Mean–risk optimization

If asked about our utility function, we would hardly be able to give a sensible answer. However, in real life, we do trade off expectations against risk; to do so, we need a way to measure risk.

DEFINITION 13.1 A risk measure is a function ρ(X), mapping a random variable X into the set of nonnegative real numbers .

In practice, the random variable X could be a random profit or a random return on an investment; generally speaking, it represents the consequence of our decision. The idea behind a risk measure is that the larger its value, the riskier our choice are. We should not confuse such a risk measure with a coefficient of risk aversion. The latter is a measure of the subjective attitude of a decision maker toward risk; the former is an objective measure of the riskiness of a lottery or a prospect, and not a measure of perceived risk. Apparently, the most natural risk measure we may come up with is related to variance or standard deviation. Standard deviation is more natural in the sense that it is measured in the same units as the corresponding expectation; however, variance may be more convenient when building optimization models. In fact, minimization of variance is equivalent to minimization of standard deviation, but the former may result in a convex optimization problem, whereas the latter does not. Later, we will see that variance and standard deviation might not make very good risk measures, but they are a suitable starting point.

Now, given a random variable X representing profit or return, we should find a tradeoff between its expected value E[X] and a risk measure ρ(X). In Section 12.3.3 we have considered multiobjective optimization, and we have defined the concept of efficient solution and efficient frontier. Even if we are not able to spot a single optimal solution trading off expectation and risk, we might trace the efficient frontier with respect to these conflicting objectives.

Example 13.8 (Mean−variance portfolio efficiency) We are already familiar with elementary portfolio choice problems.⁹ Given a universe of assets with random return R_i, i = 1, …, n, expected return μ_i, and covariance matrix Σ = [σ_ij], we should select portfolio weights ω_i adding up to one. Given a vector w representing the portfolio, its expected return and variance are given as follows:

The standard deviation of return, σ_p, might be considered as a risk measure, as it provides us with a measure of dispersion. Then, we may trade off expected return μ_p against σ_p. Of course the tradeoff may be unclear, but it can be visualized by tracing the frontier of mean–variance efficient portfolios, depicted in Fig. 13.7.¹⁰ The efficient frontier is bent toward the northwest, since we want to maximize μ_p and minimize σ_p. Portfolios on this frontier are called mean-variance efficient, since variance and standard deviation may be switched from one to another depending on convenience. A portfolio is efficient if it is not possible to obtain a higher expected return without increasing risk or, seeing things the other way around, if it is not possible to decrease risk without decreasing expected return.

It is important to emphasize that we are talking about portfolios. If we compare two assets, it is tempting to say that the decision problem is trivial when μ₁ > μ₂ and . In this case, asset 1 has a larger expected return than asset 2, and it is also less risky; hence, a naive argument would lead to the conclusion that asset 2 should not be considered at all. Actually, this may not be the case, since we have neglected the possible correlation between the two assets. The inclusion of asset 2 may, in fact, be beneficial in reducing risk, particularly if its return is negatively correlated (σ₁₂ < 0) with the return of asset 1.

Fig. 13.7 Mean–variance efficient portfolio frontier.

To trace the efficient frontier, there are two main possibilities. One is to scalarize the vector of two objectives and solve

for various values of coefficient λ. This coefficient penalizes variance, and can be interpreted as a risk aversion coefficient. By changing the value of λ, we can trace the efficient set. From a mathematical perspective, this is a quadratic programming model; since the covariance matrix is positive semidefinite, it is a convex problem. We see the reason for using variance, rather than standard deviation, in this kind of optimization models: Variance is a convex quadratic form with respect to portfolio weights; standard deviation involves a square root that makes things a bit more complicated. Incidentally, we see that the problem is somewhat related to a quadratic utility function.¹¹ Unfortunately, it is a bit difficult to get a feeling for the parameter λ. Alternatively, we may use the constraint approach, leading to the solution of the problem

for various values of the target return μ_t. Again, this is a convex quadratic programming problem. For an investor, specifying a target return may be more intuitive than struggling with risk aversion coefficient λ.¹² Anyway, common sense and experience suggest risk aversion coefficients in the range between 2 and 4.

The mean-variance efficiency framework has played a pivotal role in the development of modern financial theory, even though it has quite significant practical limitations:

• The estimation of the relevant data may not be easy.
• We are neglecting transaction costs.
• We are assuming that correlations do not change over time, even under severe market conditions.

More generally, the use of a risk measure based on variance or standard deviation is questionable. In fact, variance measures both positive and negative deviations with respect to the expected return. However, this symmetry does not make economic sense: We are certainly not annoyed by extra profits. When dealing with a symmetric distribution like the normal one, standard deviation makes sense, since the good and the bad tails of the distributions have the same shape. However, this does not necessarily apply to skewed distributions. It may also be argued that not only skewness, but also kurtosis can play a role in measuring risk: Kurtosis might do a better job at measuring extreme risks, i.e., those associated with fat tails. More recently, alternative risk measures have been proposed, based on quantiles.