13.4.1 Introduction
Copula opinion pooling (COP) and entropy pooling (EP) can both be viewed as extensions to the original BL model. These methods are due to Meucci (2006a, b, 2010a, b). The BL model and the two extensions have in common that a synthesis between an a priori distribution and a prior view distribution is accomplished in order to yield a posterior distribution. The major difference between these two methods and the BL model is in how these two ingredients are supplied or specified, and that ordinarily the posterior distributions according to the COP and EP models are retrieved by means of a Monte Carlo simulation. In the rest of this section the COP and EP models are presented.
13.4.2 The COP model
In the BL model the a priori distribution for the returns is derived from an equilib- rium model, namely the CAPM, and the views are expressed as either absolute or relative return forecasts. Furthermore, the BL model assumes that the returns follow a multivariate normal distribution. These model constituents are too restrictive, and the stylized facts of financial returns conflict with the normal assumption. In contrast, the COP method is much more flexible and less demanding with respect to its under- lying assumptions. First, the a priori distribution need not be defined in terms of the returns on a set of securities, but can also be given in terms of risk factors or a set of any other random variables, and it is not assumed that these random variables follow a multivariate normal distribution function. Instead, the prior distribution can be rep- resented by a copula and the marginal distributions can be of any kind. Second, the views are expressed as cumulative distribution functions and as such are not confined to absolute or relative return forecasts. However, confidence levels,ck,k=1,…,K, for the views have to be provided by the user, similar to the diagonal elements of the uncertainty matrix in the BL model. In addition, a “pick” matrixPis a required input.
The cost of this flexible approach is that in general no closed-form solutions can be provided, but the synthesis between the two ingredients is accomplished by means of a Monte Carlo simulation.
Deriving the posterior distribution by the COP method involves the following five steps (see Meucci 2006a):
k k 1. A rotation of the prior distribution into the views’ coordinates.
2. Computation of the views’ cumulative distribution function and the market implied prior copula.
3. Computation of the marginal posterior cumulative distribution functions of each view.
4. Computation of the joint posterior distribution of the views.
5. Computation of the joint posterior realizations of the market distributions.
Each of these five steps is now described in more detail. It is assumed that a Monte Carlo simulation,M, for theNsecurities of sizeJis given. These simulated values can be drawn, for instance, from a Student’stcopula, and any of the proposed dis- tributions in Chapter 6 could have been utilized as marginal models. But the Monte Carlo values contained inMcould also have been generated from a multiple time series model as outlined in the previous sections of this chapter.
In the first step, the simulated values contained in the(J×N)matrixMare mapped into the coordinate system pertinent to the views,V. This is accomplished by means of a simple matrix multiplication,V=MP̄′, whereP̄ denotes the (invertible) pick matrix. There is a qualitative difference between this pick matrix and the one used in the BL model. In the latter approach, the matrixPconsists ofKrows in which the securities for which views are expressed are selected, hence its dimension is(K×N).
In the COP model, however, the dimension is(N×N). The firstKrows of(P)̄ are the viewsP, and the remainingN−Krows are filled by its matrix complement,P⊥with dimension((N−K) ×N), henceP̄ = (P|P⊥)′.
In the second step, the firstKcolumns ofVare sorted in ascending order, which results in the(J×K)matrixW, the prior cumulative distribution functions. The cop- ula of the views,C, are the order statistics ofW, that is,Cj,k=Fk(Wj,k=j∕(J+1).
Next, the marginal posterior cumulative distribution functions,F, for each view are computed as a weighted average,
Fj,k=ckF̂k(Wj,k) + (1−ck) j
J+1. (13.69)
In the fourth step, the joint posterior distribution of the viewsṼ (the quantile func- tion) is retrieved by means of interpolation between the grid points(F⋅,k,W⋅,k).
In the final step, the joint posterior simulated realizations of the market distribution are recovered by inversion of the equation in the first step. Because the dimension of Ṽ is only(J×K), the matrix is amended from the right by the rightmost(N−K) columns ofV, and henceM̃ =V(̃ P̄′)−1. This set ofJjoint posterior realizations for theNassets can then be utilized in the portfolio optimization problem at hand.
