend Policy, Growth, and the Valuation of Shares,” Journal of
Business, October 1961.
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We have now valued Honda using several approaches, with estimates of intrinsic value
as follows:
Model
Two-stage dividend discount model
DDM with earnings multiple terminal value
Three-stage DDM
Free cash flow to the firm
Free cash flow to equity
Market price (from Value Line)
Intrinsic Value
$38.81
41.62
40.29
39.05
41.38
32.88
What should we make of these differences? All the estimates are somewhat higher than
Honda’s actual stock price, perhaps indicating that they use an unrealistically high value
for the ultimate constant growth rate. For example, in the long run, it seems unlikely that
Honda will be able to grow as rapidly as Value Line’s forecast for 2016 growth, 7.5%.
The two-stage dividend discount model is the most conservative of the estimates, largely
because it assumes that Honda’s dividend growth rate will fall to its terminal value after
only 3 years. In contrast, the three-stage DDM allows growth to taper off over a longer
period. Given the consistency with which all these estimates exceed market price, perhaps
the stock is indeed underpriced compared to its intrinsic value.
On balance, however, this valuation exercise suggests that finding bargains is not as easy as
it seems. While these models are easy to apply, establishing proper inputs is more of a
challenge. This should not be surprising. In even a moderately efficient market, finding
profit opportunities will be more involved than analyzing Value Line data for a few hours.
These models are extremely useful to analysts, however, because they provide ballpark
estimates of intrinsic value. More than that, they force rigorous thought about underlying
assumptions and highlight the variables with the greatest impact on value and the greatest
payoff to further analysis.
The Problem with DCF Models
Our estimates of Honda’s intrinsic value are all based on discounted cash flow (DCF) models, in which we calculate the present value of forecasted cash flows and a terminal sales
price at some future date. It is clear from the calculations for Honda that most of the action
in these models is in the terminal value and that this value can be highly sensitive to even
small changes in some input values (see, for example, Concept Check 18.4). Therefore,
you must recognize that DCF valuation estimates are almost always going to be imprecise.
Growth opportunities and future growth rates are especially hard to pin down.
For this reason, many value investors employ a hierarchy of valuation. They view the most
reliable components of value as the items on the balance sheet that allow for the most precise
estimates of market value. Real estate, plant, and equipment would fall in this category.
A somewhat less reliable component of value is the economic profit on assets already in
place. For example, a company like Intel earns a far higher ROE on its investments in chipmaking facilities than its cost of capital. The present value of these “economic profits,” or
economic value added,14 is a major component of Intel’s market value. This component
of value is less certain than its balance sheet assets, however, because there is always a
concern that new competitors will enter the market, force down prices and profit margins,
and reduce the return on Intel’s investments. Thus, one needs to carefully assess the barriers to entry that protect Intel’s pricing and profit margins. We noted some of these barriers
in the last chapter, where we discussed the role of industry analysis, market structure, and
competitive position (see Section 17.7).
14
We discuss economic value added in greater detail in the next chapter.
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Finally, the least reliable components of value are growth opportunities, the purported
ability of firms like Intel to invest in positive-NPV ventures that contribute to high market valuations today. Value investors don’t deny that such opportunities exist, but they are
skeptical that precise values can be attached to them and, therefore, tend to be less willing
to make investment decisions that depend on the value of those opportunities.
18.6 The Aggregate Stock Market
The most popular approach to valuing the overall stock market is the earnings multiplier
approach applied at the aggregate level. The first step is to forecast corporate profits for the
coming period. Then we derive an estimate of the earnings multiplier, the aggregate P/E
ratio, based on a forecast of long-term interest rates. The product of the two forecasts is the
estimate of the end-of-period level of the market.
The forecast of the P/E ratio of the market is sometimes derived from a graph similar to
that in Figure 18.8, which plots the earnings yield (earnings per share divided by price per
share, the reciprocal of the P/E ratio) of the S&P 500 and the yield to maturity on 10-year
Treasury bonds. The two series clearly move in tandem over time and suggest that this
relationship and the current yield on 10-year Treasury bonds could help in forecasting the
earnings yield on the S&P 500. Given that earnings yield, a forecast of earnings could be
used to predict the level of the S&P in some future period. Let’s consider a simple example
of this procedure.
16
14
Yield (%)
12
Treasury Yield
10
8
6
Earnings Yield
4
2
0
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Figure 18.8 Earnings yield of S&P 500 versus 10-year Treasury-bond yield
Example 18.6
Forecasting the Aggregate Stock Market
In mid-2012, the forecast for 12-month earnings per share for the S&P 500 portfolio
was about $110. The 10-year Treasury bond yield was about 2.9%. As a first approach,
we might posit that the spread between the earnings yield and the Treasury yield,
which was around 3.9%, will remain at that level by the end of the year. Given a Treasury yield of 2.9%, this would imply an earnings yield for the S&P of 6.8%, and a
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623
P/E ratio of 1/.068 5 14.7. Our forecast for the level of the S&P index would then be
14.7 3 110 5 1,617. Given a current value for the S&P 500 of 1,457, this would imply
a 1-year capital gain on the index of 160/1,457 5 11.0%.
Of course, there is uncertainty regarding all three inputs into this analysis: the actual
earnings on the S&P 500 stocks, the level of Treasury yields at year-end, and the spread
between the Treasury yield and the earnings yield. One would wish to perform sensitivity
or scenario analysis to examine the impact of changes in all of these variables. To illustrate, consider Table 18.4, which shows a simple scenario analysis treating possible effects
of variation in the Treasury bond yield. The scenario analysis shows that forecast level of
the stock market varies inversely and with dramatic sensitivity to interest rate changes.
Pessimistic
Scenario
Treasury bond yield
Earnings yield
Resulting P/E ratio
EPS forecast
Forecast for S&P 500
3.4%
7.3%
13.7
110
1,507
Most Likely
Scenario
2.9%
6.8%
14.7
110
1,617
Optimistic
Scenario
2.4%
6.3%
15.9
110
1,749
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Some analysts use an aggregate version of the dividend discount model rather than an
earnings multiplier approach. All of these models, however, rely heavily on forecasts of
such macroeconomic variables as GDP, interest rates, and the rate of inflation, which are
difficult to predict accurately.
Because stock prices reflect expectations of future dividends, which are tied to the economic fortunes of firms, it is not surprising that the performance of a broad-based stock
index like the S&P 500 is taken as a leading economic indicator, that is, a predictor of the
performance of the aggregate economy. Stock prices are viewed as embodying consensus forecasts of economic activity and are assumed to move up or down in anticipation
of movements in the economy. The government’s index of leading economic indicators,
which is taken to predict the progress of the business cycle, is made up in part of recent
stock market performance. However, the predictive value of the market is far from perfect.
A well-known joke, often attributed to Paul Samuelson, is that the market has forecast
eight of the last five recessions.
Table 18.4
S&P 500 index forecasts under various
scenarios
Forecast for the earnings yield on the S&P 500 equals Treasury bond yield plus
3.9%. The P/E ratio is the reciprocal of the forecast earnings yield.
1. One approach to firm valuation is to focus on the firm’s book value, either as it appears on the
balance sheet or as adjusted to reflect current replacement cost of assets or liquidation value.
Another approach is to focus on the present value of expected future dividends.
SUMMARY
2. The dividend discount model holds that the price of a share of stock should equal the present
value of all future dividends per share, discounted at an interest rate commensurate with the risk
of the stock.
3. Dividend discount models give estimates of the intrinsic value of a stock. If price does not equal
intrinsic value, the rate of return will differ from the equilibrium return based on the stock’s risk.
The actual return will depend on the rate at which the stock price is predicted to revert to its
intrinsic value.
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4. The constant-growth version of the DDM asserts that if dividends are expected to grow at a
constant rate forever, the intrinsic value of the stock is determined by the formula
V0 5
D1
k2g
This version of the DDM is simplistic in its assumption of a constant value of g. There are
more-sophisticated multistage versions of the model for more-complex environments. When the
constant-growth assumption is reasonably satisfied and the stock is selling for its intrinsic value,
the formula can be inverted to infer the market capitalization rate for the stock:
k5
D1
1g
P0
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5. The constant-growth dividend discount model is best suited for firms that are expected to exhibit
stable growth rates over the foreseeable future. In reality, however, firms progress through life
cycles. In early years, attractive investment opportunities are ample and the firm responds with
high plowback ratios and rapid dividend growth. Eventually, however, growth rates level off to
more sustainable values. Three-stage growth models are well suited to such a pattern. These
models allow for an initial period of rapid growth, a final period of steady dividend growth, and
a middle, or transition, period in which the dividend growth rate declines from its initial high
rate to the lower sustainable rate.
6. Stock market analysts devote considerable attention to a company’s price-to-earnings ratio. The
P/E ratio is a useful measure of the market’s assessment of the firm’s growth opportunities.
Firms with no growth opportunities should have a P/E ratio that is just the reciprocal of the capitalization rate, k. As growth opportunities become a progressively more important component of
the total value of the firm, the P/E ratio will increase.
7. The expected growth rate of earnings is related both to the firm’s expected profitability and to
its dividend policy. The relationship can be expressed as
g 5 (ROE on new investment) 3 (1 2 Dividend payout ratio)
8. You can relate any DDM to a simple capitalized earnings model by comparing the expected
ROE on future investments to the market capitalization rate, k. If the two rates are equal, then
the stock’s intrinsic value reduces to expected earnings per share (EPS) divided by k.
9. Many analysts form their estimates of a stock’s value by multiplying their forecast of next
year’s EPS by a predicted P/E multiple. Some analysts mix the P/E approach with the dividend
discount model. They use an earnings multiplier to forecast the terminal value of shares at a
future date, and add the present value of that terminal value with the present value of all interim
dividend payments.
Related Web sites
for this chapter are
available at www.
mhhe.com/bkm
KEY TERMS
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10. The free cash flow approach is the one used most often in corporate finance. The analyst first
estimates the value of the entire firm as the present value of expected future free cash flows to
the entire firm and then subtracts the value of all claims other than equity. Alternatively, the free
cash flows to equity can be discounted at a discount rate appropriate to the risk of the stock.
11. The models presented in this chapter can be used to explain and forecast the behavior of the
aggregate stock market. The key macroeconomic variables that determine the level of stock
prices in the aggregate are interest rates and corporate profits.
book value
liquidation value
replacement cost
Tobin’s q
intrinsic value of a share
market capitalization rate
dividend discount model (DDM)
constant-growth DDM
dividend payout ratio
plowback ratio
earnings retention ratio
present value of growth
opportunities (PVGO)
price–earnings multiple
earnings management
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CHAPTER 18
Intrinsic value: V0 5
Equity Valuation Models
D2
D1
D H 1 PH
1
1 c1
(1 1 k)H
11k
(1 1 k)2
Constant growth DDM: V0 5
KEY EQUATIONS
D1
k2g
Growth opportunities: Price 5
Determinant of P/E ratio:
625
E1
1 PVGO
k
P0
1
PVGO
5 ¢1 1
≤
E1
k
E1/k
Free cash flow to the firm:
FCFF 5 EBIT(1 2 tc) 1 Depreciation 2 Capital expenditures 2 Increases in NWC
1. In what circumstances would you choose to use a dividend discount model rather than a free
cash flow model to value a firm?
PROBLEM SETS
2. In what circumstances is it most important to use multistage dividend discount models rather
than constant-growth models?
3. If a security is underpriced (i.e., intrinsic value . price), then what is the relationship between
its market capitalization rate and its expected rate of return?
Basic
4. Deployment Specialists pays a current (annual) dividend of $1.00 and is expected to grow at
20% for 2 years and then at 4% thereafter. If the required return for Deployment Specialists is
8.5%, what is the intrinsic value of Deployment Specialists stock?
5. Jand, Inc., currently pays a dividend of $1.22, which is expected to grow indefinitely at 5%.
If the current value of Jand’s shares based on the constant-growth dividend discount model is
$32.03, what is the required rate of return?
6. A firm pays a current dividend of $1.00 which is expected to grow at a rate of 5% indefinitely. If
current value of the firm’s shares is $35.00, what is the required return applicable to the investment based on the constant-growth dividend discount model (DDM)?
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Free cash flow to equity: FCFE 5 FCFF 2 Interest expense 3 (1 2 tc) 1 Increases in net debt
7. Tri-coat Paints has a current market value of $41 per share with earnings of $3.64. What is the
present value of its growth opportunities (PVGO) if the required return is 9%?
8. a. Computer stocks currently provide an expected rate of return of 16%. MBI, a large computer company, will pay a year-end dividend of $2 per share. If the stock is selling at $50 per
share, what must be the market’s expectation of the growth rate of MBI dividends?
b. If dividend growth forecasts for MBI are revised downward to 5% per year, what will happen to the price of MBI stock? What (qualitatively) will happen to the company’s price–
earnings ratio?
Intermediate
9. a. MF Corp. has an ROE of 16% and a plowback ratio of 50%. If the coming year’s earnings
are expected to be $2 per share, at what price will the stock sell? The market capitalization
rate is 12%.
b. What price do you expect MF shares to sell for in 3 years?
10. The market consensus is that Analog Electronic Corporation has an ROE 5 9%, has a beta of
1.25, and plans to maintain indefinitely its traditional plowback ratio of 2/3. This year’s earnings were $3 per share. The annual dividend was just paid. The consensus estimate of the coming year’s market return is 14%, and T-bills currently offer a 6% return.
a. Find the price at which Analog stock should sell.
b. Calculate the P/E ratio.
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c. Calculate the present value of growth opportunities.
d. Suppose your research convinces you Analog will announce momentarily that it will immediately
reduce its plowback ratio to 1/3. Find the intrinsic value of the stock. The market is still unaware
of this decision. Explain why V0 no longer equals P0 and why V0 is greater or less than P0.
11. The FI Corporation’s dividends per share are expected to grow indefinitely by 5% per year.
a. If this year’s year-end dividend is $8 and the market capitalization rate is 10% per year, what
must the current stock price be according to the DDM?
b. If the expected earnings per share are $12, what is the implied value of the ROE on future
investment opportunities?
c. How much is the market paying per share for growth opportunities (i.e., for an ROE on
future investments that exceeds the market capitalization rate)?
12. The stock of Nogro Corporation is currently selling for $10 per share. Earnings per share in the
coming year are expected to be $2. The company has a policy of paying out 50% of its earnings each year in dividends. The rest is retained and invested in projects that earn a 20% rate of
return per year. This situation is expected to continue indefinitely.
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a. Assuming the current market price of the stock reflects its intrinsic value as computed using
the constant-growth DDM, what rate of return do Nogro’s investors require?
b. By how much does its value exceed what it would be if all earnings were paid as dividends
and nothing were reinvested?
c. If Nogro were to cut its dividend payout ratio to 25%, what would happen to its stock price?
What if Nogro eliminated the dividend?
13. The risk-free rate of return is 8%, the expected rate of return on the market portfolio is 15%,
and the stock of Xyrong Corporation has a beta coefficient of 1.2. Xyrong pays out 40% of its
earnings in dividends, and the latest earnings announced were $10 per share. Dividends were
just paid and are expected to be paid annually. You expect that Xyrong will earn an ROE of 20%
per year on all reinvested earnings forever.
a. What is the intrinsic value of a share of Xyrong stock?
b. If the market price of a share is currently $100, and you expect the market price to be equal
to the intrinsic value 1 year from now, what is your expected 1-year holding-period return on
Xyrong stock?
14. The Digital Electronic Quotation System (DEQS) Corporation pays no cash dividends currently
and is not expected to for the next 5 years. Its latest EPS was $10, all of which was reinvested
in the company. The firm’s expected ROE for the next 5 years is 20% per year, and during this
time it is expected to continue to reinvest all of its earnings. Starting in year 6, the firm’s ROE
on new investments is expected to fall to 15%, and the company is expected to start paying out
40% of its earnings in cash dividends, which it will continue to do forever after. DEQS’s market
capitalization rate is 15% per year.
a. What is your estimate of DEQS’s intrinsic value per share?
b. Assuming its current market price is equal to its intrinsic value, what do you expect to happen to its price over the next year? The year after?
c. What effect would it have on your estimate of DEQS’s intrinsic value if you expected DEQS
to pay out only 20% of earnings starting in year 6?
eXcel
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15. Recalculate the intrinsic value of Honda in each of the following scenarios by using the threestage growth model of Spreadsheet 18.1 (available at www.mhhe.com/bkm; link to Chapter 18
material). Treat each scenario independently.
a. ROE in the constant-growth period will be 9%.
b. Honda’s actual beta is 1.0.
c. The market risk premium is 8.5%.
eXcel
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16. Recalculate the intrinsic value of Honda shares using the free cash flow model of Spreadsheet
18.2 (available at www.mhhe.com/bkm; link to Chapter 18 material) under each of the following assumptions. Treat each scenario independently.
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a. Honda’s P/E ratio starting in 2016 will be 15.
b. Honda’s unlevered beta is 0.7.
c. The market risk premium is 9%.
17. The Duo Growth Company just paid a dividend of $1 per share. The dividend is expected to
grow at a rate of 25% per year for the next 3 years and then to level off to 5% per year forever.
You think the appropriate market capitalization rate is 20% per year.
a. What is your estimate of the intrinsic value of a share of the stock?
b. If the market price of a share is equal to this intrinsic value, what is the expected dividend
yield?
c. What do you expect its price to be 1 year from now? Is the implied capital gain consistent
with your estimate of the dividend yield and the market capitalization rate?
a. What is your estimate of GG’s intrinsic value per share?
b. Assuming its current market price is equal to its intrinsic value, what do you expect to
happen to its price over the next year?
19. The MoMi Corporation’s cash flow from operations before interest and taxes was $2 million
in the year just ended, and it expects that this will grow by 5% per year forever. To make this
happen, the firm will have to invest an amount equal to 20% of pretax cash flow each year. The
tax rate is 35%. Depreciation was $200,000 in the year just ended and is expected to grow at the
same rate as the operating cash flow. The appropriate market capitalization rate for the unleveraged cash flow is 12% per year, and the firm currently has debt of $4 million outstanding. Use
the free cash flow approach to value the firm’s equity.
20. Chiptech, Inc., is an established computer chip firm with several profitable existing products as
well as some promising new products in development. The company earned $1 a share last year,
and just paid out a dividend of $.50 per share. Investors believe the company plans to maintain
its dividend payout ratio at 50%. ROE equals 20%. Everyone in the market expects this situation to persist indefinitely.
a. What is the market price of Chiptech stock? The required return for the computer chip industry is 15%, and the company has just gone ex-dividend (i.e., the next dividend will be paid a
year from now, at t 5 1).
b. Suppose you discover that Chiptech’s competitor has developed a new chip that will eliminate Chiptech’s current technological advantage in this market. This new product, which
will be ready to come to the market in 2 years, will force Chiptech to reduce the prices of
its chips to remain competitive. This will decrease ROE to 15%, and, because of falling
demand for its product, Chiptech will decrease the plowback ratio to .40. The plowback ratio
will be decreased at the end of the second year, at t 5 2: The annual year-end dividend for
the second year (paid at t 5 2) will be 60% of that year’s earnings. What is your estimate of
Chiptech’s intrinsic value per share? (Hint: Carefully prepare a table of Chiptech’s earnings
and dividends for each of the next 3 years. Pay close attention to the change in the payout
ratio in t 5 2.)
c. No one else in the market perceives the threat to Chiptech’s market. In fact, you are confident that no one else will become aware of the change in Chiptech’s competitive status until
the competitor firm publicly announces its discovery near the end of year 2. What will be the
rate of return on Chiptech stock in the coming year (i.e., between t 5 0 and t 5 1)? In the
second year (between t 5 1 and t 5 2)? The third year (between t 5 2 and t 5 3)? (Hint: Pay
attention to when the market catches on to the new situation. A table of dividends and market
prices over time might help.)
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Challenge
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18. The Generic Genetic (GG) Corporation pays no cash dividends currently and is not
expected to for the next 4 years. Its latest EPS was $5, all of which was reinvested in the
company. The firm’s expected ROE for the next 4 years is 20% per year, during which time
it is expected to continue to reinvest all of its earnings. Starting in year 5, the firm’s ROE on
new investments is expected to fall to 15% per year. GG’s market capitalization rate is 15%
per year.
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1. At Litchfield Chemical Corp. (LCC), a director of the company said that the use of dividend
discount models by investors is “proof ” that the higher the dividend, the higher the stock price.
a. Using a constant-growth dividend discount model as a basis of reference, evaluate the director’s statement.
b. Explain how an increase in dividend payout would affect each of the following (holding all
other factors constant):
i. Sustainable growth rate.
ii. Growth in book value.
2. Helen Morgan, CFA, has been asked to use the DDM to determine the value of Sundanci, Inc.
Morgan anticipates that Sundanci’s earnings and dividends will grow at 32% for 2 years and 13%
thereafter. Calculate the current value of a share of Sundanci stock by using a two-stage dividend
discount model and the data from Tables 18A and 18B.
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3. Abbey Naylor, CFA, has been directed to determine the value of Sundanci’s stock using the Free
Cash Flow to Equity (FCFE) model. Naylor believes that Sundanci’s FCFE will grow at 27%
for 2 years and 13% thereafter. Capital expenditures, depreciation, and working capital are all
expected to increase proportionately with FCFE.
a. Calculate the amount of FCFE per share for the year 2011, using the data from Table 18A.
b. Calculate the current value of a share of Sundanci stock based on the two-stage FCFE
model.
c. i. Describe one limitation of the two-stage DDM model that is addressed by using the twostage FCFE model.
ii. Describe one limitation of the two-stage DDM model that is not addressed by using the
two-stage FCFE model.
Table 18A
Sundanci actual 2010
and 2011 financial
statements for fiscal
years ending May 31
($ million, except pershare data)
Income Statement
Revenue
Depreciation
Other operating costs
2011
$ 474
$ 598
20
23
368
460
Income before taxes
86
115
Taxes
26
35
Net income
60
80
Dividends
18
24
Earnings per share
$0.714
$0.952
Dividend per share
$0.214
$0.286
Common shares outstanding (millions)
84.0
84.0
Balance Sheet
2010
2011
Current assets
$ 201
$ 326
474
489
Total assets
675
815
Current liabilities
57
141
Long-term debt
0
0
Net property, plant and equipment
Total liabilities
Shareholders’ equity
Total liabilities and equity
Capital expenditures
bod61671_ch18_591-634.indd 628
2010
57
141
618
674
675
815
34
38
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CHAPTER 18
Required rate of return on equity
14%
Growth rate of industry
13%
Industry P/E ratio
26
Equity Valuation Models
629
Table 18B
Selected financial
information
4. Christie Johnson, CFA, has been assigned to analyze Sundanci using the constant dividend growth
price/earnings (P/E) ratio model. Johnson assumes that Sundanci’s earnings and dividends will
grow at a constant rate of 13%.
a. Calculate the P/E ratio based on information in Tables 18A and 18B and on Johnson’s assumptions for Sundanci.
b. Identify, within the context of the constant dividend growth model, how each of the following
factors would affect the P/E ratio.
5. Dynamic Communication is a U.S. industrial company with several electronics divisions. The
company has just released its 2013 annual report. Tables 18C and 18D present a summary of
Dynamic’s financial statements for the years 2012 and 2013. Selected data from the financial
statements for the years 2009 to 2011 are presented in Table 18E.
a. A group of Dynamic shareholders has expressed concern about the zero growth rate of dividends in the last 4 years and has asked for information about the growth of the company.
Calculate Dynamic’s sustainable growth rates in 2010 and 2013. Your calculations should use
beginning-of-year balance sheet data.
b. Determine how the change in Dynamic’s sustainable growth rate (2013 compared to 2010)
was affected by changes in its retention ratio and its financial leverage. (Note: Your calculations should use beginning-of-year balance sheet data.)
Table 18C
$ Million
2013
2012
Cash and equivalents
Accounts receivable
$ 149
295
$
Inventory
Total current assets
275
$ 719
285
$ 633
Gross fixed assets
Accumulated depreciation
Net fixed assets
9,350
(6,160)
$3,190
8,900
(5,677)
$3,223
Total assets
$3,909
$3,856
Accounts payable
Notes payable
Accrued taxes and expenses
$ 228
0
0
$ 220
0
0
Total current liabilities
$ 228
$ 220
Long-term debt
$1,650
$1,800
Common stock
50
50
Additional paid-in capital
Retained earnings
Total shareholders’ equity
0
1,981
$2,031
0
1,786
$1,836
Total liabilities and shareholders’ equity
$3,909
$3,856
bod61671_ch18_591-634.indd 629
83
265
Dynamic Communication balance sheets
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• Risk (beta) of Sundanci.
• Estimated growth rate of earnings and dividends.
• Market risk premium.
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PART V
Security Analysis
Table 18D
Dynamic Communication statements of
income (U.S. $ millions
except for share data)
2013
2012
Total revenues
Operating costs and expenses
Earnings before interest, taxes,
depreciation and amortization (EBITDA)
Depreciation and amortization
$3,425
2,379
$1,046
$3,300
2,319
$ 981
483
454
Operating income (EBIT)
$ 563
$ 527
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Interest expense
104
107
Income before taxes
Taxes (40%)
Net income
$ 459
184
$ 275
$ 420
168
$ 252
Dividends
Change in retained earnings
$ 80
$ 195
$ 80
$ 172
Earnings per share
Dividends per share
$ 2.75
$ 0.80
$ 2.52
$ 0.80
100
100
Number of shares outstanding (millions)
Table 18E
Dynamic Communication selected data
from financial statements (U.S. $ millions
except for share data)
Total revenues
Operating income (EBIT)
Interest expense
Net income
Dividends per share
Total assets
Long-term debt
Total shareholders’ equity
Number of shares outstanding (millions)
2011
2010
2009
$3,175
495
104
$ 235
$ 0.80
$3,625
$1,750
$1,664
$3,075
448
101
$ 208
$ 0.80
$3,414
$1,700
$1,509
$3,000
433
99
$ 200
$ 0.80
$3,230
$1,650
$1,380
100
100
100
6. Mike Brandreth, an analyst who specializes in the electronics industry, is preparing a research
report on Dynamic Communication. A colleague suggests to Brandreth that he may be able to
determine Dynamic’s implied dividend growth rate from Dynamic’s current common stock price,
using the Gordon growth model. Brandreth believes that the appropriate required rate of return
for Dynamic’s equity is 8%.
a. Assume that the firm’s current stock price of $58.49 equals intrinsic value. What sustained
rate of dividend growth as of December 2013 is implied by this value? Use the constantgrowth dividend discount model (i.e., the Gordon growth model).
b. The management of Dynamic has indicated to Brandreth and other analysts that the company’s current dividend policy will be continued. Is the use of the Gordon growth model to
value Dynamic’s common stock appropriate or inappropriate? Justify your response based on
the assumptions of the Gordon growth model.
7. Peninsular Research is initiating coverage of a mature manufacturing industry. John Jones, CFA,
head of the research department, gathered the following fundamental industry and market data to
help in his analysis:
Forecast industry earnings retention rate
Forecast industry return on equity
Industry beta
Government bond yield
Equity risk premium
bod61671_ch18_591-634.indd 630
40%
25%
1.2
6%
5%
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CHAPTER 18
Equity Valuation Models
631
a. Compute the price-to-earnings (P0 /E1) ratio for the industry based on this fundamental data.
b. Jones wants to analyze how fundamental P/E ratios might differ among countries. He gathered the following economic and market data:
Fundamental Factors
Forecast growth in real GDP
Government bond yield
Equity risk premium
Country A
Country B
5%
10%
5%
2%
6%
4%
Determine whether each of these fundamental factors would cause P/E ratios to be generally
higher for Country A or higher for Country B.
8. Janet Ludlow’s firm requires all its analysts to use a two-stage dividend discount model (DDM)
and the capital asset pricing model (CAPM) to value stocks. Using the CAPM and DDM, Ludlow
has valued QuickBrush Company at $63 per share. She now must value SmileWhite Corporation.
a. Calculate the required rate of return for SmileWhite by using the information in the following table:
Beta
Market price
Intrinsic value
Notes:
Risk-free rate
Expected market return
1.35
$45.00
$63.00
SmileWhite
1.15
$30.00
?
4.50%
14.50%
b. Ludlow estimates the following EPS and dividend growth rates for SmileWhite:
First 3 years
Years thereafter
12% per year
9% per year
Estimate the intrinsic value of SmileWhite by using the table above, and the two-stage
DDM. Dividends per share in the most recent year were $1.72.
c. Recommend QuickBrush or SmileWhite stock for purchase by comparing each company’s
intrinsic value with its current market price.
d. Describe one strength of the two-stage DDM in comparison with the constant-growth DDM.
Describe one weakness inherent in all DDMs.
9. Rio National Corp. is a U.S.-based company and the largest competitor in its industry. Tables
18F–18I present financial statements and related information for the company. Table 18J presents relevant industry and market data.
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QuickBrush
The portfolio manager of a large mutual fund comments to one of the fund’s analysts, Katrina
Shaar: “We have been considering the purchase of Rio National Corp. equity shares, so I would
like you to analyze the value of the company. To begin, based on Rio National’s past performance, you can assume that the company will grow at the same rate as the industry.”
a. Calculate the value of a share of Rio National equity on December 31, 2013, using the
Gordon constant-growth model and the capital asset pricing model.
b. Calculate the sustainable growth rate of Rio National on December 31, 2013. Use 2013
beginning-of-year balance sheet values.
10. While valuing the equity of Rio National Corp. (from the previous problem), Katrina Shaar
is considering the use of either cash flow from operations (CFO) or free cash flow to equity
(FCFE) in her valuation process.
a. State two adjustments that Shaar should make to cash flow from operations to obtain free
cash flow to equity.
b. Shaar decides to calculate Rio National’s FCFE for the year 2013, starting with net income.
Determine for each of the five supplemental notes given in Table 18H whether an adjustment
should be made to net income to calculate Rio National’s free cash flow to equity for the
year 2013, and the dollar amount of any adjustment.
c. Calculate Rio National’s free cash flow to equity for the year 2013.
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632
PART V
Security Analysis
Table 18F
Rio National Corp.
summary year-end
balance sheets (U.S.
$ millions)
2013
2012
Cash
Accounts receivable
Inventory
$ 13.00
30.00
209.06
$ 5.87
27.00
189.06
Current assets
Gross fixed assets
Accumulated depreciation
$252.06
474.47
(154.17)
$221.93
409.47
(90.00)
320.30
319.47
Total assets
Accounts payable
Notes payable
Current portion of long-term debt
$572.36
$ 25.05
0.00
0.00
$541.40
$ 26.05
0.00
0.00
Current liabilities
Long-term debt
$ 25.05
240.00
$ 26.05
245.00
Total liabilities
Common stock
Retained earnings
$265.05
160.00
147.31
$271.05
150.00
120.35
Total shareholders’ equity
$307.31
$270.35
Total liabilities and shareholders’ equity
$572.36
$541.40
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Net fixed assets
Table 18G
Rio National Corp.
summary income
statement for the year
ended December 31,
2013 (U.S. $ millions)
Revenue
Total operating expenses
Operating profit
Gain on sale
127.06
4.00
Earnings before interest, taxes, depreciation &
amortization (EBITDA)
Depreciation and amortization
131.06
Earnings before interest & taxes (EBIT)
Interest
Income tax expense
59.89
(16.80)
(12.93)
Net income
Table 18H
Rio National Corp.
supplemental notes
for 2013
Note 1:
Note 2:
Note 3:
Note 4:
Note 5:
bod61671_ch18_591-634.indd 632
$300.80
(173.74)
(71.17)
$ 30.16
Rio National had $75 million in capital expenditures during the year.
A piece of equipment that was originally purchased for $10 million was sold for
$7 million at year-end, when it had a net book value of $3 million. Equipment
sales are unusual for Rio National.
The decrease in long-term debt represents an unscheduled principal repayment;
there was no new borrowing during the year.
On January 1, 2013, the company received cash from issuing 400,000 shares of
common equity at a price of $25.00 per share.
A new appraisal during the year increased the estimated market value of land held
for investment by $2 million, which was not recognized in 2013 income.
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CHAPTER 18
Dividends paid (U.S. $ millions)
$3.20
Weighted-average shares outstanding during 2013
Dividend per share
Earnings per share
Beta
16,000,000
$0.20
$1.89
1.80
Equity Valuation Models
633
Table 18I
Rio National Corp.
common equity data
for 2013
Note: The dividend payout ratio is expected to be constant.
4.00%
9.00%
19.90
12.00%
Table 18J
Industry and market data
December 31, 2013
11. Shaar (from the previous problem) has revised slightly her estimated earnings growth rate for
Rio National and, using normalized (underlying) EPS, which is adjusted for temporary impacts
on earnings, now wants to compare the current value of Rio National’s equity to that of the
industry, on a growth-adjusted basis. Selected information about Rio National and the industry
is given in Table 18K.
Compared to the industry, is Rio National’s equity overvalued or undervalued on a P/E-togrowth (PEG) basis, using normalized (underlying) earnings per share? Assume that the risk of
Rio National is similar to the risk of the industry.
Table 18K
Rio National
Estimated earnings growth rate
11.00%
Current share price
Normalized (underlying) EPS for 2011
Weighted-average shares outstanding during 2011
Rio National
Corp. vs. industry
$25.00
$1.71
16,000,000
Industry
Estimated earnings growth rate
Median price/earnings (P/E) ratio
12.00%
19.90
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Risk-free rate of return
Expected rate of return on market index
Median industry price/earnings (P/E) ratio
Expected industry earnings growth rate
E-INVESTMENTS EXERCISES
Go to the MoneyCentral Investor page at moneycentral.msn.com/investor/home.asp.
Use the Research Wizard function under Guided Research to obtain fundamentals, price
history, price target, catalysts, and comparison for Walmart (WMT). For comparison, use
Target (TGT), BJ’s Wholesale Club (BJ), and the industry.
1. What has been the 1-year sales and income growth for Walmart?
2. What has been the company’s 5-year profit margin? How does that compare with the
other two firms’ profit margins and the industry’s profit margin?
3. What have been the percentage price changes for the last 3, 6, and 12 months? How do
they compare with the other firms’ price changes and the industry’s price changes?
4. What are the estimated high and low prices for Walmart for the coming year based on
its current P/E multiple?
5. Compare the price performance of Walmart with that of Target and BJ’s. Which of the
companies appears to be the most expensive in terms of current earnings? Which of the
companies is the least expensive in terms of current earnings?
6. What are the firms’ Stock Scouter Ratings? How are these ratings interpreted?
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634
PART V
Security Analysis
SOLUTIONS TO CONCEPT CHECKS
1. a. Dividend yield 5 $2.15/$50 5 4.3%.
Capital gains yield 5 (59.77 2 50) /50 5 19.54%.
Total return 5 4.3% 1 19.54% 5 23.84%.
b. k 5 6% 1 1.15(14% 2 6%) 5 15.2%.
c. V0 5 ($2.15 1 $59.77)/1.152 5 $53.75, which exceeds the market price. This would indicate
a “buy” opportunity.
2. a. D1/(k 2 g) 5 $2.15/(.152 2 .112) 5 $53.75.
b. P1 5 P0(1 1 g) 5 $53.75(1.112) 5 $59.77.
c. The expected capital gain equals $59.77 2 $53.75 5 $6.02, for a percentage gain of
11.2%. The dividend yield is D1/P0 5 2.15/53.75 5 4%, for a holding-period return of
4% 1 11.2% 5 15.2%.
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3. a. g 5 ROE 3 b 5 20% 3 .60 5 12%.
D1 5 .4 3 E1 5 .4 3 $5 5 $2.
P0 5 $2/(.125 2 .12) 5 $400.
b. When the firm invests in projects with ROE less than k, its stock price falls. If b 5 .60,
then g 5 10% 3 .60 5 6% and P0 5 $2/(.125 2 .06) 5 $30.77. In contrast, if b 5 0, then
P0 5 $5/.125 5 $40.
1.00 1 P2016
.78
.92
.85
1
1
1
2
3
(1.096)
(1.096)
(1.096)
(1.096)4
Now compute the sales price in 2016 using the constant-growth dividend discount model. The
growth rate will be g 5 ROE 3 b 5 9% 3 .75 5 6.75%.
1.00 3 (1 1 g)
$1.0675
P2016 5
5
5 $37.46
k2g
.096 2 .0675
4. V2012 5
Therefore, V2012 5 $28.77.
5. a. ROE 5 12%.
b 5 $.50/$2.00 5 .25.
g 5 ROE 3 b 5 12% 3 .25 5 3%.
P0 5 D1/(k 2 g) 5 $1.50/(.10 2 .03) 5 $21.43.
P0/E1 5 $21.43/$2.00 5 10.71.
b. If b 5 .4, then .4 3 $2 5 $.80 would be reinvested and the remainder of earnings, or $1.20,
would be paid as dividends.
g 5 12% 3 .4 5 4.8%.
P0 5 D1/(k 2 g) 5 $1.20/(.10 2 .048) 5 $23.08.
P0/E1 5 $23.08/$2.00 5 11.54.
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20
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CHAPTER TWENTY
Options Markets:
Introduction
crowding out the previously existing overthe-counter options market.
Option contracts are traded now on several
exchanges. They are written on common stock,
stock indexes, foreign exchange, agricultural
commodities, precious metals, and interest
rate futures. In addition, the over-the-counter
market has enjoyed a tremendous resurgence
in recent years as trading in custom-tailored
options has exploded. Popular and potent
tools in modifying portfolio characteristics,
options have become essential tools a portfolio manager must understand.
This chapter is an introduction to options
markets. It explains how puts and calls work
and examines their investment characteristics. Popular option strategies are considered
next. Finally, we examine a range of securities
with embedded options such as callable or
convertible bonds, and we take a quick look
at some so-called exotic options.
PART VI
DERIVATIVE SECURITIES, OR more simply
derivatives, play a large and increasingly
important role in financial markets. These
are securities whose prices are determined
by, or “derive from,” the prices of other
securities.
Options and futures contracts are both
derivative securities. Their payoffs depend on
the value of other securities. Swaps, which we
will discuss in Chapter 23, also are derivatives.
Because the value of derivatives depends
on the value of other securities, they can be
powerful tools for both hedging and speculation. We will investigate these applications in
the next four chapters, starting in this chapter
with options.
Trading of standardized options contracts
on a national exchange started in 1973 when
the Chicago Board Options Exchange (CBOE)
began listing call options. These contracts
were almost immediately a great success,
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CHAPTER 20
Options Markets: Introduction
679
20.1 The Option Contract
A call option gives its holder the right to purchase an asset for a specified price, called
the exercise, or strike, price, on or before some specified expiration date. For example,
a February call option on IBM stock with exercise price $195 entitles its owner to
purchase IBM stock for a price of $195 at any time up to and including the expiration
date in February. The holder of the call is not required to exercise the option. She will
choose to exercise only if the market value of the underlying asset exceeds the exercise
price. In that case, the option holder may “call away” the asset for the exercise price.
Otherwise, the option may be left unexercised. If it is not exercised before the expiration date of the contract, a call option simply expires and becomes valueless. Therefore,
if the stock price is greater than the exercise price on the expiration date, the value of
the call option equals the difference between the stock price and the exercise price; but
if the stock price is less than the exercise price at expiration, the call will be worthless.
The net profit on the call is the value of the option minus the price originally paid to
purchase it.
The purchase price of the option is called the premium. It represents the compensation
the purchaser of the call must pay for the right to exercise the option only when exercise
is desirable.
Sellers of call options, who are said to write calls, receive premium income now as payment against the possibility they will be required at some later date to deliver the asset in
return for an exercise price less than the market value of the asset. If the option is left to
expire worthless, the writer of the call clears a profit equal to the premium income derived
from the initial sale of the option. But if the call is exercised, the profit to the option writer
is the premium income minus the difference between the value of the stock that must be
delivered and the exercise price that is paid for those shares. If that difference is larger than
the initial premium, the writer will incur a loss.
Example 20.1
Profits and Losses on a Call Option
Consider the February 2013 expiration call option on a share of IBM with an exercise
price of $195 selling on January 18, 2013, for $3.65. Exchange-traded options expire on
the third Friday of the expiration month, which for this option was February 15. Until the
expiration date, the call holder may buy shares of IBM for $195. On January 18, IBM sells
for $194.47. Because the stock price is only $194.47, it clearly would not make sense at
the moment to exercise the option to buy at $195. Indeed, if IBM remains below $195
by the expiration date, the call will be left to expire worthless. On the other hand, if IBM
is selling above $195 at expiration, the call holder will find it optimal to exercise. For
example, if IBM sells for $197 on February 15, the option will be exercised, as it will give
its holder the right to pay $195 for a stock worth $197. The value of each option on the
expiration date would then be
Value at expiration 5 Stock price 2 Exercise price 5 $197 2 $195 5 $2
Despite the $2 payoff at expiration, the call holder still realizes a loss of $1.65 on the
investment because the initial purchase price was $3.65:
Profit 5 Final value 2 Original investment 5 $2.00 2 $3.65 5 2$1.65
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680
PART VI
Options, Futures, and Other Derivatives
Nevertheless, exercise of the call is optimal at expiration if the stock price exceeds the
exercise price because the exercise proceeds will offset at least part of the purchase
price. The call buyer will clear a profit if IBM is selling above $198.65 at the expiration
date. At that stock price, the proceeds from exercise will just cover the original cost of
the call.
A put option gives its holder the right to sell an asset for a specified exercise or strike
price on or before some expiration date. A February expiration put on IBM with an exercise price of $195 entitles its owner to sell IBM stock to the put writer at a price of $195 at
any time before expiration in February even if the market price of IBM is less than $195.
Whereas profits on call options increase when the asset price rises, profits on put options
increase when the asset price falls. A put will be exercised only if the exercise price is
greater than the price of the underlying asset, that is, only if its holder can deliver for the
exercise price an asset with market value less than that amount. (One doesn’t need to own
the shares of IBM to exercise the IBM put option. Upon exercise, the investor’s broker
purchases the necessary shares of IBM at the market price and immediately delivers, or
“puts them,” to an option writer for the exercise price. The owner of the put profits by the
difference between the exercise price and market price.)
