Money & Capital Markets
Review --- from Fall 2011 [this review does not cover Options]
I. Mean-Variance Analysis
A. Important Equations
1. Expected rate of return on a portfolio:
E (rp )  Ws E (rs )  WB E (rB )
Where W’s are the weights or proportion of the individual security held within the
portfolio and E(r)’s are the expected rate of return for the individual securities.
2. Variance of a portfolio:
 P2  (Ws s ) 2  (WB B ) 2  2WsWB s B 
Where ρ (rho) is the correlation coefficient between the two securities.
3. Standard deviation (risk) of a portfolio:
 P   P2
4. Numerical Example
Suppose a portfolio contains two stocks. Stock 1 has an expected rate of return of 12%
with a standard deviation of 20%. Stock 2 has an expected rate of return of 8% with a
standard deviation of 10%. The correlation between the two stocks is -.25. Assume the
portfolio is composed of 60% of stock 1 and 40% of stock 2. Calculate the expected rate
of return on the portfolio and the standard deviation (risk).
E (rp )  W1 E (r1 )  W2 E (r2 )  (.6)(12)  (.4)(8)  10.4%
 P2  (W1 1 ) 2  (W2 2 ) 2  2W1W2 1 2   [(.6)(20)] 2  [(.4)(10)] 2  2(.6)(.4)(20)(10)(.25)  136
 P   P2  136  11.66
5. Questions
a. What is the significant difference between the equation for the expected rate of return
and variance of a portfolio?
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b. What role does the correlation coefficient (or, covariance) play in the process of
diversification?
c. Where might the ‘data’ (i.e., expected rates of returns, standard deviations, correlation
coefficient) to be used in the equations come from?
d. Would every investor obtain the same results for a portfolio composed of the same
securities in the same proportion?
B. Graphical
1. Efficient Frontier
Return
Security 1
Security 2
Risk
2
2. Indifference Curves
Return
Return
Increasing
utility
Risk
“Young Executive”
Risk
“Little Old Lady”
3. Solution
Return
Risk
4. Questions
a. Why is the lower part of the efficient frontier not relevant?
b. Why do the indifference curves slope upward?
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c. Why is the point of tangency between the indifference curve and efficient frontier the
optimal solution? Provide an economic explanation of the meaning of this tangency.
d. How would the mean-variance solutions vary for the young executive and little old
lady – assuming both go to the same investment advisor (using mean-variance analysis to
construct the efficient frontier)?
e. Suppose the two securities used to construct the efficient frontier in the previous
graph, suddenly had a change in the correlation coefficient – from say -.25 to +.25?
What if the correlation coefficient became +1? Illustrate these situations.
C. Separation Theorem
1. Important Equations
You should note that these are merely special applications of the general equations used
in mean-variance analysis. Hence, you should not attempt to memorize these equations.
Expected Rate of Return on the Complete Portfolio (containing the optimal risky
portfolio and the risk-free asset)
E (rC )  WP E (rP )  W f E (r f )
Variance of the Complete Portfolio – recall the risk-free asset as a zero standard deviation
by definition.
 C2  (WP P ) 2  (W f  f ) 2  2WPW f  P f   (WP P ) 2
Standard Deviation of the Complete Portfolio
 C   C2
Reward to Variability ratio (or, Sharpe Ratio) – this is an important in the context of the
separation theorem. This provides the slope of the Capital Allocation Line – i.e., the line
from the risk-free asset just tangent to the efficient frontier.
E (rP )  r f
P
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2. Graphical
The Separation Theorem separates the decision of the optimal risky portfolio from the
investor’s particular attitude towards the trade-off between risk and return.
Return
CAL
Y
Z
E(rP)
X
rf
Risk
3. Questions
a. Consider the individual investors X and Y in the above figure. What risky portfolio is
each holding? Which investor is borrowing at the risk-free rate? Which investor is
lending at the risk-free rate?
b. Explain the expected rate of return and risk of the complete portfolios X and Y.
c. Explain why portfolio Z is the most efficient risky portfolio to hold.
d. Demonstrate what would happen if the risk-free rate decreased. What happens to the
reward-to-variability ratio?
e. Demonstrate what would happen if an investor could not borrow at precisely the riskfree rate (this is an interest rate normally reserved for the federal government after all).
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II. CAPM
A. Important Equations
1. The Equilibrium Condition
E (r1 )  r f
1

