Portfolio selection with qualitative input

Journal of Banking & Finance 36 (2012) 489–496 Contents lists available at SciVerse ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf Portfolio selection with qualitative input Anant Chiarawongse a, Seksan Kiatsupaibul b,⇑, Sunti Tirapat a, Benjamin Van Roy c a Department of Banking and Finance, Faculty of Commerce and Accountancy, Chulalongkorn University, Bangkok 10330, Thailand Department of Statistics, Faculty of Commerce and Accountancy, Chulalongkorn University, Bangkok 10330, Thailand c Department of Management Science and Engineering, Stanford University, CA 94305-4023, USA b a r t i c l e i n f o Article history: Received 27 August 2010 Accepted 22 August 2011 Available online 30 August 2011 JEL classification: G11 C11 C63 Keywords: Portfolio selection Bayesian inference Markov chain Monte Carlo Black-Litterman model Hit-and-run algorithm a b s t r a c t We formulate a mean–variance portfolio selection problem that accommodates qualitative input about expected returns and provide an algorithm that solves the problem. This model and algorithm can be used, for example, when a portfolio manager determines that one industry will benefit more from a regulatory change than another but is unable to quantify the degree of difference. Qualitative views are expressed in terms of linear inequalities among expected returns. Our formulation builds on the Black–Litterman model for portfolio selection. The algorithm makes use of an adaptation of the hitand-run method for Markov chain Monte Carlo simulation. We also present computational results that illustrate advantages of our approach over alternative heuristic methods for incorporating qualitative input. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Portfolio managers acquire and process information from multiple sources in order to form expectations of future returns, also known as alpha. The Bayesian paradigm provides a coherent framework for synthesizing such information. In this spirit, Black and Litterman (1992) propose an approach for combining multiple subjective views to generate alpha. In the Black–Litterman model, each view is expressed in terms of a linear combination of alphas. For example, such a view may assert that the alpha of stock A exceeds that of stock B by v percent. A view in the Black–Litterman model may also prescribe uncertainty to the estimate, for example, by stating that the alpha of stock A exceeds that of stock B by approximately v. In this case, the view must assign a level of variance r2 to quantify uncertainty around the estimate v. Views of the kind we have described are quantitative in that they articulate numerical estimates. In this paper, we propose a methodology that also accommodates qualitative views that are expressed in terms of linear inequalities. As an example, consider a view that the alpha of stock A exceeds that of stock B. This kind of view does not provide any quantitative estimate of degree but ⇑ Corresponding author. Tel.: +66 2 218 5657; fax: +66 2 218 5652. E-mail addresses: [email protected] (A. Chiarawongse), seksan@acc. chula.ac.th (S. Kiatsupaibul), [email protected] (S. Tirapat), [email protected] (B.V. Roy). 0378-4266/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2011.08.005 only ranks the alphas. Therefore, this model broadens the range of problems that can be solved by Bayesian methodology. Linear inequalities can be used to express a wide variety of qualitative views. Let us consider a few examples involving different asset classes: 1. Equity analysis: Analysts assess and compare the current condition and future prospects of multiple sectors of the economy. Views that certain sectors are likely to outperform or to underperform others can be encoded in terms of linear inequalities among stock alphas and used to guide sector rotation strategies. 2. Fixed income analysis: When the economy is expected to perform well, the credit spread between corporate bonds and government bonds should become narrower. As a result, corporate bonds should outperform government bonds. Such a view can be represented in terms of a linear inequality involving bond alphas. 