Example 20.2
Profits and Losses on a Put Option
Now consider the February 2013 expiration put option on IBM with an exercise
price of $195, selling on January 18 for $5.00. It entitled its owner to sell a share
of IBM for $195 at any time until February 15. If the holder of the put buys a share of
IBM and immediately exercises the right to sell it at $195, net proceeds will be
$195 2 $194.47 5 $.53. Obviously, an investor who pays $5 for the put has no intention of exercising it immediately. If, on the other hand, IBM were selling for $188 at
expiration, the put would turn out to be a profitable investment. Its value at expiration
would be
Value at expiration 5 Exercise price 2 Stock price 5 $195 2 $188 5 $7
and the investor’s profit would be $7 2 $5 5 $2. This is a holding period return of
$2/$5 5 .40, or 40%—over only 28 days! Obviously, put option sellers on January 18
(who are on the other side of the transaction) did not consider this outcome very likely.
An option is described as in the money when its exercise would produce a positive
cash flow. Therefore, a call option is in the money when the asset price is greater than
the exercise price, and a put option is in the money when the asset price is less than the
exercise price. Conversely, a call is out of the money when the asset price is less than
the exercise price; no one would exercise the right to purchase for the strike price an asset
worth less than that amount. A put option is out of the money when the exercise price
is less than the asset price. Options are at the money when the exercise price and asset
price are equal.
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CHAPTER 20
Options Markets: Introduction
681
Options Trading
Some options trade on over-the-counter markets. The OTC
PRICES AT CLOSE JANUARY 18, 2013
market offers the advantage that the terms of the option
contract—the exercise price, expiration date, and number
I B M (IBM)
Underlying Stock Price: 194.47
of shares committed—can be tailored to the needs of the
Call
Put
traders. The costs of establishing an OTC option contract,
Open
Open
Expiration Strike
Last
Volume Interest
Last Volume Interest
however, are higher than for exchange-traded options.
185
9.15
Jan
307
2431
0.76
302
2488
Options contracts traded on exchanges are standard185
10.60
Feb
299
2
1.82
710
3645
185
12.00
Apr
41
706
3.60
104
2047
ized by allowable expiration dates and exercise prices for
185
14.35
Jul
37
134
6.55
37
1354
190
4.40
Jan
815
5697
1.75
507
2496
each listed option. Each stock option contract provides
190
6.75
Feb
402
2808
3.00
3553 10377
for the right to buy or sell 100 shares of stock (except
190
8.85
Apr
107
1866
5.20
527
2177
190
10.95
Jul
15
645
8.54
6
1142
when stock splits occur after the contract is listed and
195
0.01
Jan
2451
11718
0.70
4090
8862
195
3.65
Feb
1337
11902
5.00
860
3156
the contract is adjusted for the terms of the split).
195
5.90
Apr
1785
2928
7.30
934
1141
195
8.45
Jul
13
5773
10.85
22
3419
Standardization of the terms of listed option contracts
200
1.10
Jan
1248
2966
5.55
637
6199
means all market participants trade in a limited and uni200
1.61
Feb
1053
5530
8.09
546
967
200
3.70
Apr
629
3236
10.05
375
1903
form set of securities. This increases the depth of trad200
6.10
Jul
80
1257
...
...
1105
ing in any particular option, which lowers trading costs
and results in a more competitive market. Exchanges,
therefore, offer two important benefits: ease of trading,
which flows from a central marketplace where buyFigure 20.1 Stock options on IBM closing prices
ers and sellers or their representatives congregate; and
as of January 18, 2013
a liquid secondary market where buyers and sellers of
Source: The Wall Street Journal Online, January 18, 2013.
options can transact quickly and cheaply.
Until recently, most options trading in the United
States took place on the Chicago Board Options Exchange. However, by 2003 the International Securities Exchange, an electronic exchange based in New York, displaced the
CBOE as the largest options market. Options trading in Europe is uniformly transacted in
electronic exchanges.
Figure 20.1 is a selection of listed stock option quotations for IBM. The last recorded
price on the New York Stock Exchange for IBM shares was $194.47 per share.1 The exercise (or strike) prices bracket the stock price. While exercise prices generally are set at
five-point intervals, larger intervals sometimes are set for stocks selling above $100, and
intervals of $2.50 may be used for stocks selling at low prices. If the stock price moves outside the range of exercise prices of the existing set of options, new options with appropriate
exercise prices may be offered. Therefore, at any time, both in-the-money and out-of-themoney options will be listed, as in this example.
Figure 20.1 shows both call and put options listed for each expiration date and exercise
price. The three sets of columns for each option report closing price, trading volume in
contracts, and open interest (number of outstanding contracts). When we compare prices
of call options with the same expiration date but different exercise prices in Figure 20.1,
we see that the value of a call is lower when the exercise price is higher. This makes sense,
because the right to purchase a share at a lower exercise price is more valuable than the
right to purchase at a higher price. Thus the February expiration IBM call option with
strike price $195 sells for $3.65 whereas the $200 exercise price February call sells for
1
Occasionally, this price may not match the closing price listed for the stock on the stock market page. This is
because some NYSE stocks also trade on exchanges that close after the NYSE, and the stock pages may reflect
the more recent closing price. The options exchanges, however, close with the NYSE, so the closing NYSE stock
price is appropriate for comparison with the closing option price.
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only $1.61. Conversely, put options are worth more when the exercise price is higher: You
would rather have the right to sell shares for $200 than for $195 and this is reflected in
the prices of the puts. The February expiration put option with strike price $200 sells for
$8.09, whereas the $195 exercise price February put sells for only $5.
If an option does not trade on a given day, three dots will appear in the volume and price
columns. Because trading is infrequent, it is not unusual to find option prices that appear
out of line with other prices. You might see, for example, two calls with different exercise
prices that seem to sell for the same price. This discrepancy arises because the last trades
for these options may have occurred at different times during the day. At any moment, the
call with the lower exercise price must be worth more than an otherwise-identical call with
a higher exercise price.
Expirations of most exchange-traded options tend to be fairly short, ranging up to only
several months. For larger firms and several stock indexes, however, longer-term options
are traded with expirations ranging up to several years. These options are called LEAPS
(for Long-Term Equity AnticiPation Securities).
CONCEPT CHECK
20.1
a. What will be the proceeds and net profits to an investor who purchases the February expiration IBM
calls with exercise price $195 if the stock price at expiration is $205? What if the stock price at expiration is $185?
b. Now answer part (a) for an investor who purchases a February expiration IBM put option with exercise
price $195.
American and European Options
An American option allows its holder to exercise the right to purchase (if a call) or sell (if
a put) the underlying asset on or before the expiration date. European options allow for
exercise of the option only on the expiration date. American options, because they allow
more leeway than their European counterparts, generally will be more valuable. Virtually
all traded options in the United States are American style. Foreign currency options and
stock index options are notable exceptions to this rule, however.
Adjustments in Option Contract Terms
Because options convey the right to buy or sell shares at a stated price, stock splits would
radically alter their value if the terms of the options contract were not adjusted to account
for the stock split. For example, reconsider the IBM call options in Figure 20.1. If IBM
were to announce a 2-for-1 split, its share price would fall from about $195 to about
$97.50. A call option with exercise price $195 would be just about worthless, with virtually no possibility that the stock would sell at more than $195 before the options expired.
To account for a stock split, the exercise price is reduced by a factor of the split, and the
number of options held is increased by that factor. For example, each original call option
with exercise price of $195 would be altered after a 2-for-1 split to two new options, with
each new option carrying an exercise price of $97.50. A similar adjustment is made for stock
dividends of more than 10%; the number of shares covered by each option is increased in
proportion to the stock dividend, and the exercise price is reduced by that proportion.
In contrast to stock dividends, cash dividends do not affect the terms of an option contract. Because payment of a cash dividend reduces the selling price of the stock without
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inducing offsetting adjustments in the option contract, the value of the option is affected by
dividend policy. Other things being equal, call option values are lower for high-dividend
payout policies, because such policies slow the rate of increase of stock prices; conversely,
put values are higher for high-dividend payouts. (Of course, the option values do not necessarily rise or fall on the dividend payment or ex-dividend dates. Dividend payments are
anticipated, so the effect of the payment already is built into the original option price.)
CONCEPT CHECK
20.2
Suppose that IBM’s stock price at the exercise date is $200, and the exercise price of the call is $195. What is
the payoff on one option contract? After a 2-for-1 split, the stock price is $100, the exercise price is $97.50,
and the option holder now can purchase 200 shares. Show that the split leaves the payoff from the option
unaffected.
The Options Clearing Corporation
The Options Clearing Corporation (OCC), the clearinghouse for options trading, is jointly
owned by the exchanges on which stock options are traded. Buyers and sellers of options
who agree on a price will strike a deal. At this point, the OCC steps in. The OCC places
itself between the two traders, becoming the effective buyer of the option from the writer
and the effective writer of the option to the buyer. All individuals, therefore, deal only with
the OCC, which effectively guarantees contract performance.
When an option holder exercises an option, the OCC arranges for a member firm with
clients who have written that option to make good on the option obligation. The member firm selects from its clients who have written that option to fulfill the contract. The
selected client must deliver 100 shares of stock at a price equal to the exercise price for
each call option contract written or must purchase 100 shares at the exercise price for each
put option contract written.
Because the OCC guarantees contract performance, it requires option writers to post
margin to guarantee that they can fulfill their contract obligations. The margin required is
determined in part by the amount by which the option is in the money, because that value
is an indicator of the potential obligation of the option writer. When the required margin
exceeds the posted margin, the writer will receive a margin call. In contrast, the holder of
the option need not post margin because the holder will exercise the option only if it is
profitable to do so. After purchase of the option, no further money is at risk.
Margin requirements are determined in part by the other securities held in the investor’s
portfolio. For example, a call option writer owning the stock against which the option is
written can satisfy the margin requirement simply by allowing a broker to hold that stock
in the brokerage account. The stock is then guaranteed to be available for delivery should
the call option be exercised. If the underlying security is not owned, however, the margin
requirement is determined by the value of the underlying security as well as by the amount
by which the option is in or out of the money. Out-of-the-money options require less margin from the writer, for expected payouts are lower.
Other Listed Options
Options on assets other than stocks are also widely traded. These include options on market indexes and industry indexes, on foreign currency, and even on the futures prices of
agricultural products, gold, silver, fixed-income securities, and stock indexes. We will discuss these in turn.
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Index Options An index option is a call or put based on a stock market index such
as the S&P 500 or the NASDAQ 100. Index options are traded on several broad-based
indexes as well as on several industry-specific indexes and even commodity price indexes.
We discussed many of these indexes in Chapter 2.
The construction of the indexes can vary across contracts or exchanges. For example,
the S&P 100 index is a value-weighted average of the 100 stocks in the Standard & Poor’s
100 stock group. The weights are proportional to the market value of outstanding equity
for each stock. The Dow Jones Industrial Index, by contrast, is a price-weighted average
of 30 stocks.
Option contracts on many foreign stock indexes also trade. For example, options on
the (Japanese) Nikkei Stock Index trade on the Singapore as well as the Chicago Mercantile Exchange. Options on European indexes such as the Financial Times Share Exchange
(FTSE 100) trade on the NYSE-Euronext Exchange. The Chicago Board Options Exchange
also lists options on industry indexes such as the oil or high-tech industries.
In contrast to stock options, index options do not require that the call writer actually
“deliver the index” upon exercise or that the put writer “purchase the index.” Instead, a
cash settlement procedure is used. The payoff that would accrue upon exercise of the
option is calculated, and the option writer simply pays that amount to the option holder.
The payoff is equal to the difference between the exercise price of the option and the
value of the index. For example, if the S&P index is at 1400 when a call option on the
index with exercise price 1390 is exercised, the holder of the call receives a cash payment of the difference, 1400 2 1390, times the contract multiplier of $100, or $1,000 per
contract.
Options on the major indexes, that is, the S&P 100 (often called the OEX after its
ticker symbol), the S&P 500 (the SPX), the NASDAQ 100 (the NDX), and the Dow Jones
Industrials (the DJX), are the most actively traded contracts on the CBOE. Together, these
contracts dominate CBOE volume.
Futures Options Futures options give their holders the right to buy or sell a specified futures contract, using as a futures price the exercise price of the option. Although the
delivery process is slightly complicated, the terms of futures options contracts are designed
in effect to allow the option to be written on the futures price itself. The option holder
receives upon exercise a net payoff equal to the difference between the current futures
price on the specified asset and the exercise price of the option. Thus if the futures price is,
say, $37, and the call has an exercise price of $35, the holder who exercises the call option
on the futures gets a payoff of $2.
Foreign Currency Options A currency option offers the right to buy or sell a
quantity of foreign currency for a specified amount of domestic currency. Currency
option contracts call for purchase or sale of the currency in exchange for a specified number of U.S. dollars. Contracts are quoted in cents or fractions of a cent per unit of foreign
currency.
There is an important difference between currency options and currency futures options.
The former provide payoffs that depend on the difference between the exercise price and
the exchange rate at expiration. The latter are foreign exchange futures options that provide
payoffs that depend on the difference between the exercise price and the exchange rate
futures price at expiration. Because exchange rates and exchange rate futures prices generally are not equal, the options and futures-options contracts will have different values, even
with identical expiration dates and exercise prices. Trading volume in currency futures
options dominates trading in currency options.
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Interest Rate Options Options are traded on Treasury notes and bonds, Treasury
bills, and government bonds of other major economies such as the U.K. or Japan. Options
on several interest rates also trade. Among these are contracts on Treasury bond, Treasury
note, federal funds, LIBOR, Euribor,2 and Eurodollar futures.
20.2 Values of Options at Expiration
Call Options
Recall that a call option gives the right to purchase a security at the exercise price. Suppose
you hold a call option on FinCorp stock with an exercise price of $100, and FinCorp is now
selling at $110. You can exercise your option to purchase the stock at $100 and simultaneously sell the shares at the market price of $110, clearing $10 per share. Yet if the shares
sell below $100, you can sit on the option and do nothing, realizing no further gain or loss.
The value of the call option at expiration equals
Payoff to call holder 5 b
ST 2 X
0
if ST . X
if ST # X
where ST is the value of the stock at expiration and X is the exercise price. This formula
emphasizes the option property because the payoff cannot be negative. The option is exercised only if ST exceeds X. If ST is less than X, the option expires with zero value. The loss
to the option holder in this case equals the price originally paid for the option. More generally, the profit to the option holder is the option payoff at expiration minus the original
purchase price.
The value at expiration of the call with exercise price $100 is given by the schedule:
Stock price:
Option value:
$90
0
$100
0
$110
10
$120
20
$130
30
For stock prices at or below $100, the option is worthless. Above $100, the option is worth
the excess of the stock price over $100. The option’s value increases by $1 for each dollar
increase in the stock price. This relationship can be depicted graphically as in Figure 20.2.
The solid line in Figure 20.2 is the value of the call at expiration. The net profit to the
holder of the call equals the gross payoff less the initial investment in the call. Suppose
the call cost $14. Then the profit to the call holder would be given by the dashed (bottom)
line of Figure 20.2. At option expiration, the investor suffers a loss of $14 if the stock price
is less than or equal to $100.
Profits do not become positive until the stock price at expiration exceeds $114. The breakeven point is $114, because at that price the payoff to the call, ST 2 X 5 $114 2 $100 5 $14,
equals the initial cost of the call.
Conversely, the writer of the call incurs losses if the stock price is high. In that scenario,
the writer will receive a call and will be obligated to deliver a stock worth ST for only X
dollars:
Payoff to call writer 5 b
2(ST 2 X)
0
if ST . X
if ST # X
2
The Euribor market is similar to the LIBOR market (see Chapter 2), but the interest rate charged in the Euribor
market is the interbank rate for euro-denominated deposits.
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$30
Payoff = Value at Expiration
$20
$10
0
80
−$10
90
100
Cost of Option
110
120
ST
The call writer, who is exposed to losses
if the stock price increases, is willing to bear
this risk in return for the option premium.
Figure 20.3 depicts the payoff and profit
diagrams for the call writer. These are the
mirror images of the corresponding diagrams
for call holders. The break-even point for the
option writer also is $114. The (negative) payoff at that point just offsets the premium originally received when the option was written.
Profit
Put Options
−$14
A put option is the right to sell an asset at
the exercise price. In this case, the holder
will not exercise the option unless the asset
Figure 20.2 Payoff and profit to call option at expiration
is worth less than the exercise price. For
example, if FinCorp shares were to fall to
$90, a put option with exercise price $100
could be exercised to clear $10 for its holder. The holder would purchase a share for $90
and simultaneously deliver it to the put option writer for the exercise price of $100.
The value of a put option at expiration is
Payoff to put holder 5 b
0
X 2 ST
if ST $ X
if ST , X
The solid line in Figure 20.4 illustrates the payoff at expiration to the holder of a put option
on FinCorp stock with an exercise price of $100. If the stock price at expiration is above
$100, the put has no value, as the right to sell the shares at $100 would not be exercised.
Below a price of $100, the put value at expiration increases by $1 for each dollar the stock
price falls. The dashed line in Figure 20.4 is a graph of the put option owner’s profit at
expiration, net of the initial cost of the put.
Writing puts naked (i.e., writing a put
without an offsetting short position in the
stock for hedging purposes) exposes the
writer to losses if the market falls. Writing naked, deep-out-of-the-money puts
$14
was once considered an attractive way to
generate income, as it was believed that
ST
0
$100 $114
as long as the market did not fall sharply
before the option expiration, the option
Profit
premium could be collected without the
put holder ever exercising the option
against the writer. Because only sharp
Payoff
drops in the market could result in losses
to the put writer, the strategy was not
viewed as overly risky. However, in the
wake of the market crash of October 1987,
such put writers suffered huge losses. ParFigure 20.3 Payoff and profit to call writers at expiration
ticipants now perceive much greater risk to
this strategy.
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CONCEPT CHECK
687
Options Markets: Introduction
20.3
Consider these four option strategies: (i) buy a call; (ii) write a call; (iii) buy a put; (iv) write a put.
a. For each strategy, plot both the payoff and profit diagrams as a function of the final stock price.
b. Why might one characterize both buying calls and writing puts as “bullish” strategies? What is the
difference between them?
c. Why might one characterize both buying puts and writing calls as “bearish” strategies? What is the
difference between them?
Option versus Stock Investments
Purchasing call options is a bullish strat$100
egy; that is, the calls provide profits when
stock prices increase. Purchasing puts, in
contrast, is a bearish strategy. Symmetrically, writing calls is bearish, whereas
Payoff = Value of Put at Expiration
writing puts is bullish. Because option valProfit
ues depend on the price of the underlying
stock, purchase of options may be viewed
as a substitute for direct purchase or sale of
Price of Put
a stock. Why might an option strategy be
S
0
$100
preferable to direct stock transactions?
For example, why would you purchase a
call option rather than buy shares of stock
directly? Maybe you have some information that leads you to believe the stock will
Figure 20.4 Payoff and profit to put option at expiration
increase in value from its current level,
which in our examples we will take to be
$100. You know your analysis could be incorrect, however, and that shares also could
fall in price. Suppose a 6-month maturity call option with exercise price $100 currently
sells for $10, and the interest rate for the period is 3%. Consider these three strategies for
investing a sum of money, say, $10,000. For simplicity, suppose the firm will not pay any
dividends until after the 6-month period.
T
Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100.
Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling
for $10. (This would require 10 contracts, each for 100 shares.)
Strategy C: Purchase 100 call options for $1,000. Invest your remaining $9,000 in
6-month T-bills, to earn 3% interest. The bills will grow in value from $9,000 to
$9,000 3 1.03 5 $9,270.
Let us trace the possible values of these three portfolios when the options expire in
6 months as a function of the stock price at that time:
Stock Price
Portfolio
Portfolio A: All stock
Portfolio B: All options
Portfolio C: Call plus bills
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$95
$100
$105
$110
$115
$120
$9,500
0
9,270
$10,000
0
9,270
$10,500
5,000
9,770
$11,000
10,000
10,270
$11,500
15,000
10,770
$12,000
20,000
11,270
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Portfolio A will be worth 100 times the share price. Portfolio B is worthless unless shares
sell for more than the exercise price of the call. Once that point is reached, the portfolio is
worth 1,000 times the excess of the stock price over the exercise price. Finally, portfolio C
is worth $9,270 from the investment in T-bills plus any profits from the 100 call options.
Remember that each of these portfolios involves the same $10,000 initial investment. The
rates of return on these three portfolios are as follows:
Stock Price
Portfolio
Portfolio A: All stock
Portfolio B: All options
Portfolio C: Call plus bills
$95
$100
$105
$110
$115
$120
25.0%
2100.0
27.3
0.0%
2100.0
27.3
5.0%
250.0
22.3
10.0%
0.0
2.7
15.0%
50.0
7.7
20.0%
100.0
12.7
84
86
88
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
Rate of Return (%)
These rates of return are graphed in Figure 20.5.
Comparing the returns of portfolios B and C to those of the simple investment in
stock represented by portfolio A, we see that options offer two interesting features.
First, an option offers leverage. Compare the returns of portfolios B and A. Unless
the stock increases from its initial value of $100, the value of portfolio B falls precipitously to zero—a rate of return of negative 100%. Conversely, modest increases
in the rate of return on the stock result in disproportionate increases in the option rate
of return. For example, a 4.3% increase in the stock price from $115 to $120 would
increase the rate of return on the call from 50% to 100%. In this sense, calls are a
levered investment on the stock. Their values respond more than proportionately to
changes in the stock value.
Figure 20.5 vividly illustrates this
point. The slope of the all-option portfolio is far steeper than that of the all-stock
100
portfolio, reflecting its greater propor80
tional sensitivity to the value of the
underlying security. The leverage factor
60
is the reason investors (illegally) exploit40
ing inside information commonly choose
options as their investment vehicle.
20
The potential insurance value of
ST
0
options is the second interesting feature,
as portfolio C shows. The T-bill-plus−20
option portfolio cannot be worth less
−40
than $9,270 after 6 months, as the option
can always be left to expire worthless.
−60
The worst possible rate of return on portA: All Stocks
−80
folio C is 27.3%, compared to a (theoB: All Options
retically) worst possible rate of return on
C: Call Plus Bills
−100
the stock of 2100% if the company were
−120
to go bankrupt. Of course, this insurance
comes at a price: When the share price
increases, portfolio C, the option-plusbills portfolio, does not perform as well
Figure 20.5 Rate of return to three strategies
as portfolio A, the all-stock portfolio.
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This simple example makes an important point. Although options can be used by
speculators as effectively leveraged stock positions, as in portfolio B, they also can be
used by investors who desire to tailor their risk exposures in creative ways, as in portfolio C. For example, the call-plus-bills strategy of portfolio C provides a rate of return
profile quite unlike that of the stock alone. The absolute limitation on downside risk
is a novel and attractive feature of this strategy. We next discuss several option strategies that provide other novel risk profiles that might be attractive to hedgers and other
investors.
20.3 Option Strategies
An unlimited variety of payoff patterns can be achieved by combining puts and calls with
various exercise prices. We explain in this section the motivation and structure of some of
the more popular ones.
Protective Put
Imagine you would like to invest in a stock, but you are unwilling to bear potential losses
beyond some given level. Investing in the stock alone seems risky to you because in principle you could lose all the money you invest. You might consider instead investing in
stock and purchasing a put option on the stock. Table 20.1 shows the total value of your
portfolio at option expiration: Whatever happens to the stock price, you are guaranteed a
payoff at least equal to the put option’s exercise price because the put gives you the right to
sell your shares for that price.
Example 20.3
Protective Put
Suppose the strike price is X 5 $100 and the stock is selling at $97 at option expiration.
Then the value of your total portfolio is $100. The stock is worth $97 and the value of
the expiring put option is
X 2 ST 5 $100 2 $97 5 $3
Another way to look at it is that you are holding the stock and a put contract giving you
the right to sell the stock for $100. The right to sell locks in a minimum portfolio value
of $100. On the other hand, if the stock price is above $100, say, $104, then the right
to sell a share at $100 is worthless. You allow the put to expire unexercised, ending up
with a share of stock worth ST 5 $104.
Figure 20.6 illustrates the payoff and profit to this protective put strategy. The
solid line in Figure 20.6, panel C is the total payoff. The dashed line is displaced
downward by the cost of establishing the position, S0 1 P. Notice that potential losses
are limited.
It is instructive to compare the profit on the protective put strategy with that of the
stock investment. For simplicity, consider an at-the-money protective put, so that X 5 S0.
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Table 20.1
Value of a protective
put portfolio at option
expiration
Stock
1 Put
5 TOTAL
ST " X
ST + X
ST
X 2 ST
X
ST
0
ST
Payoff of Stock
A: Stock
ST
X
Payoff of Option
+ B: Put
ST
X
Payoff of
Protective Put
Payoff
Profit
= C: Protective Put
X
X
ST
X − (S0 + P)
Figure 20.6 Value of a protective put position at option expiration
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Figure 20.7 compares the profits for the two
strategies. The profit on the stock is zero
Profits
if the stock price remains unchanged and
Profit on Stock
ST 5 S0. It rises or falls by $1 for every dollar
Profit on
Protective Put
swing in the ultimate stock price. The profit
Portfolio
on the protective put is negative and equal
to the cost of the put if ST is below S0. The
profit on the protective put increases one for
one with increases in the stock price once ST
ST
exceeds S0.
S0 = X
Figure 20.7 makes it clear that the pro−P
tective put offers some insurance against
stock price declines in that it limits losses.
Therefore, protective put strategies provide
a form of portfolio insurance. The cost of
−S0
the protection is that, in the case of stock
price increases, your profit is reduced by
the cost of the put, which turned out to be
unneeded.
This example also shows that despite
Figure 20.7 Protective put versus stock investment
the common perception that derivatives
(at-the-money option)
mean risk, derivative securities can be used
effectively for risk management. In fact,
such risk management is becoming accepted as part of the fiduciary responsibility of
financial managers. Indeed, in one often-cited court case, Brane v. Roth, a company’s
board of directors was successfully sued for failing to use derivatives to hedge the
price risk of grain held in storage. Such hedging might have been accomplished using
protective puts.
The claim that derivatives are best viewed as risk management tools may seem surprising in light of the credit crisis of the last few years. The crisis was immediately precipitated when the highly risky positions that many financial institutions had established in
credit derivatives blew up 2007–2008, resulting in large losses and government bailouts.
Still, the same characteristics that make derivatives potent tools to increase risk also make
them highly effective in managing risk, at least when used properly. Derivatives have aptly
been compared to power tools: very useful in skilled hands, but also very dangerous when
not handled with care. The nearby box makes the case for derivatives as central to risk
management.
Covered Calls
A covered call position is the purchase of a share of stock with a simultaneous sale
of a call option on that stock. The call is “covered” because the potential obligation
to deliver the stock can be satisfied using the stock held in the portfolio. Writing an
option without an offsetting stock position is called by contrast naked option writing. The value of a covered call position at expiration, presented in Table 20.2, equals
the stock value minus the value of the call. The call value is subtracted because the
covered call position involves writing a call to another investor who may exercise it at
your expense.
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The Case for Derivatives
They’ve been dubbed financial weapons of mass destruction, attacked for causing the financial turmoil sweeping
the nation and identified as the kryptonite that brought
down the global economy. Yet few Main Streeters really
know what derivatives are—namely, financial contracts
between a buyer and a seller that derive value from an
underlying asset, such as a mortgage or a stock. There
seems to be near consensus that derivatives were a source
of undue risk.
And then there’s Robert Shiller. The Yale economist
believes just the opposite is true. A champion of financial
innovation and an expert in management of risk, Shiller
contends that derivatives, far from being a problem, are
actually the solution. Derivatives, Shiller says, are merely
a risk-management tool the same way insurance is. “You
pay a premium and if an event happens, you get a payment.” That tool can be used well or, as happened recently,
used badly. Shiller warns that banishing the tool gets us
nowhere.
For all the trillions in derivative trading, there were very
few traders. Almost all the subprime mortgages that were
bundled and turned into derivatives were sold by a handful of Wall Street institutions, working with a small number of large institutional buyers. It was a huge but illiquid
and opaque market.
Meanwhile, the system was built on the myriad decisions of individual homeowners and lenders around the
world. None of them, however, could hedge their bets the
way large institutions can. Those buying a condo in Miami
had no way to protect themselves if the market went
down.
Derivatives, according to Shiller, could be used by homeowners—and, by extension, lenders—to insure themselves
against falling prices. In Shiller’s scenario, you would be
able to go to your broker and buy a new type of financial
instrument, perhaps a derivative that is inversely related to
a regional home-price index. If the value of houses in your
area declined, the financial instrument would increase in
value, offsetting the loss. Lenders could do the same thing,
which would help them hedge against foreclosures. The
idea is to make the housing market more liquid. More
buyers and sellers mean that markets stay liquid and functional even under pressure.
Some critics dismiss Shiller’s basic premise that more
derivatives would make the housing market more liquid
and more stable. They point out that futures contracts
haven’t made equity markets or commodity markets
immune from massive moves up and down. They add that
a ballooning world of home-based derivatives wouldn’t
lead to homeowners’ insurance: it would lead to a new
playground for speculators.
In essence, Shiller is laying the intellectual groundwork for the next financial revolution. We are now suffering through the first major crisis of the Information Age
economy. Shiller’s answers may be counterintuitive, but no
more so than those of doctors and scientists who centuries ago recognized that the cure for infectious diseases
was not flight or quarantine, but purposely infecting more
people through vaccinations. “We’ve had a major glitch
in derivatives and securitization,” says Shiller. “The Titanic
sank almost a century ago, but we didn’t stop sailing across
the Atlantic.”
Of course, people did think twice about getting on a
ship, at least for a while. But if we listen only to our fears,
we lose the very dynamism that has propelled us this far.
That is the nub of Shiller’s call for more derivatives and
more innovation. Shiller’s appeal is a tough sell at a time
when derivatives have produced so much havoc. But he
reminds us that the tools that got us here are not to blame;
they can be used badly and they can be used well. And
trying to stem the ineffable tide of human creativity is a
fool’s errand.
Source: Zachary Karabell, “The Case for Derivatives,” Newsweek,
February 2, 2009.
The solid line in Figure 20.8, panel C is the payoff. You see that the total position is
worth ST when the stock price at time T is below X and rises to a maximum of X when ST
exceeds X. In essence, the sale of the call options means the call writer has sold the claim
to any stock value above X in return for the initial premium (the call price). Therefore, at
expiration, the position is worth at most X. The dashed line of Figure 20.8, panel C is the
net profit to the covered call.
Writing covered call options has been a popular investment strategy among institutional
investors. Consider the managers of a fund invested largely in stocks. They might find it
appealing to write calls on some or all of the stock in order to boost income by the premiums collected. Although they thereby forfeit potential capital gains should the stock price
rise above the exercise price, if they view X as the price at which they plan to sell the stock
anyway, then the call may be viewed as a kind of “sell discipline.” The written call guarantees the stock sale will occur as planned.
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CHAPTER 20
ST " X
ST + X
ST
20
ST
ST
2(ST 2 X )
X
Payoff of stock
1 Payoff of written call
5 TOTAL
Options Markets: Introduction
693
Table 20.2
Value of a covered
call position at
option expiration
Payoff of Stock
A: Stock
ST
X
Payoff of
Written Call
ST
+ B: Write Call
Payoff of
Covered Call
= C: Covered Call
X
Payoff
X
Profit
X
ST
− (S0 − C)
Figure 20.8 Value of a covered call position at expiration
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eXcel APPLICATIONS: Spreads and Straddles
U
sing spreadsheets to analyze combinations of options is
very helpful. Once the basic models are built, it is easy
to extend the analysis to different bundles of options. The
Excel model “Spreads and Straddles” shown below can be
used to evaluate the profitability of different strategies. You
can find a link to this spreadsheet at www.mhhe.com/bkm.
A
B
C
D
1
2
3 Stock Prices
116.5
4 Beginning Market Price
Ending Market Price
130
5
6
7 Buying Options:
Call Options Strike Price
Payoff
Profit
8
110
−2.80
20.00
9
22.80
120
−6.80
10.00
10
16.80
130
−13.60
0.00
11
13.60
140
−10.30
0.00
12
10.30
13
Payoff
Profit
Put Options Strike Price
14
15
−12.60
110
12.60
0.00
16
−17.20
120
17.20
0.00
17
−23.60
130
23.60
0.00
18
−20.50
30.50
140
10.00
19
Payoff
Straddle Price
Profit
20
21
−15.40
35.40
110
20.00
22
−24.00
34.00
120
10.00
−37.20
23
37.20
130
0.00
−30.80
24
40.80
10.00
140
25
Example 20.4
Excel Question
1. Use the data in this spreadsheet to plot the profit on
a bullish spread (see Figure 20.10) with X1 5 120
and X2 5 130.
E
F
Spreads and Straddles
Return %
−12.28%
−40.48%
−100.00%
−100.00%
Return %
−100.00%
−100.00%
−100.00%
−67.21%
Return %
−43.50%
−70.59%
−100.00%
−75.49%
G
H
I
X 110 Straddle
Ending
Profit
Stock Price
−15.40
50
24.60
60
14.60
70
4.60
80
−5.40
90
−15.40
100
−25.40
110
−35.40
120
−25.40
130
−15.40
140
−5.40
150
4.60
160
14.60
170
24.60
180
34.60
190
44.60
200
54.60
210
64.60
J
K
L
X 120 Straddle
Ending
Profit
Stock Price
−24.00
36.00
50
26.00
60
16.00
70
6.00
80
−4.00
90
−14.00
100
−24.00
110
−34.00
120
−24.00
130
−14.00
140
−4.00
150
6.00
160
16.00
170
26.00
180
36.00
190
46.00
200
56.00
210
Covered Call
Assume a pension fund holds 1,000 shares of stock, with a current price of $100 per
share. Suppose the portfolio manager intends to sell all 1,000 shares if the share price
hits $110, and a call expiring in 60 days with an exercise price of $110 currently sells
for $5. By writing 10 call contracts (for 100 shares each) the fund can pick up $5,000 in
extra income. The fund would lose its share of profits from any movement of the stock
price above $110 per share, but given that it would have sold its shares at $110, it would
not have realized those profits anyway.
Straddle
A long straddle is established by buying both a call and a put on a stock, each with the
same exercise price, X, and the same expiration date, T. Straddles are useful strategies for
investors who believe a stock will move a lot in price but are uncertain about the direction of the move. For example, suppose you believe an important court case that will
make or break a company is about to be settled, and the market is not yet aware of the
situation. The stock will either double in value if the case is settled favorably or will drop
by half if the settlement goes against the company. The straddle position will do well
regardless of the outcome because its value rises when the stock price makes extreme
upward or downward moves from X.
The worst-case scenario for a straddle is no movement in the stock price. If ST equals
X, both the call and the put expire worthless, and the investor’s outlay for the purchase of
both options is lost. Straddle positions, therefore, are bets on volatility. An investor who
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695
establishes a straddle must view the stock as more volatile than the market does. Conversely, investors who write straddles—selling both a call and a put—must believe the
stock is less volatile. They accept the option premiums now, hoping the stock price will not
change much before option expiration.
The payoff to a straddle is presented in Table 20.3. The solid line of Figure 20.9,
panel C illustrates this payoff. Notice the portfolio payoff is always positive, except
at the one point where the portfolio has zero value, ST 5 X. You might wonder why
all investors don’t pursue such a seemingly “no-lose” strategy. The reason is that the
straddle requires that both the put and call be purchased. The value of the portfolio at
expiration, while never negative, still must exceed the initial cash outlay for a straddle
investor to clear a profit.
The dashed line of Figure 20.9, panel C is the profit diagram. The profit line lies below
the payoff line by the cost of purchasing the straddle, P 1 C. It is clear from the diagram that the straddle generates a loss unless the stock price deviates substantially from X.
The stock price must depart from X by the total amount
expended to purchase the call and the put for the straddle
CONCEPT CHECK 20.4
to clear a profit.
Strips and straps are variations of straddles. A strip is
Graph the profit and payoff diagrams for strips
two puts and one call on a security with the same exercise
and straps.
price and expiration date. A strap is two calls and one put.
Spreads
A spread is a combination of two or more call options (or two or more puts) on the same
stock with differing exercise prices or times to maturity. Some options are bought, whereas
others are sold, or written. A money spread involves the purchase of one option and the
simultaneous sale of another with a different exercise price. A time spread refers to the sale
and purchase of options with differing expiration dates.
Consider a money spread in which one call option is bought at an exercise price X1,
whereas another call with identical expiration date, but higher exercise price, X2, is written. The payoff will be the difference in the value of the call held and the value of the call
written, as in Table 20.4.
There are now three instead of two outcomes to distinguish: the lowest-price region
where ST is below both exercise prices, a middle region where ST is between the two exercise prices, and a high-price region where ST exceeds both exercise prices. Figure 20.10
illustrates the payoff and profit to this strategy, which is called a bullish spread because the
payoff either increases or is unaffected by stock price increases. Holders of bullish spreads
benefit from stock price increases.
One motivation for a bullish spread might be that the investor thinks one option is overpriced relative to another. For example, an investor who believes an X 5 $100 call is cheap
compared to an X 5 $110 call might establish the spread, even without a strong desire to
take a bullish position in the stock.
Collars
A collar is an options strategy that brackets the value of a portfolio between two bounds. Suppose that an investor currently is holding a large position in FinCorp stock, which is currently
selling at $100 per share. A lower bound of $90 can be placed on the value of the portfolio
by buying a protective put with exercise price $90. This protection, however, requires that the
investor pay the put premium. To raise the money to pay for the put, the investor might write a
call option, say, with exercise price $110. The call might sell for roughly the same price as the
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Table 20.3
Value of a straddle
position at option
expiration
Payoff of call
1 Payoff of put
5 TOTAL
ST * X
ST # X
0
X 2 ST
X 2 ST
ST 2 X
0
ST 2 X
Payoff of Call
Payoff
Profit
A: Call
0
ST
X
−C
Payoff of Put
X
X−P
Payoff
+ B: Put
X
0
−P
ST
Profit
Payoff of Straddle
X
Payoff
X−P−C
Profit
= C: Straddle
P+C
0
X
ST
− (P + C)
Figure 20.9 Value of a straddle at expiration
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CHAPTER 20
Options Markets: Introduction
ST " X1
X1 * ST " X2
ST # X2
0
20
0
ST 2 X1
20
S T 2 X1
ST 2 X1
2(ST 2 X2)
X 2 2 X1
Payoff of purchased call, exercise price 5 X1
1 Payoff of written call, exercise price 5 X2
5 TOTAL
697
Table 20.4
Value of a bullish
spread position at
expiration
Payoff
Payoff
Profit
A: Call Held
(Strike price 5 X1)
0
− C1
X1
X2
ST
Payoff
B: Call Written
(Strike price 5 X2)
C2
0
X1
X2
ST
Profit
Payoff
Payoff and Profit
C: Bullish
Spread
X2 − X1
Payoff
Profit
0
C2 − C1
X1
X2
ST
Figure 20.10 Value of a bullish spread position at expiration
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PART VI
Options, Futures, and Other Derivatives
put, meaning that the net outlay for the two options positions is approximately zero. Writing
the call limits the portfolio’s upside potential. Even if the stock price moves above $110, the
investor will do no better than $110, because at a higher price the stock will be called away.
Thus the investor obtains the downside protection represented by the exercise price of the put
by selling her claim to any upside potential beyond the exercise price of the call.
Example 20.5
Collars
A collar would be appropriate for an investor who has a target wealth goal in mind but
is unwilling to risk losses beyond a certain level. If you are contemplating buying a house
for $220,000, for example, you might set this figure as your goal. Your current wealth
may be $200,000, and you are unwilling to risk losing more than $20,000. A collar
established by (1) purchasing 2,000 shares of stock currently selling at $100 per share,
(2) purchasing 2,000 put options (20 options contracts) with exercise price $90, and (3)
writing 2,000 calls with exercise price $110 would give you a good chance to realize the
$20,000 capital gain without risking a loss of more than $20,000.
CONCEPT CHECK
20.5
Graph the payoff diagram for the collar described in Example 20.5.
20.4 The Put-Call Parity Relationship
We saw in the previous section that a protective put portfolio, comprising a stock position
and a put option on that position, provides a payoff with a guaranteed minimum value,
but with unlimited upside potential. This is not the only way to achieve such protection,
however. A call-plus-bills portfolio also can provide limited downside risk with unlimited
upside potential.
Consider the strategy of buying a call option and, in addition, buying Treasury bills with
face value equal to the exercise price of the call, and with maturity date equal to the expiration date of the option. For example, if the exercise price of the call option is $100, then
each option contract (which is written on 100 shares) would require payment of $10,000
upon exercise. Therefore, you would purchase a T-bill with a maturity value of $10,000.
More generally, for each option that you hold with exercise price X, you would purchase a
risk-free zero-coupon bond with face value X.
Examine the value of this position at time T, when the options expire and the zerocoupon bond matures:
Value of call option
Value of zero-coupon bond
TOTAL
ST " X
ST + X
0
X
X
ST 2 X
X
ST
If the stock price is below the exercise price, the call is worthless, but the bond matures to
its face value, X. It therefore provides a floor value to the portfolio. If the stock price
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699
exceeds X, then the payoff to the call, ST 2 X, is added to the face value of the bond to
provide a total payoff of ST . The payoff to this portfolio is precisely identical to the payoff
of the protective put that we derived in Table 20.1.
If two portfolios always provide equal values, then they must cost the same amount to
establish. Therefore, the call-plus-bond portfolio must cost the same as the stock-plus-put
portfolio. Each call costs C. The riskless zero-coupon bond costs X/(1 1 rf )T. Therefore,
the call-plus-bond portfolio costs C 1 X/(1 1 rf )T. The stock costs S0 to purchase now (at
time zero), while the put costs P. Therefore, we conclude that
C1
X
5 S0 1 P
(1 1 rf )T
(20.1)
Equation 20.1 is called the put-call parity theorem because it represents the proper
relationship between put and call prices. If the parity relation is ever violated, an arbitrage opportunity arises. For example, suppose you collect these data for a certain
stock:
Stock price
Call price (1-year expiration, X 5 $105)
Put price (1-year expiration, X 5 $105)
Risk-free interest rate
$110
$ 17
$ 5
5% per year
We can use these data in Equation 20.1 to see if parity is violated:
C1
?
X
5 S0 1 P
T
(1 1 r f )
17 1
105 ?
