With Q a risk neutral probability measure andω ∈ Ω, recall that Q(ω)/B1(ω) is some- times called the state price ofω. For this reason, the random variable
L(ω) ≡ Q(ω)
P(ω)
is called the state price vector or the state price density. The main result to be shown in this section is that the risk premium of an arbitrary portfolio is proportional to the covanance6 between a return corresponding to the state price density and the return for the portfolio, a result that resembles a principal finding of the capital asset pricing model. Assuming the time t = 0 price Sn(0) is strictly positive, the return R for risky security
n is defined to be the random variable
Rn≡ Sn(1) − Sn(0) Sn(0)
, n = 1, . . . , N
Similarly, the return corresponding to the bank account is defined by
R0≡ B1− B0
B0
= r
The returns are useful quantities for a variety of purposes, one of which is that if you know the time t = 0 prices and the returns, then you can compute time t = 1 prices. Since prices are non-negative one has Rn≥ −1, with equality if and only if Sn(1) = 0. It
is left as an exercise to verify that the gain for a portfolio can be written as
G = H0B0R0+
N
∑
n=1
HnSn(0)Rn (1.29)
Hence the gain for a portfolio is a weighted combination of the underlying returns, each weight being the amount of money invested at time t = 0 in the corresponding security.
The returns can also be used to compute risk neutral probability measures. Since
S∗n(1) − S∗n(0) = Sn(1) − B1Sn(0) B1 = [1 + Rn]Sn(0) − [1 + R0]Sn(0) 1 + R0 = Sn(0) µ Rn− R0 1 + R0 ¶
it follows from (1.15) that
If Q is a probability measure with Q(ω) > 0 for all ω ∈ Ω, then Q is a risk neutral probability measure if and only if
EQ µ Rn− R0 1 + R0 ¶ = 0, n = 1, . . . , N (1.30)
6For two random variables X and Y , the covariance cov(X,Y ) is defined to be E[XY ] − E[X]E[Y ]. Note that
cov(X − E[X],Y ) = cov(X,Y ). Moreover, given three random variables X, Y , and Z and two scalars a and b, one has cov(aX + bZ,Y ) = a cov(X,Y ) + b cov(Z,Y ).
1.6. RISK AND RETURN 29
Notice that when the interest rate R0 = r is deterministic, the equation in (1.30) be- comes simply
EQ[Rn] = r, n = 1, . . . , N
This is one example of many situations where, under the assumption of a determin- istic interest rate, one has a nice, and often important, relationship involving returns. Therefore, this assumption will be in force for the balance of this section, as will be the assumption that there exists a risk neutral probability measure Q.
The mean return for security n, denoted Rn = E[Rn], often plays an important role. For example, it is easy to see that cov(Rn, L) = E[RnL] − E[Rn]E[L] = EQ[Rn] − E[Rn] =
r − ¯Rn. In other words,
¯
Rn− r = − cov(Rn, L), n = 1, . . . , N (1.31)
The difference ¯Rn− r here is called the risk premium for the security: normally this is
positive because investors usually insist that the expected returns of risky securities be higher than the riskless return r. Thus (1.31) says that the risk premium of a security is related to the correlation7 between the security’s return and the state price density.
Consider the return R of a portfolio corresponding to an arbitrary trading strategy
H = (H0, H1, . . . , HN). Assuming V0> 0, this is
R =V1−V0 V0
Using Sn(1) = Sn(0)[1 + Rn] and the definition of Vt one obtains
R = H0 V0r + N
∑
n=1 · HnSn(0) V0 ¸ Rn (1.32)If you interpret H0/V0 as the fraction of money invested in the savings account (re- call B0 = 1) and HnSn(0)/V0 as the fraction of money invested in the nth security, then (1.32) says that the return on the portfolio is a convex combination of the returns of the individual securities. Using (1.31), (1.32), and some basic properties of the covariance, it is straightforward to verify that
¯
R − r = − cov(R, L) (1.33) where, of course, ¯R = E[R].