13.4.3 The EP model
The EP model is a further generalization of the BL and COP models. It has in common with the COP model an arbitrary model for the market prior distribution and that the
k k COP and EP models are based on a Monte Carlo simulation of the market’s prior
distribution. A qualitative difference between the COP and EP models is the kind of views expressed. In the former model class views on the dispersion, volatility, or correlations of and between securities and/or the expression of nonlinear views were not feasible. This limitation is rectified for EP models. A concise description of how to formulate these views is provided in Meucci (2010b, Section 2 and Appendix A.2). A further material difference between the COP and EP models is the form of the posterior distribution. In the latter model, the same simulated values used in the modelling of the market’s prior distribution are utilized, but with altered probabilities assigned to these random variables.
In the following, the steps for deriving the posterior distribution are presented. As stated in the previous subsection, it is assumed that the market forNassets can be modelled by a set of random variablesXwhich follow an a priori joint distribution, fM. In its simplest form,Xrepresents a sample of the securities’ returns, but is not lim- ited to these market factors. When the data has been fitted to the assumed distribution model, simulated values for this market distribution can be obtained by means of Monte Carlo, and hence one obtains a matrixMof dimension(J×N)as in the COP model. The columns inMare the marginal prior distributions and the rows are simu- lated outcomes for the market factors. Associated with each of these outcomesMj,⋅is a probabilitypj, and most easily theseJprobabilities are set equal to 1∕J, that is, each Monte Carlo draw is treated as being equally likely. Hence, the(J×1)probability vectorphas as elements the reciprocal of the size of the Monte Carlo simulation.
The second input is the formulation of the views. This is now a (K×1)vector of function valuesV=g= (g1(X),…,gk(X),…,gK(X))′with joint distributionf𝑣, a priori. The functionsgk,k=1,…,K, can be nonlinear in nature. The implied distri- bution of these views is then empirically approximated by the market simulationsM according to
Vj,k=gk(Mj,1,…,Mj,N) (13.70) for k=1,…,K andj=1,…,J, such that a (J×K)matrix results containing the empirical distribution of the views implied by the Monte Carlo simulation for the market.
Two panelsMandVhave now been created, and the question is how these can be combined to retrieve the posterior distribution of the market that obeys the views.
This problem is resolved in three steps. First, the views are expressed in terms of a set of linear inequality constraints,alower ≤Ap̄≤aupper, where now the probability vectorp̄is treated as the objective variable in the ensuing optimization. The lower and upper bounds(alower,aupper)and the matrixAare deduced fromV.
In the second step, the relative entropy—loosely speaking, a distance measure be- tween distributions—is minimized.2The (discrete) objective function is defined as
RE(p,̄ p) =
∑J
j=1
̄
pj[log(p̄j−log(pj)], (13.71)
2A detailed exposition of the entropy concept is given in Golan et al. (1996).
k k and hence the probabilities of the posterior distribution under the assumption of
perfect foresight are obtained by evaluating
̄
p= arg min
alower≤Āx≤aupper
RE(x,p). (13.72)
That is, the probabilitiesp̄under the assumption of perfect foresight are determined such that the resulting posterior distribution is least distorted by the distribution of the views, or, put differently, the view distribution is made most akin to the reference model. At first sight, the optimization stated in (13.72) seems to be a numerically demanding exercise, given the dimension of the target variable. However, in Meucci (2010b, Appendix A.3) the dual form of this mathematical program is derived from the Lagrangian function, which results in a convex program with linear constraints and with the size of the objective variable equal to the number of viewsK.
In the third step, the empirical confidence-weighted posterior distribution,(M,pc), is determined as in the COP model according to
pc= (1−c)p+cp.̄ (13.73)
The empirical posterior distribution thus constructed can then be utilized in the port- folio optimization problem at hand.