Portfolio choice
I would like to add a comment concerning portfolio theory as a part of the microeconomics of action under uncertainty. It has not always been considered so. For example, when I defended my dissertation as a student in the Economics Department of the University of Chicago, Pro- fessor argued that portfolio theory was not Economics, and that they could not award me a Ph.D. degree in Economics for a dissertation which was not in Economics. I assume that he was only half serious, since they did award me the degree without long debate. As to the merits of his arguments, at this point I am quite willing to concede: at the time I defended my dissertation, portfolio theory was not part of Economics. But now it is. – (in his 1990 Nobel prize lecture)
A portfolio is an investment consisting of several assets. Before the fundamental work of Markowitz (published in 1952), investment analysis mainly consisted of analyzing indi- vidual securities. Markowitz’s mean-variance framework exemplified the trade-off between return and risk, as well as the value of diversification. In this section, we explain some background concepts and work out the mathematics of the mean-variance framework.
6.1 Expected utility and risk-aversion
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We consider a one-period investment problem with n securities. Let their simple re- turns over the investment period be R1,…,Rn. The investor chooses portfolio weights w1,…,wn ∈Rsuchthatw1+···+wn =1andreceivesreturn
(See (2.5).) We model R = (R1,…,Rn)⊤ as a random vector, so that Rp is a random variable. If W0 > 0 is the initial wealth of the investor, the final wealth is W = W0(1+Rp). Since W is random, this is a typical problem of decision under uncertainty. If the distribution of W is known, a standard approach is expected utility theory. Given two random variables W , W ̃ , we write W ≼ W ̃ if the investor (weakly) prefers W ̃ to W . If W ≼ W ̃ and W ̃ ≼ W , then the investor is indifferent between the two choices. We call ≼ a preference relation. Under certain assumptions on the choice problem and the preference relation (in particular, the von Neumann-Morgenstern axioms which model rationality of
the economic agent), it can be shown that there exists a utility function u(·) such that W ≼ W ̃ ⇔ E[u(W)] ≤ E[u(W ̃ )].
When such a utility function exists, the choice problem is equivalent to the maximization of expected utility:
max E[u(W )], (6.2)
where the maximum is taken over the available actions. An example of u is u(w) = log w, the log utility. It has a long history originating from ’s solution (1738) to the St. Petersburg paradox (1711).
wiRi. (6.1)
Since people prefer more wealth than less wealth, it is typical to assume that the utility function u is nondecreasing; if u is differentiable, then u′ ≥ 0. It is also recognized that people are risk averse. For example, given an initial wealth w0 and x > 0, consider a final wealth W where
P(W =w0 +x)=P(W =w0 −x)= 21.
Most people prefer the constant w0 to W, i.e., w0 ≽ W. In terms of the utility function,
This motivates the general definition of risk aversion.
u(E[W ]) ≥ E[u(W )]. (6.3)
Financially, the agent is risk averse if E[W ] ≽ W for any W . From Jensen’s inequality (Theorem 2.3) we see that a concave utility function is risk averse. In fact the converse also holds, and we have:
Theorem 6.2. A utility function u is risk averse if and only if it is concave.
If u is twice differentiable, then u is risk averse if and only if u′′ ≤ 0. For example, the log utility u(w) = log w is concave and hence is risk averse. For completeness provide some common examples of risk averse utility functions.
u(w0) ≥ E[u(W)] = 12u(x0 − x) + 12u(x0 + x).
Definition 6.1 (Risk aversion). A utility function u is risk averse if for any random W
Example 6.3 (Exponential utility). The exponential utility is given by e−αw
u(w)= −α, Example 6.4 (Power utility). The power utility is given by
u(w) = w1−γ − 1, 1−γ
whereγ>0andγ̸=1. Asγ→1werecoverthelogutilityu(w)=logw.
where α > 0 is a constant.
Remark 6.5. It was shown by empirical experiments that people consistently violate the von Neumann-Morgenstern axioms. Alternative approaches to decision under uncertainy are studied in the field of behavioural economics.