E (r2 )  r f
2
   E (rM )  r f
In equilibrium, the risk premium (i.e., E(r)-rf ) adjusted for the relevant risk (i.e., beta) for
all assets are equal – and, equal to the risk premium of the entire market.
2. Equilibrium rate of return
E (r1 )  r f  1 [ E (rM )  r f ]
3. Equilibrium risk premium
E (r1 )  r f  1 [ E (rM )  r f ]
4. Numerical Example
Given an asset with a beta of 1.25, return on risk-free asset of 5%, and the expected
market return of 9%, calculated the CAPM expected rate of return.
E (r1 )  r f  1 [ E (rM )  r f ]  5%  1.25[9%  5%]  10%
What is the risk premium of the asset?
E (r1 )  r f  1 [ E (rM )  r f ] =1.25[9%-5%]=5%
If, given its current price, the expected rate of return on the asset is 12%, then we would
say that the asset is over or under priced?
We would say that the asset is underpriced. Hence, we would expect investors to
increase their demand for this asset and drive up the price, as the price increases – given
the expected payoffs – then the rate of return will fall until it reaches 10%.
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B. Graphical Representation
1. The Security Market Line (SML)
Return
SML
C
B
E(rM)
D
A
rf
1
Beta
The security market line (SML) illustrates the expected rate of return for an asset or
portfolio determined by CAPM.
2. Questions
a. Which of the assets (A, B, C, D) in the graph are overpriced? Underpriced?
A and D would be considered overpriced. We would expect the demand for these assets
to fall, causing the price to fall, and raising the expected rate of return. Graphically, the
process would be seen as a vertical movement upward toward the SML.
B and C would be considered underpriced. We would expect the demand for these assets
to increase, causing the price to rise, and lowering the expected rate of return.
Graphically, the process would be seen as a vertical movement downward toward the
SML.
b. Previously, the risk of an asset (or, portfolio) was its standard deviation. In the CAPM
world, the risk of an asset (or, portfolio) is its beta. What explains this switch in the
definition of risk?
c. Suppose an empirical estimate of the SML, found a flatter line than what would have
been predicted by CAPM. What would this imply about the behavior of investors?
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C. Applications
1. Calculating expected rates of return (and, equilibrium prices)
The risk-free rate of return and expected market rate of return are 3% and 13%. The beta
for a stock is .5. You expect the corporation issuing this stock to pay a dividend of $5 per
share next year. Use the Gordon model (Dividend Discount Model) to calculate the price
of the stock when (a) no growth is assumed for the dividend and (b) a 5% growth rate in
earnings and dividends.
We use the CAPM to determine the relevant discount rate:
E (r1 )  r f  1 [ E (rM )  r f ]  3%  .5[13%  3%]  8%
Now, we use this discount rate in the Gordon Model:
a. g=0
P0 
D0 (1  g )
D1
$5
$5



 $62.50
rg
r  g 8%  0 .08
b. g=5%
P0 
D0 (1  g )
D1
$5
$5



 $166.67
rg
r  g 8%  5% .03
Previously, we had used a simple present value calculation to translate the CAPM
expected rate of return into a determination of price. Here, we have done this using the
Gordon Model in order to illustrate its use once again.
2. Making decisions about investments in real (i.e., physical) assets
A corporation is considering forming a business unit to enter a new market. The
management has determined that the internal rate of return for the new business unit is
12% - recall, the internal rate of return is the discount rate that sets the present value of
the future earnings of a project exactly equal to the current cost of the project.
Management uses this internal rate of return has the ‘hurdle rate’ to determine whether or
not to do the project. The current risk-free rate and expected market rate of return are
8% and 14%. The corporation uses a beta of 1.5 in CAPM calculations. What is the
CAPM expected rate of return? Should the corporation do the new project?
E (r1 )  r f  1 [ E (rM )  r f ]  8%  1.5[14%  8%]  17%
Since the CAPM expected rate of return for the project exceeds the hurdle rate, the
corporation should go ahead with the project. Notice what this would say about the price
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that the corporation would be willing to pay to do the project. The internal rate of return
is the i in the formula below.
P
C3
C5
CN
C1
C2
C4