3. Currency analysis: Changes in relative economic conditions and regulations cause the value of some currencies to change relative to others. For example, rising inflation should lead to depreciation in foreign exchange rates. Such a view can be captured using linear inequalities among foreign exchange rates. The work of Black and Litterman (1992) offers an approach to incorporating external information, i.e., investors’ views of the future returns, into parameter estimation. An investor’s view 490 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 provides information in addition to the prior and observed historical returns. It can be acquired through analyses of external information and fused into a quantitative view, which is in the form of a system of linear equations of subjective forecast of returns and noises. However, a quantitative view demands details that are difficult to obtain. A qualitative view in the form of a system of linear inequalities, such as ranking or partial ranking is more natural to acquire, given that the information is not a direct observation of returns. In addition, the posterior distribution of the recent Bayesian estimations can be incorporated as the prior to the proposed model. The ability to factor qualitative views into portfolio decisions also opens the door to leveraging the growing body of work on robust comparative statics (e.g., Veinott, 1992; Vives, 1990; Topkis, 1998; Milgrom and Roberts, 1990a,b, 1994; Milgrom and Shannon, 1994). This line of economic research provides tools that can be used to draw qualitative conclusions about equilibria in games among multiple strategic agents even when parameters of the system are not known. Such tools are used, for example, to understand which companies or industries will benefit or be adversely affected by a regulatory change. Conclusions can be expressed in terms of linear inequalities and factored into portfolio decisions using our methodology. An output from our methodology is an estimate of alpha. It can be used with a mean variance portfolio model as well as other modern portfolio models that require alpha as their input to form an optimal portfolio composition (e.g., Alexander and Baptista, 2010, 2011; Brandt and Santa-Clara, 2006; Das et al., 2010; Tu and Zhou, 2011). In the next section we study a simple example involving a portfolio of two assets. The purpose is to illustrate our model and method in a simple context and to develop intuition for how qualitative input influences portfolio decisions. In this simple context, relevant calculations are amenable to closed form solution. Models involving many assets and multiple qualitative views, on the other hand, give rise to computational challenges. The primary computational challenge in applying our approach involves computing the conditional expectation of returns, conditioned on qualitative views. This is generally a difficult inference problem, requiring integration of a normal distribution over a high-dimensional polytope. To deal with this problem, we propose an adaptation of the hit-and-run algorithm (e.g., Smith, 1984; Lovász, 1999; Lovász and Vempala, 2003, 2006a,b). One contribution of this paper is to demonstrate the efficacy of this algorithm in solving our portfolio optimization problem with qualitative input. Our formulation generalizes the Black–Litterman model, which we review in Section 3. We then introduce in Section 4 our model, which incorporates qualitative input. We present in Section 5 our solution method together with computational results that demonstrate its efficacy and merits relative to some alternative heuristic approaches that deal with qualitative input. Our methodology also allows for uncertainty in qualitative views. This captures, for example, a situation where the view is based on accurate information with some probability j and irrelevant information with some probability 1  j. Here, j can be thought of as a level of confidence assigned to a qualitative view. We describe and study this model of uncertainty in Section 6. 2. A simple example Suppose we wish to construct a portfolio w 2 R2 of two stocks. Each component is the dollar position in a stock. The return is given by rTw, where the vector r of asset returns is generated according to r ¼ a þ e; T with e  N(0,I). Our prior beliefs about a = [a1, a2] take the form of  ; sIÞ, for a vector a  ¼ ½0:1; 0:1T and a a normal distribution Nða τ Fig. 1. The conditional expected return given the view (a) versus the prior variance parameter (s). small positive scalar1 s 6 0.1. (As a convention, we use bold face such as x and a to denote random quantities as opposed to their realizations x and a.) It is natural to construct a portfolio by optimizing risk-adjusted expected return:   q  T w  wT w ; max a 2 2 w2R ð1Þ where q is a risk-aversion parameter. Note that we ignore the risk introduced by our uncertainty about a. This is of negligible consequence because s is small.2 The solution to this portfolio optimization problem is given by w¼ a : q  ¼ ½0:1; 0:1T , the optimal portfolio includes a short position Since a in asset 1 and an equally sized long position in asset 2. Now suppose we obtain a new piece of exogenous information, called a view, indicating that a1 P a2. The natural optimization problem then becomes   q ^ T w  wT w ; max a 2 2 w2R ^ ¼ E½aja1 P a2 . Computing the conditional expectation where a ^ ¼ ½a; a, where gives us a a¼ a 1  a 2 2 þ 1 2 rffiffiffiffi s expðða 1  a 2 Þ2 =ð4sÞÞ pffiffiffiffiffiffi ; p ð1  Uðða 1  a 2 Þ= 2sÞÞ and the optimal portfolio becomes w¼ a^ : q Note that the conditional expected returns, and hence the portfolio composition, depend critically on s, the variance of the prior. Fig. 1 illustrates how s influences a, the magnitude of the conditional expected returns given the view. ¼ As an example, consider the case where s = 0.1. Here, a ^ ¼ ½0:147; 0:147T , and w = [0.147/q, 0.147/q]T. ½0:1; 0:1T ; a Note that this portfolio involves a long position in asset 1 and a short position in asset 2. This is opposite to the portfolio constructed in the absence of the qualitative view, which involves a short position in asset 1 and a long position in asset 2. The fact that 1 From empirical Bayes’ point of view, s is the inverse of the number of observations in prior distribution construction. Ten observations imply s = 0.1. One hundred observations imply s = 0.01. sÞ T 2 Note that the risk term technically should be qð1þ w w. 2 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 491 The above simple example illustrates that factoring a qualitative view into the estimation of alpha offers a potential benefit over the one without the view. One tends to expect a similar potential benefit in a more general setting involving multiple qualitative views and multiple assets. However, solving a portfolio selection problem with a more general information structure is not a trivial exercise. In fact, this can be accomplished by Markov chain Monte Carlo simulation in Section 5. Before presenting our general model and the computational method in detail, we describe in the next section the Black–Litterman model, casting it in a Bayesian framework upon which our model is constructed. 3. The Black–Litterman model Fig. 2. Trade-offs between the expectation and variance of portfolio payoffs with only the prior on a, with the opposing qualitative view a1 6 a2, and with the reinforcing qualitative view a1 P a2. Fig. 3. The trade-off between expected payoff and variance deteriorates as s diminishes. Black and Litterman (1991, 1992) provides a framework for synthesizing an investor’s private views with a return distribution implied by market equilibrium. Since its inception, the model receives significant attention among practitioners (Satchell and Scowcroft, 2000; Drobetz, 2001; Jones et al., 2007). We now review the Black–Litterman model, emphasizing its relation to a general Bayesian framework. This will facilitate comparison with our model for synthesizing qualitative views, which also fits in this framework. The Black–Litterman model assigns a Gaussian prior to expected returns. This prior is either implied by a market equilibrium model or estimated from historical return data. Subjective views are represented as estimates of linear combinations of returns. Provided alongside each estimate is a variance parameter that reflects confidence in the estimate. A posterior distribution is computed and its mean is then used as input to a Markowitz-style mean–variance optimization problem. Let a denote a vector of expected returns for N assets. Let the prior distribution of a be a multivariate normal distribution with  and covariance matrix sR, where s is a scalar and mean vector a R is a positive definite matrix. That is, a  Nða ; sRÞ: ð2Þ Given a, the view v is a K-dimensional random vector with these portfolios are diametrically opposed makes sense because while the prior suggests that asset 2 will outperform asset 1, the qualitative view rules this out. Consider an alternative situation where the qualitative view takes the form of an inequality a1 6 a2, reinforcing the ranking suggested by the prior. If s = 0.1, the conditional expectation of re^ ¼ ½0:220; 0:220T . turns becomes a Fig. 2 illustrates the trade-offs between the expectation and variance of portfolio payoffs conditioned on three different information sets: (1) only the prior on a, (2) the opposing qualitative view a1 6 a2, (3) the reinforcing qualitative view a1 P a2. In this simple example, conditioning on either qualitative view improves the trade-off. As one would expect, a reinforcing view is preferable to an opposing view. It is worth noting that in general qualitative views do not always improve the trade-off. Reinforcing views always improve the trade-off, but opposing views can either improve or hurt the trade-off. Fig. 3 illustrates this. Two of the plots are identical to those of Fig. 2; one based only on the prior on a and the other involving conditioning on the opposing qualitative view a1 6 a2. The other plots are also conditioned on this qualitative view, but are generated by reducing s. For smaller values of s, the trade-off deteriorates and becomes worse than the apparent trade-off in the absence of the qualitative view. Intuitively, this happens because a very small