5 110 1 5
1.05
117 2 115
This result, a violation of parity—117 does not equal 115—indicates mispricing. To exploit
the mispricing, you buy the relatively cheap portfolio (the stock-plus-put position represented on the right-hand side of the equation) and sell the relatively expensive portfolio
(the call-plus-bond position corresponding to the left-hand side). Therefore, if you buy the
stock, buy the put, write the call, and borrow $100 for 1 year (because borrowing money is
the opposite of buying a bond), you should earn arbitrage profits.
Let’s examine the payoff to this strategy. In 1 year, the stock will be worth ST. The $100
borrowed will be paid back with interest, resulting in a cash outflow of $105. The written
call will result in a cash outflow of ST 2 $105 if ST exceeds $105. The purchased put pays
off $105 2 ST if the stock price is below $105.
Table 20.5 summarizes the outcome. The immediate cash inflow is $2. In 1 year, the
various positions provide exactly offsetting cash flows: The $2 inflow is realized without any offsetting outflows. This is an arbitrage opportunity that investors will pursue on
a large scale until buying and selling pressure restores the parity condition expressed in
Equation 20.1.
Equation 20.1 actually applies only to options on stocks that pay no dividends before
the expiration date of the option. The extension of the parity condition for European call
options on dividend-paying stocks is, however, straightforward. Problem 12 at the end of
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PART VI
Options, Futures, and Other Derivatives
Table 20.5
Arbitrage
strategy
Position
Buy stock
Borrow $105/1.05 5 $100
Sell call
Buy put
TOTAL
Cash Flow in 1 Year
Immediate
Cash Flow
ST * 105
ST # 105
2110
1100
117
25
2
ST
2105
0
105 2 ST
0
ST
2105
2(ST 2 105)
0
0
the chapter leads you through the demonstration. The more general formulation of the
put-call parity condition is
P 5 C 2 S0 1 PV(X) 1 PV(dividends)
(20.2)
where PV(dividends) is the present value of the dividends that will be paid by the stock
during the life of the option. If the stock does not pay dividends, Equation 20.2 becomes
identical to Equation 20.1.
Notice that this generalization would apply as well to European options on assets other
than stocks. Instead of using dividend income in Equation 20.2, we would let any income
paid out by the underlying asset play the role of the stock dividends. For example, European
put and call options on bonds would satisfy the same parity relationship, except that the
bond’s coupon income would replace the stock’s dividend payments in the parity formula.
Even this generalization, however, applies only to European options, as the cash flow
streams from the two portfolios represented by the two sides of Equation 20.2 will match
only if each position is held until expiration. If a call and a put may be optimally exercised
at different times before their common expiration date, then the equality of payoffs cannot
be assured, or even expected, and the portfolios will have different values.
Example 20.6
Put-Call Parity
Let’s see how well parity works using the data in Figure 20.1 on the IBM options. The
February expiration call with exercise price $195 and time to expiration of 28 days cost
$3.65 while the corresponding put option cost $5. IBM was selling for $194.47, and the
annualized short-term interest rate on this date was 0.1%. IBM was expected to pay a
dividend of $.85 with an ex-dividend date of February 8, 18 days hence. According to
parity, we should find that
P 5 C 1 PV(X) 2 S0 1 PV(Dividends)
5.00 5 3.65 1
195
28/365
(1.001)
2194.47 1
.85
(1.001)18/365
5.00 5 3.65 1 194.985 2 194.47 1 .85
5.00 5 5.015
So parity is violated by about $.015 per share. Is this a big enough difference to exploit?
Almost certainly not. You have to weigh the potential profit against the trading costs of
the call, put, and stock. More important, given the fact that options trade relatively infrequently, this deviation from parity might not be “real,” but may instead be attributable
to “stale” (i.e., out-of-date) price quotes at which you cannot actually trade.
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20.5 Option-Like Securities
Suppose you never traded an option directly. Why do you need to appreciate the properties
of options in formulating an investment plan? Many financial instruments and agreements
have features that convey implicit or explicit options to one or more parties. To value and
use these securities correctly, you must understand their embedded option attributes.
Callable Bonds
You know from Chapter 14 that many corporate bonds are issued with call provisions entitling the issuer to buy bonds back from bondholders at some time in the future at a specified
call price. The bond issuer holds a call option with exercise price equal to the price at which
the bond can be repurchased. A callable bond arrangement therefore is essentially a sale of a
straight bond (a bond with no option features such as callability or convertibility) to the investor and the concurrent issuance of a call option by the investor to the bond-issuing firm.
There must be some compensation for the firm’s implicit call option. If the callable
bond were issued with the same coupon rate as a straight bond, it would sell at a lower
price than the straight bond: the price difference would equal the value of the call. To sell
callable bonds at par, firms must issue them with coupon rates higher than the coupons
on straight debt. The higher coupons are the investor’s compensation for the call option
retained by the issuer.
Figure 20.11 illustrates this optionlike property. The horizontal axis is the value of a
straight bond with otherwise identical terms to the callable bond. The dashed 45-degree
line represents the value of straight debt. The solid line is the value of the callable bond,
and the dotted line is the value of the call option retained by the firm. A callable bond’s
potential for capital gains is limited by the firm’s option to repurchase at the call price.
CONCEPT CHECK
20.6
How is a callable bond similar to a covered call strategy on a straight bond?
The option inherent in callable bonds actually is more complex than an ordinary call
option, because usually it may be exercised only after some initial period of call protection. The price at which the bond is callable may change over time also. Unlike exchangelisted options, these features are defined in the initial bond covenant and will depend on
the needs of the issuing firm and its perception of the market’s tastes.
CONCEPT CHECK
20.7
Suppose the period of call protection is extended. How will the coupon rate the
company needs to offer on its bonds change to enable the issuer to sell the bonds
at par value?
Convertible Securities
Convertible bonds and convertible preferred stock convey options to the holder of the
security rather than to the issuing firm. A convertible security typically gives its holder the
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PART VI
Options, Futures, and Other Derivatives
right to exchange each bond or share
of preferred stock for a fixed number
of shares of common stock, regardValue of Straight Debt
less of the market prices of the securities at the time.
For example, a bond with a conversion ratio of 10 allows its holder to
Value of Callable Bond
convert one bond of par value $1,000
into 10 shares of common stock.
Alternatively, we say the conversion
price in this case is $100: To receive
10 shares of stock, the investor sacrifices bonds with face value $1,000 or,
put another way, $100 of face value
Value of Firm’s
Call Option
per share. If the present value of the
bond’s scheduled payments is less
Value of
than 10 times the value of one share
Straight Debt
Call Price
of stock, it may pay to convert; that is,
the conversion option is in the money.
A bond worth $950 with a conversion
Figure 20.11 Values of callable bonds compared with straight bonds
ratio of 10 could be converted profitably if the stock were selling above
$95, as the value of the 10 shares
received for each bond surrendered would exceed $950. Most convertible bonds are issued
“deep out of the money.” That is, the issuer sets the conversion ratio so that conversion will
not be profitable unless there is a substantial increase in stock prices and/or decrease in bond
prices from the time of issue.
A bond’s conversion value equals the value it would have if you converted it into stock
immediately. Clearly, a bond must sell for at least its conversion value. If it did not, you could
purchase the bond, convert it, and clear an immediate profit. This condition could never persist,
for all investors would pursue such a strategy and ultimately would bid up the price of the bond.
The straight bond value, or “bond floor,” is the value the bond would have if it were not
convertible into stock. The bond must sell for more than its straight bond value because
a convertible bond has more value; it is in fact a straight bond plus a valuable call option.
Therefore, the convertible bond has two lower bounds on its market price: the conversion
value and the straight bond value.
CONCEPT CHECK
20.8
Should a convertible bond issued at par value have a higher or lower coupon rate
than a nonconvertible bond issued at par?
Figure 20.12 illustrates the optionlike properties of the convertible bond. Figure 20.12,
panel A shows the value of the straight debt as a function of the stock price of the issuing firm. For healthy firms, the straight debt value is almost independent of the value of
the stock because default risk is small. However, if the firm is close to bankruptcy (stock
prices are low), default risk increases, and the straight bond value falls. Panel B shows the
conversion value of the bond. Panel C compares the value of the convertible bond to these
two lower bounds.
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Straight Debt Value
A
Stock Price
Conversion Value
B
Stock Price
Convertible Bond Value
Convertible Bond Value
Conversion Value
C
Straight Debt
Stock Price
Figure 20.12 Value of a convertible bond as a function of stock price. Panel A,
Straight debt value, or bond floor. Panel B, Conversion value of the bond.
Panel C, Total value of convertible bond.
When stock prices are low, the straight bond value is the effective lower bound, and the
conversion option is nearly irrelevant. The convertible will trade like straight debt. When
stock prices are high, the bond’s price is determined by its conversion value. With conversion all but guaranteed, the bond is essentially equity in disguise.
We can illustrate with two examples:
Annual coupon
Maturity date
Quality rating
Conversion ratio
Stock price
Conversion value
Market yield on 10-year Baa-rated bonds
Value as straight debt
Actual bond price
Reported yield to maturity
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Bond A
Bond B
$80
10 years
Baa
20
$30
$600
8.5%
$967
$972
8.42%
$80
10 years
Baa
25
$50
$1,250
8.5%
$967
$1,255
4.76%
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Bond A has a conversion value of only $600. Its value as straight debt, in contrast, is
$967. This is the present value of the coupon and principal payments at a market rate for
straight debt of 8.5%. The bond’s price is $972, so the premium over straight bond value is
only $5, reflecting the low probability of conversion. Its reported yield to maturity based
on scheduled coupon payments and the market price of $972 is 8.42%, close to that of
straight debt.
The conversion option on bond B is in the money. Conversion value is $1,250, and the
bond’s price, $1,255, reflects its value as equity (plus $5 for the protection the bond offers
against stock price declines). The bond’s reported yield is 4.76%, far below the comparable yield on straight debt. The big yield sacrifice is attributable to the far greater value of
the conversion option.
In theory, we could value convertible bonds by treating them as straight debt plus call
options. In practice, however, this approach is often impractical for several reasons:
1. The conversion price frequently increases over time, which means the exercise
price of the option changes.
2. Stocks may pay several dividends over the life of the bond, further complicating the
option-valuation analysis.
3. Most convertibles also are callable at the discretion of the firm. In essence, both
the investor and the issuer hold options on each other. If the issuer exercises its
call option to repurchase the bond, the bondholders typically have a month during
which they still can convert. When issuers use a call option, knowing bondholders
will choose to convert, the issuer is said to have forced a conversion. These conditions together mean the actual maturity of the bond is indeterminate.
Warrants
Warrants are essentially call options issued by a firm. One important difference between
calls and warrants is that exercise of a warrant requires the firm to issue a new share of
stock—the total number of shares outstanding increases. Exercise of a call option requires
only that the writer of the call deliver an already-issued share of stock to discharge the
obligation. In that case, the number of shares outstanding remains fixed. Also unlike call
options, warrants result in a cash flow to the firm when the warrant holder pays the exercise price. These differences mean that warrant values will differ somewhat from the values of call options with identical terms.
Like convertible debt, warrant terms may be tailored to meet the needs of the firm. Also
like convertible debt, warrants generally are protected against stock splits and dividends
in that the exercise price and the number of warrants held are adjusted to offset the effects
of the split.
Warrants are often issued in conjunction with another security. Bonds, for example,
may be packaged together with a warrant “sweetener,” frequently a warrant that may be
sold separately. This is called a detachable warrant.
Issue of warrants and convertible securities creates the potential for an increase in outstanding shares of stock if exercise occurs. Exercise obviously would affect financial statistics that are computed on a per-share basis, so annual reports must provide earnings per
share figures under the assumption that all convertible securities and warrants are exercised. These figures are called fully diluted earnings per share.3
3
We should note that the exercise of a convertible bond need not reduce EPS. Diluted EPS will be less than undiluted EPS only if interest saved (per share) on the convertible bonds is less than the prior EPS.
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705
The executive and employee stock options that became so popular in the 1990s
actually were warrants. Some of these grants were huge, with payoffs to top executives in
excess of $100 million. Yet firms almost uniformly chose not to acknowledge these grants
as expenses on their income statements until new reporting rules that took effect in 2006
required such recognition.
Collateralized Loans
Many loan arrangements require that the borrower put up collateral to guarantee the loan
will be paid back. In the event of default, the lender takes possession of the collateral. A
nonrecourse loan gives the lender no recourse beyond the right to the collateral. That is,
the lender may not sue the borrower for further payment if the collateral turns out not to be
valuable enough to repay the loan.
This arrangement gives an implicit call option to the borrower. Assume the borrower is
obligated to pay back L dollars at the maturity of the loan. The collateral will be worth ST
dollars at maturity. (Its value today is S0.) The borrower has the option to wait until loan
maturity and repay the loan only if the collateral is worth more than the L dollars necessary
to satisfy the loan. If the collateral is worth less than L, the borrower can default on the
loan, discharging the obligation by forfeiting the collateral, which is worth only ST.4
Another way of describing such a loan is to view the borrower as turning over the collateral to the lender but retaining the right to reclaim it by paying off the loan. The transfer
of the collateral with the right to reclaim it is equivalent to a payment of S0 dollars, less a
simultaneous recovery of a sum that resembles a call option with exercise price L. In effect,
the borrower turns over collateral but keeps an option to “repurchase” it for L dollars at the
maturity of the loan if L turns out to be less than ST. This is a call option.
A third way to look at a collateralized loan is to assume that the borrower will repay the
L dollars with certainty but also retain the option to sell the collateral to the lender for L
dollars, even if ST is less than L. In this case, the sale of the collateral would generate the
cash necessary to satisfy the loan. The ability to “sell” the collateral for a price of L dollars
represents a put option, which guarantees the borrower can raise enough money to satisfy
the loan simply by turning over the collateral.
It is perhaps surprising to realize that we can describe the same loan as involving either
a put option or a call option, as the payoffs to calls and puts are so different. Yet the equivalence of the two approaches is nothing more than a reflection of the put-call parity relationship. In our call-option description of the loan, the value of the borrower’s liability is
S0 2 C: The borrower turns over the asset, which is a transfer of S0 dollars, but retains
a call worth C dollars. In the put-option description, the borrower is obligated to pay L
dollars but retains the put, which is worth P: The present value of this net obligation is
L/(1 1 rf )T 2 P. Because these alternative descriptions are equivalent ways of viewing the
same loan, the value of the obligations must be equal:
S0 2 C 5
L
2P
(1 1 rf)T
(20.3)
Treating L as the exercise price of the option, Equation 20.3 is simply the put-call parity
relationship.
4
In reality, of course, defaulting on a loan is not so simple. There are losses of reputation involved as well as
considerations of ethical behavior. This is a description of a pure nonrecourse loan where both parties agree from
the outset that only the collateral backs the loan and that default is not to be taken as a sign of bad faith if the collateral is insufficient to repay the loan.
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Payoff
A
Payoff to lender
L
L
When ST exceeds L, the loan is repaid and the collateral is reclaimed. Otherwise,
the collateral is forfeited and the total loan repayment is worth only ST
ST
Payoff
Payoff to call with exercise price L
B
L
ST dollars minus the payoff
to the implicit call option
ST
L
Payoff
C
L
Payoff to a put with
exercise price L
L dollars minus the payoff to
the implicit put option
ST
L
Figure 20.13 Collateralized loan. Panel A, Payoff to collateralized loan. Panel B,
Lender can be viewed as collecting the collateral from the borrower, but issuing
an option to the borrower to call back the collateral for the face value of the
loan. Panel C, Lender can be viewed as collecting a risk-free loan from the borrower, but issuing a put to the borrower to sell the collateral for the face value
of the loan.
Figure 20.13 illustrates this fact. Figure 20.13, panel A is the value of the payment to
be received by the lender, which equals the minimum of ST or L. Panel B shows that this
amount can be expressed as ST minus the payoff of the call implicitly written by the lender
and held by the borrower. Panel C shows it also can be viewed as a receipt of L dollars
minus the proceeds of a put option.
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707
Levered Equity and Risky Debt
Investors holding stock in incorporated firms are protected by limited liability, which
means that if the firm cannot pay its debts, the firm’s creditors may attach only the firm’s
assets, not sue the corporation’s equityholders for further payment. In effect, any time the
corporation borrows money, the maximum possible collateral for the loan is the total of
the firm’s assets. If the firm declares bankruptcy, we can interpret this as an admission that
the assets of the firm are insufficient to satisfy the claims against it. The corporation may
discharge its obligations by transferring ownership of the firm’s assets to the creditors.
Just as is true for nonrecourse collateralized loans, the required payment to the creditors represents the exercise price of the implicit option, while the value of the firm is the
underlying asset. The equityholders have a put option to transfer their ownership claims on
the firm to the creditors in return for the face value of the firm’s debt.
Alternatively, we may view the equityholders as retaining a call option. They have,
in effect, already transferred their ownership claim to the firm to the creditors but have
retained the right to reacquire that claim by paying off the loan. Hence the equityholders
have the option to “buy back” the firm for a specified price: They have a call option.
The significance of this observation is that analysts can value corporate bonds using
option-pricing techniques. The default premium required of risky debt in principle can be
estimated by using option-valuation models. We consider some of these models in the next
chapter.
20.6 Financial Engineering
One of the attractions of options is the ability they provide to create investment positions
with payoffs that depend in a variety of ways on the values of other securities. We have
seen evidence of this capability in the various options strategies examined in Section 20.4.
Options also can be used to
custom-design new securities
Rate of Return on Index-Linked CD
or portfolios with desired patterns of exposure to the price
of an underlying security. In
this sense, options (and futures
contracts, to be discussed in
Chapters 22 and 23) provide
the ability to engage in financial engineering, the creation
Slope = .7
of portfolios with specified
payoff patterns.
A simple example of a product engineered with options
is the index-linked certificate of deposit. Index-linked
CDs enable retail investors to
take small positions in index
options. Unlike conventional
CDs, which pay a fixed rate of
interest, these CDs pay deposiFigure 20.14 Return on index-linked CD
tors a specified fraction of the
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rM = Market Rate
of Return
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rate of return on a market index such as the S&P 500, while guaranteeing a minimum rate
of return should the market fall. For example, the index-linked CD may offer 70% of any
market increase, but protect its holder from any market decrease by guaranteeing at least
no loss.
The index-linked CD is clearly a type of call option. If the market rises, the depositor
profits according to the participation rate or multiplier, in this case 70%; if the market
falls, the investor is insured against loss. Just as clearly, the bank offering these CDs is in
effect writing call options and can hedge its position by buying index calls in the options
market. Figure 20.14 shows the nature of the bank’s obligation to its depositors.
How might the bank set the appropriate multiplier? To answer this, note various features of the option:
1. The price the depositor is paying for the options is the forgone interest on the
conventional CD that could be purchased. Because interest is received at the end
of the period, the present value of the interest payment on each dollar invested is
rf /(1 1 rf ). Therefore, the depositor trades a sure payment with present value per
dollar invested of rf /(1 1 rf ) for a return that depends on the market’s performance.
Conversely, the bank can fund its obligation using the interest that it would have
paid on a conventional CD.
2. The option we have described is an at-the-money option, meaning that the exercise
price equals the current value of the stock index. The option goes into the money as
soon as the market index increases from its level at the inception of the contract.
3. We can analyze the option on a per-dollar-invested basis. For example, the option
costs the depositor rf /(1 1 rf ) dollars per dollar placed in the index-linked CD. The
market price of the option per dollar invested is C/S0: The at-the-money option
costs C dollars and is written on one unit of the market index, currently at S0.
Now it is easy to determine the multiplier that the bank can offer on the CDs. It receives
from its depositors a “payment” of rf /(1 1 rf ) per dollar invested. It costs the bank C/S0 to
purchase the call option on a $1 investment in the market index. Therefore, if rf /(1 1 rf ) is,
for example, 70% of C/S0, the bank can purchase at most .7 call option on the $1 investment
and the multiplier will be .7. More generally, the break-even multiplier on an index-linked
CD is rf /(1 1 rf ) divided by C/S0.
Example 20.7
Indexed-Linked CDs
Suppose that rf 5 6% per year, and that 6-month maturity at-the-money calls on
the market index currently cost $50. The index is at 1,000. Then the option costs
50/1,000 5 $.05 per dollar of market value. The CD rate is 3% per 6 months, meaning
that rf /(1 1 rf) 5 .03/1.03 5 .0291. Therefore, the multiplier would be .0291/.05 5 .5825.
The index-linked CD has several variants. Investors can purchase similar CDs that guarantee a positive minimum return if they are willing to settle for a smaller multiplier. In this
case, the option is “purchased” by the depositor for (rf 2 rmin)/(1 1 rf ) dollars per dollar invested, where rmin is the guaranteed minimum return. Because the purchase price is
lower, fewer options can be purchased, which results in a lower multiplier. Another variant
of the “bullish” CD we have described is the bear CD, which pays depositors a fraction of
any fall in the market index. For example, a bear CD might offer a rate of return of .6 times
any percentage decline in the S&P 500.
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CONCEPT CHECK
Options Markets: Introduction
709
20.9
Continue to assume that rf 5 3% per half-year, that at-the-money calls sell for $50, and that the market
index is at 1,000. What would be the multiplier for 6-month bullish equity-linked CDs offering a guaranteed minimum return of .5% over the term of the CD?
20.7 Exotic Options
Options markets have been tremendously successful. Investors clearly value the portfolio
strategies made possible by trading options; this is reflected in the heavy trading volume
in these markets. Success breeds imitation, and in recent years we have witnessed considerable innovation in the range of option instruments available to investors. Part of this
innovation has occurred in the market for customized options, which now trade in active
over-the-counter markets. Many of these options have terms that would have been highly
unusual even a few years ago; they are therefore called exotic options. In this section we
survey a few of the more interesting variants of these new instruments.
Asian Options
You already have been introduced to American- and European-style options. Asian-style
options are options with payoffs that depend on the average price of the underlying asset
during at least some portion of the life of the option. For example, an Asian call option may
have a payoff equal to the average stock price over the last 3 months minus the strike price
if that value is positive, and zero otherwise. These options may be of interest, for example,
to firms that wish to hedge a profit stream that depends on the average price of a commodity over some period of time.
Barrier Options
Barrier options have payoffs that depend not only on some asset price at option expiration, but
also on whether the underlying asset price has crossed through some “barrier.” For example,
a down-and-out option is one type of barrier option that automatically expires worthless if
and when the stock price falls below some barrier price. Similarly, down-and-in options will
not provide a payoff unless the stock price does fall below some barrier at least once during
the life of the option. These options also are referred to as knock-out and knock-in options.
Lookback Options
Lookback options have payoffs that depend in part on the minimum or maximum price
of the underlying asset during the life of the option. For example, a lookback call option
might provide a payoff equal to the maximum stock price during the life of the option
minus the exercise price, instead of the final stock price minus the exercise price. Such an
option provides (for a price, of course) a form of perfect market timing, providing the call
holder with a payoff equal to the one that would accrue if the asset were purchased for X
dollars and later sold at what turns out to be its high price.
Currency-Translated Options
Currency-translated options have either asset or exercise prices denominated in a foreign
currency. For example, the quanto allows an investor to fix in advance the exchange rate at
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which an investment in a foreign currency can be converted back into dollars. The right to
translate a fixed amount of foreign currency into dollars at a given exchange rate is a simple foreign exchange option. Quantos are more interesting because the amount of currency
that will be translated into dollars depends on the investment performance of the foreign
security. Therefore, a quanto in effect provides a random number of options.
Digital Options
Digital options, also called binary or “bet” options, have fixed payoffs that depend on
whether a condition is satisfied by the price of the underlying asset. For example, a digital
call option might pay off a fixed amount of $100 if the stock price at maturity exceeds the
exercise price.
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SUMMARY
1. A call option is the right to buy an asset at an agreed-upon exercise price. A put option is the right
to sell an asset at a given exercise price.
2. American-style options allow exercise on or before the expiration date. European options allow
exercise only on the expiration date. Most traded options are American in nature.
3. Options are traded on stocks, stock indexes, foreign currencies, fixed-income securities, and several futures contracts.
4. Options can be used either to lever up an investor’s exposure to an asset price or to provide insurance against volatility of asset prices. Popular option strategies include covered calls, protective
puts, straddles, spreads, and collars.
5. The put-call parity theorem relates the prices of put and call options. If the relationship is violated, arbitrage opportunities will result. Specifically, the relationship that must be satisfied is
P 5 C 2 S0 1 PV(X) 1 PV(dividends)
where X is the exercise price of both the call and the put options, PV(X) is the present value of a
claim to X dollars to be paid at the expiration date of the options, and PV(dividends) is the present
value of dividends to be paid before option expiration.
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for this chapter are
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6. Many commonly traded securities embody option characteristics. Examples of these securities are
callable bonds, convertible bonds, and warrants. Other arrangements such as collateralized loans
and limited-liability borrowing can be analyzed as conveying implicit options to one or more parties.
KEY TERMS
call option
exercise or strike price
premium
put option
in the money
out of the money
KEY EQUATIONS
7. Trading in so-called exotic options now takes place in an active over-the-counter market.
at the money
American option
European option
protective put
covered call
ST 2 X
0
if ST . X
if ST # X
0
X 2 ST
if ST . X
if ST # X
Payoff to call investor 5 b
Payoff to put investor 5 b
straddle
spread
collar
put-call parity theorem
warrant
Put-call parity: P 5 C 2 S0 1 PV(X ) 1 PV(Div)
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1. We said that options can be used either to scale up or reduce overall portfolio risk. What are some
examples of risk-increasing and risk-reducing options strategies? Explain each.
711
PROBLEM SETS
2. What are the trade-offs facing an investor who is considering buying a put option on an existing
portfolio?
3. What are the trade-offs facing an investor who is considering writing a call option on an existing
portfolio?
Basic
4. Why do you think the most actively traded options tend to be the ones that are near the money?
5. Turn back to Figure 20.1, which lists prices of various IBM options. Use the data in the figure to
calculate the payoff and the profits for investments in each of the following February expiration
options, assuming that the stock price on the expiration date is $195.
Call option, X 5 $190.
Put option, X 5 $190.
Call option, X 5 $195.
Put option, X 5 $195.
Call option, X 5 $200.
Put option, X 5 $200.
6. Suppose you think FedEx stock is going to appreciate substantially in value in the next 6 months.
Say the stock’s current price, S0, is $100, and the call option expiring in 6 months has an exercise
price, X, of $100 and is selling at a price, C, of $10. With $10,000 to invest, you are considering
three alternatives.
a. Invest all $10,000 in the stock, buying 100 shares.
b. Invest all $10,000 in 1,000 options (10 contracts).
c. Buy 100 options (one contract) for $1,000, and invest the remaining $9,000 in a money market
fund paying 4% in interest over 6 months (8% per year).
What is your rate of return for each alternative for the following four stock prices 6 months
from now? Summarize your results in the table and diagram below.
Price of Stock 6 Months from Now
$80
$100
$110
$120
a. All stocks (100 shares)
b. All options (1,000 shares)
c. Bills 1 100 options
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a.
b.
c.
d.
e.
f.
Intermediate
Rate of Return
0
ST
7. The common stock of the P.U.T.T. Corporation has been trading in a narrow price range for the
past month, and you are convinced it is going to break far out of that range in the next 3 months.
You do not know whether it will go up or down, however. The current price of the stock is $100
per share, and the price of a 3-month call option at an exercise price of $100 is $10.
a. If the risk-free interest rate is 10% per year, what must be the price of a 3-month put option on
P.U.T.T. stock at an exercise price of $100? (The stock pays no dividends.)
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b. What would be a simple options strategy to exploit your conviction about the stock price’s
future movements? How far would it have to move in either direction for you to make a
profit on your initial investment?
8. The common stock of the C.A.L.L. Corporation has been trading in a narrow range around $50
per share for months, and you believe it is going to stay in that range for the next 3 months. The
price of a 3-month put option with an exercise price of $50 is $4.
a. If the risk-free interest rate is 10% per year, what must be the price of a 3-month call option
on C.A.L.L. stock at an exercise price of $50 if it is at the money? (The stock pays no
dividends.)
b. What would be a simple options strategy using a put and a call to exploit your conviction
about the stock price’s future movement? What is the most money you can make on this
position? How far can the stock price move in either direction before you lose money?
c. How can you create a position involving a put, a call, and riskless lending that would have
the same payoff structure as the stock at expiration? What is the net cost of establishing that
position now?
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9. You are a portfolio manager who uses options positions to customize the risk profile of your
clients. In each case, what strategy is best given your client’s objective?
a. • Performance to date: Up 16%.
• Client objective: Earn at least 15%.
• Your scenario: Good chance of large gains or large losses between now and end of year.
i. Long straddle.
ii. Long bullish spread.
iii. Short straddle.
b. • Performance to date: Up 16%.
• Client objective: Earn at least 15%.
• Your scenario: Good chance of large losses between now and end of year.
i. Long put options.
ii. Short call options.
iii. Long call options.
10. An investor purchases a stock for $38 and a put for $.50 with a strike price of $35. The investor sells a call for $.50 with a strike price of $40. What is the maximum profit and loss for this
position? Draw the profit and loss diagram for this strategy as a function of the stock price at
expiration.
11. Imagine that you are holding 5,000 shares of stock, currently selling at $40 per share. You are
ready to sell the shares but would prefer to put off the sale until next year for tax reasons. If
you continue to hold the shares until January, however, you face the risk that the stock will drop
in value before year-end. You decide to use a collar to limit downside risk without laying out
a good deal of additional funds. January call options with a strike of $35 are selling at $2, and
January puts with a strike price of $45 are selling at $3. What will be the value of your portfolio
in January (net of the proceeds from the options) if the stock price ends up at: (a) $30, (b) $40,
or (c) $50? Compare these proceeds to what you would realize if you simply continued to hold
the shares.
12. In this problem, we derive the put-call parity relationship for European options on stocks that
pay dividends before option expiration. For simplicity, assume that the stock makes one dividend payment of $D per share at the expiration date of the option.
a. What is the value of a stock-plus-put position on the expiration date of the option?
b. Now consider a portfolio comprising a call option and a zero-coupon bond with the same
maturity date as the option and with face value (X 1 D). What is the value of this portfolio
on the option expiration date? You should find that its value equals that of the stock-plus-put
portfolio regardless of the stock price.
c. What is the cost of establishing the two portfolios in parts (a) and (b)? Equate the costs of
these portfolios, and you will derive the put-call parity relationship, Equation 20.2.
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13. a. A butterfly spread is the purchase of one call at exercise price X1, the sale of two calls at
exercise price X2, and the purchase of one call at exercise price X3. X1 is less than X2, and
X2 is less than X3 by equal amounts, and all calls have the same expiration date. Graph the
payoff diagram to this strategy.
b. A vertical combination is the purchase of a call with exercise price X2 and a put with exercise
price X1, with X2 greater than X1. Graph the payoff to this strategy.
14. A bearish spread is the purchase of a call with exercise price X2 and the sale of a call with
exercise price X1, with X2 greater than X1. Graph the payoff to this strategy and compare it to
Figure 20.10.
a. Strategy A is to write January call options on the CSI shares with strike price $45. These
calls are currently selling for $3 each.
b. Strategy B is to buy January put options on CSI with strike price $35. These options also sell
for $3 each.
c. Strategy C is to establish a zero-cost collar by writing the January calls and buying the
January puts.
Evaluate each of these strategies with respect to Joseph’s investment goals. What are the advantages and disadvantages of each? Which would you recommend?
16. Use the spreadsheet from the Excel Application boxes on spreads and straddles (available at
www.mhhe.com/bkm; link to Chapter 20 material) to answer these questions.
a. Plot the payoff and profit diagrams to a straddle position with an exercise (strike) price of
$130. Assume the options are priced as they are in the Excel Application.
b. Plot the payoff and profit diagrams to a bullish spread position with exercise (strike) prices
of $120 and $130. Assume the options are priced as they are in the Excel Application.
17. Some agricultural price support systems have guaranteed farmers a minimum price for their
output. Describe the program provisions as an option. What is the asset? The exercise price?
eXcel
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15. Joseph Jones, a manager at Computer Science, Inc. (CSI), received 10,000 shares of company
stock as part of his compensation package. The stock currently sells at $40 a share. Joseph
would like to defer selling the stock until the next tax year. In January, however, he will need to
sell all his holdings to provide for a down payment on his new house. Joseph is worried about
the price risk involved in keeping his shares. At current prices, he would receive $400,000 for
the stock. If the value of his stock holdings falls below $350,000, his ability to come up with the
necessary down payment would be jeopardized. On the other hand, if the stock value rises to
$450,000, he would be able to maintain a small cash reserve even after making the down payment. Joseph considers three investment strategies:
18. In what ways is owning a corporate bond similar to writing a put option? A call option?
19. An executive compensation scheme might provide a manager a bonus of $1,000 for every dollar
by which the company’s stock price exceeds some cutoff level. In what way is this arrangement
equivalent to issuing the manager call options on the firm’s stock?
20. Consider the following options portfolio. You write a January expiration call option on IBM
with exercise price $195. You write a January IBM put option with exercise price $190.
a. Graph the payoff of this portfolio at option expiration as a function of IBM’s stock price at
that time.
b. What will be the profit/loss on this position if IBM is selling at $198 on the option expiration
date? What if IBM is selling at $205? Use The Wall Street Journal listing from Figure 20.1
to answer this question.
c. At what two stock prices will you just break even on your investment?
d. What kind of “bet” is this investor making; that is, what must this investor believe about
IBM’s stock price to justify this position?
21. Consider the following portfolio. You write a put option with exercise price 90 and buy a put
option on the same stock with the same expiration date with exercise price 95.
a. Plot the value of the portfolio at the expiration date of the options.
b. On the same graph, plot the profit of the portfolio. Which option must cost more?
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22. A FinCorp put option with strike price 60 trading on the Acme options exchange sells for $2. To
your amazement, a FinCorp put with the same maturity selling on the Apex options exchange
but with strike price 62 also sells for $2. If you plan to hold the options positions to expiration,
devise a zero-net-investment arbitrage strategy to exploit the pricing anomaly. Draw the profit
diagram at expiration for your position.
23. Assume a stock has a value of $100. The stock is expected to pay a dividend of $2 per share at
year-end. An at-the-money European-style put option with one-year maturity sells for $7. If the
annual interest rate is 5%, what must be the price of a 1-year at-the-money European call option
on the stock?
24. You buy a share of stock, write a 1-year call option with X 5 $10, and buy a 1-year put option
with X 5 $10. Your net outlay to establish the entire portfolio is $9.50. What is the risk-free
interest rate? The stock pays no dividends.
25. You write a put option with X 5 100 and buy a put with X 5 110. The puts are on the same stock
and have the same expiration date.
a. Draw the payoff graph for this strategy.
b. Draw the profit graph for this strategy.
c. If the underlying stock has positive beta, does this portfolio have positive or negative beta?
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26. Joe Finance has just purchased a stock index fund, currently selling at $1,200 per share.
To protect against losses, Joe also purchased an at-the-money European put option on the
fund for $60, with exercise price $1,200, and 3-month time to expiration. Sally Calm, Joe’s
financial adviser, points out that Joe is spending a lot of money on the put. She notes that
3-month puts with strike prices of $1,170 cost only $45, and suggests that Joe use the
cheaper put.
a. Analyze Joe’s and Sally’s strategies by drawing the profit diagrams for the stock-plus-put
positions for various values of the stock fund in 3 months.
b. When does Sally’s strategy do better? When does it do worse?
c. Which strategy entails greater systematic risk?
27. You write a call option with X 5 50 and buy a call with X 5 60. The options are on the same
stock and have the same expiration date. One of the calls sells for $3; the other sells for $9.
a. Draw the payoff graph for this strategy at the option expiration date.
b. Draw the profit graph for this strategy.
c. What is the break-even point for this strategy? Is the investor bullish or bearish on the stock?
28. Devise a portfolio using only call options and shares of stock with the following value (payoff)
at the option expiration date. If the stock price is currently 53, what kind of bet is the investor
making?
Payoff
50
50
Challenge
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60
110
ST
29. You are attempting to formulate an investment strategy. On the one hand, you think there is
great upward potential in the stock market and would like to participate in the upward move
if it materializes. However, you are not able to afford substantial stock market losses and so
cannot run the risk of a stock market collapse, which you think is also a possibility. Your
investment adviser suggests a protective put position: Buy both shares in a market index
stock fund and put options on those shares with 3-month expiration and exercise price of
$1,170. The stock index fund is currently selling for $1,350. However, your uncle suggests
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715
you instead buy a 3-month call option on the index fund with exercise price $1,260 and buy
3-month T-bills with face value $1,260.
Stock fund
$1,350
T-bill (face value $1,260)
Call (exercise price $1,260)
Put (exercise price $1,170)
$1,215
$ 180
$
9
Make a table of the profits realized for each portfolio for the following values of the stock
price in 3 months: ST 5 $1,000, $1,260, $1,350, $1,440.
Graph the profits to each portfolio as a function of ST on a single graph.
d. Which strategy is riskier? Which should have a higher beta?
e. Explain why the data for the securities given in part (c) do not violate the put-call parity
relationship.
30. FedEx is selling for $100 a share. A FedEx call option with one month until expiration and an
exercise price of $105 sells for $2 while a put with the same strike and expiration sells for $6.94.
What is the market price of a zero-coupon bond with face value $105 and 1 month maturity?
What is the risk-free interest rate expressed as an effective annual yield?
31. Demonstrate that an at-the-money call option on a given stock must cost more than an at-themoney put option on that stock with the same expiration. The stock will pay no dividends until
after the expiration date. (Hint: Use put-call parity.)
1. Donna Donie, CFA, has a client who believes the common stock price of TRT Materials (currently $58 per share) could move substantially in either direction in reaction to an expected court
decision involving the company. The client currently owns no TRT shares, but asks Donie for
advice about implementing a strangle strategy to capitalize on the possible stock price movement.
A strangle is a portfolio of a put and a call with a higher exercise price but the same expiration
date. Donie gathers the TRT option-pricing data:
Characteristic
Price
Strike price
Time to expiration
Call Option
Put Option
$ 5
$60
90 days from now
$ 4
$55
90 days from now
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a. On the same graph, draw the payoffs to each of these strategies as a function of the stock fund
value in 3 months. (Hint: Think of the options as being on one “share” of the stock index
fund, with the current price of each share of the fund equal to $1,350.)
b. Which portfolio must require a greater initial outlay to establish? (Hint: Does either portfolio provide a final payout that is always at least as great as the payoff of the other portfolio?)
c. Suppose the market prices of the securities are as follows:
a. Recommend whether Donie should choose a long strangle strategy or a short strangle strategy
to achieve the client’s objective.
b. Calculate, at expiration for the appropriate strangle strategy in part (a), the:
i. Maximum possible loss per share.
ii. Maximum possible gain per share.
iii. Break-even stock price(s).
2. Martin Bowman is preparing a report distinguishing traditional debt securities from structured
note securities. Discuss how the following structured note securities differ from a traditional debt
security with respect to coupon and principal payments:
a. Equity index-linked notes.
b. Commodity-linked bear bond.
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3. Suresh Singh, CFA, is analyzing a convertible bond. The characteristics of the bond and the
underlying common stock are given in the following exhibit:
Convertible Bond Characteristics
Par value
Annual coupon rate (annual pay)
Conversion ratio
Market price
Straight value
$1,000
6.5%
22
105% of par value
99% of par value
Underlying Stock Characteristics
Current market price
Annual cash dividend
$40 per share
$1.20 per share
Compute the bond’s:
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a. Conversion value.
b. Market conversion price.
4. Rich McDonald, CFA, is evaluating his investment alternatives in Ytel Incorporated by analyzing
a Ytel convertible bond and Ytel common equity. Characteristics of the two securities are given in
the following exhibit:
Characteristics
Convertible Bond
Common Equity
$1,000
4%
$980
$925
25
At any time
—
$1,125
—
—
$35 per share
—
—
—
$0
$45 per share
Par value
Coupon (annual payment)
Current market price
Straight bond value
Conversion ratio
Conversion option
Dividend
Expected market price in 1 year
a. Calculate, based on the exhibit, the:
i. Current market conversion price for the Ytel convertible bond.
ii. Expected 1-year rate of return for the Ytel convertible bond.
iii. Expected 1-year rate of return for the Ytel common equity.
One year has passed and Ytel’s common equity price has increased to $51 per share. Also, over
the year, the interest rate on Ytel’s nonconvertible bonds of the same maturity increased, while
credit spreads remained unchanged.
b. Name the two components of the convertible bond’s value. Indicate whether the value of each
component should decrease, stay the same, or increase in response to the:
i. Increase in Ytel’s common equity price.
ii. Increase in interest rates.
5. a. Consider a bullish spread option strategy using a call option with a $25 exercise price priced
at $4 and a call option with a $40 exercise price priced at $2.50. If the price of the stock
increases to $50 at expiration and each option is exercised on the expiration date, the net profit
per share at expiration (ignoring transaction costs) is:
i.
ii.
iii.
iv.
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$8.50
$13.50
$16.50
$23.50
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717
b. A put on XYZ stock with a strike price of $40 is priced at $2.00 per share, while a call
with a strike price of $40 is priced at $3.50. What is the maximum per-share loss to
the writer of the uncovered put and the maximum per-share gain to the writer of the
uncovered call?
Maximum Loss
to Put Writer
i.
ii.
iii.
iv.
Maximum Gain
to Call Writer
$38.00
$ 3.50
$38.00
$36.50
$40.00
$ 3.50
$40.00
$40.00
1. Go to www.nasdaq.com and select IBM in the quote section. Once you have the information quote, request the information on options. You will be able to access the prices
for the calls and puts that are closest to the money. For example, if the price of IBM is
$196.72, you will use the options with the $195 exercise price. Use near-term options.
For example, in February, you would select April and July expirations.
a. What are the prices for the put and call with the nearest expiration date?
b. What would be the cost of a straddle using these options?
c. At expiration, what would be the break-even stock prices for the straddle?
d. What would be the percentage increase or decrease in the stock price required to
break even?
e. What are the prices for the put and call with a later expiration date?
f. What would be the cost of a straddle using the later expiration date? At
expiration, what would be the break-even stock prices for the straddle?
g. What would be the percentage increase or decrease in the stock price required to
break even?
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E-INVESTMENTS EXERCISES
SOLUTIONS TO CONCEPT CHECKS
1. a. Denote the stock price at call option expiration by ST, and the exercise price by X. Value at
expiration 5 ST 2 X 5 ST 2 $195 if this value is positive; otherwise the call expires worthless.