Now fix two scalars a and b with b 6= 0, and assume the contingent claim a + bL is attainable, that is, suppose there exists some trading strategy H0 such that V0
1 = a +
bL. Since V00(1 + R0) = a + bL (here V0 and R0 denote the value and return processes,
7The variance, denoted var(X), of a random variable X is defined by var(X) ≡ E[X2] − (E[X])2= E[(X −
E[X])2]. The standard deviation of X isσ
x≡
p
var(X). The correlation between the random variables X and Y (assumingσX> 0 andσY> 0) is defined byρ(X,Y ) ≡ cov(X,Y )/(σXσY). Hence the risk premium for security n
respectively, corresponding to H0), one can substitute for L and use the properties of the
covariance relationship to verify that
cov(R, L) =V
0
0
b cov(R, R
0)
(R still corresponds to an arbitrary trading strategy). Hence (1.33) can be rewritten as ¯ R − r = −V 0 0 b cov(R, R 0) (1.34)
In particular, in the special case where you choose H = H0, (1.34) says that
¯ R0− r = −V 0 0 b cov(R 0, R0) = −V00 b var(R 0)
Using this to substitute for V0
0/b in (1.34), where now we are back to an arbitrary trading strategy H, we obtain the following:
Suppose for scalars a and b the contingent claim a + bL is generated by some portfolio having return R0and suppose the interest rate r is determin- istic. Let R be the return of an arbitrary portfolio. Then
¯ R − r = cov(R, R 0) var(R0) ( ¯R 0− r) (1.35)
The ratio cov(R, R0)/ var(R0) is called the beta of the trading strategy H with respect to the trading strategy H0. This result says that the risk premium of H is proportional
to the risk premium of H0, with the proportionality constant being this beta. Or from a slightly different perspective, (1.35) says that the risk premium is proportional to its beta with respect to a linear transformation of the state price density. This result re- sembles the traditional capital asset pricing model, only here H0 corresponds to a linear
transformation of the state price density instead of the market portfolio.
Notice that with a deterministic interest rate r and with arbitrary scalars a and b (b 6= 0), the contingent claim a + bL is attainable if and only if the state price density L is. This is because H0(1 + r) + ∑HnSn(1) = a + bL if and only if
1 b · H0− a 1 + r ¸ (1 + r) + N
∑
n=1 1 bHnSn(1) = LExercise 1.17. Verify equation (1.29), both in general and by applying it to example 1.1. Exercise 1.18. Assuming the time t = 0 price is strictly positive, the discounted return
R∗n is defined by R∗n ≡ [S∗n(1) − S∗n(0)]/S∗n(0) for n = 1, . . . , N. Show that (a) G∗=
N
∑
n=1
1.6. RISK AND RETURN 31
(b) R∗n= Rn− R0 1 + R0
, n = 1, . . . , N
(c) The strictly positive probability measure Q is a risk neutral probability measure if and only if EQ[R∗n] = 0 for n = 1, . . . , N.
Exercise 1.19. Analyze the risk and return properties of example 1.1 assuming P(ω1) =
p for a general parameter 0 < p < 1.
(a) What are R1and ¯R1? (b) What is L?
(c) Verify (1.31) for n = 0 and 1.
From now on suppose H = (H0, H1) = (1, 3). (d) What are R and ¯R?
(e) Verify (1.32). (f) Verify (1.33).
(g) What are H0, V , and R0? (h) Verify (1.34).
Chapter 2
Single Period Consumption and
Investment
2.1 Optimal Portfolios and Viability
This chapter is concerned with the problem of choosing the best trading strategy for the purpose of transforming wealth invested at time t = 0 into time t = 1 wealth. With some variations of this problem that will be considered in later sections, a portion of the wealth is consumed at time t = 0. The problem is to compute an optimal trading strategy, and for this a measure of performance is needed.
The measure of performance that will be used here is that of expected utility. In particular, suppose u : R × Ω → R is a function such that w → u(w, Ω) is differentiable, concave, and strictly increasing for eachω ∈ Ω. If w is the value of the portfolio at time
t = 1 andω is the state, then u(w, ω) will represent the utility of the amount w. Hence our measure of performance will be the expected utility of terminal wealth, that is,
Eu(V1) =
∑
ω∈ΩP(ω)u¡V1(ω), ω ¢
Note that the utility function u can depend explicitly on both the terminal wealth w and the stateω. However, for many applications it suffices for u to depend only on the wealth, in which case u is simply a concave, strictly increasing function with a single argument.
Let H denote the set of all trading strategies, that is, H = RN+1 the linear space of all vectors of the form (H0, H1, . . . , HN). Let v ∈ R be a specified scalar representing the initial, time t = 0 wealth. We are interested in the following optimal portfolio problem:
maximize
H∈H Eu(V1)
subject to V0 = v
(2.1) Since V1= B1V1∗ and V1∗ = V0∗+ G∗, this is the same as
maximize E£u¡B1{v + H1∆S∗1+ · · · + HN∆S∗N}
¢¤
(2.2)