value of s reflects high confidence in the prior expectation. In this case, the contradictory view results in a posterior expectation close to zero. When the stocks generate no expected return, the trade-off becomes unattractive. v  NðPa; NÞ; ð3Þ where P is a K  N matrix. The posterior density of a, conditioned on the view v, is then f ðajv Þ / f ðv jaÞf ðaÞ     1 1  ÞT ðsRÞ1 ða  a Þ / exp  ðv  PaÞT N1 ðv  PaÞ exp  ða  a 2 2  i 1h T 1  ÞT ðsRÞ1 ða  a  Þ : ð4Þ / exp  ðv  PaÞ N ðv  PaÞ þ ða  a 2 Observe that here v plays the role of an observation in the Bayesian framework. Expression (4) implies that a conditioned on the view v is normally distributed. The conditional expectation of a is given by a closed-form expression:  : E½ajv  ¼ ½PT N1 P þ ðsRÞ1 1 ½PT N1 v þ ðsRÞ1 a ð5Þ This is the Black–Litterman expected return (Black and Litterman, 1992), which synthesizes subjective views with a prior. 4. Qualitative input Assume we have a Gaussian prior over expected returns as in Relation (2). We consider qualitative views that can be expressed in terms of linear inequalities: Aa 6 b; where A is an M  N matrix and b is an M-dimensional vector. A view that ranks assets provides one example that fits this 492 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 framework. Such a view can be expressed as N  1 linear inequalities of the form aik P aikþ1 ; for k = 1, . . . , N  1, where (i1, i2, . . . , iN) is a permutation of (1, . . . , N). We propose to use the conditional expectation a^ ¼ E½ajAa 6 b in a Markowitz-style portfolio optimization problem. In particular, given this conditional expectation, the problem is to determine a vector of portfolio weights w 2 RN that attains the maximum in   q ^ T w  wT Rw : max a 2 w2RN This is entirely analogous to what is done in the Black–Litterman approach, except that we are conditioning on a different sort of ^. information when computing a Let R ¼ fa 2 RN : Aa 6 bg; and let IR be the indicator for the set R. The posterior density of returns, conditioned on qualitative views Aa 6 b, is given by  i 1h  ÞT ðsRÞ1 ða  a Þ : f ðaja 2 RÞ / IR exp  ða  a 2 ð6Þ The posterior expectation of returns is therefore a^ ¼ E½aja 2 R ¼ 1 K Z x exp x2R   1  ÞT ðsRÞ1 ðx  a  Þ dx; ðx  a 2 ð7Þ where K¼ Z x2R exp   1  ÞT ðsRÞ1 ðx  a  Þ dx: ðx  a 2 The derivations of (6) and (7) can be found in Appendix A. The above integrals are over polytopes in a potentially ^ can high-dimensional space. As such, computing the estimate a ^ can be approximated by a Markov be challenging. However, a chain Monte Carlo method (MCMC) such as the hit-and-run algorithm (Smith, 1984; Bélisle et al., 1993), the Metropolis-Hasting algorithm, or the Gibbs sampler. In general, an MCMC generates a sequence of samples {ai, i = 1, 2, . . .} that is an ergodic Markov chain with distribution converging in total variation to the posterior distribution f(ja 2 R). Furthermore, lim T!1 T 1X ai ¼ E½aja 2 R a:s: T i¼1 Therefore, an average of MCMC samples can be employed as an estimate of the alpha. In Section 5, we devise an efficient MCMC based on the hit-and-run algorithm and demonstrate its efficacy. 5. Computational study In this section, we present results from a computational study aimed at assessing the merits of our approach to synthesizing qualitative input. This study compares our approach against simpler heuristics that offer alternative ways of dealing with qualitative input. Results from experiments involving randomly sampled problem instances indicate that our approach can yield substantial benefits over these heuristics. 5.1. Generative model Our approach to estimating alpha requires the following problem parameters as input:      number of assets N, return variance R, , prior mean a prior variance scale parameter s, constraint parameters A and b. In this section, we present a generative model that is used to sample the above problem parameters. Additionally, our generative model samples the realized alpha (a) for use in assessing the quality of portfolio decisions. The generative model is itself parameterized by several inputs:      number of assets N, number of risk factors M, variance of prior mean r2a , Wishart distribution variance parameter prior variance scale parameter s. r2R , Our generative model produces a covariance matrix R = S + V D VT, with a diagonal matrix S 2 RNN ; V 2 RNM , and a diagonal matrix e is sampled from a D 2 RMM , generated as follows. First, a matrix R Wishart distribution with variance–covariance matrix parameter r2R I and degree of freedom parameter N. Then, a singular value e ¼V eD eV e T is computed, where D e contains the eigendecomposition R e values of R sorted descendingly by magnitude along the diagonal. e , while the The matrix V is taken to be the first M columns of V diagonal entries of D are taken to be the first M diagonal entries