2
3
4
5
(1  i) (1  i)
(1  i)
(1  i)
(1  i)
(1  i) N
We could now replace the internal rate of return with the higher CAPM expected rate of
return to determine the price (P) we would be willing to pay. By mere inspection, we
observe that the price we would be willing to pay exceeds the current price of the project.
Again, we are led to the same conclusion – do the project.
3. Determining ‘fair’ returns
A publicly regulated utility company has made a $500 million investment in a new plant.
The utility company has requested a rate increase for consumers that would allow it to
generate enough income to cover the cost of the new plant. The company’s requested
rate increase is expected to generate a return of 12%. The regulators plan to use CAPM
to determine whether or not the new rates are ‘fair’. The regulators find that the beta for
the utility is .4, risk-free rate of 4%, and expected market rate of return of 16%. Is the
company’s request fair?
E (r1 )  r f  1 [ E (rM )  r f ]  4%  .4[16%  4%]  8.8%
No, the company’s request is not ‘fair’ according to CAPM. The company should need
to raise rates only to the extent to generate a 8.8% return on the plant. Notice, this says
something about the financing that the company should have been able to obtain in the
markets for this type of project.
4. Evaluating performance
An investment fund management firm is deciding upon bonuses for the year. Manager 1
obtained a 8.5% rate of return on his portfolio – the portfolio had a beta of .75. Manager
2 obtained a 18% rate of return on his portfolio – the portfolio had a beta of 2.75. The
risk-free rate and market return had been 3% and 9% during the year. Use the Jensen
performance evaluation method to determine who, if any, should get a bonus?
Manager 1
_
_
_
_
_
_
_
JM  rP  [r f   P (rM  r f )]  8.5  [3  .75(9  3]  8.5%  7.5%  1%
Manager 2
_
JM  rP  [r f   P (rM  r f )]  18  [3  2.75(9  3)]  18%  19.5%  1.5%
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Manager 1 did better than expected by CAPM standards – he deservers a bonus.
Manager 2, on the other hand, did worse than expected by CAPM standards – he does not
deserve a bonus.
D. Questions
1. Write a short essay discussing the derivation of the CAPM equation. You should
avoid, or keep to an absolute minimum, using equations and graphs in your essay. Focus
attention on the intuition and economic concepts of the development.
2. What is the relationship between CAPM and the Separation Theorem?
3. What is the relationship between CAPM and the Gordon Model (or, Dividend
Discount Model)?
4. “It [CAPM] ties together into one neat package the market behavior theories of
Samuelson and Fama and the portfolio selection theories of Markowitz and Tobin. But it
would be a useless tool for investors in the market Keynes described; in Keynes’s world,
everyone is a noise trader. The model would also be misleading in markets where
obstacles like high taxes or brokerage fees stand in the way of the trading necessary to
compose optimal portfolios, and where some investors have a monopoly on information.”
Write an essay that explains the reasoning behind this passage.
5. “…stocks with low betas would appear to be the most desirable to investors. They
would be more highly valued than riskier stocks with high betas, and everyone would
want to hold them. But then the expected returns on the low-beta stocks would be lower.
… What would happen to the high-beta stocks that nobody wanted?” Answer this last
question and discuss the end result of the two processes being described.
6. “Investors demand higher returns for stocks with risks that cannot be diversified away
– stocks that move up and down in sympathy with the portfolio – but they do not expect
to earn a premium for stocks with risks that can be diversified away.” Explain why this
should be so. Also, discuss how investors can diversify risk away. What would be the
implications if this were not true?
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III. Arbitrage Pricing Theory (APT) --- FALL 2011 Class can probably skip this section
A. Important Equations
1. Factor Model
R    1 F1   2 F2     N FF  e
Where R represents the risk premium of the asset or portfolio and F’s are the ‘factors’
that determine the risk premium of the asset or portfolio. The factors can be anything
that we find empirically relevant (e.g., GDP, expected inflation, etc.).
2. APT
E (rP )  r f   [ E (rM )  r f ]
This is the ‘special’ version of the APT we developed. It is ‘special’ because we
assumed that the only factor that mattered was the market rate of return. In making this
assumption, we demonstrated that the APT could come to the same result as CAPM.
However, the APT does not require us to make this assumption. The market rate of
return could have been replaced by GDP, expected inflation, etc. Alternatively, we could
have used several factors to explain the rate of return on the portfolio. Thus, the APT is
more general than the CAPM. On the other hand, this level of generality comes at the
cost of not having any definite factor.
B. The Notion of Equilibrium
We have seen that the CAPM equilibrium occurs when the risk premiums adjusted for the
relevant risk (i.e., beta) are equal for all assets. In order for the market to arrive at this
equilibrium condition several assumptions had to be put into place (e.g., investors use
mean-variance analysis, have the same information, form the same expectations, etc.).
The APT uses a different equilibrium condition. Here, the equilibrium occurs when all
arbitrage opportunities are exhausted. Hence, we need only assume that there are some
investors in the market that will take advantage of arbitrage opportunities. In doing so,
they will drive asset prices to their equilibrium levels.
C. Questions
1. Discuss the derivation of APT.
2. In what sense is the APT more general than CAPM? Compare their equilibrium.
3. “APT fills those gaps [of CAPM] by providing a method to measure how stock prices
will respond to changes in the multitude of economic factors that influence them, such as
inflation, interest rate patterns, changing perceptions of risk, and economic growth.”
Given this statement to be true, is it a strength or weakness of APT?
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IV. Gordon Model (or, Dividend Discount Model)
A. Important Equation
P
D
rg
where D is the dividend to be paid in the next period, r the discount rate, and g the
expected growth rate of dividends (and earnings, assuming a constant pay-out ratio).
We have seen that this equation can be stated in a variety of ways. We might, for
example, solve for r and arrive at a statement of what might determine our expected rate
of return.
r
D
g
P
Alternatively, if we assume that the corporation will pay a constant fraction of its
earnings out in dividends, then we can arrive at a formula for the price-earnings ratio.
P
b