Profit 5 Final value 2 Price of call option 5 Proceeds 2 $3.65.
Proceeds
Profits
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ST 5 $185
ST 5 $205
$0
$10
$ 6.35
2$3.65
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PART VI
Options, Futures, and Other Derivatives
b. Value at expiration 5 X 2 ST 5 $195 2 ST if this value is positive; otherwise the put expires
worthless.
Profit 5 Final value 2 Price of put option 5 Proceeds 2 $5.00.
ST 5 $185
Proceeds
Profits
ST 5 $205
$10
1$ 5.00
$0
2$5.00
2. Before the split, the final payoff would have been 100 3 ($200 2 $195) 5 $500. After the split,
the payoff is 200 3 ($100 2 $97.50) 5 $500. The payoff is unaffected.
3. a.
Write Put
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Buy Call
Profit
Payoff
S
Payoff
S
Profit
Write Call
Buy Put
Profit
S
Payoff
Payoff
S
Profit
b. The payoffs and profits to both buying calls and writing puts generally are higher when the
stock price is higher. In this sense, both positions are bullish. Both involve potentially taking
delivery of the stock. However, the call holder will choose to take delivery when the stock
price is high, while the put writer is obligated to take delivery when the stock price is low.
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CHAPTER 20
Options Markets: Introduction
719
c. The payoffs and profits to both writing calls and buying puts generally are higher when the
stock price is lower. In this sense, both positions are bearish. Both involve potentially making
delivery of the stock. However, the put holder will choose to make delivery when the stock
price is low, while the call writer is obligated to make delivery when the stock price is high.
4.
Payoff to a Strip
ST " X
ST + X
2 Puts
2(X 2 ST)
0
1 Call
0
ST 2 X
Payoff and
Profit
2X
Payoff
Slope = 1
2X − 2P − C
Profit
X
− 2P − C
ST
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Slope = −2
Payoff to a Strap
ST " X
1 Put
2 Calls
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ST + X
X 2 ST
0
0
2(ST 2 X)
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PART VI
Options, Futures, and Other Derivatives
Payoff and
Profit
Payoff
Slope = 2
Profit
X
Slope = −1
X − P − 2C
ST
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X
− P − 2C
5. The payoff table on a per-share basis is as follows:
Buy put (X 5 90)
Share
Write call (X 5 110)
TOTAL
ST " 90
90 " ST " 110
ST + 110
90 2 ST
ST
0
0
0
ST
0
ST
2(ST 2 110)
90
ST
110
The graph of the payoff is as follows. If you multiply the per-share values by 2,000, you will see that
the collar provides a minimum payoff of $180,000 (representing a maximum loss of $20,000) and a
maximum payoff of $220,000 (which is the cost of the house).
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CHAPTER 20
Options Markets: Introduction
721
Payoff
$110
Collar
$90
$110
6. The covered call strategy would consist of a straight bond with a call written on the bond. The
value of the strategy at option expiration as a function of the value of the straight bond is given by
the solid colored payoff line in the following figure, which is virtually identical to Figure 20.11.
Value of Straight Bond
Payoff of Covered Call
Value of Straight Bond
X
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ST
$90
Call Written
7. The call option is worth less as call protection is expanded. Therefore, the coupon rate need not
be as high.
8. Lower. Investors will accept a lower coupon rate in return for the conversion option.
9. The depositor’s implicit cost per dollar invested is now only ($.03 2 $.005)/1.03 5 $.02427 per
6-month period. Calls cost 50/1,000 5 $.05 per dollar invested in the index. The multiplier falls
to .02427/.05 5 .4854.
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CHAPTER TWENTY-ONE
Option Valuation
PART VI
IN THE PREVIOUS chapter we examined
option markets and strategies. We noted that
many securities contain embedded options
that affect both their values and their risk–
return characteristics. In this chapter, we turn
our attention to option-valuation issues. To
understand most option-valuation models
requires considerable mathematical and statistical background. Still, many of the ideas
and insights of these models can be demonstrated in simple examples, and we will concentrate on these.
We start with a discussion of the factors
that ought to affect option prices. After
this discussion, we present several bounds
within which option prices must lie. Next we
turn to quantitative models, starting with a
simple “two-state” option-valuation model,
and then showing how this approach can be
generalized into a useful and accurate pricing tool. We then move on to one particular
valuation formula, the famous Black-Scholes
model, one of the most significant breakthroughs in finance theory in several decades.
Finally, we look at some of the more important applications of option-pricing theory in
portfolio management and control.
Option-pricing models allow us to “back
out” market estimates of stock-price volatility, and we will examine these measures of
implied volatility. Next we turn to some of the
more important applications of option-pricing
theory in risk management. Finally, we take a
brief look at some of the empirical evidence
on option pricing, and the implications of
that evidence concerning the limitations of
the Black-Scholes model.
21.1 Option Valuation: Introduction
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Intrinsic and Time Values
Consider a call option that is out of the money at the moment, with the stock price below
the exercise price. This does not mean the option is valueless. Even though immediate
exercise today would be unprofitable, the call retains a positive value because there is
always a chance the stock price will increase sufficiently by the expiration date to allow
for profitable exercise. If not, the worst that can happen is that the option will expire with
zero value.
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CHAPTER 21
Option Valuation
723
The value S0 2 X is sometimes called the intrinsic value of in-the-money call options
because it gives the payoff that could be obtained by immediate exercise. Intrinsic value is
set equal to zero for out-of-the-money or at-the-money options. The difference between the
actual call price and the intrinsic value is commonly called the time value of the option.
“Time value” is unfortunate terminology because it may confuse the option’s time value
with the time value of money. Time value in the options context refers simply to the difference between the option’s price and the value the option would have if it were expiring
immediately. It is the part of the option’s value that may be attributed to the fact that it still
has positive time to expiration.
Most of an option’s time value typically is a type of “volatility value.” Because the
option holder can choose not to exercise, the payoff cannot be worse than zero. Even if a
call option is out of the money now, it still will sell for a positive price because it offers the
potential for a profit if the stock price increases, while imposing no risk of additional loss
should the stock price fall. The volatility value lies in the value of the right not to exercise
the call if that action would be unprofitable. The option to exercise, as opposed to the obligation to exercise, provides insurance against poor stock price performance.
As the stock price increases substantially, it becomes likely that the call option will
be exercised by expiration. Ultimately, with exercise all but assured, the volatility value
becomes minimal. As the stock price gets ever larger, the option value approaches the
“adjusted” intrinsic value, the stock price minus the present value of the exercise price,
S0 2 PV(X ).
Why should this be? If you are virtually certain the option will be exercised and the
stock purchased for X dollars, it is as though you own the stock already. The stock certificate, with a value today of S0, might as well be sitting in your safe-deposit box now,
as it will be there in only a few months. You just haven’t paid for it yet. The present
value of your obligation is the present value of X, so the net value of the call option is
S0 2 PV(X).1
Figure 21.1 illustrates the call option valuation function. The value curve shows that
when the stock price is very low, the option is nearly worthless, because there is almost
no chance that it will be exercised. When the stock price is very high, the option value
approaches adjusted intrinsic value. In the midrange case, where the option is approximately at the money, the option curve diverges from the straight lines corresponding to
adjusted intrinsic value. This is because although exercise today would have a negligible
(or negative) payoff, the volatility value of the option is quite high in this region.
The call always increases in value with the stock price. The slope is greatest, however,
when the option is deep in the money. In this case, exercise is all but assured, and the
option increases in price one-for-one with the stock price.
Determinants of Option Values
We can identify at least six factors that should affect the value of a call option: the stock
price, the exercise price, the volatility of the stock price, the time to expiration, the interest rate, and the dividend rate of the stock. The call option should increase in value with
the stock price and decrease in value with the exercise price because the payoff to a call,
1
This discussion presumes that the stock pays no dividends until after option expiration. If the stock does pay
dividends before expiration, then there is a reason you would care about getting the stock now rather than at
expiration—getting it now entitles you to the interim dividend payments. In this case, the adjusted intrinsic value
of the option must subtract the value of the dividends the stock will pay out before the call is exercised. Adjusted
intrinsic value would more generally be defined as S0 2 PV(X ) 2 PV(D), where D is the dividend to be paid
before option expiration.
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PART VI
Options, Futures, and Other Derivatives
Option Value
Value of Call Option
S0 – PV(X)
Value of Option
if Now at
Expiration = S – X =
Intrinsic Value
Time Value
S0
PV(X) X
Out of the
Money
In the Money
Figure 21.1 Call option value before expiration
if exercised, equals ST 2 X. The magnitude of the expected payoff increases with the
difference S0 2 X.
Call option values also increase with the volatility of the underlying stock price. To see
why, consider circumstances where possible stock prices at expiration may range from $10
to $50 compared to a situation where they range only from $20 to $40. In both cases, the
expected, or average, stock price will be $30. Suppose the exercise price on a call option is
also $30. What are the option payoffs?
High-Volatility Scenario
Stock price
Option payoff
$10
0
$20
0
$30
0
$40
10
$50
20
$35
5
$40
10
Low-Volatility Scenario
Stock price
Option payoff
$20
0
$25
0
$30
0
If each outcome is equally likely, with probability .2, the expected payoff to the option
under high-volatility conditions will be $6, but under low-volatility conditions the expected
payoff is half as much, only $3.
Despite the fact that the average stock price in each scenario is $30, the average option
payoff is greater in the high-volatility scenario. The source of this extra value is the limited
loss an option holder can suffer, or the volatility value of the call. No matter how far below
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CHAPTER 21
Table 21.1
Determinants of call
option values
If This Variable Increases . . .
The Value of a Call Option
Stock price, S
Exercise price, X
Volatility, s
Time to expiration, T
Interest rate, rf
Dividend payouts
Increases
Decreases
Increases
Increases
Increases
Decreases
Option Valuation
725
$30 the stock price drops, the option holder will get zero. Obviously, extremely poor stock
price performance is no worse for the call option holder than moderately poor performance.
In the case of good stock performance, however, the call will expire in the money, and
it will be more profitable the higher the stock price. Thus extremely good stock outcomes
can improve the option payoff without limit, but extremely poor outcomes cannot worsen
the payoff below zero. This asymmetry means that volatility in the underlying stock price
increases the expected payoff to the option, thereby enhancing its value.2
Similarly, longer time to expiration increases the value of a call option. For more distant
expiration dates, there is more time for unpredictable future events to affect prices, and the
range of likely stock prices increases. This has an effect similar to that of increased volatility. Moreover, as time to expiration lengthens, the present value of the exercise price falls,
thereby benefiting the call option holder and increasing the option value. As a corollary
to this issue, call option values are higher when interest rates rise (holding the stock price
constant) because higher interest rates also reduce the present value of the exercise price.
Finally, the dividend payout policy of the firm affects
option values. A high-dividend payout policy puts a drag CONCEPT CHECK 21.1
on the rate of growth of the stock price. For any expected
total rate of return on the stock, a higher dividend yield Prepare a table like Table 21.1 for the determust imply a lower expected rate of capital gain. This drag minants of put option values. How should
on stock price appreciation decreases the potential pay- American put values respond to increases in
off from the call option, thereby lowering the call value. S, X, s, T, rf, and dividend payouts?
Table 21.1 summarizes these relationships.
21.2 Restrictions on Option Values
Several quantitative models of option pricing have been devised, and we will examine
some of them later in this chapter. All models, however, rely on simplifying assumptions. You might wonder which properties of option values are truly general and which
depend on the particular simplifications. To start with, we will consider some of the more
2
You should be careful interpreting the relationship between volatility and option value. Neither the focus of this
analysis on total (as opposed to systematic) volatility nor the conclusion that options buyers seem to like volatility contradicts modern portfolio theory. In conventional discounted cash flow analysis, we find the discount rate
appropriate for a given distribution of future cash flows. Greater risk implies a higher discount rate and lower
present value. Here, however, the cash flow from the option depends on the volatility of the stock. The option
value increases not because traders like risk but because the expected cash flow to the option holder increases
along with the volatility of the underlying asset.
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PART VI
Options, Futures, and Other Derivatives
important general properties of option prices. Some of these properties have important
implications for the effect of stock dividends on option values and the possible profitability
of early exercise of an American option.
Restrictions on the Value of a Call Option
The most obvious restriction on the value of a call option is that its value cannot be negative. Because the option need not be exercised, it cannot impose any liability on its holder;
moreover, as long as there is any possibility that at some point the option can be exercised
profitably, it will command a positive price. Its payoff is zero at worst, and possibly positive, so it has some positive value.
We can place another lower bound on the value of a call option. Suppose that the stock
will pay a dividend of D dollars just before the expiration date of the option, denoted by
T (where today is time 0). Now compare two portfolios, one consisting of a call option on
one share of stock and the other a leveraged equity position consisting of that share and
borrowing of (X 1 D)/(1 1 rf )T dollars. The loan repayment is X 1 D dollars, due on the
expiration date of the option. For example, for a half-year maturity option with exercise
price $70, dividends to be paid of $5, and effective annual interest of 10%, you would purchase one share of stock and borrow $75/(1.10)1/2 5 $71.51. In 6 months, when the loan
matures, the payment due is $75.
At that time, the payoff to the leveraged equity position would be
Stock value
2 Payback of loan
TOTAL
In General
Our Numbers
ST 1 D
2(X 1 D)
ST 2 X
ST 1 5
275
ST 2 70
where ST denotes the stock price at the option expiration date. Notice that the payoff to the
stock is the ex-dividend stock value plus dividends received. Whether the total payoff to
the stock-plus-borrowing position is positive or negative depends on whether ST exceeds X.
The net cash outlay required to establish this leveraged equity position is S0 2 $71.51, or,
more generally, S0 2 (X 1 D)/(1 1 rf)T, that is, the current price of the stock, S0, less the
initial cash inflow from the borrowing position.
The payoff to the call option will be ST 2 X if the option expires in the money and zero
otherwise. Thus the option payoff is equal to the leveraged equity payoff when that payoff
is positive and is greater when the leveraged equity position has a negative payoff. Because
the option payoff is always greater than or equal to that of the leveraged equity position,
the option price must exceed the cost of establishing that position.
Therefore, the value of the call must be greater than S0 2 (X 1 D)/(1 1 rf)T, or, more
generally,
C $ S0 2 PV(X ) 2 PV(D)
where PV(X) denotes the present value of the exercise price and PV(D) is the present value
of the dividends the stock will pay at the option’s expiration. More generally, we can interpret PV(D) as the present value of any and all dividends to be paid prior to the option expiration date. Because we know already that the value of a call option must be nonnegative,
we may conclude that C is greater than the maximum of either 0 or S0 2 PV(X) 2 PV(D).
We also can place an upper bound on the possible value of the call; this bound is simply
the stock price. No one would pay more than S0 dollars for the right to purchase a stock
currently worth S0 dollars. Thus C # S0.
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CHAPTER 21
0
C $ St 2 PV(X) . St 2 X
ge
S
Ra
n
5
Al
lo
w
ab
le
Bo
un
d
Up
Early Exercise and Dividends
A call option holder who wants to close out
that position has two choices: exercise the
call or sell it. If the holder exercises at time t,
the call will provide a payoff of St 2 X,
assuming, of course, that the option is in the
money. We have just seen that the option can
be sold for at least St 2 PV(X ) 2 PV(D).
Therefore, for an option on a non-dividendpaying stock, C is greater than St 2 PV(X ).
Because the present value of X is less than X
itself, it follows that
Lower Bound
5 Adjusted Intrinsic
Value
5 S0 2 PV(X) 2 PV(D)
PV(X) 1 PV(D)
S0
Figure 21.2 Range of possible call option values
C
Value of Call for
The implication here is that the proGiven Stock Price
ceeds from a sale of the option (at price C)
must exceed the proceeds from an exercise
(St 2 X). It is economically more attractive
to sell the call, which keeps it alive, than to
exercise and thereby end the option. In other
words, calls on non-dividend-paying stocks
are “worth more alive than dead.”
If it never pays to exercise a call option
before expiration, the right to exercise
early actually must be valueless. We conclude that the values of otherwise identical American and European call options
on stocks paying no dividends are equal.
This simplifies matters, because any valuPV(X) 1 PV(D)
ation formula that applies to the European
call, for which only one exercise date
need be considered, also must apply to an
Figure 21.3 Call option value as a function of the
American call.
current stock price
As most stocks do pay dividends, you
may wonder whether this result is just a theoretical curiosity. It is not: Reconsider our
argument and you will see that all that we really require is that the stock pay no dividends
until the option expires. This condition will be true for many real-world options.
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Call Value
pe
r
Figure 21.2 demonstrates graphically
the range of prices that is ruled out by these
upper and lower bounds for the value of a call
option. Any option value outside the shaded
area is not possible according to the restrictions we have derived. Before expiration, the
call option value normally will be within the
allowable range, touching neither the upper
nor lower bound, as in Figure 21.3.
Option Valuation
S0
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Options, Futures, and Other Derivatives
B: European Put Value
A: American Put Value
X
X
PV(X)
Put Value
Time
Value
S*
X
S0
PV(X)
X
S0
Figure 21.4 Put option values as a function of the current stock price
Early Exercise of American Puts
For American put options, the optimality of early exercise is most definitely a possibility. To
see why, consider a simple example. Suppose that you purchase a put option on a stock. Soon
the firm goes bankrupt, and the stock price falls to zero. Of course you want to exercise now,
because the stock price can fall no lower. Immediate exercise gives you immediate receipt of
the exercise price, which can be invested to start generating income. Delay in exercise means a
time-value-of-money cost. The right to exercise a put option before expiration must have value.
Now suppose instead that the firm is only nearly bankrupt, with the stock selling at just
a few cents. Immediate exercise may still be optimal. After all, the stock price can fall by
only a very small amount, meaning that the proceeds from future exercise cannot be more
than a few cents greater than the proceeds from immediate exercise. Against this possibility of a tiny increase in proceeds must be weighed the time-value-of-money cost of deferring exercise. Clearly, there is some stock price below which early exercise is optimal.
This argument also proves that the American put must be worth more than its European
counterpart. The American put allows you to exercise anytime before expiration. Because
the right to exercise early may be useful in some circumstances, it will command a
premium in the capital market. The American put therefore will sell for a higher price than
a European put with otherwise identical terms.
Figure 21.4, panel A illustrates the value of an American put option as a function of the
current stock price, S0. Once the stock price drops below a critical value, denoted S* in
the figure, exercise becomes optimal. At that point the option-pricing curve is tangent to the
straight line depicting the intrinsic value of the option. If and when the stock price reaches
S*, the put option is exercised and its payoff equals its intrinsic value.
In contrast, the value of the European put, which is graphed in Figure 21.4, panel B, is
not asymptotic to the intrinsic value line. Because early exercise is prohibited, the maximum value of the European put is PV(X), which occurs at the point S0 5 0. Obviously, for
a long enough horizon, PV(X) can be made arbitrarily small.
CONCEPT CHECK
21.2
In light of this discussion, explain why the put-call parity relationship is valid only for European options on
non-dividend-paying stocks. If the stock pays no dividends, what inequality for American options would
correspond to the parity theorem?
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CHAPTER 21
Option Valuation
729
21.3 Binomial Option Pricing
Two-State Option Pricing
A complete understanding of commonly used option-valuation formulas is difficult without a substantial mathematics background. Nevertheless, we can develop valuable insight
into option valuation by considering a simple special case. Assume that a stock price can
take only two possible values at option expiration: The stock will either increase to a given
higher price or decrease to a given lower price. Although this may seem an extreme simplification, it allows us to come closer to understanding more complicated and realistic models. Moreover, we can extend this approach to describe far more reasonable specifications
of stock price behavior. In fact, several major financial firms employ variants of this simple
model to value options and securities with optionlike features.
Suppose the stock now sells at S0 5 $100, and the price will either increase by a factor
of u 5 1.20 to $120 (u stands for “up”) or fall by a factor of d 5 .9 to $90 (d stands for
“down”) by year-end. A call option on the stock might specify an exercise price of $110
and a time to expiration of 1 year. The interest rate is 10%. At year-end, the payoff to the
holder of the call option will be either zero, if the stock falls, or $10, if the stock price goes
to $120.
These possibilities are illustrated by the following value “trees”:
120
100
10
C
90
Stock price
0
Call option value
Compare the payoff of the call to that of a portfolio consisting of one share of the
stock and borrowing of $81.82 at the interest rate of 10%. The payoff of this portfolio also
depends on the stock price at year-end:
Value of stock at year-end
2 Repayment of loan with interest
TOTAL
$90
290
$ 0
$120
290
$ 30
We know the cash outlay to establish the portfolio is $18.18: $100 for the stock, less the
$81.82 proceeds from borrowing. Therefore the portfolio’s value tree is
30
18.18
0
The payoff of this portfolio is exactly three times that of the call option for either value
of the stock price. In other words, three call options will exactly replicate the payoff to the
portfolio; it follows that three call options should have the same price as the cost of establishing the portfolio. Hence the three calls should sell for the same price as this replicating
portfolio. Therefore,
3C 5 $18.18
or each call should sell at C 5 $6.06. Thus, given the stock price, exercise price, interest
rate, and volatility of the stock price (as represented by the spread between the up or down
movements), we can derive the fair value for the call option.
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This valuation approach relies heavily on the notion of replication. With only two possible end-of-year values of the stock, the payoffs to the levered stock portfolio replicate the
payoffs to three call options and, therefore, command the same market price. Replication is
behind most option-pricing formulas. For more complex price distributions for stocks, the
replication technique is correspondingly more complex, but the principles remain the same.
One way to view the role of replication is to note that, using the numbers assumed for
this example, a portfolio made up of one share of stock and three call options written is
perfectly hedged. Its year-end value is independent of the ultimate stock price:
Stock value
2 Obligations from 3 calls written
Net payoff
$90
20
$90
$120
230
$ 90
The investor has formed a riskless portfolio, with a payout of $90. Its value must be the
present value of $90, or $90/1.10 5 $81.82. The value of the portfolio, which equals $100
from the stock held long, minus 3C from the three calls written, should equal $81.82.
Hence $100 2 3C 5 $81.82, or C 5 $6.06.
The ability to create a perfect hedge is the key to this argument. The hedge locks in the
end-of-year payout, which therefore can be discounted using the risk-free interest rate. To
find the value of the option in terms of the value of the stock, we do not need to know either
the option’s or the stock’s beta or expected rate of return. The perfect hedging, or replication, approach enables us to express the value of the option in terms of the current value
of the stock without this information. With a hedged position, the final stock price does
not affect the investor’s payoff, so the stock’s risk and return parameters have no bearing.
The hedge ratio of this example is one share of stock to three calls, or one-third. This
ratio has an easy interpretation in this context: It is the ratio of the range of the values of the
option to those of the stock across the two possible outcomes. The stock, which originally
sells for S0 5 100, will be worth either d 3 $100 5 $90 or u 3 $100 5 $120, for a range of
$30. If the stock price increases, the call will be worth Cu 5 $10, whereas if the stock price
decreases, the call will be worth Cd 5 0, for a range of $10. The ratio of ranges, 10/30, is
one-third, which is the hedge ratio we have established.
The hedge ratio equals the ratio of ranges because the option and stock are perfectly
correlated in this two-state example. Because they are perfectly correlated, a perfect hedge
requires that they be held in a fraction determined only by relative volatility.
We can generalize the hedge ratio for other two-state option problems as
H5
Cu 2 Cd
uS0 2 dS0
where Cu or Cd refers to the call option’s value when the stock goes up or down, respectively, and uS0 and dS0 are the stock prices in the two states. The hedge ratio, H, is the
ratio of the swings in the possible end-of-period values of the option and the stock. If the
investor writes one option and holds H shares of stock, the value of the portfolio will be
unaffected by the stock price. In this case, option pricing is easy: Simply set the value of
the hedged portfolio equal to the present value of the known payoff.
Using our example, the option-pricing technique would proceed as follows:
1. Given the possible end-of-year stock prices, uS0 5 120 and dS0 5 90, and the
exercise price of 110, calculate that Cu 5 10 and Cd 5 0. The stock price range
is 30, while the option price range is 10.
2. Find that the hedge ratio of 10/30 5 1⁄3.
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CHAPTER 21
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731
3. Find that a portfolio made up of 1⁄3 share with one written option would have an
end-of-year value of $30 with certainty.
4. Show that the present value of $30 with a 1-year interest rate of 10% is $27.27.
5. Set the value of the hedged position to the present value of the certain payoff:
1
@3 S0 2 C0 5 $27.27
$33.33 2 C0 5 $27.27
6. Solve for the call’s value, C0 5 $6.06.
What if the option is overpriced, perhaps selling for $6.50? Then you can make arbitrage profits. Here is how:
Cash Flow in 1 Year for
Each Possible Stock Price
Initial Cash Flow
S1 5 90
S1 5 120
$ 19.50
2100
80.50
$ 0
90
288.55
$230
120
288.55
$ 1.45
$
1. Write 3 options
2. Purchase 1 share
3. Borrow $80.50 at 10% interest
Repay in 1 year
TOTAL
$ 0
1.45
Although the net initial investment is zero, the payoff in 1 year is positive and riskless.
If the option were underpriced, one would simply reverse this arbitrage strategy: Buy the
option, and sell the stock short to eliminate price risk. Note, by the way, that the present
value of the profit to the arbitrage strategy above exactly equals three times the amount by
which the option is overpriced. The present value of the risk-free profit of $1.45 at a 10%
interest rate is $1.318. With three options written in the strategy above, this translates to
a profit of $.44 per option, exactly the amount by which the option was overpriced: $6.50
versus the “fair value” of $6.06.
CONCEPT CHECK
21.3
Suppose the call option had been underpriced, selling at $5.50. Formulate the arbitrage strategy to exploit
the mispricing, and show that it provides a riskless cash flow in 1 year of $.6167 per option purchased.
Compare the present value of this cash flow to the option mispricing.
Generalizing the Two-State Approach
Although the two-state stock price model seems simplistic, we can generalize it to incorporate more realistic assumptions. To start, suppose we were to break up the year into
two 6-month segments, and then assert that over each half-year segment the stock price
could take on two values. In this example, we will say it can increase 10% (i.e., u 5 1.10)
or decrease 5% (i.e., d 5 .95). A stock initially selling at 100 could follow these possible
paths over the course of the year:
121
110
100
104.50
95
90.25
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PART VI
Options, Futures, and Other Derivatives
The midrange value of 104.50 can be attained by two paths: an increase of 10% followed
by a decrease of 5%, or a decrease of 5% followed by a 10% increase.
There are now three possible end-of-year values for the stock and three for the option:
Cu
C
Cd
Cuu
Cud 5 Cdu
Cdd
Using methods similar to those we followed above, we could value Cu from knowledge of Cuu and Cud, then value Cd from knowledge of Cdu and Cdd, and finally value C
from knowledge of Cu and Cd. And there is no reason to stop at 6-month intervals. We
could next break the year into four 3-month units, or twelve 1-month units, or 365 1-day
units, each of which would be posited to have a two-state process. Although the calculations become quite numerous and correspondingly tedious, they are easy to program into
a computer, and such computer programs are used widely by participants in the options
market.
Example 21.1
Binomial Option Pricing
Suppose that the risk-free interest rate is 5% per 6-month period and we wish to
value a call option with exercise price $110 on the stock described in the two-period
price tree just above. We start by finding the value of Cu. From this point, the call can
rise to an expiration-date value of Cuu 5 $11 (because at this point the stock price is
u 3 u 3 S0 5 $121) or fall to a final value of Cud 5 0 (because at this point, the stock
price is u 3 d 3 S0 5 $104.50, which is less than the $110 exercise price). Therefore the
hedge ratio at this point is
H5
Cuu 2 Cud
$11 2 0
2
5
5
uuS0 2 udS0
$121 2 104.50
3
Thus, the following portfolio will be worth $209 at option expiration regardless of the
ultimate stock price:
udS 5 $104.50
Buy 2 shares at price uS0 5 $110
Write 3 calls at price Cu
TOTAL
$209
0
$209
uuS0 5 $121
$242
233
$209
The portfolio must have a current market value equal to the present value of $209:
2 3 110 2 3Cu 5 $209/1.05 5 $199.047
Solve to find that Cu 5 $6.984.
Next we find the value of Cd. It is easy to see that this value must be zero. If we reach
this point (corresponding to a stock price of $95), the stock price at option expiration
will be either $104.50 or $90.25; in either case, the option will expire out of the money.
(More formally, we could note that with Cud 5 Cdd 5 0, the hedge ratio is zero, and a
portfolio of zero shares will replicate the payoff of the call!)
Finally, we solve for C using the values of Cu and Cd. Concept Check 4 leads you
through the calculations that show the option value to be $4.434.
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CHAPTER 21
CONCEPT CHECK
Option Valuation
733
21.4
Show that the initial value of the call option in Example 21.1 is $4.434.
a. Confirm that the spread in option values is Cu 2 Cd 5 $6.984.
b. Confirm that the spread in stock values is uS0 2 dS0 5 $15.
c. Confirm that the hedge ratio is .4656 shares purchased for each call written.
d. Demonstrate that the value in one period of a portfolio comprised of .4656 shares and one call
written is riskless.
e. Calculate the present value of this payoff.
f. Solve for the option value.
Making the Valuation Model Practical
As we break the year into progressively finer subintervals, the range of possible year-end stock
prices expands. For example, when we increase the number of subperiods to three, the number
of possible stock prices increases to four, as demonstrated in the following stock price tree:
u2S0
S0
uS0
dS0
udS0
2
d S0
u3S0
u2dS0
ud2S0
d3S0
Thus, by allowing for an ever-greater number of subperiods, we can overcome one of the
apparent limitations of the valuation model: that the number of possible end-of-period
stock prices is small.
Notice that extreme events such as u3S0 or d3S0 are relatively rare, as they require either
three consecutive increases or decreases in the three subintervals. More moderate, or midrange, results such as u2dS0 can be arrived at by more than one path—any combination of
two price increases and one decrease will result in stock price u2dS0. There are three of
these paths: uud, udu, duu. In contrast, only one path, uuu, results in a stock price of u3S0.
Thus midrange values are more likely. As we make the model more realistic and break
up the option maturity into more and more subperiods, the probability distribution for the
final stock price begins to resemble the familiar bell-shaped curve with highly unlikely
extreme outcomes and far more likely midrange outcomes. The probability of each outcome is given by the binomial probability distribution, and this multiperiod approach to
option pricing is therefore called the binomial model.
But we still need to answer an important practical question. Before the binomial
model can be used to value actual options, we need a way to choose reasonable values for u and d. The spread between up and down movements in the price of the stock
reflects the volatility of its rate of return, so the choice for u and d should depend on
that volatility. Call s your estimate of the standard deviation of the stock’s continuously
compounded annualized rate of return, and ∆t the length of each subperiod. To make
the standard deviation of the stock in the binomial model match your estimate of s, it
turns out that you can set u 5 exp(s"Dt) and d 5 exp(2s"Dt).3 You can see that the
3
Notice that d 5 1/u. This is the most common, but not the only, way to calibrate the model to empirical
volatility. For alternative methods, see Robert L. McDonald, Derivatives Markets, 3rd ed., Pearson/AddisonWesley, Boston: 2013, Ch. 10.
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PART VI
Options, Futures, and Other Derivatives
proportional difference between u and d increases with both annualized volatility as well
as the duration of the subperiod. This makes sense, as both higher s and longer holding
periods make future stock prices more uncertain. The following example illustrates how
to use this calibration.
Example 21.2
Calibrating u and d to Stock Volatility
Suppose you are using a 3-period model to value a 1-year option on a stock with
volatility (i.e., annualized standard deviation) of s 5 .30. With a time to expiration of
T 5 1 year, and three subperiods, you would calculate Dt 5 T/n 5 1/3, u 5 exp(s"Dt) 5
exp (.30"1/3) 5 1.189 and d 5 exp(2s"Dt) 5 exp(2.30"1/3) 5 .841. Given the
probability of an up movement, you could then work out the probability of any final
stock price. For example, suppose the probability that the stock price increases is .554
and the probability that it decreases is .446.4 Then the probability of stock prices at the
end of the year would be as follows:
Possible
Paths
Event
3 down
movements
2 down
and 1 up
1 down
and 2 up
3 up
movements
Probability
.4463 5 0.089
ddd
ddu, dud,
udd
uud, udu,
duu
Final Stock Price
59.48 5 100 3 .8413
3 3 .4462 3 .554 5 0.330
84.10 5 100 3 1.189 3 .8412
3 3 .446 3 .5542 5 0.411
118.89 5 100 3 1.1892 3 .841
.5543 5 0.170
uuu
168.09 5 100 3 1.1893
We plot this probability distribution in Figure 21.5, panel A. Notice that the two middle
end-of-period stock prices are, in fact, more likely than either extreme.
Now we can extend Example 21.2 by breaking up the option maturity into ever-shorter
subintervals. As we do, the stock price distribution becomes increasingly plausible, as we
demonstrate in Example 21.3.
Example 21.3
Increasing the Number of Subperiods
In Example 21.2, we broke up the year into three subperiods. Let’s now look at the cases
of six and 20 subperiods.
Subperiods, n
Dt 5 T/n
u 5 exp ( s"Dt)
d 5 exp (2s"Dt)
3
6
20
.333
.167
.015
exp(.173) 5 1.189
exp(.123) 5 1.130
exp(.067) 5 1.069
exp(2.173) 5 .841
exp(2.095) 5 .885
exp(2.067) 5 .935
4
Using this probability, the continuously compounded expected rate of return on the stock is .10. In general,
the formula relating the probability of an upward movement with the annual expected rate of return, r, is
exp(rDt) 2 d
p5
.
u2d
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CHAPTER 21
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We plot the resulting probability distributions in panels B and C of Figure 21.5.5
Notice that the right tail of the distribution in panel C is noticeably longer than
the left tail. In fact, as the number of intervals increases, the distribution progressively
approaches the skewed log-normal (rather than the symmetric normal) distribution. Even
if the stock price were to decline in each subinterval, it can never drop below zero. But
there is no corresponding upper bound on its potential performance. This asymmetry
gives rise to the skewness of the distribution.
.45
.40
.35
.30
.25
.20
.15
.10
.05
.00
A
B
.35
.30
Probability
Probability
Eventually, as we divide the option maturity into an ever-greater number of subintervals,
each node of the event tree would correspond to an infinitesimally small time interval. The
possible stock price movement within that time interval would be correspondingly small.
As those many intervals passed, the end-of-period stock price would more and more closely
.25
.20
.15
.10
.05
.00
25
50
75
100 125 150 175 200 225 250
Final Stock Price
50
75
100 125 150 175 200 225 250
Final Stock Price
C
.20
Probability
25
.15
.10
.05
.00
25
50
75 100 125 150 175 200 225 250
Final Stock Price
Figure 21.5 Probability distributions for final stock price. Possible outcomes and associated probabilities.
In each panel, the stock’s annualized, continuously compounded expected rate of return is 10% and its
standard deviation is 30%. Panel A. Three subintervals. In each subinterval, the stock can increase by 18.9%
or fall by 15.9%. Panel B. Six subintervals. In each subinterval, the stock can increase by 13.0% or fall by
11.5%. Panel C. Twenty subintervals. In each subinterval, the stock can increase by 6.9% or fall by 6.5%.
5
We adjust the probabilities of up versus down movements using the formula in footnote 4 to make the distributions in Figure 21.5 comparable. In each panel, p is chosen so that the stock’s expected annualized, continuously
compounded rate of return is 10%.
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A Risk-Neutral Shortcut
We pointed out earlier in the chapter that the binomial
model valuation approach is arbitrage-based. We can value
the option by replicating it with shares of stock plus borrowing. The ability to replicate the option means that its
price relative to the stock and the interest rate must be
based only on the technology of replication and not on risk
preferences. It cannot depend on risk aversion or the capital asset pricing model or any other model of equilibrium
risk-return relationships.
This insight—that the pricing model must be independent of risk aversion—leads to a very useful shortcut
to valuing options. Imagine a risk-neutral economy, that
is, an economy in which all investors are risk-neutral. This
hypothetical economy must value options the same as our
real one because risk aversion cannot affect the valuation
formula.
In a risk-neutral economy, investors would not
demand risk premiums and would therefore value all
assets by discounting expected payoffs at the risk-free
rate of interest. Therefore, a security such as a call option
would be valued by discounting its expected cash flow
“E”(CF)
at the risk-free rate: C 5
. We put the expectation
1 1 rf
operator E in quotation marks to signify that this is not
the true expectation, but the expectation that would
prevail in the hypothetical risk-neutral economy. To be
consistent, we must calculate this expected cash flow
using the rate of return the stock would have in the riskneutral economy, not using its true rate of return. But
if we successfully maintain consistency, the value derived
for the hypothetical economy should match the one in
our own.
How do we compute the expected cash flow from the
option in the risk-neutral economy? Because there are no
risk premiums, the stock’s expected rate of return must
equal the risk-free rate. Call p the probability that the
stock price increases. Then p must be chosen to equate the
expected rate of increase of the stock price to the risk-free
rate (we ignore dividends here):
“E”(S1) 5 p(uS) 1 (1 2 p)dS 5 (1 1 rf)S
This implies that p 5
1 1 rf 2 d
. We call p a risk-neutral
u2d
probability to distinguish it from the true, or “objective,”
probability. To illustrate, in our two-state example at the
beginning of Section 21.2, we had u 5 1.2, d 5 .9, and
1 1 .10 2 .9
2
5 .
rf 5 .10. Given these values, p 5
1.2 2 .9
3
Now let’s see what happens if we use the discounted
cash flow formula to value the option in the risk-neutral
economy. We continue to use the two-state example from
Section 21.2. We find the present value of the option payoff using the risk-neutral probability and discount at the
risk-free interest rate:
C5
“E”(CF)
1 1 rf
5
p Cu 1 (12p) Cd
1 1 rf
5
2/3 3 10 1 1/3 3 0
1.10
5 6.06
This answer exactly matches the value we found using our
no-arbitrage approach!
We repeat: This is not truly an expected discounted value.
•
The numerator is not the true expected cash flow from
the option because we use the risk-neutral probability,
p, rather than the true probability.
•
The denominator is not the proper discount rate for
option cash flows because we do not account for
the risk.
•
In a sense, these two “errors” cancel out. But this is
not just luck: We are assured to get the correct result
because the no-arbitrage approach implies that risk
preferences cannot affect the option value. Therefore,
the value computed for the risk-neutral economy must
equal the value that we obtain in our economy.
When we move to the more realistic multiperiod
model, the calculations are more cumbersome, but the
idea is the same. Footnote 4 shows how to relate p to any
expected rate of return and volatility estimate. Simply set
the expected rate of return on the stock equal to the riskfree rate, use the resulting probability to work out the
expected payoff from the option, discount at the risk-free
rate, and you will find the option value. These calculations
are actually fairly easy to program in Excel.
resemble a lognormal distribution.6 Thus the apparent oversimplification of the two-state
model can be overcome by progressively subdividing any period into many subperiods.
At any node, one still could set up a portfolio that would be perfectly hedged over the
next tiny time interval. Then, at the end of that interval, on reaching the next node, a new
hedge ratio could be computed and the portfolio composition could be revised to remain
6
Actually, more complex considerations enter here. The limit of this process is lognormal only if we assume
also that stock prices move continuously, by which we mean that over small time intervals only small price
movements can occur. This rules out rare events such as sudden, extreme price moves in response to dramatic
information (like a takeover attempt). For a treatment of this type of “jump process,” see John C. Cox and Stephen
A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics 3
(January–March 1976), pp. 145–66, or Robert C. Merton, “Option Pricing When Underlying Stock Returns Are
Discontinuous,” Journal of Financial Economics 3 (January–March 1976), pp. 125–44.
736
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CHAPTER 21
Option Valuation
737
hedged over the coming small interval. By continuously revising the hedge position, the
portfolio would remain hedged and would earn a riskless rate of return over each interval. This is called dynamic hedging, the continued updating of the hedge ratio as time
passes. As the dynamic hedge becomes ever finer, the resulting option-valuation procedure becomes more precise. The nearby box offers further refinements on the use of the
binomial model.
CONCEPT CHECK
21.5
In the table in Example 21.3, u and d both get closer to 1 (u is smaller and d is larger) as the time interval
∆t shrinks. Why does this make sense? Does the fact that u and d are each closer to 1 mean that the total
volatility of the stock over the remaining life of the option is lower?
21.4 Black-Scholes Option Valuation
While the binomial model is extremely flexible, a computer is needed for it to be useful in
actual trading. An option-pricing formula would be far easier to use than the tedious algorithm involved in the binomial model. It turns out that such a formula can be derived if one
is willing to make just two more assumptions: that both the risk-free interest rate and stock
price volatility are constant over the life of the option. In this case, as the time to expiration is divided into ever-more subperiods, the distribution of the stock price at expiration
progressively approaches the lognormal distribution, as suggested by Figure 21.5. When
the stock price distribution is actually lognormal, we can derive an exact option-pricing
formula.
The Black-Scholes Formula
Financial economists searched for years for a workable option-pricing model before
Black and Scholes7 and Merton8 derived a formula for the value of a call option. Scholes
and Merton shared the 1997 Nobel Prize in Economics for their accomplishment.9 Now
widely used by options market participants, the Black-Scholes pricing formula for a
call option is
C0 5 S0N(d1) 2 Xe2rTN(d2)
(21.1)
where
d1 5
ln(S0 /X) 1 (r 1 s2 /2)T
s"T
d2 5 d1 2 s"T
7
Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political
Economy 81 (May–June 1973).
8
Robert C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science 4
(Spring 1973).
9
Fischer Black died in 1995.
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738
PART VI
Options, Futures, and Other Derivatives
and
C0 5 Current call option value.
S0 5 Current stock price.
N(d) 5 The probability that a random draw from a standard normal distribution will
be less than d. This equals the area under the normal curve up to d, as in the
shaded area of Figure 21.6. In Excel, this function is called NORMSDIST( ).
X 5 Exercise price.
e 5 The base of the natural log function, approximately 2.71828. In Excel, ex can
be evaluated using the function EXP(x).
r 5 Risk-free interest rate (the annualized continuously compounded rate on a safe
asset with the same maturity as the expiration date of the option, which is to be
distinguished from rf, the discrete period interest rate).
T 5 Time to expiration of option, in years.
ln 5 Natural logarithm function. In Excel, ln(x) can be calculated as LN(x).
s 5 Standard deviation of the annualized continuously compounded rate of return
of the stock.
