Hedge Portfolios, the No Arbitrage Condition & Arbitrage Pricing Theory
Hedge Portfolios
A portfolio that has zero risk is said to be "perfectly hedged" or, in the jargon of
Economics and Finance, is referred to as a hedge portfolio. An investor who invests
100% of wealth in risk-free debt has obviously procured a hedge portfolio but this is not
the only way to achieve this result. Consider, for example, a portfolio made up of some
number k of risky securities where αi is used to denote the share of the portfolio invested
in the risky asset with rate of return ri . Then the rate of return on the portfolio is rp such
that
rp = Σ αi ri and Σ αi = 1 , where both summations are from i = 1 to k.
If the investor chooses values for the αi such that Var[rp] = 0, the portfolio will be a
hedge portfolio with a rate of return E[rp] = Σ αi E[ ri ] with absolute certainty.
The No Arbitrage Condition
A necessary condition for financial markets to be in equilibrium is something economists
have termed the no arbitrage condition. In words it says that any investor who incurs
zero risk and invests zero wealth must earn zero profits. A few examples will suffice to
convince students that financial markets cannot be in equilibrium if the no arbitrage
condition is not satisfied.
Example 1 -- the foreign exchange market.
Suppose the $US/$CAD exchange rate is simultaneously 0.75 in New York City and 0.76
in Toronto. This means that an investor can instruct her broker to purchase 1 Canadian
dollar in New York for a price of $0.75 US and to simultaneously sell 1 Canadian dollar
in Toronto for $0.76. The investor faces no risk and invests none of her own wealth and
yet earns a profit of $0.01 US for every Canadian dollar simultaneously bought and sold.
Clearly the no arbitrage condition is not satisfied here and just as clearly the foreign
exchange market is not in equilibrium. What will happen is that large numbers of
investors will try to buy Canadian dollars in New York and sell them in Toronto. This
will cause exchange rate to swiftly rise in New York and fall in Toronto until both
exchange rates are identical. The market will then be in equilibrium and the no arbitrage
condition will be satisfied.
Example 2 -- a hedge portfolio
Consider the hedge portfolio of risky assets described in the previous subsection and
suppose that the dollar cost of acquiring this risk-free portfolio is W. Instead of
purchasing the portfolio outright, the investor could acquire it by borrowing W dollars at
the risk-free rate of interest rf . If the investor does borrow, she will have invested zero of
her own wealth in a portfolio that has zero risk. She will receive the following profit:
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Profit = W (1+ E[rp] ) - W (1+rf ).
= W (E[rp] - rf ).
The first term on the RHS of the upper equation represents the gross return on the W
dollars invested in the hedge portfolio and is known with certainty. The second term on
the RHS represents the repayment of interest and principal on the investor's borrowing of
W dollars and is also known with certainty.
For the no arbitrage condition to hold, it must be the case that the certain rate of return
on the hedge portfolio be equal to the risk free rate; i.e. that E[rp] = rf . This makes
intuitive sense; a portfolio with no risk should earn the risk-free rate of interest.
But suppose that the no arbitrage condition did not hold here and that E[rp] > rf . Then
the security market can not be in equilibrium. A great many investors would wish to
borrow to buy the hedge portfolio and this would bid up the prices of the securities that
comprise it, causing the expected rates of return on these securities to decline. Since
E[rp] = Σ αi E[ ri ], the value of E[rp] must also decline until E[rp] = rf , at which point
security prices would stabilize and the market would reach equilibrium.
Example 3 -- a problem for students to solve
Consider a setting with 1 risk-free asset and 2 risky securities, A and B. There are only 2
possible states of the world in the future as described by the following table.
Probability of occurrence
Payoff of Security A
Payoff of Security B
State 1
1/2
$2.00
$1.50
State 2
1/2
$0.00
$0.50
In words: If State 1 occurs next time period, Security A will pay $2 and Security B will
pay $1.50. If, instead, State 2 occurs, Security A will pay nothing and Security B will
pay $0.50. Each State has an equal probability of occurring.
