We considered portfolio optimization in Example 12.5 and in Section 13.2.2. For the sake of convenience, let us reconsider the problem here. We must allocate our wealth among n risky assets and a risk-free one. The returns of the risky assets are a vector of random variables with expected value μ and covariance matrix Σ; let rf be the return of the risk-free asset. Let w0 be the weight of the risk free asset in the portfolio, whereas the weights of the risky assets are denoted by wi, i= 1, …, n, collected into vector 
. We assume that short sales and cash borrowing are possible; hence, we do not enforce any nonnegativity restriction on portfolio weights. Then, the expected value of portfolio return is
and its variance is wTΣw. Note that w0 does not affect variance. If we assume a quadratic utility function, we are essentially lead to consider a mean-variance objective, with risk aversion coefficient λ, resulting in the following quadratic programming problem:
The factor 
is not really essential, as it may be included in the risk aversion coefficient, but it simplifies the derivatives we take below. This constrained optimization problem may be transformed into an unconstrained one by eliminating w0. Plugging
into the objective, we obtain
where μe,i ≡ μi − rf is the expected excess return of asset i. Since the leading rf term is inconsequential, the portfolio choice problem can be restated as
which is a convex optimization problem. Then, to find optimal portfolio weights, we just enforce stationarity conditions:
Solving a system of linear equations is easy enough, and we can find the optimal portfolio weights wi, i = 1, …, n, for the risky assets; then, using Eq. (14.31) we also get the weight of the risk-free asset. One of the difficulties of this simple framework is the estimation of problem inputs,31 such as the vector of expected returns μ and the covariance matrix Σ. Practical experience shows that if we take a simplistic approach and use straightforward sample estimates of these parameters, quite unreasonable portfolios are obtained. Indeed, they may include very large positions in a few assets, possibly with large short positions as well.
Example 14.16 Consider a universe consisting of seven assets. The correlations among their excess returns, over an investment horizon, are given by the following symmetric table:32
Let us assume that the vector of volatilities is
Based on these data, we may easily build the covariance matrix Σ. If the risk aversion coefficient λ is set to 2.5 and the expected excess return is 7% for each asset, the resulting portfolio weights are
This solution might look unreasonable because weights do not add up to 1, but we should remember that there is a weight w0 for the risk free asset, and therefore
The portfolio looks a bit extreme, as there are quite large weights. This is partially a consequence of a relatively small risk aversion; experience suggests that λ can be in the range from 2 to 4. In practice, by forbidding short sales and adding policy constraints bounding portfolio weights, we might get a more sensible portfolio, but doing so essentially means that we are shaping our strategy using constraints.
Fig. 14.6 Portfolio instability with respect to estimates of expected return.
A very surprising finding is the dramatic effect of changing estimates of expected excess returns on portfolio weights. Let us assume that the vector of expected (excess) returns is changed to
This could be a way to translate an investor’s view, feeling that some assets will outperform other assets. The new portfolio is
The change in portfolio composition is illustrated in Fig. 14.6. For each asset we display a pair of bars corresponding to portfolio weights; the second portfolio is associated with the right (white) bar within each pair. The dramatic change in the portfolio weights is rather evident and can be explained by the change in expected return of asset 3, 4, and 6, with respect to the first portfolio choice problem. But is this change justified, or is just due to estimation errors? The problem that we are highlighting is that an error in the estimate of expected value may have a significant impact on portfolio choice.
It is a common opinion that estimating expected return is even more troublesome than estimating covariances, in the sense that it has a stronger effect on optimized portfolios. A cynical view states that portfolio optimization is the best way to maximize the effect of estimation errors. Even if we refrain from being so drastic, it is certainly true that most financial analysts would feel comfortable only with the estimation of a few expected returns, for those market segments on which they have the most experience. In practice, analysts would not just estimate a parameter based on past observations; what about predicting an expected return for the future? There is no doubt that portfolio management should look forward, rather than backward, but again, analysts may do this for limited market segments. For the rest, they might just share the market consensus and go for a passive management strategy. We also recall that a passive management strategy is a consequence of the capital asset pricing model (CAPM).33 According to a passive strategy, one should just hold the market portfolio, i.e., a portfolio with stock share weights shaped after market capitalization of the corresponding firms. If we accept the validity of CAPM, we may even find expected excess returns implied by the market portfolio. According to CAPM, for each asset i = l, …, n, we have
where μm is the expected return of the market portfolio, which can be approximated by a broad index. This relationship may be restated in terms of expected excess returns:
We also note that expected excess returns can be interpreted as risk premia, as they state by how much the expected return of a risky asset exceeds the risk-free return rf. The coefficient βi looks much like the slope of a linear regression, as it is given by
where the numerator is the covariance between the return of asset i and the return of the market portfolio, and 
is the variance of market portfolio return. If we denote by wm the weights of the market portfolio, its return is
Hence, we have
Thus, we may collect the “asset betas” βi into vector β and rewrite CAPM as
trades off the risk premium (μm − rf) against squared market volatility. Incidentally, a comparison between Eqs. (14.32) and (14.33) suggests some link between δ and an average risk aversion coefficient. The important contribution of Eq. (14.33) is that it yields a consensus market view of the expected excess returns, implied by equilibrium. This might be considered as a starting point of an estimation process, which is not just backward-looking and based on historical data, but also forward-looking.
