8 Robust optimization for utilizing forecasted returns in institutional
8.4 Extensions to the theory
There is an academic approach to robustness that equates robustness with ambiguity aversion. The ideas are of great interest, and we present them below. The original contri
bution in this field is due to Ellsberg (1961) who presented a fascinating decision-theoretic paradox, which we will describe next.
To quote Anderson et al. (2000: 7):
Ellsberg (1961) created an example that challenged the Bayesian–Savage model of decision- making. Ellsberg (1961) considered a choice between bets on two urns. In Urn A, it is known in advance that there are fifty red balls and fifty black balls. In Urn B, the fractions of red and black balls are not known in advance. A ball will be drawn randomly from each urn.
At no cost a decision maker is permitted to guess the colour of the ball drawn from one and only one of the urns. If the decision maker guesses the colour drawn from the chosen urn, he receives a positive payoff. The Bayesian–Savage model predicts either indifference between urns or a preference for Urn B, depending on the prior probability assignment. But Ellsberg (1961) argued that a preference for Urn A is reasonable. To Ellsberg, there is an important distinction between urns for which you are informed about probabilities vis-à-vis ones for which you are not � � �The ambiguity about the assignment of priors is a way to make Knight’s (1921) notion of uncertainty operational. An aversion to such uncertainty can rationalise a preference for Urn A.
Just to clarify why it is ‘rational’ to pick Urn B, note that, assuming Urn B has anything other than an equal number of red and black balls, it will lead to better odds than Urn A, where we know in advance that our chance of winning is exactly 50 per cent. Urn B can never be worse than that, yet we choose Urn A because, psychologically, it is better the devil you know.
This notion of ambiguity aversion was captured by Gilboa and Schmeidler (1989).
Gilboa and Schmeidler (1989: 142, paras 1–2) present an ingenious argument relating ambiguity aversion to uncertainty in the presence of multiple priors. Considering the worst case for Urn B, you may bet on red when there are zero red balls in the urn. Likewise, you may bet on black when there are zero black balls in the urn. Such worst-case scenarios will have a payout of zero. This is obviously worse than the $50 you would expect to get from Urn A if the prize is $100.
This resolution is a situation where uncertainty is dealt with by multiple priors and some version of maximin expected utility used to determine one’s final decision. The more
185 Robust optimization for utilizing forecasted returns in institutional investment
ambiguity-averse the investor, the more priors he or she would entertain and, generically, the lower the expected utility the investor would end up with. This is an exciting idea, because it gives a theoretical structure to the well-known practice of scenario-analysis.
In the context of mean-variance analysis, this would mean that we would consider the minimum expected utility function taken over a number of scenarios, such as high/low risk and high/low return, for example. In this case, the scenario of low return, high risk would always give the lowest expected utility. But we can easily imagine examples where, in moving over different portfolio combinations, the investor moves from one regime to the other. Once we have determined the minimum of these regimes, the optimal portfolio is calculated by choosing those portfolio weights that maximize utility over this minimum path. The essential difference between the above and conventional Bayesian analysis is that we do not assign probabilities to the scenarios.
To move the discussion from mathematics to institutional investment, consider the decision by a pension fund trustee to invest in a hedge fund. This is clearly fraught with ambiguity aversion, as we can imagine many scenarios in the mind of the trustee. We can also see that regulated, exchange-traded assets will have, in the eyes of the investor, much less ambiguity aversion. Likewise, operational risk is an ambiguity-aversion situation. You may be able to compute most of the scenarios, but you cannot assign prior probabilities to their likely occurrence, either through lack of data or through failure to clarify what the appropriate states of the world are.
A discussion of mean-variance analysis, and the explicit form of the min utility function that arises in the two scenarios, one risky asset case, is presented in the appendix of Lutgens and Schotman (2004).
To illustrate this further, consider the case of two scenarios and one risky asset (see Figure 8.1). The graphs show expected utility as we vary the weight w of the risky asset in the portfolio. Instead of choosing portfolio AA or BB, we prefer portfolio CC since this is the maximum of the minimum function. Notice that this is not the same as the worst-case scenario, which would be BB, since BB has lower expected utility than AA. Notice also that the chosen portfolio is not necessarily terribly conservative, in our example, we hold more of the risky asset, e.g. equity, than AA but less than BB.