6.2 The mean-variance framework
For a generic utility function, the expected utility maximization problem
max Eu(W ) (6.6)
requires knowing the joint distribution of R = (R1, . . . , Rn)⊤. The mean-variance frame- work may be regarded as a simplification. Instead of the joint distribution fo R, we “only” require knowledge of the mean vector μ = E[R] = (E[R1], . . . , E[Rn])⊤ and the covariance
matrix Σ = (Cov(Ri,Rj)), i.e., the first two moments of the distribution. (We will see that the estimation of μ and Σ is highly nontrivial.) From these, we may compute (as done in Section 2.2.3) the mean and variance of the portfolio:
μp = E[Rp] = w⊤μ, σp2 = Var(Rp) = w⊤Σw.
The main idea is that the portfolio variance σp2 is a measure of the risk of the portfolio. It is desirable to maximize μp while keeping σp2 as small as possible. Let 1 = (1, . . . , 1)⊤ be the vector of ones. Given a required expected return m, the basic version of the mean-variance portfolio optimization problem is
min 1w⊤Σw, such that 1⊤w = 1 and μ⊤w = m. (6.7) w∈Rn 2
Remark 6.6.
(i) The mean-variance optimization problem can be stated in other ways. One example
min μ⊤w − λw⊤Σw, w∈Rn ,1⊤ w=1 2
where λ > 0 is a fixed constant. One may interpret the objective function as a utility function, where λ is a measure of risk aversion. It can be shown that the solutions (for varying λ) trace out the same efficient frontier (see Definition 6.11).
(ii) In practice, one includes in (6.8) other constraints to control the portfolio weights. An example is w ≥ 0 (i.e., wi ≥ 0 for all i) which implies no short selling. When extra constraints are included, the problem has to be solved numerically. There are many available packages based on quadratic programming. Additional constraints will be explored in the assignment.
A closely related problem is to minimize the variance without any constraints on the expected return. This problem is meaningful because in practice the mean vector is very difficult to estimate.
Definition 6.7 (Global minimum variance portfolio). The global minimum variance port- folio is the solution to the problem
min 1w⊤Σw, such that 1⊤w = 1. (6.8) w∈Rn 2
The optimization problems (6.8) and (6.8) can be solved using the method of Lagrange multipliers. In what follows we assume Σ is invertible.
Proposition 6.8. The solution to (6.8) is
wGMV = Σ−11 .
1⊤Σ−11 The minimum variance is σ2 = 1 .
GMV 1⊤Σ−11
Proof. Consider the Lagrangian
L(w,λ)= 12w⊤Σw−λ(1−1⊤w).
Differentiating with respect to w, we have the first order condition ∂L =Σw−λ1=0⇒w=λΣ−11.
Next we choose λ so that the constraint 1⊤w = 1 is satisfied. Setting 1⊤(λΣ−11) = 1,
we have λ = (1⊤Σ1)−1. Plugging this into the above gives (6.9) and the expression for σG2MV .
Example 6.9 (Value of diversification). Suppose Σ = diag(σ2, . . . , σ2) = σ2I is a diagonal
matrix, i.e., the stock returns are pathwise uncorrelated. Also let μ = c1 be a constant
vector. Then Σ−1 = 1 I and the minimum variance portfolio is σ2
1I1 wGMV = σ2 = 1,
1⊤1I1 n σ2
which is the equal-weighted portfolio. The corresponding variance is
2 σ2⊤ σ2 σGMV =n21 I1= n.
when n is large (and σ2 is kept fixed), this variance tends to zero. When the stocks are not strongly correlated, diversification can lead to significant reduction of risk as measured by the variance.
By a similar approach, we can explicitly solve (6.8), but the resulting expressions are more complicated.
Proposition 6.10. Consider the optimization problem (6.8). Define the following quan- tities:
Assume ∆ > 0. Let
λ=C−Bm, γ=Am−B. ∆∆
A=1⊤Σ−11, B=1⊤Σ−1μ, C=μ⊤Σ−1μ, ∆=AC−B2.
Then the optimal solution to (6.8) is
wopt = λΣ−11 + γΣ−1μ,
and the corresponding variance is
σo2pt(m) = ∆1 (Am2 − 2Bm + C).
Proof. Exercise.
From (6.11), we see that the optimal σp2 traces out a parabola as a function of m. (We have a hyperbola if we look instead at σp.) The vertex of the parabola corresponds to the global minimum variance portfolio.
and estimate Σ by
(R(t) − μˆ)(R(t) − μˆ)⊤.
mean-variance problem as if μˆ are Σˆ are the true parameters. 48
Figure 6.1: Efficient frontier for a hypothetical example with n stocks. The black dots represent initial assets, and the global minimum variance portfolio is shown by the red dot.