Notice a surprising feature of Equation 21.1: The option value does not depend on the
expected rate of return on the stock. In a sense, this information is already built into the
formula with the inclusion of the stock price, which itself depends on the stock’s risk
and return characteristics. This version of the Black-Scholes formula is predicated on the
assumption that the stock pays no dividends.
Although you may find the Black-Scholes formula intimidating, we can explain it at
a somewhat intuitive level. The trick is to view the N(d ) terms (loosely) as risk-adjusted
probabilities that the call option will expire in the money. First, look at Equation 21.1
assuming both N(d ) terms are close to 1.0, that is, when there is a very high probability
the option will be exercised. Then the call option value is equal to S0 2 Xe2rT, which
is what we called earlier the adjusted intrinsic value, S0 2 PV(X). This makes sense; if
exercise is certain, we have a claim on a stock with current value S0, and an obligation
with present value PV(X ), or, with continuous compounding, Xe2rT.
Now look at Equation 21.1 assuming the
N(d) = Shaded area
N(d ) terms are close to zero, meaning the
option almost certainly will not be exercised.
Then the equation confirms that the call is
worth nothing. For middle-range values of
N(d ) between 0 and 1, Equation 21.1 tells us
that the call value can be viewed as the present
value of the call’s potential payoff adjusting
for the probability of in-the-money expiration.
How do the N(d) terms serve as riskadjusted probabilities? This question quickly
leads us into advanced statistics. Notice,
d
however, that ln(S0/X), which appears in the
0
numerator of d1 and d2, is approximately the
percentage amount by which the option is currently in or out of the money. For example, if
Figure 21.6 A standard normal curve
S0 5 105 and X 5 100, the option is 5% in the
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CHAPTER 21
Option Valuation
739
money, and ln(105/100) 5 .049. Similarly, if S0 5 95, the option is 5% out of the money,
and ln(95/100) 5 2.051. The denominator, s"T, adjusts the amount by which the option
is in or out of the money for the volatility of the stock price over the remaining life of the
option. An option in the money by a given percent is more likely to stay in the money
if both stock price volatility and time to expiration are low. Therefore, N(d1) and N(d2)
increase with the probability that the option will expire in the money.
Example 21.4
Black-Scholes Valuation
You can use the Black-Scholes formula fairly easily. Suppose you want to value a call
option under the following circumstances:
Stock price:
S0 5 100
Exercise price:
X 5 95
Interest rate:
r 5 .10 (10% per year)
Time to expiration:
T 5 .25 (3 months or one-quarter of a year)
Standard deviation:
s 5 .50 (50% per year)
First calculate
d1 5
ln(100 / 95) 1 (.10 1 .52 / 2).25
.5√.25
5 .43
d2 5 .43 2 .5".25 5 .18
Next find N(d1) and N(d2). The values of the normal distribution are tabulated and may
be found in many statistics textbooks. A table of N(d ) is provided here as Table 21.2.
The normal distribution function, N(d ), is also provided in any spreadsheet program. In
Microsoft Excel, for example, the function name is NORMSDIST. Using either Excel or
Table 21.2 we find that
N(.43) 5 .6664
N(.18) 5 .5714
Thus the value of the call option is
C 5 100 3 .6664 2 95e2.103.25 3 .5714
5 66.64 2 52.94 5 $13.70
CONCEPT CHECK
21.6
Recalculate the value of the call option in Example 21.4 using a standard deviation of .6 instead of .5.
Confirm that the option is worth more using the higher stock-return volatility.
What if the option price in Example 21.4 were $15 rather than $13.70? Is the option
mispriced? Maybe, but before betting your fortune on that, you may want to reconsider
the valuation analysis. First, like all models, the Black-Scholes formula is based on some
simplifying abstractions that make the formula only approximately valid.
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740
PART VI
Options, Futures, and Other Derivatives
d
N(d )
d
N(d )
d
N(d )
d
N(d )
d
N(d )
d
N(d )
23.00
22.95
22.90
22.85
22.80
22.75
22.70
22.65
22.60
22.55
22.50
22.45
22.40
22.35
22.30
22.25
22.20
22.15
22.10
22.05
22.00
21.98
21.96
21.94
21.92
21.90
21.88
21.86
21.84
21.82
21.80
21.78
21.76
21.74
21.72
21.70
21.68
21.66
21.64
21.62
21.60
.0013
.0016
.0019
.0022
.0026
.0030
.0035
.0040
.0047
.0054
.0062
.0071
.0082
.0094
.0107
.0122
.0139
.0158
.0179
.0202
.0228
.0239
.0250
.0262
.0274
.0287
.0301
.0314
.0329
.0344
.0359
.0375
.0392
.0409
.0427
.0446
.0465
.0485
.0505
.0526
.0548
21.58
21.56
21.54
21.52
21.50
21.48
21.46
21.44
21.42
21.40
21.38
21.36
21.34
21.32
21.30
21.28
21.26
21.24
21.22
21.20
21.18
21.16
21.14
21.12
21.10
21.08
21.06
21.04
21.02
21.00
20.98
20.96
20.94
20.92
20.90
20.88
20.86
20.84
20.82
20.80
20.78
.0571
.0594
.0618
.0643
.0668
.0694
.0721
.0749
.0778
.0808
.0838
.0869
.0901
.0934
.0968
.1003
.1038
.1075
.1112
.1151
.1190
.1230
.1271
.1314
.1357
.1401
.1446
.1492
.1539
.1587
.1635
.1685
.1736
.1788
.1841
.1894
.1949
.2005
.2061
.2119
.2177
20.76
20.74
20.72
20.70
20.68
20.66
20.64
20.62
20.60
20.58
20.56
20.54
20.52
20.50
20.48
20.46
20.44
20.42
20.40
20.38
20.36
20.34
20.32
20.30
20.28
20.26
20.24
20.22
20.20
20.18
20.16
20.14
20.12
20.10
20.08
20.06
20.04
20.02
0.00
0.02
0.04
.2236
.2297
.2358
.2420
.2483
.2546
.2611
.2676
.2743
.2810
.2877
.2946
.3015
.3085
.3156
.3228
.3300
.3373
.3446
.3520
.3594
.3669
.3745
.3821
.3897
.3974
.4052
.4129
.4207
.4286
.4365
.4443
.4523
.4602
.4681
.4761
.4841
.4920
.5000
.5080
.5160
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0.78
0.80
0.82
0.84
.5239
.5319
.5398
.5478
.5557
.5636
.5714
.5793
.5871
.5948
.6026
.6103
.6179
.6255
.6331
.6406
.6480
.6554
.6628
.6700
.6773
.6844
.6915
.6985
.7054
.7123
.7191
.7258
.7324
.7389
.7454
.7518
.7580
.7642
.7704
.7764
.7823
.7882
.7939
.7996
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.30
1.32
1.34
1.36
1.38
1.40
1.42
1.44
1.46
1.48
1.50
1.52
1.54
1.56
1.58
1.60
1.62
1.64
.8051
.8106
.8159
.8212
.8264
.8315
.8365
.8414
.8461
.8508
.8554
.8599
.8643
.8686
.8729
.8770
.8810
.8849
.8888
.8925
.8962
.8997
.9032
.9066
.9099
.9131
.9162
.9192
.9222
.9251
.9279
.9306
.9332
.9357
.9382
.9406
.9429
.9452
.9474
.9495
1.66
1.68
1.70
1.72
1.74
1.76
1.78
1.80
1.82
1.84
1.86
1.88
1.90
1.92
1.94
1.96
1.98
2.00
2.05
2.10
2.15
2.20
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.05
.9515
.9535
.9554
.9573
.9591
.9608
.9625
.9641
.9656
.9671
.9686
.9699
.9713
.9726
.9738
.9750
.9761
.9772
.9798
.9821
.9842
.9861
.9878
.9893
.9906
.9918
.9929
.9938
.9946
.9953
.9960
.9965
.9970
.9974
.9978
.9981
.9984
.9986
.9989
Table 21.2
Cumulative normal distribution
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CHAPTER 21
Option Valuation
741
Some of the important assumptions underlying the formula are the following:
1. The stock will pay no dividends until after the option expiration date.
2. Both the interest rate, r, and variance rate, s2, of the stock are constant (or in
slightly more general versions of the formula, both are known functions of time—
any changes are perfectly predictable).
3. Stock prices are continuous, meaning that sudden extreme jumps such as those in
the aftermath of an announcement of a takeover attempt are ruled out.
Variants of the Black-Scholes formula have been developed to deal with many of these
limitations.
Second, even within the context of the Black-Scholes model, you must be sure of
the accuracy of the parameters used in the formula. Four of these—S0, X, T, and r—are
straightforward. The stock price, exercise price, and time to expiration are readily determined. The interest rate used is the money market rate for a maturity equal to that of the
option, and the dividend payout is reasonably predictable, at least over short horizons.
The last input, though, the standard deviation of the stock return, is not directly observable. It must be estimated from historical data, from scenario analysis, or from the prices
of other options, as we will describe momentarily.
We saw in Chapter 5 that the historical variance of stock market returns can be calculated from n observations as follows:
s2 5
n (r 2 r)2
n
t
a
n
n 2 1 t51
where r is the average return over the sample period. The rate of return on day t is defined
to be consistent with continuous compounding as rt 5 ln(St/St21). [We note again that the
natural logarithm of a ratio is approximately the percentage difference between the numerator and denominator so that ln(St/St21) is a measure of the rate of return of the stock from
time t 2 1 to time t.] Historical variance commonly is computed using daily returns over
periods of several months. Because the volatility of stock returns must be estimated, however, it is always possible that discrepancies between an option price and its Black-Scholes
value are simply artifacts of error in the estimation of the stock’s volatility.
In fact, market participants often give the option-valuation problem a different twist.
Rather than calculating a Black-Scholes option value for a given stock’s standard deviation, they ask instead: What standard deviation would be necessary for the option price that
I observe to be consistent with the Black-Scholes formula? This is called the implied volatility of the option, the volatility level for the stock implied by the option price.10 Investors
can then judge whether they think the actual stock standard deviation exceeds the implied
volatility. If it does, the option is considered a good buy; if actual volatility seems greater
than the implied volatility, its fair price would exceed the observed price.
Another variation is to compare two options on the same stock with equal expiration
dates but different exercise prices. The option with the higher implied volatility would be
considered relatively expensive, because a higher standard deviation is required to justify
its price. The analyst might consider buying the option with the lower implied volatility
and writing the option with the higher implied volatility.
The Black-Scholes valuation formula, as well as the implied volatility, is easily calculated using an Excel spreadsheet like Spreadsheet 21.1. The model inputs are provided in
10
This concept was introduced in Richard E. Schmalensee and Robert R. Trippi, “Common Stock Volatility
Expectations Implied by Option Premia,” Journal of Finance 33 (March 1978), pp. 129–47.
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742
PART VI
Options, Futures, and Other Derivatives
A
B
C
1
INPUTS
2
Standard deviation (annual)
3
Maturity (in years)
4
Risk-free rate (annual)
0.06
5
Stock price
6
Exercise price
7
Dividend yield (annual)
D
E
F
OUTPUTS
G
H
I
J
FORMULA FOR OUTPUT IN COLUMN E
0.2783
d1
0.0029
0.5
d2
–0.1939
E2–B2*SQRT(B3)
N(d1)
0.5012
NORMSDIST(E2)
100
N(d2)
0.4231
NORMSDIST(E3)
105
B/S call value
7.0000
B5*EXP(–B7*B3)*E4–B6*EXP(–B4*B3)*E5
0
B/S put value
8.8968
B6*EXP(–B4*B3)*(1–E5)–B5*EXP(–B7*B3)*(1–E4)
(LN(B5/B6)+(B4–B7+.5*B2^2)*B3)/(B2*SQRT(B3))
Spreadsheet 21.1
eXcel
Please visit us at
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Spreadsheet to calculate Black-Scholes call option values
column B, and the outputs are given in column E. The formulas for d1 and d2 are provided
in the spreadsheet, and the Excel formula NORMSDIST(d1) is used to calculate N(d1).11
Cell E6 contains the Black-Scholes formula. (The formula in the spreadsheet actually
includes an adjustment for dividends, as described in the next section.)
To compute an implied volatility, we can use the Goal Seek command from the What-If
Analysis menu (which can be found under the Data tab) in Excel. See Figure 21.7 for an
illustration. Goal Seek asks us to change the value of one cell to make the value of another
cell (called the target cell) equal to a specific value. For example, if we observe a call
option selling for $7 with other inputs as given in the spreadsheet, we can use Goal Seek
to change the value in cell B2 (the standard deviation of the stock) to set the option value
in cell E6 equal to $7. The target cell, E6, is the call price, and the spreadsheet manipulates
cell B2. When you click OK, the spreadsheet finds that a standard deviation equal to .2783
is consistent with a call price of $7; this would be the option’s implied volatility if it were
selling at $7.
A
C
B
D
E
OUTPUTS
F
G
H
I
J
1
INPUTS
2
Standard deviation (annual)
3
Maturity (in years)
4
Risk-free rate (annual)
0.06
N(d1)
0.5012
NORMSDIST(E2)
5
Stock price
100
N(d2)
0.4231
NORMSDIST(E3)
6
Exercise price
105
B/S call value
7.0000
B5*EXP(2B7*B3)*E42B6*EXP(2B4*B3)*E5
7
Dividend yield (annual)
0
B/S put value
8.8968
B6*EXP(2B4*B3)*(12E5) 2 B5*EXP(2B7*B3)*(12E4)
K
FORMULA FOR OUTPUT IN COLUMN E
0.2783
d1
0.0029
0.5
d2
20.1939
(LN(B5/B6)1(B42B71.5*B2^2)*B3)/(B2*SQRT(B3))
E22B2*SQRT(B3)
8
9
10
11
12
13
14
15
16
17
Figure 21.7 Using Goal Seek to find implied volatility
eXcel
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11
In some versions of Excel, the function is NORM.S.DIST(d,TRUE).
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CHAPTER 21
Option Valuation
743
Jan-90
Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Jan-99
Jan-00
Jan-01
Jan-02
Jan-03
Jan-04
Jan-05
Jan-06
Jan-07
Jan-08
Jan-09
Jan-10
Jan-11
Jan-12
Jan-13
Implied Volatility (%)
The Chicago Board Options Exchange
70
regularly computes the implied volatilSubprime and
Credit Crises
60
ity of major stock indexes. Figure 21.8 is
U.S. Debt
a graph of the implied (30-day) volatility
50
LTCM
Downgrade
Iraq
of the S&P 500 since 1990. During peri40
9/11
Gulf War
ods of turmoil, implied volatility can spike
30
quickly. Notice the peaks in January 1991
20
(Gulf War), August 1998 (collapse of Long10
Term Capital Management), September 11,
0
2001, 2002 (build-up to invasion of Iraq),
and, most dramatically, during the credit crisis of 2008. Because implied volatility correlates with crisis, it is sometimes called an
“investor fear gauge.”
Figure 21.8 Implied volatility of the S&P 500 (VIX index)
A futures contract on the 30-day implied
Source: Chicago Board Options Exchange, www.cboe.com.
volatility of the S&P 500 has traded on the
CBOE Futures Exchange since 2004. The
payoff of the contract depends on market
implied volatility at the expiration of the contract. The ticker symbol of the contract is VIX.
As the nearby box makes clear, observers use it to infer the market’s assessment of possible
stock price swings in coming months. In this case, the article questioned the relatively low level
of the VIX in light of tense political negotiations at the end of 2012 over the so-called fiscal
cliff. The question was whether the price of the VIX contract indicated that investors were being
too complacent about the potential for market disruption if those negotiations were to fail.
Figure 21.8 reveals an awkward empirical fact. While the Black-Scholes formula is
derived assuming that stock volatility is constant, the time series of implied volatilities
derived from that formula is in fact far from constant. This contradiction reminds us that
the Black-Scholes model (like all models) is a simplification that does not capture all
aspects of real markets. In this particular context, extensions of the pricing model that
allow stock volatility to evolve randomly over time would be desirable, and, in fact, many
extensions of the model along these lines have been developed.12
The fact that volatility changes unpredictably means that it can be difficult to choose
the proper volatility input to use in any option-pricing model. A considerable amount of
recent research has been devoted to techniques to predict changes in volatility. These techniques, known as ARCH and stochastic volatility models, posit that changes in volatility
are partially predictable and that by analyzing recent levels and trends in volatility, one can
improve predictions of future volatility.13
CONCEPT CHECK
21.7
Suppose the call option in Spreadsheet 21.1 actually is selling for $8. Is its implied volatility more or less
than 27.83%? Use the spreadsheet (available at the Online Learning Center) and Goal Seek to find its
implied volatility at this price.
12
Influential articles on this topic are J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” Journal of Finance (June 1987), pp. 281–300; J. Wiggins, “Option Values under Stochastic Volatility,” Journal of Financial Economics (December 1987), pp. 351–72; and S. Heston, “A Closed-Form Solution for Options
with Stochastic Volatility with Applications to Bonds and Currency Options,” Review of Financial Studies 6 (1993),
pp. 327–43. For a more recent review, see E. Ghysels, A. Harvey, and E. Renault, “Stochastic Volatility,” in Handbook of Statistics, Vol. 14: Statistical Methods in Finance, ed. G. S. Maddala (Amsterdam: North Holland, 1996).
13
For an introduction to these models see C. Alexander, Market Models (Chichester, England: Wiley, 2001).
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“Fear” Gauge Showing Little of It
On the floor of the Chicago Board Options Exchange
(CBOE), those who trade in fear have seen little but calm.
Measures of volatility in U.S. markets are pointing to
relative calm, but some investors say the low readings are
a sign of complacency. Unlike the rocky ride in the stock
market since the U.S. presidential election, the CBOE’s
Volatility Index, or VIX, has registered tranquility. For four
months, the so-called fear gauge of financial markets has
traded below its two-decade historical average of 20, its
longest such streak in more than 5 years.
Some investors worry that the low readings are a sign
of complacency, and that the potential for further declines
in response to unexpected bad news isn’t reflected in stock
prices. These investors say various barometers of skittishness could tick higher as the year-end deadline nears for an
agreement on taxes and spending. That anxiety also would
likely be reflected in increased volatility in the stock and
commodities markets.
The VIX is an index calculated from the prices investors
are willing to pay for options tied to the Standard & Poor’s
500-stock index. As investors become nervous, they are willing to pay more for options, driving up the value of the VIX.
As market watchers search for clues about whether the
relative calm can last, some of them are looking back to
the early summer of 2011. Back then, the VIX was trading
at levels close to today’s, even though market pundits worried that lawmakers wouldn’t agree to raise the debt ceiling. That scenario could have led the U.S. government to
default on its debt.
Within a few weeks, as the debt-ceiling wrangling was
going down to the wire, Standard & Poor’s cut the U.S.’s
long-term triple-A credit rating. The VIX nearly tripled to
48 in a span of two weeks.
“The whole situation leads me to believe that this lack of
a cushion in the market will lead to wilder shocks” to markets if negative events happen, says Michael Palmer of Group
One Trading who trades the VIX on the floor of the CBOE.
Source: Steven Russolillo and Kaitlyn Kiernan, The Wall Street Journal,
November 26, 2012. Reprinted with permission. © 2012 Dow Jones
& Company, Inc. All Rights Reserved Worldwide.
Dividends and Call Option Valuation
We noted earlier that the Black-Scholes call option formula applies to stocks that do not
pay dividends. When dividends are to be paid before the option expires, we need to adjust
the formula. The payment of dividends raises the possibility of early exercise, and for most
realistic dividend payout schemes the valuation formula becomes significantly more complex than the Black-Scholes equation.
We can apply some simple rules of thumb to approximate the option value, however.
One popular approach, originally suggested by Black, calls for adjusting the stock price
downward by the present value of any dividends that are to be paid before option expiration.14 Therefore, we would simply replace S0 with S0 2 PV(dividends) in the BlackScholes formula. Such an adjustment will take dividends into account by reflecting their
eventual impact on the stock price. The option value then may be computed as before,
assuming that the option will be held to expiration.
In one special case, the dividend adjustment takes a simple form. Suppose the underlying
asset pays a continuous flow of income. This might be a reasonable assumption for options
on a stock index, where different stocks in the index pay dividends on different days, so that
dividend income arrives in a more or less continuous flow. If the dividend yield, denoted d, is
constant, one can show that the present value of that dividend flow accruing until the option
expiration date is S0 (1 2 e2dT). (For intuition, notice that e2dT approximately equals 1 2 dT,
so the value of the dividend is approximately dTS0.) In this case, S0 2 PV(Div) 5 S0 e2dT, and
we can derive a Black-Scholes call option formula on the dividend-paying asset simply by
substituting S0 e2dT for S0 in the original formula. This approach is used in Spreadsheet 21.1.
These procedures yield a very good approximation of option value for European call
options that must be held until expiration, but they do not allow for the fact that the holder
of an American call option might choose to exercise the option just before a dividend. The
current value of a call option, assuming that it will be exercised just before the ex-dividend
date, might be greater than the value of the option assuming it will be held until expiration.
Although holding the option until expiration allows greater effective time to expiration,
14
Fischer Black, “Fact and Fantasy in the Use of Options,” Financial Analysts Journal 31 (July–August 1975).
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CHAPTER 21
Option Valuation
745
which increases the option value, it also entails more dividend payments, lowering the
expected stock price at expiration and thereby lowering the current option value.
For example, suppose that a stock selling at $20 will pay a $1 dividend in 4 months,
whereas the call option on the stock does not expire for 6 months. The effective annual
interest rate is 10%, so that the present value of the dividend is $1/(1.10)1/3 5 $0.97. Black
suggests that we can compute the option value in one of two ways:
1. Apply the Black-Scholes formula assuming early exercise, thus using the actual stock
price of $20 and a time to expiration of 4 months (the time until the dividend payment).
2. Apply the Black-Scholes formula assuming no early exercise, using the dividendadjusted stock price of $20 2 $.97 5 $19.03 and a time to expiration of 6 months.
The greater of the two values is the estimate of the option value, recognizing that early
exercise might be optimal. In other words, the so-called pseudo-American call option
value is the maximum of the value derived by assuming that the option will be held until
expiration and the value derived by assuming that the option will be exercised just before
an ex-dividend date. Even this technique is not exact, however, for it assumes that the
option holder makes an irrevocable decision now on when to exercise, when in fact the
decision is not binding until exercise notice is given.15
Put Option Valuation
We have concentrated so far on call option valuation. We can derive Black-Scholes
European put option values from call option values using the put-call parity theorem.
To value the put option, we simply calculate the value of the corresponding call option in
Equation 21.1 from the Black-Scholes formula, and solve for the put option value as
P 5 C 1 PV(X ) 2 S0
5 C 1 Xe2rT 2 S0
(21.2)
We calculate the present value of the exercise price using continuous compounding to be
consistent with the Black-Scholes formula.
Sometimes, it is easier to work with a put option valuation formula directly. If we substitute the Black-Scholes formula for a call in Equation 21.2, we obtain the value of a
European put option as
P 5 Xe2rT 3 1 2 N(d2) 4 2 S0 3 1 2 N(d1) 4
Example 21.5
(21.3)
Black-Scholes Put Valuation
Using data from Example 21.4 (C 5 $13.70, X 5 $95, S 5 $100, r 5 .10, s 5 .50, and
T 5 .25), Equation 21.3 implies that a European put option on that stock with identical
exercise price and time to expiration is worth
$95e2.10 3 .25(1 2 .5714) 2 $100(1 2 .6664) 5 $6.35
15
An exact formula for American call valuation on dividend-paying stocks has been developed in Richard Roll,
“An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends,”
Journal of Financial Economics 5 (November 1977). The technique has been discussed and revised in
Robert Geske, “A Note on an Analytical Formula for Unprotected American Call Options on Stocks with Known
Dividends,” Journal of Financial Economics 7 (December 1979), and Robert E. Whaley, “On the Valuation of
American Call Options on Stocks with Known Dividends,” Journal of Financial Economics 9 (June 1981).
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746
PART VI
Options, Futures, and Other Derivatives
Notice that this value is consistent with put-call parity:
P 5 C 1 PV(X ) 2 S0 5 13.70 1 95e2.10 3 .25 2 100 5 6.35
As we noted traders can do, we might then compare this formula value to the actual put
price as one step in formulating a trading strategy.
Dividends and Put Option Valuation
Equation 21.2 or 21.3 is valid for European puts on non-dividend-paying stocks. As we did
for call options, if the underlying asset pays a dividend, we can find European put values
by substituting S0 2 PV(Div) for S0. Cell E7 in Spreadsheet 21.1 allows for a continuous
dividend flow with a dividend yield of d. In that case S0 2 PV(Div) 5 S0e2dT.
However, listed put options on stocks are American options that offer the opportunity of
early exercise, and we have seen that the right to exercise puts early can turn out to be valuable. This means that an American put option must be worth more than the corresponding
European option. Therefore, Equation 21.2 or 21.3 describes only the lower bound on the
true value of the American put. However, in many applications the approximation is very
accurate.16
21.5 Using the Black-Scholes Formula
Hedge Ratios and the Black-Scholes Formula
In the last chapter, we considered two investments in FinCorp stock: 100 shares or 1,000
call options. We saw that the call option position was more sensitive to swings in the stock
price than was the all-stock position. To analyze the overall exposure to a stock price more
precisely, however, it is necessary to quantify these relative sensitivities. We can summarize the overall exposure of portfolios of options with various exercise prices and times to
expiration using the hedge ratio, the change in option price for a $1 increase in the stock
price. A call option, therefore, has a positive hedge ratio and a put option a negative hedge
ratio. The hedge ratio is commonly called the option’s delta.
If you were to graph the option value as a function of the stock value, as we have done
for a call option in Figure 21.9, the hedge ratio is simply the slope of the curve evaluated
at the current stock price. For example, suppose the slope of the curve at S0 5 $120 equals .60.
As the stock increases in value by $1, the option increases by approximately $.60, as the
figure shows.
For every call option written, .60 share of stock would be needed to hedge the investor’s
portfolio. If one writes 10 options and holds six shares of stock, according to the hedge ratio
of .6, a $1 increase in stock price will result in a gain of $6 on the stock holdings, whereas
the loss on the 10 options written will be 10 3 $.60, an equivalent $6. The stock price
movement leaves total wealth unaltered, which is what a hedged position is intended to do.
Black-Scholes hedge ratios are particularly easy to compute. The hedge ratio for a call is
N(d1), whereas the hedge ratio for a put is N(d1) 2 1. We defined N(d1) as part of the BlackScholes formula in Equation 21.1. Recall that N(d) stands for the area under the standard
normal curve up to d. Therefore, the call option hedge ratio must be positive and less than
1.0, whereas the put option hedge ratio is negative and of smaller absolute value than 1.0.
16
For a more complete treatment of American put valuation, see R. Geske and H. E. Johnson, “The American Put
Valued Analytically,” Journal of Finance 39 (December 1984), pp. 1511–24.
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CHAPTER 21
Option Valuation
Figure 21.9 verifies that the slope of the
call option valuation function is less than
Value of a Call (C)
1.0, approaching 1.0 only as the stock price
becomes much greater than the exercise price.
This tells us that option values change less
40
than one-for-one with changes in stock prices.
Why should this be? Suppose an option is so
far in the money that you are absolutely certain it will be exercised. In that case, every
20
dollar increase in the stock price would
Slope 5 .6
increase the option value by $1. But if there is
a reasonable chance the call option will expire
out of the money, even after a moderate stock
0
price gain, a $1 increase in the stock price will
not necessarily increase the ultimate payoff
120
to the call; therefore, the call price will not
respond by a full dollar.
The fact that hedge ratios are less than 1.0
Figure 21.9 Call option value and hedge ratio
does not contradict our earlier observation
that options offer leverage and disproportionate sensitivity to stock price movements. Although dollar movements in option prices
are less than dollar movements in the stock price, the rate of return volatility of options
remains greater than stock return volatility because options sell at lower prices. In our
example, with the stock selling at $120, and a hedge ratio of .6, an option with exercise
price $120 may sell for $5. If the stock price increases to $121, the call price would be
expected to increase by only $.60 to $5.60. The percentage increase in the option value is
$.60/$5.00 5 12%, however, whereas the stock price increase is only $1/$120 5 .83%. The
ratio of the percentage changes is 12%/.83% 5 14.4. For every 1% increase in the stock
price, the option price increases by 14.4%. This ratio, the percentage change in option
price per percentage change in stock price, is called the option elasticity.
The hedge ratio is an essential tool in portfolio management and control. An example
will show why.
Example 21.6
747
S0
Hedge Ratios
Consider two portfolios, one holding 750 FinCorp calls and 200 shares of FinCorp and
the other holding 800 shares of FinCorp. Which portfolio has greater dollar exposure to
FinCorp price movements? You can answer this question easily by using the hedge ratio.
Each option changes in value by H dollars for each dollar change in stock price,
where H stands for the hedge ratio. Thus, if H equals .6, the 750 options are equivalent
to .6 3 750 5 450 shares in terms of the response of their market value to FinCorp
stock price movements. The first portfolio has less dollar sensitivity to stock price change
because the 450 share-equivalents of the options plus the 200 shares actually held are
less than the 800 shares held in the second portfolio.
This is not to say, however, that the first portfolio is less sensitive to the stock’s rate of
return. As we noted in discussing option elasticities, the first portfolio may have lower
total value than the second, so despite its lower sensitivity in terms of total market
value, it might have greater rate of return sensitivity. Because a call option has a lower
market value than the stock, its price changes more than proportionally with stock price
changes, even though its hedge ratio is less than 1.0.
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eXcel APPLICATIONS: Black-Scholes Option
Valuation
T
analysis of calls while the second workbook presents similar analysis for puts. You can find these spreadsheets at the
Online Learning Center at www.mhhe.com/bkm.
he spreadsheet below can be used to determine option
values using the Black-Scholes model. The inputs are
the stock price, standard deviation, expiration of the
option, exercise price, risk-free rate, and dividend yield.
The call option is valued using Equation 21.1 and the put
is valued using Equation 21.3. For both calls and puts, the
dividend-adjusted Black-Scholes formula substitutes Se2dT
for S, as outlined on page 744. The model also calculates
the intrinsic and time value for both puts and calls.
Further, the model presents sensitivity analysis using
the one-way data table. The first workbook presents the
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
A
B
Chapter 21- Black-Scholes Option Pricing
Call Valuation & Call Time Premiums
Standard deviation (σ)
Variance (annual, σ2)
Time to expiration (years, T)
Risk-free rate (annual, r)
Current stock price (S0)
Exercise price (X)
Dividend yield (annual, δ)
d1
d2
N(d1)
N(d2)
Black-Scholes call value
Black-Scholes put value
0.27830
0.07745
0.50
6.00%
$100.00
$105.00
0.00%
0.0029095
−0.193878
0.50116
0.42314
$6.99992
$8.89670
Intrinsic value of call
Time value of call
$0.00000
6.99992
Intrinsic value of put
Time value of put
$5.00000
3.89670
CONCEPT CHECK
21.8
C
D
E
Call
Standard Option
Deviation Value
7.000
0.15
3.388
0.18
4.089
0.20
4.792
0.23
5.497
0.25
6.202
0.28
6.907
0.30
7.612
0.33
8.317
0.35
9.022
0.38
9.726
0.40 10.429
0.43 11.132
0.45 11.834
0.48 12.536
0.50 13.236
Excel Questions
1. Find the value of the call and put options using the parameters given in this box but changing the standard deviation
to .25. What happens to the value of each option?
2. What is implied volatility if the call option is selling for $9?
F
H
G
LEGEND:
Enter data
Value calculated
See comment
Standard
Deviation
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.400
0.425
0.450
0.475
0.500
Call
Time
Value
7.000
3.388
4.089
4.792
5.497
6.202
6.907
7.612
8.317
9.022
9.726
10.429
11.132
11.834
12.536
13.236
I
J
Stock
Price
$60
$65
$70
$75
$80
$85
$90
$95
$100
$105
$110
$115
$120
$125
$130
$135.00
K
Call
Option
Value
7.000
0.017
0.061
0.179
0.440
0.935
1.763
3.014
4.750
7.000
9.754
12.974
16.602
20.572
24.817
29.275
33.893
L
M
Stock
Price
$60
$65
$70
$75
$80
$85
$90
$95
$100
$105
$110
$115
$120
$125
$130
$135
N
Call
Time
Value
7.000
0.017
0.061
0.179
0.440
0.935
1.763
3.014
4.750
7.000
9.754
7.974
6.602
5.572
4.817
4.275
3.893
Portfolio Insurance
In Chapter 20, we showed that protective put strategies
offer a sort of insurance policy on an asset. The
What is the elasticity of a put option currently
protective
put has proven to be extremely popular with
selling for $4 with exercise price $120 and hedge
investors. Even if the asset price falls, the put conveys
ratio 2.4 if the stock price is currently $122?
the right to sell the asset for the exercise price, which
is a way to lock in a minimum portfolio value. With an
at-the-money put (X 5 S0), the maximum loss that can be realized is the cost of the put.
The asset can be sold for X, which equals its original value, so even if the asset price
falls, the investor’s net loss over the period is just the cost of the put. If the asset value
increases, however, upside potential is unlimited. Figure 21.10 graphs the profit or loss
on a protective put position as a function of the change in the value of the underlying
asset, P.
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CHAPTER 21
Option Valuation
749
While the protective put is
a simple and convenient way
Change in Value
of Protected Position
to achieve portfolio insurance,
that is, to limit the worst-case
portfolio rate of return, there are
practical difficulties in trying
to insure a portfolio of stocks.
First, unless the investor’s portfolio corresponds to a standard
0
Change in Value
market index for which puts are
0
of Underlying Asset
traded, a put option on the portCost of Put
folio will not be available for
2P
purchase. And if index puts are
used to protect a non-indexed
portfolio, tracking error can
result. For example, if the portfolio falls in value while the
market index rises, the put will
Figure 21.10 Profit on a protective put strategy
fail to provide the intended protection. Moreover, the maturities of traded options may not match the investor’s horizon. Therefore, rather than using
option strategies, investors may use trading strategies that mimic the payoff to a protective
put option.
Here is the general idea. Even if a put option on the desired portfolio does not exist, a
theoretical option-pricing model (such as the Black-Scholes model) can be used to determine how that option’s price would respond to the portfolio’s value if it did trade. For
example, if stock prices were to fall, the put option would increase in value. The option
model could quantify this relationship. The net exposure of the (hypothetical) protective
put portfolio to swings in stock prices is the sum of the exposures of the two components
of the portfolio, the stock and the put. The net exposure of the portfolio equals the equity
exposure less the (offsetting) put option exposure.
We can create “synthetic” protective put positions by holding a quantity of stocks with
the same net exposure to market swings as the hypothetical protective put position. The
key to this strategy is the option delta, or hedge ratio, that is, the change in the price of the
protective put option per change in the value of the underlying stock portfolio.
Example 21.7
Synthetic Protective Put Options
Suppose a portfolio is currently valued at $100 million. An at-the-money put option
on the portfolio might have a hedge ratio or delta of 2.6, meaning the option’s value
swings $.60 for every dollar change in portfolio value, but in an opposite direction. Suppose the stock portfolio falls in value by 2%. The profit on a hypothetical protective put
position (if the put existed) would be as follows (in millions of dollars):
Loss on stocks:
Gain on put:
Net loss
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2% of $100 5 $2.00
.6 3 $2.00 5 1.20
5 $ .80
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750
PART VI
Options, Futures, and Other Derivatives
We create the synthetic option position by selling a proportion of shares equal to the
put option’s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free
T-bills. The rationale is that the hypothetical put option would have offset 60% of any
change in the stock portfolio’s value, so one must reduce portfolio risk directly by selling 60% of the equity and putting the proceeds into a risk-free asset. Total return on a
synthetic protective put position with $60 million in risk-free investments such as T-bills
and $40 million in equity is
Loss on stocks:
1 Loss on bills:
Net loss
2% of $40 5 $.80
5
0
5 $.80
The synthetic and actual protective put positions have equal returns. We conclude
that if you sell a proportion of shares equal to the put option’s delta and place the
proceeds in cash equivalents, your exposure to the stock market will equal that of the
desired protective put position.
The challenge with this procedure is that deltas constantly change. Figure 21.11
shows that as the stock price falls, the magnitude of the appropriate hedge ratio increases.
Therefore, market declines require extra hedging, that is, additional conversion of equity
into cash. This constant updating of the hedge ratio is called dynamic hedging (alternatively, delta hedging).
Dynamic hedging is one reason portfolio insurance has been said to contribute to market volatility. Market declines trigger additional sales of stock as portfolio insurers strive to
increase their hedging. These additional sales
are seen as reinforcing or exaggerating marValue of a Put (P)
ket downturns.
In practice, portfolio insurers often do
Higher Slope 5
not actually buy or sell stocks directly when
High Hedge Ratio
they update their hedge positions. Instead,
they minimize trading costs by buying or
selling stock index futures as a substitute
for sale of the stocks themselves. As you
will see in the next chapter, stock prices and
index futures prices usually are very tightly
linked by cross-market arbitrageurs so that
futures transactions can be used as reliable
Low Slope 5
proxies for stock transactions. Instead of
Low Hedge Ratio
selling equities based on the put option’s
S0
delta, insurers will sell an equivalent number
0
of futures contracts.17
Several portfolio insurers suffered great setbacks during the market crash of October 19,
Figure 21.11 Hedge ratios change as the stock price
1987, when the market suffered an unprecefluctuates
dented 1-day loss of about 20%. A description
17
Notice, however, that the use of index futures reintroduces the problem of tracking error between the portfolio
and the market index.
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CHAPTER 21
Option Valuation
751
of what happened then should let you appreciate the complexities of applying a seemingly
straightforward hedging concept.
1. Market volatility at the crash was much greater than ever encountered before. Put
option deltas based on historical experience were too low; insurers underhedged,
held too much equity, and suffered excessive losses.
2. Prices moved so fast that insurers could not keep up with the necessary rebalancing.
They were “chasing deltas” that kept getting away from them. The futures market
also saw a “gap” opening, where the opening price was nearly 10% below the previous day’s close. The price dropped before insurers could update their hedge ratios.
3. Execution problems were severe. First, current market prices were unavailable, with
trade execution and the price quotation system hours behind, which made computation of correct hedge ratios impossible. Moreover, trading in stocks and stock futures
ceased during some periods. The continuous rebalancing capability that is essential
for a viable insurance program vanished during the precipitous market collapse.
4. Futures prices traded at steep discounts to their proper levels compared to reported
stock prices, thereby making the sale of futures (as a proxy for equity sales) seem
expensive. Although you will see in the next chapter that stock index futures prices
normally exceed the value of the stock index, Figure 21.12 shows that on October 19,
futures sold far below the stock index level. When some insurers gambled that the
futures price would recover to its usual premium over the stock index, and chose to
defer sales, they remained underhedged. As the market fell farther, their portfolios
experienced substantial losses.
Although most observers at the time believed that the portfolio insurance industry
would never recover from the market crash, delta hedging is still alive and well on Wall
Street. Dynamic hedges are widely used by large firms to hedge potential losses from
options positions. For example, the nearby box notes that when Microsoft ended its
employee stock option program and J. P. Morgan purchased many already-issued options
10
0
210
220
230
240
10
11
12
October 19
1
2
3
4 10
11
12
October 20
1
2
3
4
Figure 21.12 S&P 500 cash-to-futures spread in points at 15-minute intervals
Note: Trading in futures contracts halted between 12:15 and 1:05.
Source: The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1987 Dow Jones &
Company, Inc. All rights reserved worldwide.
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J. P. Morgan Rolls Dice on Microsoft Options
Microsoft, in a shift that could be copied throughout the
technology business, said yesterday that it plans to stop
issuing stock options to its employees, and instead will provide them with restricted stock.
Though details of the plan still aren’t clear, J. P. Morgan
effectively plans to buy the options from Microsoft employees who opt for restricted stock instead. Employee stock
options are granted as a form of compensation and allow
employees the right to exchange the options for shares of
company stock.
The price offered to employees for the options presumably will be lower than the current value, giving
J. P. Morgan a chance to make a profit on the deal. Rather
than holding the options, and thus betting Microsoft’s
stock will rise, people familiar with the bank’s strategy
say J. P. Morgan probably will match each option it buys
from the company’s employees with a separate trade in the
stock market that both hedges the bet and gives itself a
margin of profit.
For Wall Street’s so-called rocket scientists who do complicated financial transactions such as this one, the strategy behind
J. P. Morgan’s deal with Microsoft isn’t particularly unique or
sophisticated. They add that the bank has several ways to deal
with the millions of Microsoft options that could come its way.
The bank, for instance, could hedge the options by shorting, or betting against, Microsoft stock. Microsoft has the
largest market capitalization of any stock in the market, and
its shares are among the most liquid, meaning it would be
easy to hedge the risk of holding those options. J. P. Morgan
also could sell the options to investors, much as they would
do with a syndicated loan, thereby spreading the risk.
Source: Jathon Sapsford and Ken Brown, The Wall Street Journal,
July 9, 2003. Reprinted with permission. © 2003 Dow Jones & Company, Inc. All rights reserved.
of Microsoft employees, it was widely expected that Morgan would protect its options
position by selling shares in Microsoft in accord with a delta hedging strategy.
Option Pricing and the Crisis of 2008–2009
Merton18 shows how option pricing models can provide insight into the financial crisis of
2008–2009. The key to understanding his argument is to remember that when banks lend
to or buy the debt of firms with limited liability, they implicitly write a put option to the
borrower (see Chapter 20, Section 20.5). If the borrower has sufficient assets to pay off
the loan when it comes due, it will do so, and the lender will be fully repaid. But if the
borrower has insufficient assets, it can declare bankruptcy and discharge its obligations by
transferring ownership of the firm to its creditors. The borrower’s ability to satisfy the loan
by transferring ownership is equivalent to the right to “sell” itself to the creditor for the
face value of the loan. This arrangement is therefore just like a put option on the firm with
exercise price equal to the stipulated loan repayment.
Consider the payoff to the lender at loan maturity (time T) as a function of the value of
the borrowing firm, VT, when the loan, with face value L, comes due. If VT $ L, the lender
is paid off in full. But if VT , L, the lender gets the firm, which is worth less than the
promised payment L.
