Methodological Concepts

Seen in this light, risk budgeting is a method for allocating active risk. In times in which valuations in general are falling or stagnant and plans must face funding shortfalls, active risk is a commodity that can be allocated along with the remaining assets in the portfolio.

What does it mean to be confident in a view that a particular manager will generate relatively superior active returns? Budgeting active risk is predicated on ranking relative likelihoods of success among active managers. On an intuitive level, confidence in a view is inversely related to the level of active risk (or just returns volatility). More formally, confidences can be implied using the estimated moments of the returns distribution (sample covariances and mean returns) in conjunction with a set of views concerning relative asset performance (for example, small cap stocks will outperform large cap stocks by 50 basis points). Similarly, a set of views and their respective confidences will produce a set of expected returns that reflect a mix of sample information (covariances and mean returns) and prior information (views and associated confidences). Black and Litterman (1992) describe these expected returns as the mean of the posterior distribution in a Bayesian setup in which sample information is mixed with prior information in an optimal way. We cover this topic in more detail further on. This so-called Black-Litterman model was refined further in He and Litterman (1999); more accessible versions can be found in Idzorek (2003). We use the Black-Litterman model as well in our approach to risk budgeting as a formal method of incorporating views and risks in a risk budgeting framework. Mean-variance optimizers require a covariance matrix and a vector of expected returns. Up to now, we have seen how variances can be decomposed to highlight the impact of small changes in individual asset contributions to portfolio risk. We now develop in more detail the expected return assumption. In the introduction, we discussed briefly an example in which we wanted to gain exposure to the active risk in a hypothetical small cap strategy by porting its alpha onto a large cap futures position. The important point to make here is the belief that this active strategy would work.

Mixing Subjective Beliefs with the Historical Record

All portfolio managers have beliefs about forward expected returns, and these beliefs do not necessarily need to be consistent with the historical record. (The manager may have insight into future developments not in the historical record.) And because expected returns are generally more difficult to estimate than volatilities and covariances, debate centers less on covariance estimates and more on return estimation. One question that arises is how a meaningful discussion can be facilitated concerning differences in expected returns and their impact on portfolio allocation and risk. The Black-Litterman model provides a platform for constructive debate.

Suppose we have a vector of expected returns generated from either historical data or a factor model and a covariance matrix. Assume covariances are taken as given. Now, suppose that, independent of expected returns, a manager opines the view that asset class X (for example, the small cap strategy) will outperform asset class Y by some stated amount. As a consequence of this view, he proposes an increase in the allocation to asset X. (Forget about their respective volatilities for now and just assume that the volatility for X is no more than for Y.)

How do we assess this view? Is it risky? How confident should we be in the view itself? What are the implications for plan-wide risk? To answer this question, we must consider first the relative confidence in the view and the impact this has on our existing covariance matrix. The covariance matrix summarizes our current beliefs about risk. Views complicate that picture. More specifically, unless we are 100 percent confident in the view(s), then we must update our estimates of risk by adding to our covariance matrix another matrix summarizing confidences in our views. If we have only a single view (there are no limits to the number of views, however), then that simply means we have zero views on the remaining asset returns, and the matrix summarizing confidences in views is populated mostly by zeros. The point is that the revised covariance matrix is a combination of the original covariances and the view confidences. Lower confidences imply revising the original covariances upward by proportionately more.

What does this have to do with assessing the impact of the view? The view itself almost never replaces the expected return unless we have 100 percent confidence. Rather, the view modifies the expected return. Black-Litterman provides an optimal method for modification. More specifically, the modified expected return vector is a weighted average of the original expected return and the view(s). The weights are a function of the original covariance matrix and the matrix of view confidences. In Bayesian probability theory, the revised returns are the means of the posterior distribution. The views can be thought of as being combined with the priors (original covariances and expected returns) to form the posterior. It is the mean of this posterior and the posterior covariance matrix that find their way into the mean-variance optimization problem. Solving this yields an allocation reflecting the views and the risk or tracking error on the revised portfolio along with associated measures of risk decomposition and attribution discussed in the previous section.