of e nn  PM Dmm V 2 . Note e Finally, for n = 1, . . . , N, we let Snn ¼ R D. m¼1 nm that R reflects risks associated with M factors plus idiosyncratic asset-specific risks. The number of risk factors M will typically be much smaller than N.  and realized Our generative model produces the prior mean a  is drawn from a Nð0; r2a IÞ. Then, a is alpha a as follows. First, a  ; sRÞ. drawn from Nða For constraints, we assume a particular kind of view that reflects a ranking of assets. In particular, let i1, . . . , iN be a permutation of 1, . . . , N such that ai1 P ai2 P    P aiN . We consider N  1 constraints: ai1 P ai2 ; ai2 P ai3 ; . . . ; aiN1 P aiN . Such a complete ranking view may reflect practical contexts and also presents a challenging computational problem that serves to stress test our inference algorithm. In particular, as the number of assets increases, the number of constraints also increases, making the support of the posterior distribution increasingly complex. These constraints are encoded in terms of the matrix A and vector b. Note that these constraints offer no information about alpha beyond restriction to a certain polytope. In particular, density of a conditioned on observing such constraints is a multiple of the prior density for points that satisfy the constraints and zero for points that do not. To the best of our knowledge, no prior work addresses how to incorporate such qualitative ranking information into a portfolio decision. With extant methods, one can at best form a simple heuristic to adjust alpha estimates in response to ranking information. On the other hand, our proposed approach leads to coherent fusion of ranking information into the alpha estimation process. The estimated alpha is then used in a portfolio selection model to guide investment decisions. The estimation algorithm is introduced in the next section followed by computational results designed to assess the performance of our approach relative to simple heuristics. 5.2. Solution method Algorithm 1 computes a posterior expectation based on the model proposed in Section 4. Based on our computational experience, this algorithm efficiently processes asset rankings. It combines the hit-and-run algorithm with the Metropolis-Hasting algorithm and operates on polar coordinates. The algorithm takes A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496  and as input parameters the number of assets N, the expectation a the variance sR of the density f, parameters A and b that characterize the polytope R, an initial point a0 2 R, and a number of iterations niters. Algorithm 1. Polar Metropolis Hit-and-Run (PMHR) 1: for i = 0 to niters 2: Let kaik be the length of ai and define the current direction h = ai/kaik. Let L be a line segment defined by L ¼ fl : l ¼ ai þ sh; s 2 Rg \ R; and let S be the surface of the intersection between the hypersphere with radius kaik and the polytope R. 3: Randomly select either an L move or an S move with equal probability. L move: Sample a from the conditional density f( j a 2 L) and accept it as a(i+1) with probability min{1, (kak/kaik)(N1)}. In case of rejection, let a(i+1) = ai. S move: Sample a from the uniform distribution on S and accept it as a(i+1) with probability min{1, f(a)/ f(ai)}. In case of rejection, let a(i+1) = ai. 4: end for We measure the efficiency of Algorithm 1, Polar Metropolis Hitand-Run (PMHR), via Brooks and Gelman’s (1998) convergence criterion for an MCMC. For a Bayesian estimation problem, five sequences of samples are generated from the MCMC with five different starting points and are terminated when the estimates from the five sequences converge. The estimates from the five sequences are considered to converge when their potential scale reduction factor or PSRF is close to one. The number of iterations (niters) required until the five sequences converge determines the efficiency; the smaller the niters the more efficient the algorithm. To measure relative performance, we compare PMHR against the simple hit-and-run algorithm and the Gibbs sampler, two of the most commonly used MCMC methods in Bayesian inference literature. We try the three algorithms on problems generated by the generative model described above. For a number of assets N, 12 problems are generated. For each problem, we run the three algorithms to estimate the posterior expectation of the alpha of the first asset. Each algorithm proceeds in batches of 1000 iterations and terminates when either the PSRF is less than 1.2 or 2500 batches have been reached. Upon termination, we record the terminal niters in a unit of 1000 iterations. The average of the 12 niters of each algorithm for each problem dimension N is shown in Table 1. Observe that, at each N, PMHR offers an average niters that is much smaller than those offered by the simple hit-and-run algorithm or the