consumption is exactly equal to the marginal loss of expected utility at date t+ 1 resulting from the smaller amount of money for consumption available by selling the position in that bond at timet+1. Let us consider a one-period case. If we denote by C0 (known at time 0) and C1(stochastic viewed from time 0) the optimal consumption of the agent at dates 0 and 1,it will be the case in equilibrium that the price of the i0th asset satisfies
Pi(0)u0(C0) =E0P
Pi(1)u0(C1) i.e.
Pi(0) = E0P
Pi(1)u0(C1) u0(C0)
= EtP
Pi(1)Z1 1 +ρ0
where
Z1 = u0(C1) E0Pu0(C1) 1 +ρ0 = u0(C0)
EtPu0(C1) and this we may then write as
Pi(0) =EtQ
Pi(1) 1 +ρ0
where Qis defined by
Q(A) =EP(1AZ1).
This establishes the connection between utility maximization and the equiv-alent martingale measure. An agent who is risk neutral will have an affine utility function, and hence for such an agentu0(C1) is constant (i.e. does not vary with ωas C1 does). In that case Z1 = 1 andP =Q.
It is clear that P =Qis sufficient for the local expectation hypothesis to hold but it may seem to be too strong a requirement. After all, it is only an expectation of one random variable that we are referring to and one could imagine that a measure change would not alter this particular expectation.
To analyze this question a little further, consider the fundamental definition of a new measure through the random variable Z1 :
Q(A) =EP(1AZ1).
128 CHAPTER 8. STOCHASTIC INTEREST RATES For some random variable X,which could be the short rate at some future date, we have
EQ(X) =EP(XZ) and thereforeEQ(X) =EP(X) if and only if
EP(X(Z−1)) = 0.
SinceE(Z −1) = 0 this is the same as requiring Cov(X, Z) = 0.
Therefore, for the change of measure to preserve a mean value we must have that the variable in question is uncorrelated with the change of measure variable Z,and this will typically not hold in the term structure models we consider.
Chapter 9
Portfolio Theory
Matrix Algebra
First we need a few things about matrices. (A very useful reference for mathematical results in the large class imprecisely defined as “well-known”is Berck & Sydsæter (1992), “Economists’ Mathematical Manual”, Springer.)
• When x∈Rn and V∈Rn×n then
∂
∂x(x>Vx) = (V+V>)x
• A matrix V ∈ Rn×n is said to be positive definite if z>Vz > 0 for all z 6= 0. If V is positive definite then V−1 exists and is also positive definite.
• Multiplying (appropriately) partitioned matrices is just like multiplying 2×2-matrices.
• Covariance is bilinear. Or more specifically: WhenXis ann-dimensional random variable with covariance matrix Σthen
Cov(AX+B,CX+D) =AΣC>,
whereA,B, C, andD are deterministic matrices such that the multi-plications involved are well-defined.
Basic Definitions and Justification of Mean/Variance Analysis We will consider an agent who wants to invest in the financial markets. We look at a simple model with only two time-points, 0 and 1. The agent has an initial wealth of W0 to invest. We are not interested in how the agent
129
130 CHAPTER 9. PORTFOLIO THEORY rate of return
relative portfolio weights
determined this amount, it’s just there. There arenfinancial assets to choose from and these have prices
Si,t for i= 1, . . . , nand t = 0,1,
where Si,1 is stochastic and not known until time 1. The rate of return on asset i is defined as
ri = Si,1−Si,0
Si,0
,
andr= (r1, . . . , rn)>is the vector of rates of return. Note thatris stochastic.
At time 0 the agent chooses a portfolio, that is he buys ai units of asset i and since all in all W0 is invested we have
W0 = Xn
i=1
aiSi,0.
(Ifai < 0 the agent is selling some of asset i; in most of our analysis short-selling will be allowed.)
Rather than working with the absolute number of assets held, it is more convenient to work with relative portfolio weights. This means that for the ith asset we measure the value of the investment in that asset relative to total investment and call this wi, i.e.
wi = aiSi,0 Pn
i=1aiSi,0
= aiSi,0 W0
.
We put w = (w1, . . . wn)>, and have that w>1 = 1. In fact, any vector satisfying this condition identifies an investment strategy. Hence in the fol-lowing a portfolio is a vector whose coordinate sum to 1. Note that in this one period model a portfolio w is not a stochastic variable (in the sense of being unknown at time 0).
The terminal wealth is W1 =
Xn i=1
aiSi,1 = Xn
i=1
ai(Si,1−Si,0) + Xn
i=1
aiSi,0
= W0 1 + Xn
i=1
Si,0ai
W0
Si,1−Si,0
Si,0
!
= W0(1 +w>r), (9.1)
so if we know the relative portfolio weights and the realized rates of return, we know terminal wealth. We also see that
E(W1) =W0(1 +w>E(r))
131
utility function and
Var(W1) = W02Cov(w>r,w>r) = W02w>Cov(r)
| {z }
n×n
w.
In this chapter we will look at how agents should choose w. We will focus on how to choose wsuch that for a given expected rate of return, the variance on the rate of return is minimized. This is called mean-variance analysis. Intuitively, it sounds reasonable enough, but can it be justified?
An agent has a utility function, u, and let us for simplicity say that he derives utility from directly from terminal wealth. (So in fact we are saying that we can eat money.) We can expand u in a Taylor series around the expected terminal wealth,
u(W1) = u(E(W1)) +u0(E(W1))(W1−E(W1)) +1
2u00(E(W1))(W1−E(W1))2+R3, where the remainder term R3 is
R3 = X∞
i=3
1
i!u(i)(E(W1))(W1−E(W1))i,
“and hopefully small”. With appropriate (weak) regularity condition this means that expected terminal wealth can be written as
E(u(W1)) =u(E(W1)) + 1
2u00(E(W1))Var(W1) +E(R3),
where the remainder term involves higher order central moments. As usual we consider agents with increasing, concave (i.e. u00 < 0) utility functions who maximize expected wealth. This then shows that to a second order approximation there is a preference for expected wealth (and thus, by (9.1), to expected rate of return), and an aversion towards variance of wealth (and thus to variance of rates of return).
But we also see that mean/variance analysis cannot be a completely gen-eral model of portfolio choice. A sensible question to ask is: What restrictions can we impose (on u and/or on r) to ensure that mean-variance analysis is fully consistent with maximization of expected utility?
An obvious way to do this is to assume that utility is quadratic. Then the remainder term is identically 0. But quadratic utility does not go too well with the assumption that utility is increasing and concave. If u is concave (which it has to be for mean-variance analysis to hold ; otherwise our interest
132 CHAPTER 9. PORTFOLIO THEORY mean/variance
analysis Markowitz
analysis
would be in maximizing variance) there will be a point of satiation beyond which utility decreases. Despite this, quadratic utility is often used with a
“happy-go-lucky” assumption that when maximizing, we do not end up in an area where it is decreasing.
We can also justify mean-variance analysis by putting distributional re-strictions on rates of return. If rates of return on individual assets are nor-mally distributed then the rate of return on a portfolio is also normal, and the higher order moments in the remainder can be expressed in terms of the variance. In general we are still not sure of the signs and magnitudes of the higher order derivatives of u, but for large classes of reasonable utility functions, mean-variance analysis can be formally justified.