Some detail will illuminate these ideas. Since the Black-Litterman method is covered in so many other citations, for example, Idzorek, that we leave many of the details to those papers. Think of priors as composed of the sample vector of expected returns and the covariance matrix that are estimated from historical returns. Expected returns, for example, could be the plan's long-run forward returns expectations. Denote these as Π. Therefore, observed returns are distributed around Π with error ε_u, according to:

and assume that returns are normally distributed, that is, , where V is the covariance matrix of returns and the parameter τ simply scales this matrix. (We can ignore τ's contribution in the discussion that follows, so let us set it equal to 1.)

As an aside, Black and Litterman argue that market capitalization weights wm satisfy demand and supply equilibrium at any moment in time, thus suggesting a set of implied returns. That is, if the objective is to find the mean-variance efficient portfolio by maximizing:

(λ is the risk aversion parameter) with respect to wm, yielding Π ^∧ = λVw_m. The Π ^∧ are implied by the equilibrium cap weights (hence, implied returns). As a further aside, if V is ill conditioned, then small changes in wm will generate large swings in implied returns. This suggests that close attention be paid to the eigenvalues in V.

Note that expected returns—our forward-looking returns expectations that appear under the heading E(r) in the risk budgeting spreadsheet are the same as Π; these are the returns we expect to prevail in equilibrium.

Views can be captured in an m × k matrix P (m may be any size), which has one row for each view. For example, with k = 3, a view about the sum of the returns to assets one and two is captured by putting the first row in P equal to (1 – 1 0). A second view that targets the average return to assets two and three is represented by setting the second row in P equal to (0, .5, .5). The vector μ′ is (μ₁, μ₂, μ₃) for this example and the product Pμ is therefore a 2 × 1 vector of returns consistent with these views. For example, if view one is that the sum of the returns to assets one and two is zero and the second view is that the average return across assets two and three is 3 percent, then we have Pμ = Z = (0, 3)′. Now model view uncertainty by including a normally distributed error term such that:

Then, Z|μ is distributed N(Pμ, Ω), meaning that the conditional distribution of view-consistent returns has mean Pμ and that ε_v has covariance matrix Ω. This last matrix captures the confidences associated with the views that we discussed earlier. Application of Bayes's rule yields the posterior distribution of returns with mean and variance given by:

The first term in μ∗ is V∗—the posterior covariance matrix. Notice that it includes the old covariance matrix (the prior) and is supplemented by a matrix-weighted average of view confidences given by Ω. The second term is a combination of the prior returns given in Π weighted by their prior covariances plus the views Z weighted by their covariances. Thus μ∗ is the sum of two expected returns: the prior Π and the views Z with weights given by:

Hence, the revised (posterior) mean reflects the prior information and the views in an optimal way in the sense that the weighting maximizes the likelihood of seeing μ∗.

Altering views changes P (and therefore Z) while view confidences are adjusted within Ω. In this way, the Black-Litterman methodology supports any number of views and the effect on the expected return and risk that optimally incorporates both view information and risk. Portfolio implications are the direct result of solving the mean-variance optimization problem with these new parameters. MCAR and attribution statistics can then be extracted in the manner outlined in the previous section. In sum, management has at its disposal, a platform for analyzing differing risk and return assumptions.

That leaves one additional point, which pertains to calibrating the values in the matrix Ω.

Intuitively, one would believe that view confidence would be inversely related to the underlying volatilities in the assets comprising the view. For example, our earlier view that the sum of the returns to the first two assets would be zero would ostensibly depend inversely upon their relative volatilities. Typically, these confidence are captured by making Ω a diagonal matrix with elements p_iVp_i′ = Ω_i. For example, in the two-asset case with a single view indicating that the sum of returns to assets X and Y is zero, this entry would consist of a function of the differences between the variances and covariances between the first two assets (since p_i is a row vector associated with this one view and has all other elements equal to zero). Specifically, p_iVp_i′ is now:

which, upon solving, yields:

This is intuitively appealing; the risk of being wrong is linear in the sum of the variances of the underlying returns. As this quantity gets large, the weight on the view, that is,

in forming the prior goes to zero and the posterior expected return μ∗ converges to the prior. Increasing confidence, on the other hand, drives the weight on the prior to zero, leaving the view to determine the posterior μ∗. This is an appealing result. Nevertheless, there is no requirement that Ω reflect linear combinations of the underlying asset covariations. Alternatively, this matrix might reflect a more independently determined set of distributional assumptions and we leave that open to further debate, as it is beyond the scope of this book.