In Chapter 10 it was stated that the weight solution is sensitive to the input parameters for Markowitz solutions. In that chapter the focus was on robust estimation techniques for the dispersion matrix to rectify this problem. The sensitivity of the weight solu- tion to the assumed return vector was neglected, but references to the literature were provided.
In this and the next section, approaches are presented that deal indirectly with this issue. The first of these is the Black–Litterman (BL) model (see Black and Litterman 1990, 1991, 1992; He and Litterman 2002).
This consists of five building blocks, with a Bayesian estimation of equilibrium and expected returns at its core. The latter lets a portfolio manager directly include his return expectations into the portfolio solution and it is this characteristic that has led to the ubiquitous application of this method for tactical asset allocation. It will be assumed that the returns are joint normally distributed. The five building blocks are
•the capital asset pricing model (CAPM; Sharpe 1964),
•reverse optimization (Sharpe 1974),
•mixed (Bayesian) estimation,
•the concept of a universal hedge ratio (Black 1989),
•mean-variance optimization (Markowitz 1952);
these will be discussed in turn.
k k The CAPM is used as an equilibrium model, where the supply and demand of assets
are equated. Recall Figure 5.1, in which this point is depicted as the locus where the capital market line is tangent to the curve of efficient portfolios. At equilibrium the efficient market portfolio consists, therefore, of an efficient allocation of risky assets and the risk-free asset. This point is further characterized by the market capitaliza- tion of each asset. The relative market capitalizations represent the assets’ weights at equilibrium and are denoted by𝝎MKT. If one assumes that the unconstrained portfolio optimization problem can be stated as
arg max
𝝎inΩ 𝝎′𝝁−𝜆
2𝝎′Σ𝝎, (13.59)
where the expected excess returns,𝝁, are balanced against the portfolio’s riskiness, 𝝎′Σ𝝎, for a risk aversion parameter,𝜆 >0, then the optimal (equilibrium) weight vector is given by
𝝎OPT = [𝜆Σ]−1𝝁. (13.60)
For arbitrary expectations for the assets’ excess returns, the allocation𝝎OPT will generally differ from𝝎MKT, only coinciding if𝝁is equal to the excess return expec- tations implied by the relative market capitalizations. By reverse optimization, that is, left-multiplying (13.60) by𝜆Σ, one obtains the implied excess returns,𝛑, for a given market capitalization structure:
𝛑=𝜆Σ𝝎MKT. (13.61)
The implied equilibrium returns, the vector elements of𝛑, are employed as the market neutral starting point in the BL model, that is, the returns are a priori dis- tributed with an expected mean of𝛑. In the absence of any views on the future values of the assets, an investor is therefore best advised to allocate his wealth according to 𝝎MKT, thus implicitly sharing the market expectation for the excess returns as stated in (13.61). The question of how the investor’s expectations about the excess returns can be included in this model arises next. In the BL model the inclusion of these
“views” is accomplished by deriving the Bayesian posterior distribution for the re- turns. LetAdenote the (multivariate) return expectation(s) andBthe implied excess returns at equilibrium. The joint likelihood functionP(A,B)can then be expressed by Bayes’ theorem as
P(A,B) =P(A|B)P(B) (13.62a)
=P(B|A)P(A), (13.62b)
and because one is interested in the conditional return distribution in terms of the equilibrium returns, one obtains, from the right-hand equality in (13.62),
P(A|B) = P(B|A)P(A)
P(B) . (13.63)
k k If the expected excess returns are stated asE(r), then (13.63) is written as
P(E(r)|𝛑) = P(𝛑|E(r))P(E(r))
P(𝛑) . (13.64)
A feature of the BL model is that expectations do not have to be supplied for all market assets and/or that these return expectations can be either expressed as absolute or relative return targets. Therefore, the views can be modelled as:
PE(r) ∼N(q,Ω), (13.65)
wherePdenotes a(K×N)pick matrix,qis the(K×1)vector of absolute/relative return expectations, andΩ a(K×K) diagonal matrix expressing the uncertainties about each view. Given the normality assumption, the latter assumption implies in- dependence between the expressed views. The implied independence of views is awkward. It does not make much sense to assume that return forecasts for various as- sets are formed independently of each other, in particular when these are derived from a multivariate statistical model, so this assumption will be relaxed in the next section on copula opinion pooling, but retained for the time being. Finally, the conditional equilibrium returns for the expected returns are distributed as
𝛑|E(r) ∼N(E(r), 𝜏Σ), (13.66) where a homogeneity assumption for the market participants has been exploited such thatE(𝛑) =E(r)and𝜏is a scalar for scaling the uncertainty around the implicit mar- ket equilibrium excess returns.
The posterior probability density function of the conditional random variable r=E(r)|𝛑is multivariate normally distributed with parameters
E(r) = [(𝜏Σ)−1+P′Ω−1P]−1[(𝜏Σ)−1𝛑+P′Ω−1q], (13.67) VAR(r) = [(𝜏Σ)−1+P′Ω−1P]−1. (13.68) In essence, (13.67) is a risk-scaled weighted average of the market equilibrium returns and the view returns. The latter two are scaled by the inverse of the variance-covariance matrix, (𝜏Σ)−1, and by the confidence of the views, P′Ω−1, respectively. The relative importance of the market equilibrium return and the view return expressed in thekth row ofPis determined by the ratioΩkk∕𝜏=p′kΣp. The term on the right-hand side of this equation is the variance of the view portfolio.
This equation can also be used to determine implicit confidence levels Ωkk for a given value of 𝜏. If this route is followed, then the point estimates for E(r) are unaffected by the chosen value for 𝜏 (see He and Litterman 2002). In the limiting case of no expressed views (P=0), the point estimates for E(r) are the equilibrium returns. At the other extreme, when the views are expressed without prediction error, the point estimates forE(r)are identical to the view expectations if specified and otherwise are equal to the market returns. If these point estimates are used in an unconstrained Markowitz-type portfolio optimization, then only the
k k weights will differ from the relative market capitalization ones, for which a return
expectation has been formulated. To summarize, the BL method, in its original form, consists of a Bayesian-derived vector of expected returns, which is then utilized in a Markowitz-type portfolio optimization. Of course, the Bayesian estimates can also be used in any other kind of portfolio problem formulation, for example, with respect to optimizing a downside-risk measure.