We can write the payoff in a way that emphasizes the implicit put option:
Payoff 5 b
L
0
5L2 b
VT
L 2 VT
if VT $ L
if VT , L
(21.4)
Equation 21.4 shows that the payoff on the loan equals L (when the firm has sufficient
assets to pay off the debt), minus the payoff of a put option on the value of the firm (VT)
with an exercise price of L. Therefore, we may view risky lending as a combination of safe
lending, with a guaranteed payoff of L, combined with a short position in a put option on
the borrower.
18
This material is based on a lecture given by Robert Merton at MIT in March 2009. You can find the lecture
online at http://mitworld.mit.edu/video/659.
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753
Value of Put as % of Face
Value of Debt
When firms sell credit default swaps
50
(see Chapter 14, Section 14.5), the
implicit put option is even clearer. Here,
40
the CDS seller agrees to make up any
losses due to the insolvency of a bond
30
issuer. If the issuer goes bankrupt, leav20
ing assets of only VT for the creditors, the
CDS seller is obligated to make up the
10
difference, L 2 VT. This is in essence a
pure put option.
0
50
75
100
125
150
175
200
225
250
Now think about the exposure of these
210
implicit put writers to changes in the
Firm Value as Percent of Face Value of Debt
financial health of the underlying firm.
The value of a put option on VT appears
in Figure 21.13. When the firm is finanFigure 21.13 Value of implicit put option on a loan guarantee as
cially strong (i.e., V is far greater than L),
a percentage of the face value of debt (Debt maturity 5 1 year;
the slope of the curve is nearly zero,
Standard deviation of value of firm 5 40%; Risk free rate 5 6%)
implying that there is little exposure of the
implicit put writer (either the bank or the
CDS writer) to the value of the borrowing firm. For example, when firm value is 1.75 times
the value of the debt, the dashed line drawn tangent to the put value curve has a slope of only
2.040. But if there is a big shock to the economy, and firm value falls, not only does the
value of the implicit put rise, but its slope is now steeper, implying that exposure to further
shocks is now far greater. When firm value is only 75% of the value of the loan, the slope of
the line tangent to the put value valuation curve is far steeper, 2.644. You can see how as you
get closer to the edge of the cliff, it gets easier and easier to slide right off.
We often hear people say that a shock to asset values of the magnitude of the financial
crisis was a 10-sigma event, by which they mean that such an event was so extreme that
it would be 10 standard deviations away from an expected outcome, making it virtually
inconceivable. But this analysis shows that standard deviation may be a moving target,
increasing dramatically as the firm weakens. As the economy falters and put options go
further into the money, their sensitivity to further shocks increases, increasing the risk
that even worse losses may be around the corner. The built-in instability of risk exposures
makes a scenario like the crisis more plausible and should give us pause when we discount
an extreme scenario as “almost impossible.”
Option Pricing and Portfolio Theory
We’ve just seen that the option pricing model predicts that security risk characteristics can
be unstable. For example, as the firm weakens, the risk of its debt can quickly accelerate.
So, too, can equity risk change dramatically as the firm’s financial condition deteriorates.
We know from the last chapter (Section 20.5) that equity in a levered firm is like a call
option on the value of the firm. If firm value exceeds the value of the firm’s maturing debt,
the firm can choose to pay off the debt, retaining the difference between firm value and the
face value of its debt. If not, the firm can default on the loan, turning the firm over to its
creditors, and the equity holders get nothing. In this sense, equity is a call option, and the
firm’s total value is the underlying asset.
In Section 21.5, we saw that the elasticity of an option measures the sensitivity of its rate
of return to the rate of return on the underlying asset. For example, if a call option’s elasticity
is five, its rate of return will swing five times as widely as the rate of return on the underlying
asset. This would imply that both the option’s beta and its standard deviation are five times the
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PART VI
Options, Futures, and Other Derivatives
beta and standard deviation of
the underlying asset.
Therefore, when compil30
ing the “input list” for creating
an efficient portfolio, we may
25
wish to think of equity as an
implicit call option and com20
pute its elasticity with respect
15
to the total value of the firm.
For example, if the covariance
10
of the firm’s assets with other
securities is stable, then we
5
can use elasticity to find the
covariance of the firm’s equity
0
25
50
75
100
125
150
175
200
with those securities. This will
Stock Price
allow us to calculate beta and
standard deviation.
Unfortunately, elasticity can
Figure 21.14 Call option elasticity as a function of stock price
itself be a moving target. As
(Parameters: s 5 .25; T 5 .5, r 5 .06; X 5 100)
the firm gets weaker, its elasticity will increase, potentially
very quickly. Figure 21.14 uses
the Black-Scholes model to plot call option elasticity as a function of the value of the
underlying stock. Notice that as the option goes out of the money (the stock price falls
below 100), elasticity increases rapidly and without limit. Similarly, as the firm gets closer
to insolvency (the value of firm assets falls below the face value of debt), equity elasticity
shoots up, and even small changes in financial condition can lead to major changes in risk.
Elasticity is far more stable (and closer to 1) when the firm is healthy, i.e., the implicit call
option is deep in the money. Similarly, equity risk characteristics will be far more stable for
healthy firms than for precarious ones.
Elasticity
35
Hedging Bets on Mispriced Options
Suppose you believe that the standard deviation of IBM stock returns will be 35% over the
next few weeks, but IBM put options are selling at a price consistent with a volatility of
33%. Because the put’s implied volatility is less than your forecast of the stock volatility,
you believe the option is underpriced. Using your assessment of volatility in an optionpricing model like the Black-Scholes formula, you would estimate that the fair price for
the puts exceeds the actual price.
Does this mean that you ought to buy put options? Perhaps it does, but by doing so,
you risk losses if IBM stock performs well, even if you are correct about the volatility. You
would like to separate your bet on volatility from the “attached” bet inherent in purchasing
a put that IBM’s stock price will fall. In other words, you would like to speculate on the
option mispricing by purchasing the put option, but hedge the resulting exposure to the
performance of IBM stock.
The option delta can be interpreted as a hedge ratio that can be used for this purpose.
The delta was defined as
Delta 5
Change in value of option
Change in value of stock
(21.5)
Therefore, delta is the slope of the option-pricing curve.
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CHAPTER 21
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755
This ratio tells us precisely how many shares of stock we must hold to offset our
exposure to IBM. For example, if the delta is 2.6, then the put will fall by $.60 in value
for every one-point increase in IBM stock, and we need to hold .6 share of stock to hedge
each put. If we purchase 10 option contracts, each for 100 shares, we would need to buy
600 shares of stock. If the stock price rises by $1, each put option will decrease in value by
$.60, resulting in a loss of $600. However, the loss on the puts will be offset by a gain on
the stock holdings of $1 per share 3 600 shares.
To see how the profits on this strategy might develop, let’s use the following example.
Example 21.8
Speculating on Mispriced Options
Suppose option expiration T is 60 days; put price P is $4.495; exercise price X is $90;
stock price S is $90; and the risk-free rate r is 4%. We assume that the stock will not pay
a dividend in the next 60 days. Given these data, the implied volatility on the option is
33%, as we posited. However, you believe the true volatility is 35%, implying that the
fair put price is $4.785. Therefore, if the market assessment of volatility is revised to the
value you believe is correct, your profit will be $.29 per put purchased.
Recall that the hedge ratio, or delta, of a put option equals N(d1) 2 1, where N (•) is
the cumulative normal distribution function and
d1 5
ln(S/X ) 1 (r 1 s2 / 2)T
s√T
Using your estimate of s 5 .35, you find that the hedge ratio N(d1) 2 1 5 2.453.
Suppose, therefore, that you purchase 10 option contracts (1,000 puts) and purchase
453 shares of stock. Once the market “catches up” to your presumably better volatility estimate, the put options purchased will increase in value. If the market assessment
of volatility changes as soon as you purchase the options, your profits should equal
1,000 3 $.29 5 $290. The option price will be affected as well by any change in the
stock price, but this part of your exposure will be eliminated if the hedge ratio is chosen
properly. Your profit should be based solely on the effect of the change in the implied
volatility of the put, with the impact of the stock price hedged away.
Table 21.3 illustrates your profits as a function of the stock price assuming that the
put price changes to reflect your estimate of volatility. Panel B shows that the put option
alone can provide profits or losses depending on whether the stock price falls or rises.
We see in panel C, however, that each hedged put option provides profits nearly equal
to the original mispricing, regardless of the change in the stock price.
CONCEPT CHECK
21.9
Suppose you bet on volatility by purchasing calls instead of puts. How would you
hedge your exposure to stock-price fluctuations? What is the hedge ratio?
Notice in Example 21.8 that the profit is not exactly independent of the stock price. This
is because as the stock price changes, so do the deltas used to calculate the hedge ratio. The
hedge ratio in principle would need to be continually adjusted as deltas evolve. The sensitivity of the delta to the stock price is called the gamma of the option. Option gammas are
analogous to bond convexity. In both cases, the curvature of the value function means that
hedge ratios or durations change with market conditions, making rebalancing a necessary
part of hedging strategies.
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PART VI
Table 21.3
Profit on hedged
put portfolio
Options, Futures, and Other Derivatives
A. Cost to establish hedged position
1,000 put options @ $4.495/option
453 shares @ $90/share
TOTAL outlay
$ 4,495
40,770
$45,265
B. Value of put option as a function of the stock price at implied volatility of 35%
Stock Price:
Put price
Profit (loss) on each put
89
$ 5.254
0.759
90
$ 4.785
0.290
91
$ 4.347
(0.148)
Stock Price:
Value of 1,000 put options
Value of 453 shares
89
$ 5,254
40,317
90
$ 4,785
40,770
91
$ 4,347
41,223
TOTAL
Profit (5 Value 2 Cost from panel A)
$45,571
306
$45,555
290
$45,570
305
C. Value of and profit on hedged put portfolio
A variant of the strategy in Example 21.8 involves cross-option speculation. Suppose
you observe a 45-day expiration call option on IBM with strike price 95 selling at a price
consistent with a volatility of s 5 33% while another 45-day call with strike price 90 has
an implied volatility of only 27%. Because the underlying asset and expiration date are
identical, you conclude that the call with the higher implied volatility is relatively overpriced. To exploit the mispricing, you might buy the cheap calls (with strike price 90 and
implied volatility of 27%) and write the expensive calls (with strike price 95 and implied
volatility 33%). If the risk-free rate is 4% and IBM is selling at $90 per share, the calls
purchased will be priced at $3.6202 and the calls written will be priced at $2.3735.
Despite the fact that you are long one call and short another, your exposure to IBM
stock-price uncertainty will not be hedged using this strategy. This is because calls with
different strike prices have different sensitivities to the price of the underlying asset. The
lower-strike-price call has a higher delta and therefore greater exposure to the price of
IBM. If you take an equal number of positions in these two options, you will inadvertently
establish a bullish position in IBM, as the calls you purchase have higher deltas than the
calls you write. In fact, you may recall from Chapter 20 that this portfolio (long call with
low exercise price and short call with high exercise price) is called a bullish spread.
To establish a hedged position, we can use the hedge ratio approach as follows.
Consider the 95-strike-price options you write as the asset that hedges your exposure to the
90-strike-price options you purchase. Then the hedge ratio is
H5
5
Change in value of 90-strike-price call for $1 change in IBM
Change in value of 95-strike-price call for $1 change in IBM
Delta of 90-strike-price call
.1
Delta of 95-strike-price call
You need to write more than one call with the higher strike price to hedge the purchase of
each call with the lower strike price. Because the prices of higher-strike-price calls are less
sensitive to IBM prices, more of them are required to offset the exposure.
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CHAPTER 21
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757
Suppose the true annual volatility of the stock is midway between the two implied volatilities, so s 5 30%. We know that the delta of a call option is N(d1). Therefore, the deltas
of the two options and the hedge ratio are computed as follows:
Option with strike price 90:
d1 5
ln(90/90) 1 (.04 1 .302 /2) 3 45/365
.30"45/365
5 .0995
N(d1) 5 .5396
Option with strike price 95:
d1 5
ln(90/95) 1 (.04 1 .302 /2) 3 45/365
.30"45/365
5 2.4138
N(d1) 5 .3395
Hedge ratio:
.5396
5 1.589
.3395
Therefore, for every 1,000 call options purchased with strike price 90, we need to write
1,589 call options with strike price 95. Following this strategy enables us to bet on the relative mispricing of the two options without taking a position on IBM. Panel A of Table 21.4
shows that the position will result in a cash inflow of $151.30. The premium income on the
calls written exceeds the cost of the calls purchased.
When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral,
meaning that the portfolio has no tendency to either increase or decrease in value when the
stock price fluctuates.
Let’s check that our options position is in fact delta neutral. Suppose that the implied
volatilities of the two options come back into alignment just after you establish your
Table 21.4
A. Cost flow when portfolio is established
Purchase 1,000 calls (X 5 90) @ $3.6202
(option priced at implied volatility of 27%)
Write 1,589 calls (X 5 95) @ $2.3735
(option priced at implied volatility of 33%)
TOTAL
Profits on deltaneutral options
portfolio
$ 3,620.20 cash outflow
3,771.50 cash inflow
$ 151.30 net cash inflow
B. Option prices at implied volatility of 30%
Stock Price:
90-strike-price calls
95-strike-price calls
89
90
91
$3.478
1.703
$3.997
2.023
$4.557
2.382
C. Value of portfolio after implied volatilities converge to 30%
Stock Price:
Value of 1,000 calls held
2 Value of 1,589 calls written
TOTAL
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89
$3,478
2,705
$ 773
90
$3,997
3,214
$ 782
91
$4,557
3,785
$ 772
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PART VI
Options, Futures, and Other Derivatives
position, so that both options are priced at implied volatilities of 30%. You expect to
profit from the increase in the value of the call purchased as well as from the decrease
in the value of the call written. The option prices at 30% volatility are given in panel
B of Table 21.4 and the values of your position for various stock prices are presented
in panel C. Although the profit or loss on each option is affected by the stock price,
the value of the delta-neutral option portfolio is positive and essentially independent
of the price of IBM. Moreover, we saw in panel A that the portfolio would have been
established without ever requiring a cash outlay. You would have cash inflows both
when you establish the portfolio and when you liquidate it after the implied volatilities
converge to 30%.
This unusual profit opportunity arises because you have identified prices out of alignment. Such opportunities could not arise if prices were at equilibrium levels. By exploiting
the pricing discrepancy using a delta-neutral strategy, you should earn profits regardless of
the price movement in IBM stock.
Delta-neutral hedging strategies are also subject to practical problems, the most important of which is the difficulty in assessing the proper volatility for the coming period. If the
volatility estimate is incorrect, so will be the deltas, and the overall position will not truly
be hedged. Moreover, option or option-plus-stock positions generally will not be neutral
with respect to changes in volatility. For example, a put option hedged by a stock might be
delta neutral, but it is not volatility neutral. Changes in the market assessments of volatility
will affect the option price even if the stock price is unchanged.
These problems can be serious, because volatility estimates are never fully reliable.
First, volatility cannot be observed directly and must be estimated from past data which
imparts measurement error to the forecast. Second, we’ve seen that both historical and
implied volatilities fluctuate over time. Therefore, we are always shooting at a moving
target. Although delta-neutral positions are hedged against changes in the price of the
underlying asset, they still are subject to volatility risk, the risk incurred from unpredictable changes in volatility. The sensitivity of an option price to changes in volatility
is called the option’s vega. Thus, although delta-neutral option hedges might eliminate
exposure to risk from fluctuations in the value of the underlying asset, they do not eliminate volatility risk.
21.6 Empirical Evidence on Option Pricing
The Black-Scholes option-pricing model has been subject to an enormous number of
empirical tests. For the most part, the results of the studies have been positive in that the
Black-Scholes model generates option values fairly close to the actual prices at which
options trade. At the same time, some regular empirical failures of the model have been
noted.
The biggest problem concerns volatility. If the model were accurate, the implied volatility of all options on a particular stock with the same expiration date would be equal—after
all, the underlying asset and expiration date are the same for each option, so the volatility
inferred from each also ought to be the same. But in fact, when one actually plots implied
volatility as a function of exercise price, the typical results appear as in Figure 21.15, which
treats S&P 500 index options as the underlying asset. Implied volatility steadily falls as the
exercise price rises. Clearly, the Black-Scholes model is missing something.
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19
Mark Rubinstein, “Implied Binomial Trees,” Journal of Finance 49 (July 1994), pp. 771–818.
For an extensive discussion of these more general models, see R. L. McDonald, Derivatives Markets, 3rd ed.
(Boston: Pearson Education [Addison-Wesley], 2013).
20
1. Option values may be viewed as the sum of intrinsic value plus time or “volatility” value. The
volatility value is the right to choose not to exercise if the stock price moves against the holder.
Thus the option holder cannot lose more than the cost of the option regardless of stock price
performance.
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Implied Volatility (%)
Rubinstein19 suggests that the problem with the model has to do with fears of
25
a market crash like that of October 1987.
The idea is that deep out-of-the-money puts
20
would be nearly worthless if stock prices
15
evolve smoothly, because the probability of the stock falling by a large amount
10
(and the put option thereby moving into
the money) in a short time would be very
5
small. But a possibility of a sudden large
downward jump that could move the puts
0
into the money, as in a market crash, would
0.84
0.89
0.94
0.99
1.04
1.09
impart greater value to these options. Thus,
Ratio
of
Exercise
Price
to
Current
Value
of
Index
the market might price these options as
though there is a bigger chance of a large
drop in the stock price than would be suggested by the Black-Scholes assumptions.
Figure 21.15 Implied volatility of the S&P 500 index as a
The result of the higher option price is a
function of exercise price
greater implied volatility derived from the
Source: Mark Rubinstein, “Implied Binomial Trees,” Journal of Finance (July
1994), pp. 771–818.
Black-Scholes model.
Interestingly, Rubinstein points out
that prior to the 1987 market crash, plots
of implied volatility like the one in Figure 21.15 were relatively flat, consistent with the
notion that the market was then less attuned to fears of a crash. However, postcrash plots
have been consistently downward sloping, exhibiting a shape often called the option smirk.
When we use option-pricing models that allow for more general stock price distributions,
including crash risk and random changes in volatility, they generate downward-sloping
implied volatility curves similar to the one observed in Figure 21.15.20
SUMMARY
2. Call options are more valuable when the exercise price is lower, when the stock price is higher,
when the interest rate is higher, when the time to expiration is greater, when the stock’s volatility
is greater, and when dividends are lower.
3. Call options must sell for at least the stock price less the present value of the exercise price and
dividends to be paid before expiration. This implies that a call option on a non-dividend-paying
stock may be sold for more than the proceeds from immediate exercise. Thus European calls are
worth as much as American calls on stocks that pay no dividends, because the right to exercise the
American call early has no value.
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PART VI
Options, Futures, and Other Derivatives
4. Options may be priced relative to the underlying stock price using a simple two-period, twostate pricing model. As the number of periods increases, the binomial model can approximate
more realistic stock price distributions. The Black-Scholes formula may be seen as a limiting
case of the binomial option model, as the holding period is divided into progressively smaller
subperiods when the interest rate and stock volatility are constant.
5. The Black-Scholes formula applies to options on stocks that pay no dividends. Dividend
adjustments may be adequate to price European calls on dividend-paying stocks, but the proper
treatment of American calls on dividend-paying stocks requires more complex formulas.
6. Put options may be exercised early, whether the stock pays dividends or not. Therefore,
American puts generally are worth more than European puts.
7. European put values can be derived from the call value and the put-call parity relationship. This
technique cannot be applied to American puts for which early exercise is a possibility.
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8. The implied volatility of an option is the standard deviation of stock returns consistent with
an option’s market price. It can be backed out of an option-pricing model by finding the stock
volatility that makes the option’s value equal to its observed price.
9. The hedge ratio is the number of shares of stock required to hedge the price risk involved
in writing one option. Hedge ratios are near zero for deep out-of-the-money call options and
approach 1.0 for deep in-the-money calls.
10. Although their hedge ratios are less than 1.0, call options have elasticities greater than 1.0. The
rate of return on a call (as opposed to the dollar return) responds more than one-for-one with
stock price movements.
11. Portfolio insurance can be obtained by purchasing a protective put option on an equity position.
When the appropriate put is not traded, portfolio insurance entails a dynamic hedge strategy
where a fraction of the equity portfolio equal to the desired put option’s delta is sold and placed
in risk-free securities.
12. The option delta is used to determine the hedge ratio for options positions. Delta-neutral portfolios are independent of price changes in the underlying asset. Even delta-neutral option portfolios are subject to volatility risk, however.
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for this chapter are
available at www.
mhhe.com/bkm
13. Empirically, implied volatilities derived from the Black-Scholes formula tend to be lower on
options with higher exercise prices. This may be evidence that the option prices reflect the possibility of a sudden dramatic decline in stock prices. Such “crashes” are inconsistent with the
Black-Scholes assumptions.
KEY TERMS
intrinsic value
time value
binomial model
Black-Scholes pricing formula
implied volatility
KEY EQUATIONS
pseudo-American call option
value
hedge ratio
delta
option elasticity
Binomial model: u 5 exp(s"Dt); d 5 exp(2s"Dt); p 5
portfolio insurance
dynamic hedging
gamma
delta neutral
vega
exp(rDt) 2 d
u2d
Put-call parity: P 5 C 1 PV(X) 2 S0 1 PV(Dividends)
Black-Scholes formula (no dividend case): SN(d1) 2 Xe2rTN(d2)
where d1 5
ln(S/X) 1 (r 1 ½ s2)T
s"T
Delta (or hedge ratio): H 5
bod61671_ch21_722-769.indd 760
; d2 5 d1 2 s"T
Change in option value
Change in stock value
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CHAPTER 21
Option Valuation
761
1. We showed in the text that the value of a call option increases with the volatility of the stock. Is
this also true of put option values? Use the put-call parity theorem as well as a numerical example
to prove your answer.
PROBLEM SETS
2. Would you expect a $1 increase in a call option’s exercise price to lead to a decrease in the
option’s value of more or less than $1?
Basic
3. Is a put option on a high-beta stock worth more than one on a low-beta stock? The stocks have
identical firm-specific risk.
4. All else equal, is a call option on a stock with a lot of firm-specific risk worth more than one on a
stock with little firm-specific risk? The betas of the two stocks are equal.
5. All else equal, will a call option with a high exercise price have a higher or lower hedge ratio than
one with a low exercise price?
6. In each of the following questions, you are asked to compare two options with parameters as
given. The risk-free interest rate for all cases should be assumed to be 6%. Assume the stocks on
which these options are written pay no dividends.
A
B
T
X
s
.5
.5
50
50
.20
.25
Price of Option
$10
$10
Which put option is written on the stock with the lower price?
i. A.
ii. B.
iii. Not enough information.
b. Put
T
X
s
A
B
.5
.5
50
50
.2
.2
Price of Option
$10
$12
Which put option must be written on the stock with the lower price?
i. A.
ii. B.
iii. Not enough information.
c. Call
S
X
s
Price of Option
A
50
50
.20
$12
B
55
50
.20
$10
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a. Put
Intermediate
Which call option must have the lower time to expiration?
i. A.
ii. B.
iii. Not enough information.
d. Call
T
X
S
Price of Option
A
B
.5
.5
50
50
55
55
$10
$12
Which call option is written on the stock with higher volatility?
i. A.
ii. B.
iii. Not enough information.
e. Call
T
X
S
Price of Option
A
B
.5
.5
50
50
55
55
$10
$ 7
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PART VI
Options, Futures, and Other Derivatives
Which call option is written on the stock with higher volatility?
i. A.
ii. B.
iii. Not enough information.
7. Reconsider the determination of the hedge ratio in the two-state model (see page 730), where
we showed that one-third share of stock would hedge one option. What would be the hedge ratio
for the following exercise prices: 120, 110, 100, 90? What do you conclude about the hedge
ratio as the option becomes progressively more in the money?
8. Show that Black-Scholes call option hedge ratios also increase as the stock price increases.
Consider a 1-year option with exercise price $50, on a stock with annual standard deviation
20%. The T-bill rate is 3% per year. Find N(d1) for stock prices $45, $50, and $55.
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9. We will derive a two-state put option value in this problem. Data: S0 5 100; X 5 110; 1 1 r 5
1.10. The two possibilities for ST are 130 and 80.
a. Show that the range of S is 50, whereas that of P is 30 across the two states. What is the
hedge ratio of the put?
b. Form a portfolio of three shares of stock and five puts. What is the (nonrandom) payoff to
this portfolio? What is the present value of the portfolio?
c. Given that the stock currently is selling at 100, solve for the value of the put.
10. Calculate the value of a call option on the stock in the previous problem with an exercise price
of 110. Verify that the put-call parity theorem is satisfied by your answers to Problems 9 and 10.
(Do not use continuous compounding to calculate the present value of X in this example because
we are using a two-state model here, not a continuous-time Black-Scholes model.)
11. Use the Black-Scholes formula to find the value of a call option on the following stock:
Time to expiration
Standard deviation
Exercise price
Stock price
Interest rate
6 months
50% per year
$50
$50
3%
12. Find the Black-Scholes value of a put option on the stock in the previous problem with the same
exercise price and expiration as the call option.
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13. Recalculate the value of the call option in Problem 11, successively substituting one of the
changes below while keeping the other parameters as in Problem 11:
a.
b.
c.
d.
e.
Time to expiration 5 3 months.
Standard deviation 5 25% per year.
Exercise price 5 $55.
Stock price 5 $55.
Interest rate 5 5%.
Consider each scenario independently. Confirm that the option value changes in accordance
with the prediction of Table 21.1.
14. A call option with X 5 $50 on a stock currently priced at S 5 $55 is selling for $10. Using a
volatility estimate of s 5 .30, you find that N(d1) 5 .6 and N(d2) 5 .5. The risk-free interest rate
is zero. Is the implied volatility based on the option price more or less than .30? Explain.
15. What would be the Excel formula in Spreadsheet 21.1 for the Black-Scholes value of a straddle
position?
Use the following case in answering Problems 16–21: Mark Washington, CFA, is an analyst with
BIC. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a
slight downturn and suggested delta-hedging the BIC portfolio. As predicted, the U.S. equity markets
did indeed experience a downturn of approximately 4% over a 12-month period. However, portfolio
performance for BIC was disappointing, lagging its peer group by nearly 10%. Washington has been
told to review the options strategy to determine why the hedged portfolio did not perform as expected.
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CHAPTER 21
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16. Which of the following best explains a delta-neutral portfolio? A delta-neutral portfolio is
perfectly hedged against:
a. Small price changes in the underlying asset.
b. Small price decreases in the underlying asset.
c. All price changes in the underlying asset.
17. After discussing the concept of a delta-neutral portfolio, Washington determines that he needs
to further explain the concept of delta. Washington draws the value of an option as a function of
the underlying stock price. Using this diagram, indicate how delta is interpreted. Delta is the:
a. Slope in the option price diagram.
b. Curvature of the option price graph.
c. Level in the option price diagram.
18. Washington considers a put option that has a delta of 20.65. If the price of the underlying asset
decreases by $6, then what is the best estimate of the change in option price?
19. BIC owns 51,750 shares of Smith & Oates. The shares are currently priced at $69. A call option
on Smith & Oates with a strike price of $70 is selling at $3.50 and has a delta of .69. What is the
number of call options necessary to create a delta-neutral hedge?
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20. Return to the previous problem. Will the number of call options written for a delta-neutral
hedge increase or decrease if the stock price falls?
21. Which of the following statements regarding the goal of a delta-neutral portfolio is most accurate? One example of a delta-neutral portfolio is to combine a:
a. Long position in a stock with a short position in call options so that the value of the portfolio
does not change with changes in the value of the stock.
b. Long position in a stock with a short position in a call option so that the value of the portfolio changes with changes in the value of the stock.
c. Long position in a stock with a long position in call options so that the value of the portfolio
does not change with changes in the value of the stock.
22. Should the rate of return of a call option on a long-term Treasury bond be more or less sensitive
to changes in interest rates than is the rate of return of the underlying bond?
23. If the stock price falls and the call price rises, then what has happened to the call option’s
implied volatility?
24. If the time to expiration falls and the put price rises, then what has happened to the put option’s
implied volatility?
25. According to the Black-Scholes formula, what will be the value of the hedge ratio of a call
option as the stock price becomes infinitely large? Explain briefly.
26. According to the Black-Scholes formula, what will be the value of the hedge ratio of a put
option for a very small exercise price?
27. The hedge ratio of an at-the-money call option on IBM is .4. The hedge ratio of an at-the-money
put option is 2.6. What is the hedge ratio of an at-the-money straddle position on IBM?
28. Consider a 6-month expiration European call option with exercise price $105. The underlying
stock sells for $100 a share and pays no dividends. The risk-free rate is 5%. What is the implied
volatility of the option if the option currently sells for $8? Use Spreadsheet 21.1 (available at
www.mhhe.com/bkm; link to Chapter 21 material) to answer this question.
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a. Go to the Data tab of the spreadsheet and select Goal Seek from the What-If menu. The dialog box will ask you for three pieces of information. In that dialog box, you should set cell
E6 to value 8 by changing cell B2. In other words, you ask the spreadsheet to find the value
of standard deviation (which appears in cell B2) that forces the value of the option (in cell
E6) equal to $8. Then click OK, and you should find that the call is now worth $8, and the
entry for standard deviation has been changed to a level consistent with this value. This is the
call’s implied standard deviation at a price of $8.
b. What happens to implied volatility if the option is selling at $9? Why has implied volatility
increased?
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PART VI
Options, Futures, and Other Derivatives
c. What happens to implied volatility if the option price is unchanged at $8, but option expiration is lower, say, only 4 months? Why?
d. What happens to implied volatility if the option price is unchanged at $8, but the exercise
price is lower, say, only $100? Why?
e. What happens to implied volatility if the option price is unchanged at $8, but the stock price
is lower, say, only $98? Why?
29. A collar is established by buying a share of stock for $50, buying a 6-month put option with
exercise price $45, and writing a 6-month call option with exercise price $55. On the basis of
the volatility of the stock, you calculate that for a strike price of $45 and expiration of 6 months,
N(d1) 5 .60, whereas for the exercise price of $55, N(d1) 5 .35.
a. What will be the gain or loss on the collar if the stock price increases by $1?
b. What happens to the delta of the portfolio if the stock price becomes very large? Very small?
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30. These three put options are all written on the same stock. One has a delta of 2.9, one a delta of
2.5, and one a delta of 2.1. Assign deltas to the three puts by filling in this table.
Put
X
A
B
C
10
20
30
Delta
31. You are very bullish (optimistic) on stock EFG, much more so than the rest of the market. In
each question, choose the portfolio strategy that will give you the biggest dollar profit if your
bullish forecast turns out to be correct. Explain your answer.
a. Choice A: $10,000 invested in calls with X 5 50.
Choice B: $10,000 invested in EFG stock.
b. Choice A: 10 call option contracts (for 100 shares each), with X 5 50.
Choice B: 1,000 shares of EFG stock.
32. You would like to be holding a protective put position on the stock of XYZ Co. to lock in a guaranteed minimum value of $100 at year-end. XYZ currently sells for $100. Over the next year
the stock price will increase by 10% or decrease by 10%. The T-bill rate is 5%. Unfortunately,
no put options are traded on XYZ Co.
a. Suppose the desired put option were traded. How much would it cost to purchase?
b. What would have been the cost of the protective put portfolio?
c. What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that
would be provided by a protective put with X 5 100? Show that the payoff to this portfolio
and the cost of establishing the portfolio matches that of the desired protective put.
33. Return to Example 21.1. Use the binomial model to value a 1-year European put option with
exercise price $110 on the stock in that example. Does your solution for the put price satisfy
put-call parity?
34. Suppose that the risk-free interest rate is zero. Would an American put option ever be exercised
early? Explain.
35. Let p(S, T, X) denote the value of a European put on a stock selling at S dollars, with time to
maturity T, and with exercise price X, and let P(S, T, X) be the value of an American put.
a.
b.
c.
d.
e.
Evaluate p(0, T, X).
Evaluate P(0, T, X).
Evaluate p(S, T, 0).
Evaluate P(S, T, 0).
What does your answer to (b) tell you about the possibility that American puts may be exercised early?
36. You are attempting to value a call option with an exercise price of $100 and 1 year to expiration.
The underlying stock pays no dividends, its current price is $100, and you believe it has a 50%
chance of increasing to $120 and a 50% chance of decreasing to $80. The risk-free rate of interest is 10%. Calculate the call option’s value using the two-state stock price model.
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CHAPTER 21
Option Valuation
765
37. Consider an increase in the volatility of the stock in the previous problem. Suppose that if the
stock increases in price, it will increase to $130, and that if it falls, it will fall to $70. Show that
the value of the call option is now higher than the value derived in the previous problem.
38. Calculate the value of a put option with exercise price $100 using the data in Problem 36. Show
that put-call parity is satisfied by your solution.
39. XYZ Corp. will pay a $2 per share dividend in 2 months. Its stock price currently is $60 per
share. A call option on XYZ has an exercise price of $55 and 3-month time to expiration. The
risk-free interest rate is .5% per month, and the stock’s volatility (standard deviation) 5 7% per
month. Find the pseudo-American option value. (Hint: Try defining one “period” as a month,
rather than as a year.)
40. “The beta of a call option on General Electric is greater than the beta of a share of General
Electric.” True or false?
41. “The beta of a call option on the S&P 500 index with an exercise price of 1,330 is greater than
the beta of a call on the index with an exercise price of 1,340.” True or false?
43. Goldman Sachs believes that market volatility will be 20% annually for the next 3 years. Threeyear at-the-money call and put options on the market index sell at an implied volatility of 22%.
What options portfolio can Goldman establish to speculate on its volatility belief without taking
a bullish or bearish position on the market? Using Goldman’s estimate of volatility, 3-year atthe-money options have N(d1) 5 .6.
44. You are holding call options on a stock. The stock’s beta is .75, and you are concerned that the
stock market is about to fall. The stock is currently selling for $5 and you hold 1 million options
on the stock (i.e., you hold 10,000 contracts for 100 shares each). The option delta is .8. How
much of the market-index portfolio must you buy or sell to hedge your market exposure?
45. Imagine you are a provider of portfolio insurance. You are establishing a 4-year program. The
portfolio you manage is currently worth $100 million, and you hope to provide a minimum return
of 0%. The equity portfolio has a standard deviation of 25% per year, and T-bills pay 5% per year.
Assume for simplicity that the portfolio pays no dividends (or that all dividends are reinvested).
a. How much should be placed in bills? How much in equity?
b. What should the manager do if the stock portfolio falls by 3% on the first day of trading?
46. Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike
price $90 are selling at an implied volatility of 30%. ExxonMobil stock currently is $90 per
share, and the risk-free rate is 4%. If you believe the true volatility of the stock is 32%, how can
you trade on your belief without taking on exposure to the performance of ExxonMobil? How
many shares of stock will you hold for each option contract purchased or sold?
Challenge
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42. What will happen to the hedge ratio of a convertible bond as the stock price becomes very large?
47. Using the data in the previous problem, suppose that 3-month put options with a strike price of
$90 are selling at an implied volatility of 34%. Construct a delta-neutral portfolio comprising
positions in calls and puts that will profit when the option prices come back into alignment.
48. Suppose that JPMorgan Chase sells call options on $1.25 million worth of a stock portfolio
with beta 5 1.5. The option delta is .8. It wishes to hedge out its resultant exposure to a market
advance by buying a market-index portfolio.
a. How many dollars’ worth of the market-index portfolio should it purchase to hedge its position?
b. What if it instead uses market index puts to hedge its exposure? Should it buy or sell puts?
Each put option is on 100 units of the index, and the index at current prices represents
$1,000 worth of stock.
49. Suppose you are attempting to value a 1-year maturity option on a stock with volatility (i.e.,
annualized standard deviation) of s 5 .40. What would be the appropriate values for u and d if
your binomial model is set up using:
a. 1 period of 1 year.
b. 4 subperiods, each 3 months.
c. 12 subperiods, each 1 month.
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PART VI
Options, Futures, and Other Derivatives
50. You build a binomial model with one period and assert that over the course of a year, the stock
price will either rise by a factor of 1.5 or fall by a factor of 2/3. What is your implicit assumption
about the volatility of the stock’s rate of return over the next year?
51. Use the put-call parity relationship to demonstrate that an at-the-money call option on a
nondividend-paying stock must cost more than an at-the-money put option. Show that the prices
of the put and call will be equal if S 5 (1 1 r)T.
52. Return to Problem 36. Value the call option using the risk-neutral shortcut described in the
box on page 736. Confirm that your answer matches the value you get using the two-state
approach.
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53. Return to Problem 38. What will be the payoff to the put, Pu, if the stock goes up? What will
be the payoff, Pd, if the stock price falls? Value the put option using the risk-neutral shortcut
described in the box on page 736. Confirm that your answer matches the value you get using the
two-state approach.
1. The board of directors of Abco Company is concerned about the downside risk of a $100
million equity portfolio in its pension plan. The board’s consultant has proposed temporarily (for
1 month) hedging the portfolio with either futures or options. Referring to the following table, the
consultant states:
a. “The $100 million equity portfolio can be fully protected on the downside by selling
(shorting) 4,000 futures contracts.”
b. “The cost of this protection is that the portfolio’s expected rate of return will be zero percent.”
Market, Portfolio, and Contract Data
Equity index level
Equity futures price
Futures contract multiplier
Portfolio beta
Contract expiration (months)
99.00
100.00
$250
1.20
3
Critique the accuracy of each of the consultant’s two statements.
2. Michael Weber, CFA, is analyzing several aspects of option valuation, including the determinants
of the value of an option, the characteristics of various models used to value options, and the
potential for divergence of calculated option values from observed market prices.
a. What is the expected effect on the value of a call option on common stock if the volatility of
the underlying stock price decreases? If the time to expiration of the option increases?
b. Using the Black-Scholes option-pricing model, Weber calculates the price of a 3-month call
option and notices the option’s calculated value is different from its market price. With respect
to Weber’s use of the Black-Scholes option-pricing model,
i. Discuss why the calculated value of an out-of-the-money European option may differ
from its market price.
ii. Discuss why the calculated value of an American option may differ from its market
price.
3. Joel Franklin is a portfolio manager responsible for derivatives. Franklin observes an Americanstyle option and a European-style option with the same strike price, expiration, and underlying stock. Franklin believes that the European-style option will have a higher premium than the
American-style option.
a. Critique Franklin’s belief that the European-style option will have a higher premium. Franklin
is asked to value a 1-year European-style call option for Abaco Ltd. common stock, which last
traded at $43.00. He has collected the information in the following table.
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CHAPTER 21
Closing stock price
Call and put option exercise price
1-year put option price
$43.00
45.00
4.00
1-year Treasury bill rate
Time to expiration
5.50%
One year
Option Valuation
767
b. Calculate, using put-call parity and the information provided in the table, the European-style
call option value.
c. State the effect, if any, of each of the following three variables on the value of a call option.
(No calculations required.)
i. An increase in short-term interest rate.
ii. An increase in stock price volatility.
iii. A decrease in time to option expiration.
a.
b.
c.
d.
Construct a two-period binomial tree for the value of the stock index.
Calculate the value of a European call option on the index with an exercise price of 60.
Calculate the value of a European put option on the index with an exercise price of 60.
Confirm that your solutions for the values of the call and the put satisfy put-call parity.
5. Ken Webster manages a $200 million equity portfolio benchmarked to the S&P 500 index.
Webster believes the market is overvalued when measured by several traditional fundamental/
economic indicators. He is concerned about potential losses but recognizes that the S&P 500
index could nevertheless move above its current 1136 level.
Webster is considering the following option collar strategy:
• Protection for the portfolio can be attained by purchasing an S&P 500 index put with a strike
price of 1130 (just out of the money).
• The put can be financed by selling two 1150 calls (farther out-of-the-money) for every put purchased.
• Because the combined delta of the two calls (see following table) is less than 1 (that is,
2 3 .36 5 .72), the options will not lose more than the underlying portfolio will gain if the
market advances.
The information in the following table describes the two options used to create the collar.
Characteristics
Option price
Option implied volatility
Option’s delta
Contracts needed for collar
1150 Call
1130 Put
$8.60
22%
0.36
602
$16.10
24%
20.44
301
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4. A stock index is currently trading at 50. Paul Tripp, CFA, wants to value 2-year index options using
the binomial model. The stock will either increase in value by 20% or fall in value by 20%. The annual
risk-free interest rate is 6%. No dividends are paid on any of the underlying securities in the index.
Notes:
• Ignore transaction costs.
• S&P 500 historical 30-day volatility 5 23%.
• Time to option expiration 5 30 days.
a. Describe the potential returns of the combined portfolio (the underlying portfolio plus the
option collar) if after 30 days the S&P 500 index has:
i. risen approximately 5% to 1193.
ii. remained at 1136 (no change).
iii. declined by approximately 5% to 1080.
(No calculations are necessary.)
b. Discuss the effect on the hedge ratio (delta) of each option as the S&P 500 approaches the
level for each of the potential outcomes listed in part (a).
c. Evaluate the pricing of each of the following in relation to the volatility data provided:
i. the put
ii. the call
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PART VI
Options, Futures, and Other Derivatives
E-INVESTMENTS EXERCISES
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Select a stock for which options are listed on the CBOE Web site (www.cboe.com). The
price data for captions can be found on the “delayed quotes” menu option. Enter a ticker
symbol for a stock of your choice and pull up its option price data.
Using daily price data from finance.yahoo.com calculate the annualized standard
deviation of the daily percentage change in the stock price. Create a Black-Scholes
option-pricing model in a spreadsheet, or use our Spreadsheet 21.1, available at www.
mhhe.com/bkm (link to the Chapter 21 material). Using the standard deviation and
a risk-free rate found at www.bloomberg.com/markets/rates/index.html, calculate
the value of the call options.
How do the calculated values compare to the market prices of the options? On the
basis of the difference between the price you calculated using historical volatility and
the actual price of the option, what do you conclude about expected trends in market
volatility?
SOLUTIONS TO CONCEPT CHECKS
1. If This Variable Increases . . . The Value of a Put Option
S
X
Decreases
Increases
Increases
s
T
rf
Dividend payouts
Increases*
Decreases
Increases
*For American puts, increase in time to expiration must increase value. One can always choose to exercise early if
this is optimal; the longer expiration date simply expands the range of alternatives open to the option holder which
must make the option more valuable. For a European put, where early exercise is not allowed, longer time to
expiration can have an indeterminate effect. Longer expiration increases volatility value because the final stock price
is more uncertain, but it reduces the present value of the exercise price that will be received if the put is exercised.