Gibbs sampler. On several problems, with N equal to 50 and 100, both the simple hit-and-run and Gibbs sampler reach 2500 batches before the convergence criteria are met. On the other hand, PMHR manages to converge on all problems. Table 1 Relative efficiency between the simple hit-and-run, Gibbs sampler and PMHR as measured by the average number of iterations required for the algorithms to converge to the solution (unit in 1000 iterations) at different numbers of assets N. N Simple hit-and-run Gibbs PMHR 10 50 100 8.83 2411.83 2500.00 5.25 243.25 976.42 3.67 3.83 17.17 493 5.3. Estimation of alpha We assess the advantages of our approach by comparing against simple heuristics that factor in the side information of asset ranking. Each heuristic differs in the way it estimates alpha from the prior and side information. We now describe each method that we consider and how it estimates alpha based on data produced by the generative model: 1. Bayesian estimation. This is the approach we have proposed. ^  E½a j Aa 6 b, we execute PMHR To compute an estimate a ^ be the algorithm with niters suggested by Table 1 and let a sample average of iterates generated in the course of running the algorithm. 2. Projection. A simple alternative to Bayesian estimation is to  onto the polytope defined by our project the prior mean a constraints according to a^ ¼ argminAa6b ða  a ÞT R1 ða  a Þ: ^ is the mode of the posterior distriThe resulting estimate a bution used in our Bayesian estimation algorithm. The Bayesian estimation algorithm computes the mean, which poses a far greater computational challenge than computing the mode. 3. Black–Litterman adaptation. The Black–Litterman model does not accommodate qualitative views that rank assets. However, we will consider an adaptation that does. Observe that the asset ranking can be represented by N  1 inequalities of the form ain  ainþ1 P 0. Such an inequality can be rewritten as ain  ainþ1 ¼ v n with vn P 0. Consider now relaxing the inequality vn P 0 and instead letting v n ¼ E½ai n  ainþ1 jain  ainþ1 P 0: ð8Þ Further, let r2v n ¼ Var½ain  ainþ1 jain  ainþ1 P 0: ð9Þ Now we can apply the Black–Litterman approach with N  1 views, each view vn representing an estimate of ain  ainþ1 with ^ . In essence, this variance r2v n , leading to an alpha estimate a approach infers Black–Litterman’s quantitative views from an asset ranking. 4. Prior. To provide a sanity check for other approaches, we consider the option of ignoring the views provided as side infor^ be equal to our prior mean a  . We mation and simply letting a expect this to perform worse than other methods that leverage the side information. 5. Clairvoyance. As an upper-bound on performance, we con^ is taken to be the realized alpha sider also the case where a a. This is not a practical approach because in practice we would not know the value of a. However, a is produced as a by-product from our generative model and we can use it in this context as a tool for our conceptual study. 5.4. Results We carry out a computational study with generative model inputs set to M ¼ 3; r2a ¼ 2:5  107 ; r2R ¼ 103 , and s = 0.1. The risk aversion parameter q is 4. We vary the number of assets N across experiments. Algorithm 2 specifies the procedure used for evaluating any given method of estimating alpha. This procedure samples a hundred problem instances, and in each case computes the portfolio that would be selected and the certain equivalent payoff given knowledge of a. These values are averaged over the problem in- 494 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 stances. We use common random numbers so that, for any fixed number of assets, the same problem instances are generated when evaluating each of the different methods for estimating a. 1.40 Certainty equivalent of the payoff (a) Clairvoyance Algorithm 2. Evaluation procedure 1: for i = 1 to 100 do  ; a; A, and b, based on the generative model 2: Sample R; a ^ 3: Compute alpha estimate a 4: Solve portfolio optimization problem   q ^ T w  wT Rw w ¼ argmaxP wn 61 a 2 n P (Note that w0 ¼ 1  n wn is assumed to be invested in a risk free asset with no expected return.) 