If we look forward, rather than backward, we are immediately lead to a question: How can we include subjective views that an investor might have? For instance, an investor could think that an asset will outperform another one by, say, 5%. Black and Litterman proposed an approach whereby expected returns are estimated based on two primary inputs:
- A forecast implied by market equilibrium
- A set of subjective views on expected returns
The way the two ingredients are blended depends on the uncertainty that is associated with each of them, and can be interpreted as a Bayesian estimate blending subjective expectations with an outside input.34
One way of stating the model is the following:
- The vector of expected excess returns is considered as a vector of random variables θ with multivariate normal distribution:
where μe is implied by market equilibrium and Σ is the estimated co-variance matrix. Note that we are trusting the estimate of covariance matrix in providing us the covariance matrix TΣ of the prior distribution; in fact, one of the main difficulties of the Bayesian framework is specifying sensible multivariate priors. The rationale here is that if assets are correlated, the error in estimating their risk premia will be as well. The parameter T can be used to fine tune the degree of confidence in the market view. - Subjective views about risk premia and expected returns can be expressed as linear relationships. For instance, say that we believe that asset 2 will outperform asset 5 by 2%. Then, we could write a condition such as
In this case, using expected return or risk premia is inconsequential. Several similar views could be expressed in terms of excess returns and collected in the following matrix form:
The view above would correspond to a row in matrix P, with elements set to 1 in columns 2 and 5, zero otherwise; the corresponding element in vector q would be 0.02. Of course, subjective views are uncertain as well, and we might express this as
where Ω is typically a diagonal matrix whose elements are related to the confidence in subjective views.
The Black–Litterman model, relying on Bayesian estimation, yields the following estimate of risk premia:
The proof of this relationship is definitely beyond the scope but it is quite instructive to note its similarity with Eq. (14.30). The interpretation is again that the estimate blends subjective views and objective data, in a way that reflects their perceived reliability.35 Several variations of the Black–Litterman model have been proposed, but the essential message is that subjective and more or less objective views may be blended within a Bayesian framework. In fact, Bayesian approaches have also been proposed in marketing, where they are most relevant to cope with brand-new products and markets, where past data are quite scarce or irrelevant.
Problems
14.1 Find the Nash equilibria in the games in Tables 14.2 and 14.4. Are they unique?
14.2 Consider the Cournot competition outcome of Eqs. (14.12) and (14.13). Analyze the sensitivity of the solution with respect to innovation in production technology, i.e., how a reduction in production cost ci for firm i affects equilibrium quantities, price, and profit.
14.3 Two firms have the same production technology, represented by the cost function:
The cost function involves a fixed cost F and a squared term implying a diseconomy of scale; a firm will produce only when its profit is positive. The two firms compete on quantity, and price is related to total quantity Q = q1 + q2 by the linear function P(Q) = 100 − Q. Find the maximum F such that both firms engage in production, assuming a Cournot competition.
14.4 Two firms have a production technology involving a fixed cost and constant marginal cost, as represented by the cost function:
Analyze the von Stackelberg equilibrium when firm 1 is the leader and the two firms compete on quantities.
14.5 Consider the data of the Braess’ paradox example in Section 14.5, but imagine that a central planner can assign routes to drivers, in order to minimize total travel cost. Check that adding the new link e, as in Fig. 14.4(b), cannot make the total cost worse.
14.6 Consider again the Bayesian coin flipping experiment of Example 14.14, where the prior is uniform. If we use Eq. (14.21) to find the Bayesian estimator, what is the estimate of θ after the first head? And after the second head?
14.7 Consider the fraction θ of defective items in a batch of manufactured parts. Say that the prior distribution of θ is a beta distribution with parameters α1 = 5 and α2 = 10 (see Section 7.6.2 for a description of the beta distribution). A new batch of 30 items is manufactured and two of them are found defective. What is the posterior estimat of θ, if we use Eq. (14.21)?
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