The net effect on put value can be positive or negative.
To understand the impact of higher volatility, consider the same scenarios as for the call. The lowvolatility scenario yields a lower expected payoff.
High
volatility
Low
volatility
Stock price
Put payoff
Stock price
Put payoff
$10
$20
$20
$10
$20
$10
$25
$ 5
$30
$ 0
$30
$ 0
$40
$ 0
$35
$ 0
$50
$ 0
$40
$ 0
2. The parity relationship assumes that all options are held until expiration and that there are no cash
flows until expiration. These assumptions are valid only in the special case of European options
on non-dividend-paying stocks. If the stock pays no dividends, the American and European calls
are equally valuable, whereas the American put is worth more than the European put. Therefore,
although the parity theorem for European options states that
P 5 C 2 S0 1 PV(X)
in fact, P will be greater than this value if the put is American.
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CHAPTER 21
Option Valuation
769
3. Because the option now is underpriced, we want to reverse our previous strategy.
Cash Flow in 1 Year for
Each Possible Stock Price
Initial Cash Flow
Buy 3 options
Short-sell 1 share; repay in 1 year
Lend $83.50 at 10% interest rate
TOTAL
S 5 90
0
290
91.85
1.85
216.50
100
283.50
0
S 5 120
30
2120
91.85
1.85
The riskless cash flow in 1 year per option is $1.85/3 5 $.6167, and the present value is
$.6167/1.10 5 $.56, precisely the amount by which the option is underpriced.
4. a. Cu 2 Cd 5 $6.984 2 0
b. uS0 2 dS0 5 $110 2 $95 5 $15
c. 6.984/15 5 .4656
Action Today (time 0)
dS0 5 $95
uS0 5 $110
Buy .4656 shares at price S0 5 $100
Write 1 call at price C0
$44.232
$51.216
0
TOTAL
$44.232
26.984
$44.232
The portfolio must have a market value equal to the present value of $44.232.
e. $44.232/1.05 5 $42.126
f. .4656 3 $100 2 C0 5 $42.126
C0 5 $46.56 2 $42.126 5 $4.434
5. When ∆t shrinks, there should be lower possible dispersion in the stock price by the end of
the subperiod because each shorter subperiod offers less time in which new information can
move stock prices. However, as the time interval shrinks, there will be a correspondingly greater
number of these subperiods until option expiration. Thus, total volatility over the remaining life
of the option will be unaffected. In fact, take another look at Figure 21.2. There, despite the fact
that u and d each get closer to 1 as the number of subintervals increases and the length of each
subinterval falls, the total volatility of the stock return until option expiration is unaffected.
Visit us at www.mhhe.com/bkm
Value in Next Period as
Function of Stock Price
d.
6. Because s 5 .6, s2 5 .36.
d1 5
ln(100/95) 1 (.10 1 .36/2).25
.6".25
5 .4043
d2 5 d1 2 .6".25 5 .1043
Using Table 21.2 and interpolation, or from a spreadsheet function:
N(d1) 5 .6570
N(d2) 5 .5415
C 5 100 3 .6570 2 95e2.103.25 3 .5415 5 15.53
7. Implied volatility exceeds .2783. Given a standard deviation of .2783, the option value is $7. A
higher volatility is needed to justify an $8 price. Using Spreadsheet 21.1 and Goal Seek, you can
confirm that implied volatility at an option price of $8 is .3138.
8. A $1 increase in stock price is a percentage increase of 1/122 5 .82%. The put option will fall by
(.4 3 $1) 5 $.40, a percentage decrease of $.40/$4 5 10%. Elasticity is 210/.82 5 212.2.
9. The delta for a call option is N(d1), which is positive, and in this case is .547. Therefore, for every
10 option contracts purchased, you would need to short 547 shares of stock.
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CHAPTER TWENTY-TWO
PART VI
Futures Markets
FUTURES AND FORWARD contracts are like
options in that they specify purchase or sale
of some underlying security at some future
date. The key difference is that the holder
of an option is not compelled to buy or sell,
and will not do so if the trade is unprofitable.
A futures or forward contract, however, carries the obligation to go through with the
agreed-upon transaction.
A forward contract is not an investment
in the strict sense that funds are paid for an
asset. It is only a commitment today to transact
in the future. Forward arrangements are part
of our study of investments, however, because
they offer powerful means to hedge other
investments and generally modify portfolio
characteristics.
Forward markets for future delivery of
various commodities go back at least to
ancient Greece. Organized futures markets,
though, are a relatively modern development, dating only to the 19th century. Futures
markets replace informal forward contracts
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with highly standardized, exchange-traded
securities.
While futures markets have their roots in
agricultural products and commodities, the
markets today are dominated by trading in
financial futures such as those on stock indexes,
interest-rate-dependent securities such as government bonds, and foreign exchange. The
markets themselves also have changed, with
trading today largely taking place in electronic
markets.
This chapter describes the workings of
futures markets and the mechanics of trading
in these markets. We show how futures contracts are useful investment vehicles for both
hedgers and speculators and how the futures
price relates to the spot price of an asset. We
also show how futures can be used in several
risk-management applications. This chapter
deals with general principles of future markets. Chapter 23 describes specific futures
markets in greater detail.
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CHAPTER 22
Futures Markets
771
22.1 The Futures Contract
To see how futures and forwards work and how they might be useful, consider the portfolio diversification problem facing a farmer growing a single crop, let us say wheat. The
entire planting season’s revenue depends critically on the highly volatile crop price. The
farmer can’t easily diversify his position because virtually his entire wealth is tied up in
the crop.
The miller who must purchase wheat for processing faces a risk management problem
that is the mirror image of the farmer’s. He is subject to profit uncertainty because of the
unpredictable cost of the wheat.
Both parties can hedge their risk by entering into a forward contract requiring the
farmer to deliver the wheat when harvested at a price agreed upon now, regardless of the
market price at harvest time. No money need change hands at this time. A forward contract
is simply a deferred-delivery sale of some asset with the sales price agreed on now. All that
is required is that each party be willing to lock in the ultimate delivery price. The contract
protects each party from future price fluctuations.
Futures markets formalize and standardize forward contracting. Buyers and sellers
trade in a centralized futures exchange. The exchange standardizes the types of contracts that may be traded: It establishes contract size, the acceptable grade of commodity,
contract delivery dates, and so forth. Although standardization eliminates much of the
flexibility available in forward contracting, it has the offsetting advantage of liquidity
because many traders will concentrate on the same small set of contracts. Futures contracts also differ from forward contracts in that they call for a daily settling up of any
gains or losses on the contract. By contrast, no money changes hands in forward contracts
until the delivery date.
The centralized market, standardization of contracts, and depth of trading in each contract allows futures positions to be liquidated easily rather than renegotiated with the other
party to the contract. Because the exchange guarantees the performance of each party,
costly credit checks on other traders are not necessary. Instead, each trader simply posts a
good-faith deposit, called the margin, to guarantee contract performance.
The Basics of Futures Contracts
The futures contract calls for delivery of a commodity at a specified delivery or maturity
date, for an agreed-upon price, called the futures price, to be paid at contract maturity. The
contract specifies precise requirements for the commodity. For agricultural commodities,
the exchange sets allowable grades (e.g., No. 2 hard winter wheat or No. 1 soft red wheat).
The place or means of delivery of the commodity is specified as well. Delivery of agricultural commodities is made by transfer of warehouse receipts issued by approved warehouses. For financial futures, delivery may be made by wire transfer; for index futures,
delivery may be accomplished by a cash settlement procedure such as those for index
options. Although the futures contract technically calls for delivery of an asset, delivery
rarely occurs. Instead, parties to the contract much more commonly close out their positions before contract maturity, taking gains or losses in cash.
Because the futures exchange specifies all the terms of the contract, the traders need
bargain only over the futures price. The trader taking the long position commits to purchasing the commodity on the delivery date. The trader who takes the short position commits to delivering the commodity at contract maturity. The trader in the long position is
said to “buy” a contract; the short-side trader “sells” a contract. The words buy and sell
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PART VI
Options, Futures, and Other Derivatives
are figurative only, because a contract is not really bought or sold like a stock or bond; it
is entered into by mutual agreement. At the time the contract is entered into, no money
changes hands.
Figure 22.1 shows prices for several futures contracts as they appear in The Wall Street
Journal. The boldface heading lists in each case the commodity, the exchange where the
futures contract is traded, the contract size, and the pricing unit. The first agricultural contract listed is for corn, traded on the Chicago Board of Trade (CBT). (The CBT merged
with the Chicago Mercantile Exchange in 2007 but, for now, maintains a separate identity.)
Each contract calls for delivery of 5,000 bushels, and prices in the entry are quoted in cents
per bushel.
The next several rows detail price data for contracts expiring on various dates. The
March 2013 maturity corn contract, for example, opened during the day at a futures price
of 720.25 cents per bushel. The highest futures price during the day was 726, the lowest
was 714.50, and the settlement price (a representative trading price during the last few
minutes of trading) was 724.25. The settlement price increased by 3.50 cents from the
previous trading day. Finally, open interest, or the number of outstanding contracts, was
494,588. Similar information is given for each maturity date.
The trader holding the long position, that is, the person who will purchase the good,
profits from price increases. Suppose that when the contract matures in March, the price
of corn turns out to be 729.25 cents per bushel. The long-position trader who entered the
contract at the futures price of 724.25 cents therefore earns a profit of 5 cents per bushel.
As each contract calls for delivery of 5,000 bushels, the profit to the long position equals
5,000 3 $.05 5 $250 per contract. Conversely, the short position loses 5 cents per bushel.
The short position’s loss equals the long position’s gain.
To summarize, at maturity:
Profit to long 5 Spot price at maturity 2 Original futures price
Profit to short 5 Original futures price 2 Spot price at maturity
where the spot price is the actual market price of the commodity at the time of the delivery.
The futures contract, therefore, is a zero-sum game, with losses and gains netting out to
zero. Every long position is offset by a short position. The aggregate profits to futures trading, summing over all investors, also must be zero, as is the net exposure to changes in the
commodity price. For this reason, the establishment of a futures market in a commodity
should not have a major impact on prices in the spot market for that commodity.
Figure 22.2, panel A is a plot of the profits realized by an investor who enters the long
side of a futures contract as a function of the price of the asset on the maturity date. Notice
that profit is zero when the ultimate spot price, PT, equals the initial futures price, F0.
Profit per unit of the underlying asset rises or falls one-for-one with changes in the final
spot price. Unlike the payoff of a call option, the payoff of the long futures position can
be negative: This will be the case if the spot price falls below the original futures price.
Unlike the holder of a call, who has an option to buy, the long futures position trader cannot simply walk away from the contract. Also unlike options, in the case of futures there is
no need to distinguish gross payoffs from net profits. This is because the futures contract
is not purchased; it is simply a contract that is agreed to by two parties. The futures price
adjusts to make the present value of entering into a new contract equal to zero.
The distinction between futures and options is highlighted by comparing panel A of
Figure 22.2 to the payoff and profit diagrams for an investor in a call option with exercise
price, X, chosen equal to the futures price F0 (see panel C). The futures investor is exposed
to considerable losses if the asset price falls. In contrast, the investor in the call cannot lose
more than the cost of the option.
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Futures Contracts | WSJ.com/commodities
Open
Contract
High hi lo low
Settle
Copper-High (CMX)-25,000 Ibs.; $ per Ib.
Jan
March
3.6580
3.6725
3.6760
3.6965
Gold (CMX)-100 troy oz.; $ per troy oz.
Chg
3.6645 20.0065
3.6765 20.0080
3.6520
3.6610
Feb
April
June
Aug
Oct
Dec
1683.80 1685.80
1664.20
1669.90
1686.40 1688.10
1666.40
1672.10
1688.10 1689.60
1669.50
1674.20
1689.00 1689.00
1670.60
1676.00
1689.50 1690.20
1673.10
1677.70
1694.70 1694.70
1674.40
1679.60
miNY Gold (CMX) -50 troy oz.; $ per troy oz.
Feb
1684.25 1684.50
1664.75
1670.00
April
1683.50 1684.00
1667.75
1672.00
June
1684.00 1684.00
1674.50
1674.25
Palladium (NYM)-50 troy oz.; $ per troy oz.
Feb
718.80
718.80
718.70
726.30
March
725.20
728.80
716.60
726.70
June
725.55
729.90
719.20
728.15
Platinum (NYM)-50 troy oz.; $ per troy oz.
Jan
1681.80 1681.80
1674.00
1682.80
Feb
1673.00 1680.40
1673.00
1680.80
Silver (CMX)-5,000 troy oz.; $ per troy oz.
Jan
31.735
31.750
31.735
31.695
March
32.240
32.245
31.605
31.722
miNY Silver (CMX)-2500 troy oz.; $ per troy oz.
March
32.138
32.438
31.613
31.725
May
31.863
31.900
31.838
31.775
Sept
32.038
32.038
32.038
31.875
Crude Oil, Light Sweet (NYM)-1,000 bbls.; $ per bbl.
March
95.50
96.68
95.12
95.95
April
95.89
97.16
95.60
96.45
May
96.38
97.58
96.00
96.90
Aug
96.42
97.57
96.32
97.31
Sept
96.08
97.58
96.07
97.07
Oct
96.10
96.96
96.04
96.77
Heating Oil No.2 (NYM)-42,000 gal.; $ per gal.
Feb
3.0795
3.0958
3.0693
3.0864
March
3.0674
3.0848
3.0561
3.0764
Gasoline-NY RBOB (NYM)-42,000 gal.; $ per gal.
Feb
2.8335
2.8762
2.8171
2.8629
March
2.8495
2.8882
2.8316
2.8767
Natural Gas (NYM)-10,000 MMBtu.; $ per MMBtu.
Feb
3.570
3.592
3.441
3.446
March
3.569
3.588
3.447
3.454
April
3.597
3.613
3.482
3.490
May
3.653
3.667
3.540
3.549
Oct
3.804
3.826
3.703
3.715
Jan’14
4.218
4.219
4.123
4.135
Open
interest
411
103,266
216.80
216.80
216.80
216.80
216.90
216.90
160,918
149,804
38,688
24,841
11,949
26,055
216.70
216.90
216.75
2,282
96
12
0.50
0.50
0.65
4
30,654
1,480
28.90
28.00
47
13
20.714
20.717
17
78,766
20.714
20.722
20.717
295
11
1
0.72
0.70
0.69
0.73
0.71
0.69
295,685
102,709
86,130
38,272
51,513
41,212
.0083
.0112
46,296
83,625
.0291
.0278
39,099
121,519
2.108
2.099
2.091
2.088
2.080
2.068
47,921
320,563
159,548
94,821
104,406
78,522
Agriculture Futures
Corn (CBT)-5,000 bu.; cents per bu.
March
Dec
720.25
589.75
726.00
591.00
Ethanol (CBT)-29,000 gal.; $ per gal.
724.25
585.50
714.50
583.75
Feb
2.367
2.383
2.367
March
2.378
2.397
2.374
Oats (CBT)-5,000 bu.; cents per bu.
March
360.00
361.25
355.50
May
365.50
367.75
362.75
Soybeans (CBT)-5,000 bu.; cents per bu.
March
1437.00 1439.00
1415.00
May
1426.75 1428.25
1404.50
Soybean Meal (CBT)-100 tons.; $ per ton.
March
416.50
417.00
408.00
MAy
412.10
412.50
404.20
Soybean Oil (CBT)-60,000 lbs.; cents per lb.
March
52.04
52.20
51.50
May
52.40
52.60
51.87
Rough Rice (CBT)-2,000 cwt.; cents per cwt.
March
1530.50 1537.00
1523.00
May
1562.00 1562.00
1554.50
Wheat (CBT)-5,000 bu.; cents per bu.
March
775.50
775.75
763.00
July
788.25
788.75
777.00
Wheat (KC)-5,000 bu.; cents per bu.
March
819.00
820.00
816.75
July
n.a.
n.a.
835.25
Wheat (MPLS)-5,000 bu.; cents per bu.
March
860.00
860.75
852.25
May
872.25
872.25
864.50
Cattle-Feeder (CME)-50,000 lbs.; cents per lb.
Jan
144.425 145.000
144.100
March
147.300 148.575
146.725
Cattle-Live (CME)-40,000 lbs.; cents per lb.
Feb
126.100 126.875
125.625
April
130.650 131.425
130.175
Contract
High hi lo low
Chg
Open
interest
146.55
149.50
23.85
23.85
90,658
29,315
18.49
18.46
2.01
2.07
341,806
167,304
20.88
21.23
2.67
2.43
958
4,057
82.89
82.20
2.41
1.53
132,025
34,978
113.20
114.30
22.40
22.20
14,247
4,127
146-050 146-150
145-090
145-270
144-170 144-300
143-300
144-120
Treasury Notes (CBT)-$100,000; pts 32nds of 100%
March
132-120 132-175
131-315
132-075
June
131-125 131-130
131-000
131-060
5 Yr. Treasury Notes (CBT)-$100,000; pts 32nds of 100%
March
124-082 124-102
124-020
124-060
June
123-255 123-267
123-207
123-245
2 Yr. Treasury Notes (CBT)-$200,000; pts 32nds of 100%
March
110-077 110-080
110-070
110-075
June
110-070 110-075
110-065
110-067
30 Day Federal Funds (CBT)-$5,00,000; 100-daily avg.
Jan
99.858
99.858
99.855
99.858
March
...
99.875
99.865
99,870
10 Yr. Int. Rate Swaps (CBT)-$100,000; pts 32nds of 100%
March
118.922 118.922
118.422
118.609
1 Month Libor (CME)-$3,000,000; pts of 100%
Feb
...
...
...
99.7975
Eurodollar (CME)-$1,000,000; pts of 100%
Feb
99.7025 99.7050
99.7000
99.7025
March
99.7000 99.7050
99.7000
99.7000
June
99.6850 99.6850
99.6800
99.6800
Dec
99.6400 99.6400
99.6250
99.6300
24.0
24.0
548,615
747
22.0
22.0
1,865,707
24,269
21.0
21.0
1,527,948
3,958
...
...
1,007,447
1,956
...
...
57,169
36,813
2.078
9,349
...
3,622
.0025
...
...
2.0050
53,143
835,707
706,194
728,161
Open
Metal & Petroleum Futures
3.50
24.75
494,588
223,643
2.382
2.396
.01
.01
1,131
1,605
361.00
367.75
1.25
...
7,404
2,438
1435.25
1423.75
21.75
23.00
226,757
140,830
414.70
410.60
21.80
21.50
127,823
56,552
52.11
52.49
.08
.00
155,915
59,010
1528.50
1560.50
22.00
22.00
12,472
1,490
768.50
782.25
26.25
26.50
226.800
83,615
821.50
840.50
28.75
28.00
91,904
35,595
855.50
867.50
25.00
24.75
20,338
9,776
144.600
147.950
.400
.800
2,205
14,972
125.875
130.350
.100
2.100
47,761
155,046
Coffee (ICE-US)-37,500 lbs.; cents per Ib.
March
May
150.65
151.30
146.10
153.75
154.20
149.10
Sugar-World (ICE-US)-112,000 lbs.; cents per Ib.
March
18.54
18.59
18.31
May
18.59
18.59
18.30
Sugar-Domestic (ICE-US)-112,000 lbs.; cents per Ib.
March
21.00
21.00
20.55
May
21.45
21.45
21.10
Cotton (ICE-US)-50,000 Ibs.; cents per Ib.
March
80.10
84.00
80.00
May
80.35
82.95
80.15
Orange Juice (ICE-US)-15,000 Ibs.; cents per Ib.
March
115.60
116.80
113.00
May
116.55
117.40
114.30
Settle
Interest Rate Futures
Treasury Bonds (CBT)-$100,000; pts 32nds of 100%
March
June
Currency Futures
Japanese Yen (CME)-¥12,500,000; $ per 100¥
March
June
1.1296
1.1310
1.1313
1.1310
1.1045
1.1056
1.1119
1.1127
2.0163
2.0164
204,630
991
March
June
.9992
.9970
.9997
.9978
.9952
.9934
.9958
.9938
2.0036
2.0036
139,041
2,204
March
June
1.5828
1.5830
1.5848
1.5837
1.5752
1.5750
1.5787
1.5781
2.0049
2.0049
158,952
236
March
1.0758
1.0782
1.0730
1.0779
.0016
40,548
1.0495
1.0509
1.0405
1.0433
1.0405
1.0441
1.0355
1.0366
1.0320
1.0320
1.0310
1.0303
Mexican Peso (CME)-MXN 500,000; $ per 10MXN
March
.07830
.07893
.07820
.07865
June
.07758
.07783
.07758
.07798
Euro (CME)-€125,000; $ per €
March
1.3317
1.3398
1.3291
1.3378
June
1.3329
1.3400
1.3300
1.3385
Euro/Japanese Yen (ICE-US)-€125,000; ¥ per €
March
117.675 11.6750
11.6750
120.3200
Euro/British Pound (ICE-US)-€125,000; £ per €
March
...
...
...
0.8474
Euro/Swiss France (ICE-US)-€125,000; CHF per €
March
1.2376
1.2445
1.2360
1.2411
2.0074
2.0073
2.0073
204,989
331
1
.00015
.00015
187,352
122
.0053
.0052
216,667
1,777
2.2100
4,186
.0060
2,992
.0031
8,588
Canadian Dollar (CME)-CAD 100,000; $ per CAD
British Pound (CME)-£62,500; $ per £
Swiss Franc (CME)-CHF 125,000; $ per CHF
Australian Dollar (CME)-AUD 100,000; $ per AUD
March
June
Sept
Index Futures
DJ Industrial Average (CBT)-$10 3 index
March
13703
13781
62
10,749
13702
13818
13701
S&P 500 Index (CME)-$250 3 index
March
1483.80 1497.50
1483.30
June
1482.90 1491.60
1480.60
Mini S&P 500 (CME)-$50 3 index
March
1483.25 1497.75
1482.75
June
1477.25 1491.25
1476.25
Mini S&P Midcap 400 (CME)-$100 3 index
March
1077.50 1089.20
1075.70
Nasdaq 100 (CME)-$100 3 index
March
2716.75 2743.50
2709.75
13781
62
112,004
1491.80
1485.10
1.50
1.50
194,350
4,188
1491.75
1485.00
1.50
1.50
2,972,476
18,475
1086.30
4.70
116,953
2718.25
240.75
10,891
13706
13815
Mini DJ Industrial Average (CBT)-$5 3 index
March
Figure 22.1 Futures listings
Source: The Wall Street Journal, January 25, 2013. Reprinted by permission of The Wall Street Journal, © 2013
Dow Jones & Company, Inc. All Rights Reserved Worldwide.
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Options, Futures, and Other Derivatives
Profit
Profit
Payoff
Profit
PT
F0
F0
A: Long Futures Profit = PT – F0
PT
B: Short Futures Profit = F0 – PT
X
PT
C: Buy a Call Option
Figure 22.2 Profits to buyers and sellers of futures and options contracts
Figure 22.2, panel B is a plot of the profits realized by an investor who enters the short
side of a futures contract. It is the mirror image of the profit diagram for the long position.
CONCEPT CHECK
22.1
a. Compare the profit diagram in Figure 22.2, panel B to the payoff diagram for a long position in a put
option. Assume the exercise price of the option equals the initial futures price.
b. Compare the profit diagram in Figure 22.2, panel B to the payoff diagram for an investor who writes a
call option.
Existing Contracts
Futures and forward contracts are traded on a wide variety of goods in four broad categories: agricultural commodities, metals and minerals (including energy commodities), foreign currencies, and financial futures (fixed-income securities and stock market indexes).
In addition to indexes on broad stock indexes, one can now trade single-stock futures on
individual stocks and narrowly based indexes. OneChicago has operated an entirely electronic market in single-stock futures since 2002. The exchange maintains futures markets
in actively traded stocks with the most liquidity as well as in some popular ETFs such as
those on the S&P 500 (ticker SPY), the NASDAQ-100 (QQQ), and the Dow Jones Industrial Average (DIA). However, trading volume in single-stock futures has to date been
somewhat disappointing.
Table 22.1 offers a sample of the various contracts trading in 2013. Contracts now trade
on items that would not have been considered possible only a few years ago, such as electricity as well as weather futures and options contracts. Weather derivatives (which trade
on the Chicago Mercantile Exchange) have payoffs that depend on average weather conditions, for example, the number of degree-days by which the temperature in a region
exceeds or falls short of 65 degrees Fahrenheit. The potential use of these derivatives in
managing the risk surrounding electricity or oil and natural gas use should be evident.
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Foreign
Currencies
British pound
Canadian dollar
Japanese yen
Euro
Swiss franc
Australian dollar
Mexican peso
Brazilian real
Agricultural
Corn
Oats
Soybeans
Soybean meal
Soybean oil
Wheat
Barley
Flaxseed
Canola
Rye
Cattle
Hogs
Pork bellies
Cocoa
Coffee
Cotton
Milk
Orange juice
Sugar
Lumber
Rice
Metals and
Energy
Copper
Aluminum
Gold
Platinum
Palladium
Silver
Crude oil
Heating oil
Gas oil
Natural gas
Gasoline
Propane
Commodity index
Electricity
Weather
Futures Markets
Interest Rate Futures
Equity Indexes
Eurodollars
Euroyen
Euro-denominated bond
Euroswiss
Sterling
British government bond
German government bond
Italian government bond
Canadian government bond
Treasury bonds
Treasury notes
Treasury bills
LIBOR
EURIBOR
Municipal bond index
Federal funds rate
Bankers’ acceptance
Interest rate swaps
S&P 500 index
Dow Jones Industrials
S&P Midcap 400
NASDAQ 100
NYSE index
Russell 2000 index
Nikkei 225 (Japanese)
FTSE index (British)
CAC-40 (French)
DAX-30 (German)
All ordinary (Australian)
Toronto 35 (Canadian)
Dow Jones Euro STOXX 50
Industry indexes, e.g.,
Banking
Telecom
Utilities
Health care
Technology
775
Table 22.1
Sample of futures contracts
While Table 22.1 includes many contracts, the large and ever-growing array of markets
makes this list necessarily incomplete. The nearby box discusses some comparatively fanciful futures markets, sometimes called prediction markets, in which payoffs may be tied
to the winner of presidential elections, the box office receipts of a particular movie, or
anything else in which participants are willing to take positions.
Outside the futures markets, a well-developed network of banks and brokers has established a forward market in foreign exchange. This forward market is not a formal exchange
in the sense that the exchange specifies the terms of the traded contract. Instead, participants in a forward contract may negotiate for delivery of any quantity of goods, whereas
in the formal futures markets contract size and delivery dates are set by the exchange. In
forward arrangements, banks and brokers simply negotiate contracts for clients (or themselves) as needed. This market is huge. In London alone, the largest currency market,
around $2 trillion of currency trades each day.
22.2 Trading Mechanics
The Clearinghouse and Open Interest
Until about 10 years ago, most futures trades in the United States occurred among floor
traders in the “trading pit” for each contract. Today, however, trading is overwhelmingly
conducted through electronic networks, particularly for financial futures.
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Prediction Markets
If you find S&P 500 or T-bond contracts a bit dry, perhaps
you’d be interested in futures contracts with payoffs that
depend on the winner of the next presidential election, or
the severity of the next influenza season, or the host city
of the 2024 Olympics. You can now find “futures markets”
in these events and many others.
For example, both Intrade (www.intrade.com) and Iowa
Electronic Markets (www.biz.uiowa.edu/iem) maintain
presidential futures markets. In July 2011, you could have
purchased a contract that would pay off $1 in November 2012
if the Republican candidate won the presidential race but
nothing if he lost. The contract price (expressed as a percentage of face value) therefore may be viewed as the probability
of a Republican victory, at least according to the consensus
view of market participants at the time. If you believed in
July that the probability of a Republican victory was 55%, you
would have been prepared to pay up to $.55 for the contract.
Alternatively, if you had wished to bet against the Republicans, you could have sold the contract. Similarly, you could
bet on (or against) a Democrat victory using the Democrat
contract. (When there are only two relevant parties, betting
on one is equivalent to betting against the other, but in other
elections, such as primaries where there are several viable
candidates, selling one candidate’s contract is not the same
as buying another’s.)
The accompanying figure shows the price of each contract from July 2011 through Election Day 2012. The price
clearly tracks each candidate’s perceived prospects. You
can see Obama’s price rise dramatically in the days shortly
before the election as it became ever clearer that he would
win the election.
Interpreting prediction market prices as probabilities
actually requires a caveat. Because the contract payoff is
risky, the price of the contract may reflect a risk premium.
Therefore, to be precise, these probabilities are actually
risk-neutral probabilities (see Chapter 21). In practice, however, it seems unlikely that the risk premium associated
with these contracts is substantial.
Prediction markets for the 2012 presidential election.
the election. Price is in cents.
Contract on each party pays $1 if the party wins
100
Democrat
90
Republican
80
70
Price
60
50
40
30
20
10
08-Nov-12
11-Oct-12
23-Sep-12
05-Sep-12
18-Aug-12
31-Jul-12
13-Jul-12
25-Jun-12
07-Jun-12
20-May-12
02-May-12
14-Apr-12
27-Mar-12
20-Feb-12
09-Mar-12
15-Jan-12
02-Feb-12
28-Dec-11
10-Dec-11
22-Nov-11
04-Nov-11
17-Oct-11
29-Sep-11
11-Sep-11
24-Aug-11
19-Jul-11
06-Aug-11
01-Jul-11
0
Date
Source: Iowa Electronic Markets, downloaded January 25, 2013.
The impetus for this shift originated in Europe, where electronic trading is the norm.
Eurex, which is jointly owned by the Deutsche Börse and Swiss exchange, is among the
world’s largest derivatives exchanges. It operates a fully electronic trading and clearing
platform and, in 2004, received clearance from regulators to list contracts in the U.S. In
response, the Chicago Board of Trade adopted an electronic platform provided by Eurex’s
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European rival Euronext.liffe,1 and the CBOT’s Treasury contracts are now traded electronically. The Chicago Mercantile Exchange maintains another electronic trading system
called Globex. The electronic markets enable trading around the clock.
The CBOT and CME merged in 2007 into one combined company, named the CME
Group, with all electronic trading from both exchanges moving onto Globex. It seems
inevitable that electronic trading will continue to displace floor trading.
Once a trade is agreed to, the clearinghouse enters the picture. Rather than having
the long and short traders hold contracts with each other, the clearinghouse becomes the
seller of the contract for the long position and the buyer of the contract for the short position. The clearinghouse is obligated to deliver the commodity to the long position and to
pay for delivery from the short; consequently, the clearinghouse’s position nets to zero.
This arrangement makes the clearinghouse the trading partner of each trader, both long
and short. The clearinghouse, bound to perform on its side of each contract, is the only
party that can be hurt by the failure of any trader to observe the obligations of the futures
contract. This arrangement is necessary because a futures contract calls for future performance, which cannot be as easily guaranteed as an immediate stock transaction.
Figure 22.3 illustrates the role of the clearinghouse. Panel A shows what would happen
in the absence of the clearinghouse. The trader in the long position would be obligated to
pay the futures price to the short-position trader, and the trader in the short position would
be obligated to deliver the commodity. Panel B shows how the clearinghouse becomes
an intermediary, acting as the trading partner for each side of the contract. The clearinghouse’s position is neutral, as it takes a long and a short position for each transaction.
The clearinghouse makes it possible for traders to liquidate positions easily. If you are
currently long in a contract and want to undo your position, you simply instruct your broker to enter the short side of a contract to close out your position. This is called a reversing
trade. The exchange nets out your long and short positions, reducing your net position to
zero. Your zero net position with the clearinghouse eliminates the need to fulfill at maturity
either the original long or reversing
short position.
The open interest on the conA
Money
tract is the number of contracts outShort
Long
standing. (Long and short positions
Position
Position
are not counted separately, meaning
that open interest can be defined
Commodity
either as the number of long or short
contracts outstanding.) The clearB
inghouse’s position nets out to zero,
Money
Money
and so is not counted in the compuLong
Short
Clearinghouse
tation of open interest. When conPosition
Position
tracts begin trading, open interest
is zero. As time passes, open interCommodity
Commodity
est increases as progressively more
contracts are entered.
There are many apocryphal stoFigure 22.3 Panel A, Trading without a clearinghouse. Panel B,
ries about futures traders who wake
Trading with a clearinghouse.
up to discover a small mountain of
1
Euronext.liffe is the international derivatives market of Euronext. It resulted from Euronext’s purchase of LIFFE
(the London International Financial Futures and Options Exchange) and a merger with the Lisbon exchange in
2002. Euronext was itself the result of a 2000 merger of the exchanges of Amsterdam, Brussels, and Paris.
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wheat or corn on their front lawn. But the truth is that futures contracts rarely result in
actual delivery of the underlying asset. Traders establish long or short positions in contracts that will benefit from a rise or fall in the futures price and almost always close out,
or reverse, those positions before the contract expires. The fraction of contracts that result
in actual delivery is estimated to range from less than 1% to 3%, depending on the commodity and activity in the contract. In the unusual case of actual deliveries of commodities,
they occur via regular channels of supply, most often warehouse receipts.
You can see the typical pattern of open interest in Figure 22.1. In the copper contract,
for example, the January delivery contract is approaching maturity, and open interest is
small; most contracts have been reversed already. The greatest open interest is in the March
contract. For other contracts, for example, gold, for which the nearest maturity date isn’t
until February, open interest is typically highest in the nearest contract.
The Margin Account and Marking to Market
The total profit or loss realized by the long trader who buys a contract at time 0 and closes,
or reverses, it at time t is just the change in the futures price over the period, Ft 2 F0.
Symmetrically, the short trader earns F0 2 Ft.
The process by which profits or losses accrue to traders is called marking to market. At
initial execution of a trade, each trader establishes a margin account. The margin is a security account consisting of cash or near-cash securities, such as Treasury bills, that ensures
the trader is able to satisfy the obligations of the futures contract. Because both parties to
a futures contract are exposed to losses, both must post margin. To illustrate, return to the
first corn contract listed in Figure 22.1. If the initial required margin on corn, for example,
is 10%, then the trader must post $3,620 per contract of the margin account. This is 10% of
the value of the contract, $7.24 per bushel 3 5,000 bushels per contract.
Because the initial margin may be satisfied by posting interest-earning securities, the
requirement does not impose a significant opportunity cost of funds on the trader. The initial margin is usually set between 5% and 15% of the total value of the contract. Contracts
written on assets with more volatile prices require higher margins.
On any day that futures contracts trade, futures prices may rise or may fall. Instead of waiting until the maturity date for traders to realize all gains and losses, the clearinghouse requires
all positions to recognize profits as they accrue daily. If the futures price of corn rises from
724 to 726 cents per bushel, the clearinghouse credits the margin account of the long position
for 5,000 bushels times 2 cents per bushel, or $100 per contract. Conversely, for the short
position, the clearinghouse takes this amount from the margin account for each contract held.
This daily settling is called marking to market. It means the maturity date of the contract does not govern realization of profit or loss. Instead, as futures prices change, the proceeds accrue to the trader’s margin account immediately. We will provide a more detailed
example of this process shortly.
Marking to market is the major way in which futures
and forward contracts differ, besides contract standardCONCEPT CHECK 22.2
ization. Futures follow this pay-(or receive-)as-you-go
method. Forward contracts are simply held until maturity,
What must be the net inflow or outlay from
and no funds are transferred until that date, although the
marking to market for the clearinghouse?
contracts may be traded.
If a trader accrues sustained losses from daily marking
to market, the margin account may fall below a critical
value called the maintenance margin. If the value of the account falls below this value,
the trader receives a margin call, requiring that the margin account be replenished or the
position be reduced to a size commensurate with the remaining funds. Margins and
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margin calls safeguard the position of the clearinghouse. Positions are closed out before
the margin account is exhausted—the trader’s losses are covered, and the clearinghouse is
not put at risk.
Example 22.1
Maintenance Margin
Suppose the maintenance margin is 5% while the initial margin was 10% of the value
of the corn, or $3,620. Then a margin call will go out when the original margin account
has fallen in half, by about $1,810. Each 1-cent decline in the corn price results in a $50
loss to the long position. Therefore, the futures price need only fall by 37 cents (or 5%
of its current value) to trigger a margin call.
On the contract maturity date, the futures price will equal the spot price of the commodity. As a maturing contract calls for immediate delivery, the futures price on that
day must equal the spot price—the cost of the commodity from the two competing
sources is equalized in a competitive market.2 You may obtain delivery of the commodity either by purchasing it directly in the spot market or by entering the long side of a
futures contract.
A commodity available from two sources (spot or futures market) must be priced identically, or else investors will rush to purchase it from the cheap source in order to sell it in
the high-priced market. Such arbitrage activity could not persist without prices adjusting
to eliminate the arbitrage opportunity. Therefore, the futures price and the spot price must
converge at maturity. This is called the convergence property.
For an investor who establishes a long position in a contract now (time 0) and holds that
position until maturity (time T), the sum of all daily settlements will equal FT 2 F0, where
FT stands for the futures price at contract maturity. Because of convergence, however, the
futures price at maturity, FT, equals the spot price, PT, so total futures profits also may be
expressed as PT 2 F0. Thus we see that profits on a futures contract held to maturity perfectly track changes in the value of the underlying asset.
Example 22.2
Marking to Market
Assume the current futures price for silver for delivery 5 days from today is $30.10 per
ounce. Suppose that over the next 5 days, the futures price evolves as follows:
Day
0 (today)
1
2
3
4
5 (delivery)
Futures Price
$30.10
30.20
30.25
30.18
30.18
30.21
2
Small differences between the spot and futures price at maturity may persist because of transportation costs, but
this is a minor factor.
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The daily mark-to-market settlements for each contract held by the long position will
be as follows:
Day
1
2
3
4
5
Profit (Loss) per Ounce 3 5,000 Ounces/Contract 5 Daily Proceeds
30.20 2 30.10 5 .10
30.25 2 30.20 5 .05
30.18 2 30.25 5 2.07
30.18 2 30.18 5
0
30.21 2 30.18 5 .03
$500
250
2350
0
150
Sum 5 $550
The profit on Day 1 is the increase in the futures price from the previous day, or
($30.20 2 $30.10) per ounce. Because each silver contract on the Commodity Exchange
(CMX) calls for purchase and delivery of 5,000 ounces, the total profit per contract is
5,000 times $.10, or $500. On Day 3, when the futures price falls, the long position’s
margin account will be debited by $350. By Day 5, the sum of all daily proceeds is $550.
This is exactly equal to 5,000 times the difference between the final futures price of
$30.21 and original futures price of $30.10. Because the final futures price equals the
spot price on that date, the sum of all the daily proceeds (per ounce of silver held long)
also equals PT 2 F0.
Cash versus Actual Delivery
Most futures contracts call for delivery of an actual commodity such as a particular grade
of wheat or a specified amount of foreign currency if the contract is not reversed before
maturity. For agricultural commodities, where quality of the delivered good may vary, the
exchange sets quality standards as part of the futures contract. In some cases, contracts
may be settled with higher- or lower-grade commodities. In these cases, a premium or discount is applied to the delivered commodity to adjust for the quality difference.
Some futures contracts call for cash settlement. An example is a stock index futures
contract where the underlying asset is an index such as the Standard & Poor’s 500 or the
New York Stock Exchange Index. Delivery of every stock in the index clearly would be
impractical. Hence the contract calls for “delivery” of a cash amount equal to the value
that the index attains on the maturity date of the contract. The sum of all the daily settlements from marking to market results in the long position realizing total profits or losses of
ST 2 F0, where ST is the value of the stock index on the maturity date T and F0 is the original futures price. Cash settlement closely mimics actual delivery, except the cash value of
the asset rather than the asset itself is delivered.
More concretely, the S&P 500 index contract calls for delivery of $250 times the value
of the index. At maturity, the index might list at 1,200, the market-value-weighted index
of the prices of all 500 stocks in the index. The cash settlement contract calls for delivery
of $250 3 1,200, or $300,000 cash in return for $250 times the futures price. This yields
exactly the same profit as would result from directly purchasing 250 units of the index for
$300,000 and then delivering it for $250 times the original futures price.
Regulations
Futures markets are regulated by the federal Commodities Futures Trading Commission.
The CFTC sets capital requirements for member firms of the futures exchanges, authorizes
trading in new contracts, and oversees maintenance of daily trading records.
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The futures exchange may set limits on the amount by which futures prices may
change from one day to the next. For example, if the price limit on silver contracts were
set at $1 and silver futures close today at $22.10 per ounce, then trades in silver tomorrow may vary only between $21.10 and $23.10 per ounce. The exchanges may increase
or reduce price limits in response to perceived changes in price volatility of the contract.
Price limits are often eliminated as contracts approach maturity, usually in the last month
of trading.
Price limits traditionally are viewed as a means to limit violent price fluctuations. This
reasoning seems dubious. Suppose an international monetary crisis overnight drives up the
spot price of silver to $30. No one would sell silver futures at prices for future delivery as
low as $22.10. Instead, the futures price would rise each day by the $1 limit, although the
quoted price would represent only an unfilled bid order—no contracts would trade at the
low quoted price. After several days of limit moves of $1 per day, the futures price would
finally reach its equilibrium level, and trading would occur again. This process means no
one could unload a position until the price reached its equilibrium level. We conclude that
price limits offer no real protection against fluctuations in equilibrium prices.
Taxation
Because of the mark-to-market procedure, investors do not have control over the tax year
in which they realize gains or losses. Instead, price changes are realized gradually, with
each daily settlement. Therefore, taxes are paid at year-end on cumulated profits or losses
regardless of whether the position has been closed out. As a general rule, 60% of futures
gains or losses are treated as long term, and 40% as short term.
22.3 Futures Markets Strategies
Hedging and Speculation
Hedging and speculating are two polar uses of futures markets. A speculator uses a futures
contract to profit from movements in futures prices, a hedger to protect against price
movement.
If speculators believe prices will increase, they will take a long position for expected
profits. Conversely, they exploit expected price declines by taking a short position.
Example 22.3
Speculating with Oil Futures
Suppose you believe that crude oil prices are going to increase and therefore decide to
purchase crude oil futures. Each contract calls for delivery of 1,000 barrels of oil, so for
every dollar increase in the futures price of crude, the long position gains $1,000 and the
short position loses that amount.