5: Assess performance ui ¼ aT w  q 2 1.20 (b) Bayesian Ranking (c) Black-Litterman 1.00 (d) Projection Ranking (e) Prior 0.80 0.60 0.40 0.20 0.00 10 50 100 Number of assets (N) Fig. 4. Certainty equivalent of payoff generated using: (a) clairvoyance, (b) Bayesian estimation, (c) the adapted Black–Litterman approach, (d) projection and (e) only prior information. wT Rw 6: end for P i ^ ¼ ð1=100Þ 100 7: Let u i¼1 u We write the procedure in the R statistical software platform and execute it on a PC with an Intel Core i5 2.67 GHz CPU and Windows 7 operating system. Table 2 reports the compute time (in seconds) required for Bayesian estimation, the adapted Black–Litterman approach, and the projection method. Assuming clairvoyance or just using the prior eliminates the need for computational inference, and as such, we do not report the compute times for these approaches. Bayesian estimation takes the longest time on average, while the adapted Black–Litterman and the projection method require very little time. Nevertheless, the time taken by the Bayesian approach is acceptable for practical use. Table 3 presents results from evaluating each method with 10, 50, and 100 assets. The columns are sorted so better performing methods appear to the left. As one would expect, clairvoyance leads to superior performance. Our Bayesian estimation approach has the next best performance, offering a certain equivalent payoff several times greater than that offered by the adapted Black–Litterman approach. The benefit of adopting the Bayesian approach over the adapted Black–Litterman approach, as measured by the ratio between the performance of the former and that of the latter, Table 2 Average compute time (in seconds) per problem required by the Bayesian estimation, the adapted Black–Litterman, and the projection method. N Bayesian Black–Litterman Projection 10 50 100 1.65 3.62 52.50 <0.001 <0.001 0.001 <0.01 <0.01 0.02 increases as the number of assets increases. The projection method looks far worse. As one would expect, ignoring side information and using only the prior leads to the worst performance. Fig. 4 plots the data from Table 3. These plots help to illustrate the large performance differences and how they grow with the number of assets in the portfolio. The average performance with clairvoyance increases as the number of assets increases due to the expansion of the opportunity set. This expansion is not so useful, however, given only the prior, as that is not sufficient to identify which opportunities are more desirable. A ranking, especially when the number of assets is large, greatly reduces this degree of uncertainty since it forms a narrow support for the posterior distribution of alpha. As opposed to other heuristics, our approach makes coherent use of this ranking information. It is worth noting that the benefits realized by our approach partly rely on an assumption that the ranking information is correct with certainty. In the next section, we further investigate the situation when there is uncertainty in the ranking view. 6. Uncertain qualitative views When an analyst offers a qualitative view but is uncertain about its validity, it is useful for the decision maker to be provided with a measure of confidence. This could take the form of a probability that the view is valid. In particular, an uncertain qualitative view may be represented in terms of a polytope R ¼ fa 2 RN : Aa 6 bg: together with a probability j 2 [0, 1]. To facilitate a coherent decision process we must have a precise way of interpreting the probability j. For this purpose, we treat the observed view R as a random variable. Letting R denote a realization, we define j as follows j¼ Pða 2 XjR ¼ RÞ  Pða 2 XÞ : Pða 2 Xja 2 RÞ  Pða 2 XÞ Our definition is best understood through an equivalent relation: Table 3 Performance of alternative methods for estimating alpha measured in terms of the certain equivalent payoff. N Clairvoyance Bayesian Black–Litterman Projection Prior 10 50 100 0.12850 0.64026 1.26021 0.09683 0.45056 0.76391 0.03369 0.05691 0.06435 0.00190 0.00197 0.00203 0.00004 0.00014 0.00013 Pða 2 XjR ¼ RÞ ¼ jPða 2 Xja 2 RÞ þ ð1  jÞPða 2 XÞ: ð10Þ To interpret this relation, first consider the case of j = 1, where it reduces to Pða 2 XjR ¼ RÞ ¼ Pða 2 Xja 2 RÞ, which says that the probability distribution of a conditioned on the observed view is equal to the distribution conditioned on the view being correct. This is equivalent to our earlier interpretation of the view in the absence of uncertainty. At the other extreme, when j = 0, we have 495 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 Pða 2 XjR ¼ RÞ ¼ Pða 2 XÞ. In this case, the view does not influence the conditional distribution of a, which remains equal to the prior distribution. When j 2 (0,1), the conditional distribution lies between the extremes. The following result provides a simple formula for computing the conditional expectation of a given an uncertain qualitative view R and the associated degree of confidence j. As in the previ^ ¼ E½aja 2 R. Further, we let a ~ ¼ E½ajR ¼ R. ous sections, we let a Theorem 1. For any view R with confidence j, a~ ¼ ja^ þ ð1  jÞa : ð11Þ This result follows immediately from Eq. (10). Theorem 1 establishes that the Bayesian estimate with an uncertain qualitative view is a convex combination between the estimate that would be generated if the view were certain and the estimate that would be generated in the absence of any view. This convex combination can be computed with virtually no effort beyond what is required in the case of a certain qualitative view. To assess the influence of the confidence parameter j on the value of the view, we carried out experiments using data sampled in a way similar to that described in Section 5.1. In particular, to generate each realization, we simulate the generative model described in that section twice to obtain two independent data instances:  ; a; A; b,  base instance: N; R; a  ; ay ; Ay ; by .  