Observe from their respective payoffs that a portfolio composed of +2 units of Security B
and -1 unit of Security A is a hedge portfolio that will deliver a payoff of $1.00 in each of
the 2 states.
Currently the risk-free rate of interest is rf = 0.10 and the price of Security A is
PA = $0.80. You are asked to use the no arbitrage condition to determine the value of PB
(the current price of Security B).
{The solution will be revealed at a later date.)
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Observe that in each of the examples the no arbitrage condition imposes some
restrictions on the relationships that exist among various asset prices and/or rates of
return. In the foreign exchange example there are 2 asset prices -- the US dollar price of
1 Canadian dollar in New York and in Toronto. The no arbitrage condition requires that
these 2 asset prices be identical. In Example 2, the no arbitrage condition requires that
the rate of return on a hedge portfolio be equal to the risk-free rate. This imposes some
restrictions on the expected rates of return of the risky assets that make up a hedge
portfolio, though it is unclear from our example what form these restrictions might take.
Arbitrage Pricing Theory (APT) spells out the nature of these restrictions and it is to that
theory that we now turn.
A Short Introduction to Arbitrage Pricing Theory
APT is the impressive creation of Steve Ross. It is a much more general theory of the
pricing of risky securities than the CAPM. (Indeed, the CAPM can be shown to be a
special case of APT). APT requires no assumptions about investor's preferences other
than that they are risk averse and does not require special assumptions (e.g. normality)
about the probability distributions of rates of return. The rate of return on the market
portfolio plays no special role in the APT. In addition, APT readily generalizes to
multiple time periods. What follows is a derivation of APT in as simple a setting as
possible.
APT begins by trying to identify the underlying sources of uncertainty that make
securities risky. Any source of uncertainty that creates risk among many securities is
called a factor. There may be many factors in the real world (more on this later), but for
simplicity I will derive the APT under the assumption that there is a single factor
affecting the returns on risky securities and that factor is uncertainty about the overall
performance of the economy. (A not totally implausible assumption). Let the variable G
denote the rate of growth of real GDP during any time period. Then we will use as a
measure the performance of the economy the factor F1 defined as
F1 = G - E[G] .
Observe that F1 is the difference between the actual rate of growth in any time period and
the rate of growth that had been expected to occur a priori. In other words, F1 here is the
unexpected rate of growth in real GDP. If F1 > 0 in some time period, this means that
the economy grew faster than expected; if F1 < 0, then growth was less than expected.
Finally, observe that E[F1] = 0.
We can express the relationship between the rate of return on any security j and the
performance of the economy via the following equation:
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(1)
rj = E[rj ] + bj F1 + εj ,
where bj = Cov(rj ,F1) / Var(F1) and Cov( εj , F1 ) = 0.
Equation (1) says that there is a systematic relationship between the return on any
security and the performance of the economy. If bj > 0, then the return on security j tends
to be above / below its expected value as F1 is above / below zero. [We might expect the
value of b to be positive for most securities but there will be securities for which b is 0 or
negative.] Uncertainty about the value of F1 gives rise to systematic risk in all securities
for which the value of bj ≠ 0. The equation also says that rj has an unsystematic
component in the form of the random εj , which gives rise to unsystematic (diversifiable)
risk..
Var[rj] = bj2 Var[F1] + Var[εj]
systematic risk
unsystematic risk
We know that the unsystematic risk can be eliminated in a portfolio involving a large
number k of risky assets.
rp = Σ αj E[rj] + F1 Σ αj bj + Σ αj εj , with the summation from j = 1 to k .
When k is large, Σ αj εj will equal zero and the portfolio return can be expressed as
rp = E[rp] + bp F1 , where E[rp] = Σ αj E[rj] and bp = Σ αj bj .
Now consider 2 large portfolios that we shall refer to as portfolios A and B.
rA = E[rA] + bA F1
rB = E[rB] + bB F1
Let's combine these 2 portfolios into a hedge portfolio. Let w denote the share of A in the
hedge portfolio and (1-w) denote the share of B. We choose w so that
w bA + (1-w) bB = 0
(zero systematic risk);
hence w = bB / (bB -bA) .