0 0.25 0.5 0.75 1
Weight of risky asset (w) 0.01
AA CC BB
0.005
0
Expected utility of portfolio E (U(w)) –0.005
–0.01
Figure 8.1 Maximin expected utility.
�
�
186 Forecasting Expected Returns in the Financial Markets
Up until now we have framed robustness solely in terms of ambiguity aversion, but there are numerous other approaches that have been considered, including in a portfolio construction framework.
Lutgens and Schotman (2004) consider an analysis based on linear factor models and use earlier work by Pastor and Stambaugh (2000). Their model essentially assigns priors to the covariance matrices of the factors and of the errors. They treat the factor exposures and factor returns as known. Other work in this area focuses on unconditional distributions but does not look at diversity of views about factor returns.
We shall consider a number of experts with differing views about implied factor returns, and ask how this would impact our optimal portfolio construction. If the experts agreed about all other aspects of the model, then we would have diversity of opinion on stock expected returns but nothing else.
Following the notation of Lutgens and Schotman (2004), we consider a k-factor model for returns as:
yt =�ti+�tVt +ut (8.10)
E�utut� = D
where yt and ut are �N ×1� vectors, i is a �N ×1� vector of ones, �t is an �N ×k� matrix of (known) exposures, whilst Vt is the �k×1� vector of (unknown) implied factor returns.
The matrix D is a diagonal �N ×N� covariance matrix and �t is an unknown scalar. For the purpose of stock selection, �t, being the same for all stocks, is unimportant, and to simplify matters we shall just consider:
yt =�tVt +ut (8.11)
The weighted least squares estimator Vˆt is given by Vˆt = ��t Dˆ−1�t�−1��t Dˆ−1 yt�
where Dˆ−1 is estimated by using residuals of previous periods to get estimates of idiosyn
cratic variance; it is recommended that rolling windows be used for this purpose. We write the ‘true’ value of Vt Vtp. Furthermore, Vˆt ∼�Vtp� ��tDˆ−1�t�−1�. For the case of mul
tiple priors, we can conceive of each expert j proposing Vtj ∼ N�V0j� �j�. In a Bayesian framework, we would assign probability �j to the expert, based on past performance or pedigree or introspection.
The overall prior distribution is then a mixture of normals:
pdf�Vtp
�= �jN�Voj� �j�
If we combine this with the sample distribution of Vˆt, we see that:
pdf�Vˆt� Vt p� = pdf�Vˆ tVtp�pdf�Vt p�= �jN��j� �j� where
Dˆ−1�t�+�−1 Dˆ−1�t�−1Vt +�−1 j
�j =���t j �−1���t j V0 � (8.12)
�
187 Robust optimization for utilizing forecasted returns in institutional investment
It can be shown that the mean of a mixture of normals is equal to ��j�j and the covariance matrix is equal to ��j�j +��j�j�j −���j�j����j�j� (see for example Satchell and Scowcroft (2000), reprinted in this book as Chapter 3).
In the case that we have a simple prior, we recapture the Black–Litterman model (1990, 1992; this has been briefly discussed above). In the case of ambiguity aversion, we would consider each separate distribution for Vt ∼ N��j� �j�, and calculate expected utility, as discussed before. Specifically, our measure of expected stock return would be �t�j, where
�j is given by equation (8.12), and our measure of covariance �t��t +D, that is we allow experts to agree on risk but differ on alpha. Based on these different alphas, we would compute �j�w� = w�t�j −�w��t��t +D�w. Note that � is an estimate of the factor covariance matrix. We then compute U�w� = minj�Uj�w�� and optimize. The solution to this mean-variance problem will be like Theorem 1 in Lutgens and Schotman (2004).
That is, our optimal portfolio will be of the form:
w ∗= ��t��t +D�−1� �j�r�j� (8.13)
where the �j’s are Lagrange multipliers essentially determining which of the experts’
opinions is the most pessimistic for the particular combination of weights. These may be very difficult to compute numerically.
There is no agreed way of updating non-unique priors, although there is a huge literature in decision theory (see Gilboa and Schmeidler (1993) for some results and references). Our procedure of including all scenarios (priors) in the update can be seen as extreme in that we do not use sample information to gain information about the possible probabilities of priors. Gilboa and Schmeidler (1993) refer to work by Fagin and Halpern (1991) that follows the procedure we have adopted.