auxiliary instance: N; R; a From this data, we sample a realization. With probability 1  j this realization is the base instance. With probability j this realization is the base instance but with the view parameters A and b replaced by A and b . It is easy to see that a realization generated in this way satisfies Eq. (10). To evaluate the effect of the confidence level j on the performance of the Bayesian inequalities model for portfolio optimization problem, we perform experiments on a special case when the side information we obtain is a complete ranking as in Section 5. We set up the simulation environment as that in Section 5 and modify Step 2, the estimation step, as follows. ~ by first Given the simulated instance, we obtain the estimate a ^ using PMHR algorithm as described earlier. The resultcomputing a ^ represents what the expectation would be if we were ing estimate a ~ given the uncercertain about the view. To obtain the expectation a ~ ¼ ja . ^ þ ð1  jÞa tainty, we compute the convex combination a We simulate and evaluate the certain equivalent payoffs resulting from our estimation procedure for simulated instances using the same generative model parameter values as in Section 5.4. and with numbers of assets N and values of j. Results are plotted ^ which in Fig. 5. When j = 1, results are based on our expectation a assigns certainty to the view. When j = 0, results are identical to  . For intermediate values, those obtained by our prior expectation a performance gains from incorporating the view increases with j. The slope also appears to increase, which indicates that performance is most sensitive to j when j is close to one. 7. Conclusion We introduce a new approach for incorporating qualitative views, taking the form of linear inequalities, into a mean–variance portfolio optimization framework. Our model and algorithm enable a portfolio manager to incorporate qualitative views into portfolio decisions. For example, in the recent credit crisis, an expert may have formed a qualitative view that assets in Asian countries should outperform assets in the US or European countries. Without our model, a portfolio manager needs to resort to other heuristics. We show that our approach, which directly and coherently incorporates such views into a portfolio decision, achieves superior performance to other natural heuristics. Our approach involves computing the expectation of alpha conditioned on qualitative views, which can be provided together with a degree of confidence. Computing the conditional expectation is a challenging problem, but our algorithm addresses this problem effectively. In conclusion, our framework provides an effective practical mechanism to utilize this form of information. Acknowledgements We thank participants of the Chulalongkorn Accounting and Finance Symposium 2010, especially Wolf Wagner, the discussant, for their helpful comments. This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, administered by the Office of the Higher Education Commission (HS1149A). The fourth author also acknowledges generous support from the Chin Sophonpanich Foundation in the form of endowed chair while he served as a Visiting Professor at Chulalongkorn Business School from 2007 to 2008. Appendix A. Derivations of Eqs. (6) and (7) Let R ¼ fa 2 RN : Aa 6 bg; and let IR be the indicator for the set R. The posterior density of returns, conditioned on qualitative views Aa 6 b, is given by f ðaja 2 RÞ ¼ ( f ðaÞ ; Pða2RÞ if a 2 R; 0; otherwise IR f ðaÞ ¼ ; Pða 2 RÞ ;  h i  ÞT ðsRÞ1 ða  a Þ IR ð2pÞN=2 jRj1=2 exp  12 ða  a   ; ¼R  ÞT ðsRÞ1 ðx  a  Þ dx ð2pÞN=2 jRj1=2 exp 12 ðx  a x2R  h i  ÞT ðsRÞ1 ða  a Þ IR exp  12 ða  a ; ¼ K where K ¼ κ Fig. 5. Certainty equivalent of the payoff as a function of the confidence level j. R x2R exp  1 ðx 2   ÞT ðsRÞ1 ðx  a  Þ dx. Hence, a  i 1h  ÞT ðsRÞ1 ða  a Þ : f ða j a 2 RÞ / IR exp  ða  a 2 496 A. Chiarawongse et al. / Journal of Banking & Finance 36 (2012) 489–496 The posterior expectation of returns is therefore a^ ¼ E½aja 2 R R ¼ xf ðxja 2 RÞ dx; ¼ K1 R x2R x exp  1 ðx 2   ÞT ðsRÞ1 ðx  a  Þ dx: a References Alexander, G.J., Baptista, A.M., 2010. Active portfolio management with benchmarking: a frontier based on alpha. Journal of Banking and Finance 34 (9), 2185–2197. Alexander, G.J., Baptista, A.M., 2011. 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