Then the expected return on the hedge portfolio is
(2)
E[rH] = [bB / (bB -bA )]E[rA] + [-bA / (bB -bA )] E[rB]
with 100% certainty.
The no arbitrage condition (from which APT derives its name) requires that rH = rf , the
risk-free rate and this, along with Equation (2) implies
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(3)
(E[rA]- rf ) / bA = (E[rB]- rf ) / bB .
Equation (3) was derived directly for portfolios in which unsystematic risk is zero but it
applies as well to all securities because the unsystematic risk of a security can have no
influence on its expected return; in other words, (E[rj]- rf ) / bj must have the same
constant value for all securities. [If bj = 0 for some security, the value of (E[rj]- rf ) / bj is
undefined, but any security with bj = 0 has no systematic risk and must, therefor, have an
expected rate of return equal to the risk-free rate rf.
Let E[R1] denote the expected rate of return on a security that has a value of b=1. Then
APT implies that the following holds for all securities.
(4)
E[rj]- rf = bj (E[R1] - rf ) .
Observe that although our measure of overall economic performance F1 does not appear
directly in Equation (4) it does appear indirectly since bj = Cov(rj ,F1) / Var(F1) .
Now let's step back and take an intuitive look at what Equation (4) says. It says that in a
world in which the only source of overall uncertainty is unexpected growth in real GDP
(as measured by the factor F1) the risk of any security is proportional to the Covariance
between the rate of return on the security and F1. The larger is this Covariance, the
higher will be the expected rate of return on the security. [An implication of the Equation
is that if we could find a way to predict rates of growth of real GDP with 100% accuracy,
there would be no systematic risk and the expected return on all securities would be equal
to rf.]
It is easy to see from this result that CAPM is just a special case of APT. Had we not
chosen overall economic performance as our explanatory factor, but had instead chosen
the return on the market portfolio, then F1 would have been (rM - E[rM] ) and
Equation (4) would reduce to the familiar CAPM relationship.
Our derivation was purposely based on only one factor for simplicity, but APT readily
generalizes to any number of factors. Remember that a factor is an aggregate source of
uncertainty, and it is easy to think of things other than uncertainty about the rate of
growth of real GDP that might affect security returns. Empirical investigations using
APT have identified a total of 4 factors that appear to be important sources of systematic
risk for US securities. These are:
F1 -- unexpected growth in real GDP (as in our 1-factor derivation)
F2 -- unexpected price inflation
F3 -- differences between the yields on AAA versus Baa corporate bonds
F4 -- differences between the yields on long- versus short-term government bonds.
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The 4-factor counterpart to Equation (1) is
(5)
rj = E[ri ] + bj1 F1 + bj2 F2 + bj3 F3 + bj4 F4 + εj ,
where bjk = Cov( rj , Fk ) / Var(Fk ) for k = 1,2,3,4 and the factors are measured in such
a way that Cov( Fi , Fk ) = 0 for all i,k pairings.
Here there are four types of systematic risk -- one for each of the factors:
Var[rj] = bj12 Var[F1] + bj22 Var[F2] + bj32 Var[F3] + bj42 Var[F4] + Var[εj] .
systematic risk 1
systematic risk 2
systematic risk 3
systematic risk 4
unsystematic risk
We can create several large portfolios that each eliminate unsystematic risk, then
combine these into a hedge portfolio that eliminates all 4 types of systematic risk.
Applying the no arbitrage condition to the hedge portfolio yiels the following counterpart
to Equation 4.
(6)
E[rj]- rf = bj1 (E[R1] - rf ) + bj2 (E[R2] - rf )+ bj3 (E[R3] - rf )+ bj4 (E[R4] - rf ),
where E[Ri] = the expected rate of return on a security that has b=1 for the ith factor and
b = 0 for the other 3 factors.
For a couple of fairly simple examples of numerical applications of APT, students
should complete Problems 7.20 and 7.21 at the end of Chapter 7 in our textbook.
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