E rg
where E is the earnings of the corporation and b the constant pay-out ratio. All of these
merely follow from a rearrangement of the first equation – hence, there should be no need
to memorize them.
B. Applications
1. The most recent dividend paid by a firm was $2. You expect the dividends to grow by
2% and assign a required return to this stock of 8%. According to the Gordon Model,
what would be the price of this stock?
The price should be $34.
P
D0 (1  g ) $2(1  .02)

 $34
rg
.08  .02
Notice, the question provides you with the most recent dividend. This explains why we
do not simply have the dividend in the numerator.
2. A news report uncovers numerous irregularities in the accounting at several large
corporations. What impact would this report likely have on stock prices?
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As seen recently in the U.S., accounting scandals can have a depressing impact on stock
prices. How might you use the Gordon Model to predict a decrease in stock prices due to
the accounting scandals?
P
D
rg
The accounting scandals may cause investors to increase the required return on equities
(r) due to a perception of greater risk. Alternatively, the accounting scandals may cause
investors to lower their expected growth rate of dividends (earnings as well). Either of
these would cause a decrease in the price of stocks.
3. The required return on a particular stock is 8% with an expected dividend growth rate
of 4%. Market analysts expect the firm to pay a $1 dividend per share in the coming
year. However, the firm has announced a 99 cent dividend for the coming year. (A)
Calculate the price of the stock based on this information. (B) Suppose investors revise
the estimated growth rate down to 3%, calculate the price.
(A) The price of the stock, according to the Gordon Model, will fall from $25 to $24.75.
$1
 $25
.08  .04
$0.99
P
 $24.75
.08  .04
P
(B) The price of the stock, according to the Gordon Model, will fall much further, from
$25 to $19.80.
P
$0.99
 $19.80
.08  .03
Some researchers believe that this type of change in expectations may explain how a
changed dividend payment can cause large fluctuations in price – much larger than what
appears to be justified based solely on the changed dividend payment as in (A).
4. The price of a particular stock is currently $50. The stock is expected to pay a
dividend of $2 in the coming year with a growth rate of 4%. Suppose you have studied
this firm in detail and determined – based on risk and projected growth - that you would
require a return of 8% before purchasing this stock. Based on this information, would
you purchase the stock?
Based on this information, and rearranging the Gordon Model, the expected return on this
stock would be
D
$2
r g 
 .02  .04  .02  .06  6%
P
$50
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Thus, you would not purchase this stock. But, suppose you wanted to let your
broker know when to purchase the stock. At what price would you tell your broker to
buy the stock?
In order to determine the price to tell the broker to begin buying, we simply go back to
the Gordon Model and use the 8% required return.
P
$2
 $33.33
.08  .02
If the price fell to $33.33 or below, then the broker should begin purchasing the
stock for us. In other words, given the dividend and growth rate, the price would need to
fall in order to increase the return.
5. Historically, the P/E ratio for a particular stock has been around 15. Currently, the
P/E ratio has dropped to 12. The dividend and its growth has been what the market has
expected. What would explain the decrease in the P/E ratio?
Recall the equation for the P/E derived from the Gordon Model:
P
b

E rg
If the dividends have been what the market expected – implying the payout fraction has
probably not changed – then the market must be requiring a higher return (k) on this
stock. Some may use this information (after convincing themselves that the market is
wrong on its assessment of the required return), to conclude the stock is under-priced.
6. You read a story in the newspaper announcing the proposed merger of Dell Computer
and Gateway. The merger is expected to greatly increase Gateway’s profitability. If you
decide to invest in Gateway stock, you can expect to earn
a. Above average returns since you will share in the higher profits.
b. Above average returns since your stock price will definitely appreciate as higher
profits are earned.
c. Below average returns since computer makers have low profit rates.
d. A normal return since stock prices adjust to reflect expected changes in profitability
almost immediately.
The correct answer is (d). This is an application of the efficient market hypothesis. How
would you relate this to the discussion of the return on a stock?
Recall, that by rearranging the Gordon Model, we can obtain a general statement for the
return on a stock:
r
D
g
P
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When the news of the merger is announced – with the recognition of higher
profitability – their will be an immediate rush to buy the stock. The price of the stock
will increase, driving down the dividend yield and lowering the return until it reaches a
‘normal’ level.
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