Conversely, suppose you think that prices are heading lower and therefore sell a
contract. If crude oil prices do in fact fall, then you will gain $1,000 per contract for
every dollar that prices decline.
If the futures price for delivery in February is $91.86 and crude oil is selling at the
contract maturity date for $93.86, the long side will profit by $2,000 per contract purchased. The short side will lose an identical amount on each contract sold. On the other
hand, if oil has fallen to $89.86, the long side will lose, and the short side will gain,
$2,000 per contract.
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Why does a speculator buy a futures contract? Why not buy the underlying asset
directly? One reason lies in transaction costs, which are far smaller in futures markets.
Another important reason is the leverage that futures trading provides. Recall that
futures contracts require traders to post margin considerably less than the value of the asset
underlying the contract. Therefore, they allow speculators to achieve much greater leverage than is available from direct trading in a commodity.
Example 22.4
Futures and Leverage
Suppose the initial margin requirement for the oil contract is 10%. At a current
futures price of $91.86, and contract size of 1,000 barrels, this would require margin
of .10 3 91.86 3 1,000 5 $9,186. A $2 jump in oil prices represents an increase of
2.18%, and results in a $2,000 gain on the contract for the long position. This is a
percentage gain of 21.8% in the $9,186 posted as margin, precisely 10 times the percentage increase in the oil price. The 10-to-1 ratio of percentage changes reflects the
leverage inherent in the futures position, because the contract was established with an
initial margin of one-tenth the value of the underlying asset.
Hedgers, by contrast, use futures to insulate themselves against price movements. A firm
planning to sell oil, for example, might anticipate a period of market volatility and wish to
protect its revenue against price fluctuations. To hedge the total revenue derived from the sale,
the firm enters a short position in oil futures. As the following example illustrates, this locks in
its total proceeds (i.e., revenue from the sale of the oil plus proceeds from its futures position).
Example 22.5
Hedging with Oil Futures
Consider an oil distributor planning to sell 100,000 barrels of oil in February that wishes
to hedge against a possible decline in oil prices. Because each contract calls for delivery
of 1,000 barrels, it would sell 100 contracts that mature in February. Any decrease in
prices would then generate a profit on the contracts that would offset the lower sales
revenue from the oil.
To illustrate, suppose that the only three possible prices for oil in February are $89.86,
$91.86, and $93.86 per barrel. The revenue from the oil sale will be 100,000 times the
price per barrel. The profit on each contract sold will be 1,000 times any decline in the
futures price. At maturity, the convergence property ensures that the final futures price
will equal the spot price of oil. Therefore, the profit on the 100 contracts sold will equal
100,000 3 (F 0 2 PT), where PT is the oil price on the delivery date, and F 0 is the original
futures price, $91.86.
Now consider the firm’s overall position. The total revenue in February can be computed as follows:
Oil Price in February, PT
Revenue from oil sale: 100,000 3 PT
1 Profit on futures: 100,000 3 (F0 2 PT)
TOTAL PROCEEDS
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$89.86
$91.86
$93.86
$8,986,000
200,000
$9,186,000
$9,186,000
0
$9,186,000
$9,386,000
2200,000
$9,186,000
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The revenue from the oil sale plus the proceeds from the contracts equals the current
futures price, $91.86 per barrel. The variation in the price of the oil is precisely offset by
the profits or losses on the futures position. For example, if oil falls to $89.86 a barrel,
the short futures position generates $200,000 profit, just enough to bring total revenues
to $9,186,000. The total is the same as if one were to arrange today to sell the oil in
February at the futures price.
Figure 22.4 illustrates the nature of the hedge in Example 22.5. The upward-sloping
line is the revenue from the sale of oil. The downward-sloping line is the profit on the
futures contract. The horizontal line is the sum of sales revenue plus futures profits. This
line is flat, as the hedged position is independent of oil prices.
To generalize Example 22.5, note that oil will sell for PT per barrel at the maturity of
the contract. The profit per barrel on the futures will be F0 2 PT. Therefore, total revenue
is PT 1 (F0 2 PT) 5 F0, which is independent of the eventual oil price.
The oil distributor in Example 22.5 engaged in a short hedge, taking a short futures
position to offset risk in the sales price of a particular asset. A long hedge is the analogous
hedge for someone who wishes to eliminate the risk of an uncertain purchase price. For
example, a power supplier planning to purchase oil may be afraid that prices might rise by
the time of the purchase. As the following Concept Check illustrates, the supplier might
buy oil futures to lock in the net purchase price at the time of the transaction.
CONCEPT CHECK
22.3
Suppose as in Example 22.5 that oil will be selling in February for $89.86, $91.86, or $93.86 per barrel. Consider a firm that plans to buy 100,000 barrels of oil in February. Show that if the firm buys 100 oil contracts
today, its net expenditures in February will be hedged and equal to $9,186,000.
120
Hedged revenues are constant at $91.86 per barrel,
equal to the futures price
Proceeds (per Barrel)
100
80
Sales revenue increases with oil price
60
40
Profit on short futures position falls with oil price
20
0
85
–20
90
91.86
95
100
105
Oil Price
Sales Revenue per Barrel
Futures Profits per Barrel
Total Proceeds
Figure 22.4 Hedging revenues using futures, Example 22.5 (Futures price 5 $91.86)
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Exact futures hedging may be impossible for some goods because the necessary futures
contract is not traded. For example, a portfolio manager might want to hedge the value
of a diversified, actively managed portfolio for a period
of time. However, futures contracts are listed only on
CONCEPT CHECK 22.4
indexed portfolios. Nevertheless, because returns on the
manager’s diversified portfolio will have a high correWhat are the sources of risk to an investor who
lation with returns on broad-based indexed portfolios,
uses stock index futures to hedge an actively
an effective hedge may be established by selling index
managed stock portfolio? How might you estifutures contracts. Hedging a position using futures on
mate the magnitude of that risk?
another asset is called cross-hedging.
Basis Risk and Hedging
The basis is the difference between the futures price and the spot price.3 As we have noted,
on the maturity date of a contract, the basis must be zero: The convergence property implies
that FT 2 PT 5 0. Before maturity, however, the futures price for later delivery may differ
substantially from the current spot price.
In Example 22.5 we discussed the case of a short hedger who manages risk by entering
a short position to deliver oil in the future. If the asset and futures contract are held until
maturity, the hedger bears no risk. Risk is eliminated because the futures price and spot
price at contract maturity must be equal: Gains and losses on the futures and the commodity position will exactly cancel. However, if the contract and asset are to be liquidated
early, before contract maturity, the hedger bears basis risk, because the futures price and
spot price need not move in perfect lockstep at all times before the delivery date. In this
case, gains and losses on the contract and the asset may not exactly offset each other.
Some speculators try to profit from movements in the basis. Rather than betting on
the direction of the futures or spot prices per se, they bet on the changes in the difference
between the two. A long spot–short futures position will profit when the basis narrows.
Example 22.6
Speculating on the Basis
Consider an investor holding 100 ounces of gold, who is short one gold-futures contract. Suppose that gold today sells for $1,591 an ounce, and the futures price for June
delivery is $1,596 an ounce. Therefore, the basis is currently $5. Tomorrow, the spot
price might increase to $1,595, while the futures price increases to $1,599, so the basis
narrows to $4.
The investor’s gains and losses are as follows:
Gain on holdings of gold (per ounce): $1,595 2 $1,591 5 $4
Loss on gold futures position (per ounce): $1,599 2 $1,596 5 $3
The net gain is the decrease in the basis, or $1 per ounce.
A related strategy is a calendar spread position, where the investor takes a long position in a futures contract of one maturity and a short position in a contract on the same
commodity, but with a different maturity.4 Profits accrue if the difference in futures prices
3
Usage of the word basis is somewhat loose. It sometimes is used to refer to the futures-spot difference F 2 P, and
sometimes to the spot-futures difference P 2 F. We will consistently call the basis F 2 P.
4
Yet another strategy is an intercommodity spread, in which the investor buys a contract on one commodity and
sells a contract on a different commodity.
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between the two contracts changes in the hoped-for direction, that is, if the futures price
on the contract held long increases by more (or decreases by less) than the futures price on
the contract held short.
Example 22.7
Speculating on the Spread
Consider an investor who holds a September maturity contract long and a June contract
short. If the September futures price increases by 5 cents while the June futures price
increases by 4 cents, the net gain will be 5 cents 2 4 cents, or 1 cent. Like basis strategies, spread positions aim to exploit movements in relative price structures rather than to
profit from movements in the general level of prices.
22.4 Futures Prices
The Spot-Futures Parity Theorem
We have seen that a futures contract can be used to hedge changes in the value of the
underlying asset. If the hedge is perfect, meaning that the asset-plus-futures portfolio has
no risk, then the hedged position must provide a rate of return equal to the rate on other
risk-free investments. Otherwise, there will be arbitrage opportunities that investors will
exploit until prices are brought back into line. This insight can be used to derive the theoretical relationship between a futures price and the price of its underlying asset.
Suppose for simplicity that the S&P 500 index currently is at 1,000 and an investor
who holds $1,000 in a mutual fund indexed to the S&P 500 wishes to temporarily hedge
her exposure to market risk. Assume that the indexed portfolio pays dividends totaling
$20 over the course of the year, and for simplicity, that all dividends are paid at year-end.
Finally, assume that the futures price for year-end delivery of the S&P 500 contract is
1,010.5 Let’s examine the end-of-year proceeds for various values of the stock index if the
investor hedges her portfolio by entering the short side of the futures contract.
Final value of stock portfolio, ST
Payoff from short futures position
(equals F0 2 FT 5 $1,010 2 ST)
Dividend income
TOTAL
$ 970
40
$ 990
20
$1,010
0
$1,030
220
$1,050
240
$1,070
260
20
$1,030
20
$1,030
20
$1,030
20
$1,030
20
$1,030
20
$1,030
The payoff from the short futures position equals the difference between the original
futures price, $1,010, and the year-end stock price. This is because of convergence: The
futures price at contract maturity will equal the stock price at that time.
Notice that the overall position is perfectly hedged. Any increase in the value of the
indexed stock portfolio is offset by an equal decrease in the payoff of the short futures
position, resulting in a final value independent of the stock price. The $1,030 total payoff is
5
Actually, the futures contract calls for delivery of $250 times the value of the S&P 500 index, so that each
contract would be settled for $250 times the index. With the index at 1,000, each contract would hedge about
$250 3 1,000 5 $250,000 worth of stock. Of course, institutional investors would consider a stock portfolio of
this size to be quite small. We will simplify by assuming that you can buy a contract for one unit rather than 250
units of the index.
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the sum of the current futures price, F0 5 $1,010, and the $20 dividend. It is as though the
investor arranged to sell the stock at year-end for the current futures price, thereby eliminating price risk and locking in total proceeds equal to the futures price plus dividends paid
before the sale.
What rate of return is earned on this riskless position? The stock investment requires an
initial outlay of $1,000, whereas the futures position is established without an initial cash
outflow. Therefore, the $1,000 portfolio grows to a year-end value of $1,030, providing a
rate of return of 3%. More generally, a total investment of S0, the current stock price, grows
to a final value of F0 1 D, where D is the dividend payout on the portfolio. The rate of
return is therefore
Rate of return on hedged stock portfolio 5
(F0 1 D) 2 S0
S0
This return is essentially riskless. We observe F0 at the beginning of the period when
we enter the futures contract. While dividend payouts are not perfectly riskless, they are
highly predictable over short periods, especially for diversified portfolios. Any uncertainty
is extremely small compared to the uncertainty in stock prices.
Presumably, 3% must be the rate of return available on other riskless investments. If
not, then investors would face two competing risk-free strategies with different rates of
return, a situation that could not last. Therefore, we conclude that
(F0 1 D) 2 S0
5 rf
S0
Rearranging, we find that the futures price must be
F0 5 S0(1 1 rf) 2 D 5 S0(1 1 rf 2 d)
(22.1)
where d is the dividend yield on the stock portfolio, defined as D/S0. This result is called
the spot-futures parity theorem. It gives the normal or theoretically correct relationship
between spot and futures prices. Any deviation from parity would give rise to risk-free
arbitrage opportunities.
Example 22.8
Futures Market Arbitrage
Suppose that parity were violated. For example, suppose the risk-free interest rate
were only 1% so that according to Equation 22.1, the futures price should be
$1,000(1.01) 2 $20 5 $990. The actual futures price, F0 5 $1,010, is $20 higher than
its “appropriate” value. This implies that an investor can make arbitrage profits by shorting the relatively overpriced futures contract and buying the relatively underpriced stock
portfolio using money borrowed at the 1% market interest rate. The proceeds from this
strategy would be as follows:
Action
Borrow $1,000, repay with interest in 1 year
Buy stock for $1,000
Enter short futures position (F0 5 $1,010)
TOTAL
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Initial
Cash Flow
Cash Flow in
1 Year
11,000
21,000
0
0
21,000(1.01) 5 2$1,010
ST 1 $20 dividend
$1,010 2 ST
$20
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The net initial investment of the strategy is zero. But its cash flow in 1 year is $20 regardless of the stock price. In other words, it is riskless. This payoff is precisely equal to the
mispricing of the futures contract relative to its parity value, 1,010 2 990.
When parity is violated, the strategy to exploit the mispricing produces an arbitrage
profit—a riskless profit requiring no initial net investment. If such an opportunity existed, all
market participants would rush to take advantage of it. The results? The stock price would
be bid up, and/or the futures price offered down until Equation 22.1 is satisfied. A similar
analysis applies to the possibility that F0 is less than $990. In this case, you simply reverse the
strategy above to earn riskless profits. We conclude, therefore, that in a well-functioning
market in which arbitrage opportunities are competed away, F0 5 S0(1 1 rf) 2 D.
CONCEPT CHECK
22.5
Return to the arbitrage strategy laid out in Example 22.8. What would be the three steps of the strategy
if F0 were too low, say, $980? Work out the cash flows of the strategy now and in 1 year in a table like the
one in the example. Confirm that your profits equal the mispricing of the contract.
The arbitrage strategy of Example 22.8 can be represented more generally as follows:
Action
1. Borrow S0 dollars
2. Buy stock for S0
3. Enter short futures position
TOTAL
Initial Cash Flow
Cash Flow in 1 Year
S0
2S0
0
0
2S0 (1 1 rf)
ST 1 D
F0 2 ST
F0 2 S0(1 1 rf) 1 D
The initial cash flow is zero by construction: The money necessary to purchase the stock
in step 2 is borrowed in step 1, and the futures position in step 3, which is used to hedge
the value of the stock position, does not require an initial outlay. Moreover, the total cash
flow at year-end is riskless because it involves only terms that are already known when the
contract is entered. If the final cash flow were not zero, all investors would try to cash in
on the arbitrage opportunity. Ultimately prices would change until the year-end cash flow
is reduced to zero, at which point F0 would equal S0(1 1 rf) 2 D.
The parity relationship also is called the cost-of-carry relationship because it asserts
that the futures price is determined by the relative costs of buying a stock with deferred
delivery in the futures market versus buying it in the spot market with immediate delivery
and “carrying” it in inventory. If you buy stock now, you tie up your funds and incur a
time-value-of-money cost of rf per period. On the other hand, you receive dividend payments with a current yield of d. The net carrying cost advantage of deferring delivery of the
stock is therefore rf 2 d per period. This advantage must be offset by a differential between
the futures price and the spot price. The price differential just offsets the cost-of-carry
advantage when F0 5 S0(1 1 rf 2 d).
The parity relationship is easily generalized to multiperiod applications. We simply recognize that the difference between the futures and spot price will be larger as the maturity
of the contract is longer. This reflects the longer period to which we apply the net cost of
carry. For contract maturity of T periods, the parity relationship is
F0 5 S0(1 1 rf 2 d)T
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(22.2)
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Options, Futures, and Other Derivatives
Dividend Yield (Annualized, %)
7
6
5
4
3
2
1
Jul-12
Jan-13
Jul-11
Jan-12
Jul-10
Jan-11
Jan-10
Jul-09
Jul-08
Jan-09
Jul-07
Jan-08
Jul-06
Jan-07
Jul-05
Jan-06
Jul-04
Jan-05
Jul-03
Jan-04
Jul-02
Jan-03
Jul-01
Jan-02
Jan-01
0
Figure 22.5 S&P 500 monthly dividend yield
Notice that when the dividend yield is less than the risk-free rate, Equation 22.2 implies
that futures prices will exceed spot prices, and by greater amounts for longer times to contract
maturity. But when d > rf, as is the case today, the income yield on the stock exceeds the forgone (risk-free) interest that could be earned on the money invested; in this event, the futures
price will be less than the current stock price, again by greater amounts for longer maturities.
You can confirm that this is so by examining the S&P 500 contract listings in Figure 22.1.
Although dividends of individual securities may fluctuate unpredictably, the annualized
dividend yield of a broad-based index such as the S&P 500 is fairly stable, recently in the
neighborhood of a bit more than 2% per year. The yield is seasonal, however, with regular
peaks and troughs, so the dividend yield for the relevant months must be the one used.
Figure 22.5 illustrates the yield pattern for the S&P 500. Some months, such as January or
April, have consistently low yields, while others, such as May, have consistently high ones.6
We have described parity in terms of stocks and stock index futures, but it should be
clear that the logic applies as well to any financial futures contract. For gold futures, for
example, we would simply set the dividend yield to zero. For bond contracts, we would
let the coupon income on the bond play the role of dividend payments. In both cases, the
parity relationship would be essentially the same as Equation 22.2.
The arbitrage strategy described above should convince you that these parity relationships are more than just theoretical results. Any violations of the parity relationship give
rise to arbitrage opportunities that can provide large profits to traders. We will see in the
next chapter that index arbitrage in the stock market is a tool to exploit violations of the
parity relationship for stock index futures contracts.
Spreads
Just as we can predict the relationship between spot and futures prices, there are similar
ways to determine the proper relationships among futures prices for contracts of different
6
The very high dividend yield in November 2004 was due to Microsoft’s special one-time dividend of $3 per share.
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CHAPTER 22
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789
maturity dates. Equation 22.2 shows that the futures price is in part determined by time
to maturity. If the risk-free rate is greater than the dividend yield (i.e., rf > d), then the
futures price will be higher on longer-maturity contracts and if rf < d, longer-maturity
futures prices will be lower. You can confirm from Figure 22.1 that in 2013, when the
risk-free rate was below the dividend yield, the longer-maturity S&P 500 contract did
have a lower futures price than the shorter term contract. For futures on assets like gold,
which pay no “dividend yield,” we can set d 5 0 and conclude that F must increase as
time to maturity increases.
To be more precise about spread pricing, call F(T1) the current futures price for delivery
at date T1, and F(T2) the futures price for delivery at T2. Let d be the dividend yield of the
stock. We know from the parity Equation 22.2 that
F(T1) 5 S0(1 1 rf 2 d)T1
F(T2) 5 S0(1 1 rf 2 d)T2
As a result,
F(T2) /F(T1) 5 (1 1 rf 2 d)(T2 2T1)
Therefore, the basic parity relationship for spreads is
F(T2) 5 F(T1)(1 1 rf 2 d)(T2 2T1)
(22.3)
Equation 22.3 should remind you of the spot-futures parity relationship. The major difference
is in the substitution of F(T1) for the current spot price. The intuition is also similar. Delaying
delivery from T1 to T2 assures the long position that the stock will be purchased for F(T2) dollars at T2 but does not require that money be tied up in the stock until T2. The savings realized
are the net cost of carry between T1 and T2. Delaying delivery from T1 until T2 frees up F(T1)
dollars, which earn risk-free interest at rf. The delayed delivery of the stock also results in the
lost dividend yield between T1 and T2. The net cost of carry saved by delaying the delivery is
thus rf 2 d. This gives the proportional increase in the futures price that is required to compensate market participants for the delayed delivery of the stock and postponement of the payment of the futures price. If the parity condition for spreads is violated, arbitrage opportunities
will arise. (Problem 19 at the end of the chapter explores this possibility.)
Example 22.9
Spread Pricing
To see how to use Equation 22.3, consider the following data for a hypothetical contract:
Contract Maturity Data
January 15
March 15
Futures Price
$105.00
104.75
Suppose that the effective annual T-bill rate is 1% and that the dividend yield is 2%
per year. The “correct” March futures price relative to the January price is, according to
Equation 22.3,
105(1 1 .01 2 .02)1/6 5 104.82
The actual March futures price is 104.75, meaning that the March futures price is slightly
underpriced compared to the January futures and that, aside from transaction costs, an
arbitrage opportunity seems to be present.
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eXcel APPLICATIONS: Parity and Spreads
T
he parity spreadsheet allows you to calculate futures
prices corresponding to a spot price for different
maturities, interest rates, and income yields. You can use
the spreadsheet to see how prices of more distant contracts will fluctuate with spot prices and the cost of carry.
You can learn more about this spreadsheet by using the
version available on our Web site at www.mhhe.com/bkm.
(the difference in futures prices for the long versus short
maturity contracts) if the interest rate increases by 2%?
2. What happens to the time spread if the income yield
increases by 2%?
3. What happens to the spread if the income yield equals the
interest rate?
Excel Questions
1. Experiment with different values for both income yield and
interest rate. What happens to the size of the time spread
A
B
C
D
E
1
Spot Futures Parity and Time Spreads
2
3
4
5
6
7
8
9
10
Spot price
Income yield (%)
Interest rate (%)
Today’s date
Maturity date 1
Maturity date 2
Maturity date 3
100
2
4.5
5/14/09
11/17/09
1/2/10
6/7/10
Futures prices versus maturity
100.00
101.26
101.58
102.66
Spot price
Futures 1
Futures 2
Futures 3
11
12
13
14
Time to maturity 1
Time to maturity 2
Time to maturity 3
0.51
0.63
1.06
Equation 22.3 shows that futures prices with different maturities should all move
together. This is not surprising because all are linked to the same spot price through the
parity relationship. Figure 22.6 plots futures prices on gold for three maturity dates. It is
apparent that the prices move in virtual lockstep and that the more distant delivery dates
command higher futures prices, as Equation 22.3 predicts.
Futures Price, $/ounce
1660
1640
1620
1600
1580
1560
Dec ‘12 delivery
June ‘13 delivery
Dec ‘13 delivery
1540
1520
1500
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
Date in July 2012
Figure 22.6 Gold futures prices
790
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Forward versus Futures Pricing
Until now we have paid little attention to the differing time profile of returns of futures and
forward contracts. Instead, we have taken the sum of daily mark-to-market proceeds to the
long position as PT 2 F0 and assumed for convenience that the entire profit accrues on the
delivery date. Our parity theorems apply only to forward pricing because they assume that
contract proceeds are in fact realized only on delivery. In contrast, the actual timing of cash
flows conceivably might affect the futures price.
Futures prices will deviate from parity when marking to market gives a systematic
advantage to either the long or short position. If marking to market tends to favor the long
position, for example, the futures price should exceed the forward price, because the long
position will be willing to pay a premium for the advantage of marking to market.
When will marking to market favor either a long or short trader? A trader will benefit
if daily settlements are received (and can be invested) when the interest rate is high and
are paid (and can be financed) when the interest rate is low. Because long positions will
benefit if futures prices tend to rise when interest rates are high, they will be willing to
accept a higher futures price. Therefore, a positive correlation between interest rates and
changes in futures prices implies that the “fair” futures price will exceed the forward price.
Conversely, a negative correlation means that marking to market favors the short position
and implies that the equilibrium futures price should be below the forward price.
For most contracts, the covariance between futures prices and interest rates is so low
that the difference between futures and forward prices will be negligible. However, contracts on long-term fixed-income securities are an important exception to this rule. In this
case, because prices have a high correlation with interest rates, the covariance can be large
enough to generate a meaningful spread between forward and future prices.
22.5 Futures Prices versus Expected Spot Prices
So far we have considered the relationship between futures prices and the current spot
price. What about the relationship between the futures price and the expected value of
the spot price? In other words, how well does the futures price forecast the ultimate spot
price? Three traditional theories have been put forth: the expectations hypothesis, normal
backwardation, and contango. Today’s consensus is that all of these traditional hypotheses
are subsumed by modern portfolio theory. Figure 22.7 shows the expected path of futures
under the three traditional hypotheses.
Expectations Hypothesis
The expectations hypothesis is the simplest theory of futures pricing. It states that the
futures price equals the expected value of the future spot price: F0 5 E(PT). Under this
theory the expected profit to either position of a futures contract would equal zero: The
short position’s expected profit is F0 2 E(PT), whereas the long’s is E(PT) 2 F0. With
F0 5 E(PT), the expected profit to either side is zero. This hypothesis relies on a notion
of risk neutrality. If all market participants are risk neutral, they should agree on a futures
price that provides an expected profit of zero to all parties.
The expectations hypothesis resembles market equilibrium in a world with no uncertainty;
that is, if prices of goods at all future dates were currently known, then the futures price for
delivery at any particular date would equal the currently known future spot price for that date.
It is a tempting but incorrect leap to then assert that under uncertainty the futures price should
equal the currently expected spot price. This view ignores the risk premiums that must be
built into futures prices when ultimate spot prices are uncertain.
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Normal Backwardation
Future Prices
This theory is associated with the
famous British economists John
Maynard Keynes and John Hicks.
Contango
They argued that for most commodities there are natural hedgers
who wish to shed risk. For examExpectations Hypothesis
E(PT)
ple, wheat farmers desire to shed
the risk of uncertain wheat prices.
These farmers will take short posiNormal Backwardation
tions to deliver wheat at a guaranteed price; they will short hedge.
To induce speculators to take the
corresponding long positions, the
farmers need to offer them an
Time
expectation of profit. They will
Delivery
enter the long side of the contract
Date
only if the futures price is below the
expected spot price of wheat, for
an expected profit of E(PT) 2 F0.
Figure 22.7 Futures price over time, in the special case that the
The speculators’ expected profit
expected spot price remains unchanged
is the farmers’ expected loss, but
farmers are willing to bear this
expected loss to avoid the risk of uncertain wheat prices. The theory of normal backwardation thus suggests that the futures price will be bid down to a level below the expected spot
price and will rise over the life of the contract until the maturity date, at which point FT 5 PT.
Although this theory recognizes the important role of risk premiums in futures markets,
it is based on total variability rather than on systematic risk. (This is not surprising, as
Keynes wrote almost 40 years before the development of modern portfolio theory.) The
modern view refines the measure of risk used to determine appropriate risk premiums.
Contango
The polar hypothesis to backwardation holds that the natural hedgers are the purchasers of
a commodity, rather than the suppliers. In the case of wheat, for example, we would view
grain processors as willing to pay a premium to lock in the price that they must pay for
wheat. These processors hedge by taking a long position in the futures market; therefore,
they are called long hedgers, whereas farmers are short hedgers. Because long hedgers
will agree to pay high futures prices to shed risk, and because speculators must be paid a
premium to enter the short position, the contango theory holds that F0 must exceed E(PT).
It is clear that any commodity will have both natural long hedgers and short hedgers.
The compromise traditional view, called the “net hedging hypothesis,” is that F0 will be
less than E(PT) when short hedgers outnumber long hedgers and vice versa. The strong
side of the market will be the side (short or long) that has more natural hedgers. The strong
side must pay a premium to induce speculators to enter into enough contracts to balance
the “natural” supply of long and short hedgers.
Modern Portfolio Theory
The three traditional hypotheses all envision a mass of speculators willing to enter either
side of the futures market if they are sufficiently compensated for the risk they incur.
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Modern portfolio theory fine-tunes this approach by refining the notion of risk used in the
determination of risk premiums. Simply put, if commodity prices pose positive systematic
risk, futures prices must be lower than expected spot prices.
To illustrate this approach, consider once again a stock paying no dividends. If E(PT)
denotes the expected time-T stock price and k denotes the required rate of return on the
stock, then the price of the stock today must equal the present value of its expected future
payoff as follows:
E(PT)
P0 5
(22.4)
(1 1 k)T
We also know from the spot-futures parity relationship that
F0
P0 5
(22.5)
(1 1 rf)T
Therefore, the right-hand sides of Equations 22.4 and 22.5 must be equal. Equating these
terms allows us to solve for F0:
1 1 rf
11k
≤
T
(22.6)
You can see immediately from Equation 22.6 that F0 will be less than the expectation of PT
whenever k is greater than rf, which will be the case for any positive-beta asset. This means
that the long side of the contract will make an expected profit [F0 will be lower than E(PT)]
when the commodity exhibits positive systematic risk (k is greater than rf).
Why should this be? A long futures position will provide a profit (or loss) of PT 2 F0. If
the ultimate value of PT entails positive systematic risk, so will the profit to the long position. Speculators with well-diversified portfolios will be willing to enter long futures positions only if they receive compensation for bearing that risk in the form of positive expected
profits. Their expected profits will be positive only if E(PT) is greater than F0. The short
position’s profit is the negative of the long’s and will have
negative systematic risk. Diversified investors in the short
CONCEPT CHECK 22.6
position will be willing to suffer that expected loss to
lower portfolio risk and will be willing to enter the conWhat must be true of the risk of the spot price of
tract even when F0 is less than E(PT). Therefore, if PT has
an asset if the futures price is an unbiased estimate of the ultimate spot price?
positive beta, F0 must be less than the expectation of PT.
The analysis is reversed for negative-beta commodities.
1. Forward contracts call for future delivery of an asset at a currently agreed-on price. The long
trader purchases the good, and the short trader delivers it. If the price of the asset at the maturity
of the contract exceeds the forward price, the long side benefits by virtue of acquiring the good at
the contract price.
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F0 5 E(PT) ¢
SUMMARY
2. A futures contract is similar to a forward contract, differing most importantly in the aspects of
standardization and marking to market, which is the process by which gains and losses on futures
contract positions are settled daily. In contrast, forward contracts call for no cash transfers until
contract maturity.
3. Futures contracts are traded on organized exchanges that standardize the size of the contract, the
grade of the deliverable asset, the delivery date, and the delivery location. Traders negotiate only
over the contract price. This standardization increases liquidity and means that buyers and sellers
can easily find many traders for a desired purchase or sale.
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4. The clearinghouse steps in between each pair of traders, acting as the short position for each long
and as the long position for each short. In this way traders need not be concerned about the performance of the trader on the opposite side of the contract. In turn, traders post margins to guarantee
their own performance.
5. The long position’s gain or loss between time 0 and time t is Ft 2 F0. Because FT 5 PT, the long’s
profit if the contract is held until maturity is PT 2 F0, where PT is the spot price at time T and F0
is the original futures price. The gain or loss to the short position is F0 2 PT.
6. Futures contracts may be used for hedging or speculating. Speculators use the contracts to take a
stand on the ultimate price of an asset. Short hedgers take short positions in contracts to offset any
gains or losses on the value of an asset already held in inventory. Long hedgers take long positions to offset gains or losses in the purchase price of a good.
7. The spot-futures parity relationship states that the equilibrium futures price on an asset providing
no service or payments (such as dividends) is F0 5 P0(1 1 rf)T. If the futures price deviates from
this value, then market participants can earn arbitrage profits.
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8. If the asset provides services or payments with yield d, the parity relationship becomes
F0 5 P0(1 1 rf 2 d)T. This model is also called the cost-of-carry model, because it states that
futures price must exceed the spot price by the net cost of carrying the asset until maturity date T.
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9. The equilibrium futures price will be less than the currently expected time T spot price if the spot
price exhibits systematic risk. This provides an expected profit for the long position who bears the
risk and imposes an expected loss on the short position who is willing to accept that expected loss
as a means to shed systematic risk.
KEY TERMS
forward contract
futures price
long position
short position
single-stock futures
clearinghouse
KEY EQUATIONS
Spot-futures parity: F0(T ) 5 S0 (1 1 r 2 d )T
open interest
marking to market
maintenance margin
convergence property
cash settlement
basis
basis risk
calendar spread
spot-futures parity theorem
cost-of-carry relationship
Futures spread parity: F0(T2) 5 F0(T1) (1 1 r 2 d)(T2 2T1)
Futures vs. expected spot prices: F0 5 E(PT) ¢
PROBLEM SETS
1 1 rf
11k
T
≤
1. Why is there no futures market in cement?
2. Why might individuals purchase futures contracts rather than the underlying asset?
Basic
3. What is the difference in cash flow between short-selling an asset and entering a short futures
position?
4. Are the following statements true or false? Why?
a. All else equal, the futures price on a stock index with a high dividend yield should be higher
than the futures price on an index with a low dividend yield.
b. All else equal, the futures price on a high-beta stock should be higher than the futures price on
a low-beta stock.
c. The beta of a short position in the S&P 500 futures contract is negative.
5. What is the difference between the futures price and the value of the futures contract?
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6. Evaluate the criticism that futures markets siphon off capital from more productive uses.
7. a. Turn to the S&P 500 contract in Figure 22.1. If the margin requirement is 10% of the futures
price times the multiplier of $250, how much must you deposit with your broker to trade the
March maturity contract?
b. If the March futures price were to increase to 1,498, what percentage return would you earn
on your net investment if you entered the long side of the contract at the price shown in the
figure?
c. If the March futures price falls by 1%, what is your percentage return?
Intermediate
8. a. A single-stock futures contract on a non-dividend-paying stock with current price $150 has
a maturity of 1 year. If the T-bill rate is 3%, what should the futures price be?
b. What should the futures price be if the maturity of the contract is 3 years?
c. What if the interest rate is 6% and the maturity of the contract is 3 years?
a. You own a large position in a relatively illiquid bond that you want to sell.
b. You have a large gain on one of your Treasuries and want to sell it, but you would like to
defer the gain until the next tax year.
c. You will receive your annual bonus next month that you hope to invest in long-term corporate bonds. You believe that bonds today are selling at quite attractive yields, and you are
concerned that bond prices will rise over the next few weeks.
10. Suppose the value of the S&P 500 stock index is currently 1,400. If the 1-year T-bill rate is 3%
and the expected dividend yield on the S&P 500 is 2%, what should the 1-year maturity futures
price be? What if the T-bill rate is less than the dividend yield, for example, 1%?
11. Consider a stock that pays no dividends on which a futures contract, a call option, and a put
option trade. The maturity date for all three contracts is T, the exercise price of both the put and
the call is X, and the futures price is F. Show that if X 5 F, then the call price equals the put
price. Use parity conditions to guide your demonstration.
12. It is now January. The current interest rate is 2%. The June futures price for gold is $1,500,
whereas the December futures price is $1,510. Is there an arbitrage opportunity here? If so, how
would you exploit it?
13. OneChicago has just introduced a single-stock futures contract on Brandex stock, a company
that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in
1 year. The T-bill rate is 6% per year.
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9. How might a portfolio manager use financial futures to hedge risk in each of the following
circumstances:
a. If Brandex stock now sells at $120 per share, what should the futures price be?
b. If the Brandex price drops by 3%, what will be the change in the futures price and the change
in the investor’s margin account?
c. If the margin on the contract is $12,000, what is the percentage return on the investor’s
position?
14. The multiplier for a futures contract on a stock market index is $250. The maturity of the
contract is 1 year, the current level of the index is 1,300, and the risk-free interest rate is .5% per
month. The dividend yield on the index is .2% per month. Suppose that after 1 month, the stock
index is at 1,320.
a. Find the cash flow from the mark-to-market proceeds on the contract. Assume that the parity
condition always holds exactly.
b. Find the holding-period return if the initial margin on the contract is $13,000.
15. You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in
3 months. You believe rates will soon fall, and you would like to repurchase the company’s
sinking fund bonds (which currently are selling below par) in advance of requirements. Unfortunately, you must obtain approval from the board of directors for such a purchase, and this can
take up to 2 months. What action can you take in the futures market to hedge any adverse movements in bond yields and prices until you can actually buy the bonds? Will you be long or short?
Why? A qualitative answer is fine.
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16. The S&P portfolio pays a dividend yield of 1% annually. Its current value is 1,500. The T-bill
rate is 4%. Suppose the S&P futures price for delivery in 1 year is 1,550. Construct an arbitrage
strategy to exploit the mispricing and show that your profits 1 year hence will equal the mispricing in the futures market.
eXcel
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17. The Excel Application box in the chapter (available at www.mhhe.com/bkm; link to Chapter
22 material) shows how to use the spot-futures parity relationship to find a “term structure of
futures prices,” that is, futures prices for various maturity dates.
a. Suppose that today is January 1, 2013. Assume the interest rate is 3% per year and a stock
index currently at 1,500 pays a dividend yield of 1.5%. Find the futures price for contract
maturity dates of February 14, 2013, May 21, 2013, and November 18, 2013.
b. What happens to the term structure of futures prices if the dividend yield is higher than the
risk-free rate? For example, what if the dividend yield is 4%?
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Challenge
18. a. How should the parity condition (Equation 22.2) for stocks be modified for futures contracts
on Treasury bonds? What should play the role of the dividend yield in that equation?
b. In an environment with an upward-sloping yield curve, should T-bond futures prices on
more-distant contracts be higher or lower than those on near-term contracts?
c. Confirm your intuition by examining Figure 22.1.
19. Consider this arbitrage strategy to derive the parity relationship for spreads: (1) enter a long
futures position with maturity date T1 and futures price F(T1); (2) enter a short position with
maturity T2 and futures price F(T2); (3) at T1, when the first contract expires, buy the asset and
borrow F(T1) dollars at rate rf; (4) pay back the loan with interest at time T2.
a. What are the total cash flows to this strategy at times 0, T1, and T2?
b. Why must profits at time T2 be zero if no arbitrage opportunities are present?
c. What must the relationship between F(T1) and F(T2) be for the profits at T2 to be equal to
zero? This relationship is the parity relationship for spreads.
1. Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated by the following information:
Spot price for commodity
Futures price for commodity expiring in 1 year
Interest rate for 1 year
$120
$125
8%
a. Describe the transactions necessary to take advantage of this specific arbitrage opportunity.
b. Calculate the arbitrage profit.
2. Michelle Industries issued a Swiss franc–denominated 5-year discount note for SFr200 million.
The proceeds were converted to U.S. dollars to purchase capital equipment in the United States.
The company wants to hedge this currency exposure and is considering the following alternatives:
• At-the-money Swiss franc call options.
• Swiss franc forwards.
• Swiss franc futures.
a. Contrast the essential characteristics of each of these three derivative instruments.
b. Evaluate the suitability of each in relation to Michelle’s hedging objective, including both
advantages and disadvantages.
3. Identify the fundamental distinction between a futures contract and an option contract, and
briefly explain the difference in the manner that futures and options modify portfolio risk.
4. Maria VanHusen, CFA, suggests that using forward contracts on fixed-income securities can
be used to protect the value of the Star Hospital Pension Plan’s bond portfolio against the possibility of rising interest rates. VanHusen prepares the following example to illustrate how such
protection would work:
• A 10-year bond with a face value of $1,000 is issued today at par value. The bond pays
an annual coupon.
• An investor intends to buy this bond today and sell it in 6 months.
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• The 6-month risk-free interest rate today is 5% (annualized).
• A 6-month forward contract on this bond is available, with a forward price of $1,024.70.
• In 6 months, the price of the bond, including accrued interest, is forecast to fall to
$978.40 as a result of a rise in interest rates.
a. Should the investor buy or sell the forward contract to protect the value of the bond against
rising interest rates during the holding period?
b. Calculate the value of the forward contract for the investor at the maturity of the forward contract if VanHusen’s bond-price forecast turns out to be accurate.
c. Calculate the change in value of the combined portfolio (the underlying bond and the appropriate forward contract position) 6 months after contract initiation.
5. Sandra Kapple asks Maria VanHusen about using futures contracts to protect the value of the Star
Hospital Pension Plan’s bond portfolio if interest rates rise. VanHusen states:
a. “Selling a bond futures contract will generate positive cash flow in a rising interest rate environment prior to the maturity of the futures contract.”
b. “The cost of carry causes bond futures contracts to trade for a higher price than the spot price
of the underlying bond prior to the maturity of the futures contract.”
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Comment on the accuracy of each of VanHusen’s two statements.
E-INVESTMENTS EXERCISES
Go to the Chicago Mercantile Exchange site at www.cme.com. From the Products &
Trading tab, select the link to Equity Index, and then link to the NASDAQ-100 E-mini contract.
Now find the tab for Contract Specifications.
1. What is the contract size for the futures contract?
2. What is the settlement method for the futures contract?
3. For what months are the futures contracts available?
4. Click the link to View Price Limits and then U.S. Equity Price Limits. What is the current
value of the 10% price limit for this contract?
5. Click on View Calendar. What is the settlement date of the shortest-maturity outstanding contract? The longest-maturity contract?
SOLUTIONS TO CONCEPT CHECKS
1.
Short Futures
F0
Write (Sell) Call
Buy (Long) Put
Payoff, Profit
Payoff
Payoff
PT
X
PT
X
PT
2. The clearinghouse has a zero net position in all contracts. Its long and short positions are
offsetting, so that net cash flow from marking to market must be zero.
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PART VI
Options, Futures, and Other Derivatives
3.
Oil Price in February, PT
Cash flow to purchase oil: 2100,000 3 PT
1 Profit on long futures: 100,000 3 (PT 2 F0)
TOTAL CASH FLOW
$89.86
$91.86
$93.86
2$8,986,000
2$9,186,000
0
2$9,386,000
2200,000
2$9,186,000
2$9,186,000
2$9,186,000
1200,000
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4. The risk would be that the index and the portfolio do not move perfectly together. Thus basis
risk involving the spread between the futures price and the portfolio value could persist even
if the index futures price were set perfectly relative to the index itself. You can measure this
risk using the index model. If you regress the return of the active portfolio on the return of the
index portfolio, the regression R-square will equal the proportion of the variance of the active
portfolio’s return that could have been hedged using the index futures contract. You can also
measure the risk of the imperfectly hedged position using the standard error of the regression,
which tells you the standard deviation of the residuals in the index model. Because these are the
components of the risky returns that are independent of the market index, this standard deviation
measures the variability of the portion of the active portfolio’s return that cannot be hedged using
the index futures contract.
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5. The futures price, $980, is $10 below the parity value of $990. The cash flow to the following
strategy is riskless and equal to this mispricing.
Action
Initial Cash Flow
Cash Flow in 1 Year
Lend S0 dollars
21,000
1,000(1.01) 5 1,010
Sell stock short
11,000
0
Long futures
TOTAL
0
2ST 2 20
ST 2 980
$10 risklessly
6. It must have zero beta. If the futures price is an unbiased estimator, then we infer that it has a zero
risk premium, which means that beta must be zero.
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