Definition 6.11 (Efficient frontier). The efficient frontier is the locus of (σo2pt(m), m) for m ≥ μ⊤wGMV .
For a mean-variance investor, the portfolio choice must lie on the efficient frontier. The exact location depends on the risk aversion.
An example of the efficient frontier is shown in Figure 6.1. We consider a hypothetical market with n = 10 stocks. We generate randomly and then fix a mean vector μ and a covariance matrix Σ. The points (σi2, μi), i = 1, . . . , n, are shown by the dots. The efficient frontier is shown by solid back curve (the lower half is shown by the dashed curve). The global minimum variance portfolio is shown by the red dot. By combining the assets, it is possible to produce portfolios that are more efficient (i.e., having a higher expected return and a smaller variance) than each of the assets.
6.3 Naive implementation and its limitations
To implement the mean-variance optimization problem (6.8), we need estimates of the mean vector μ and the covariance matrix Σ. Suppose we have observations R(1), . . . , R(T ) of the simple returns. A naive method is to use the sample moments. That is, we estimate μ by
μˆ = T R(t),
Then (provided that Σˆ is invertible) we may plug these values into (6.8) and solve the
expected return
−0.05 0.05 0.10 0.15 0.20
0 20 40 60 80 100
0 20 40 60 80 100
0 20 40 60 80 100
0 20 40 60 80 100
Figure 6.2: Top: Expected return and variance for wˆopt which is estimated using 60 samples (repeated 100 times). Bottom: The estimated weights for the first 4 assets.
If R(1),R(2),… are i.i.d. (and the distribution has finite second moments), then μˆ and Σˆ are consistent estimators of the respective quantities. In practice, the number of samples cannot be very large because market conditions change over time. Thus, one is subject to estimation errors even if the i.i.d. assumption holds.
To give a concrete example, consider the context depicted in Figure 6.1. We assume in addition that R ∼ N(μ,Σ) follows the multivariate normal distribution. Suppose we
−0.1 0.0 0.1 0.2 0.3 −0.1 0.0 0.1 0.2 0.3
−0.1 0.0 0.1 0.2 −0.1 0.0 0.1 0.2
expected return
0.0 0.1 0.2 0.3
observe T = 60 i.i.d. observations from R. If R represents the monthly return, the sample period is 5 years. We compute μˆ and Σˆ and compute the optimal portfolio wˆopt with required return m = 0.1 (the numerical values may not be realistic but this is nothing but a toy example). For this portfolio, we compute the expected return and variance using the true parameters μ and Σ. We repeat this procedure 100 times. We show the results in Figure 6.2. The estimated portfolios are typically inefficient, some significantly so. We also show the estimated portfolio weights for the first 4 assets over the 100 batches. We see that the portfolio weights can be significantly different from the true optimal portfolios.
Remark 6.12. Another technical issue is that when n (the number of assets) is large, T (the number of samples) is typically smaller than n. When T ≤ n, the sample covariance matrix Σˆ is not invertible.
The covariance matrix has n(n + 1)/2 unknown parameters (note that Σ is symmet- ric). In the literature, academic researchers and practitioners have considered various approaches to improve the estimation of Σ. One approach, motivated by Bayesian statis- tics, is to use a skrinkage method which replaces the sample covariance matrix Σˆ by
Σ ̃ = αΣˆ + (1 − α)Dˆ , (6.14)
where Dˆ is the diagonal matrix of Σˆ and α ∈ (0,1) is a tuning parameter. Another approach, considered later, is to impose structural conditions on the covariance matrix. For example, suppose that the asset returns satisfy the single index model
Ri = αi + βiRM + εi, (6.15) where RM is the “market return” and the noises εi have mean zero and are pairwise
uncorrelated. Under this assumption, we have
Cov(Ri,Rj) = βiβjσM2 , (6.16)
where σM2 = Var(RM ). In matrix form, we have
Σ = σ M2 β β ⊤ ,
where β = (β1, . . . , βn)⊤. This reduces the number of parameters to n + 1. The single index model is closely related to the capital asset pricing model (CAPM) which will be studied in the next section.
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