A Broader View of the Mean-Variance Optimization Framework

Transcription

1 A Broader View of the Mean-Variance Optimization Framework Christopher J. Donohue 1 Global Association of Risk Professionals January 15, 2008 Abstract In theory, mean-variance optimization provides a rich and elegant framework for asset allocation. Given a practical representation of the possible distribution of returns for a set of assets, mean-variance optimization defines the optimal asset weights to maximize a reasonable objective of balancing an investor s desire for high returns with their aversion to risk. In practice, the assumptions that underly the meanvariance framework are overly restrictive and almost certainly violated. Further, mean-variance optimization commonly produces asset allocations that are extreme, counter-intuitive and highly sensitive to small changes to input parameters. In this paper, we explore the relative mismatch between the coarseness of the inputs to mean-variance optimization and the precision of the output, and demonstrate that greater insight may be gained from mean-variance optimization by decreasing the precision of the optimization. 1 Corresponding Address: Christopher J. Donohue, Global Association of Risk Professionals, 111 Pavonia Avenue, Suite 1215, Jersey City, NJ USA, chris.donohue@garp.com ( ), +1 (201) (phone), +1 (201) (fax).

2 A Broader View of the Mean-Variance Optimization Framework ABSTRACT In theory, mean-variance optimization provides a rich and elegant framework for asset allocation. Given a practical representation of the possible distribution of returns for a set of assets, mean-variance optimization defines the optimal asset weights to maximize a reasonable objective of balancing an investor s desire for high returns with their aversion to risk. In practice, the assumptions that underly the meanvariance framework are overly restrictive and almost certainly violated. Further, mean-variance optimization commonly produces asset allocations that are extreme, counter-intuitive and highly sensitive to small changes to input parameters. In this paper, we explore the relative mismatch between the coarseness of the inputs to mean-variance optimization and the precision of the output, and demonstrate that greater insight may be gained from mean-variance optimization by decreasing the precision of the optimization. Keywords: Portfolio optimization, asset allocation, mean-variance optimization, sensitivity analysis, portfolio management. EFM Classification: 370, 450, 380, 760

3 1 Introduction The general premise of the mean-variance framework (Markowitz [7]) is to balance an investor s goals of maximizing the expected return of a portfolio while simultaneously minimizing its volatility. The significant contribution of mean-variance optimization is an algorithmic approach to deriving the asset weights which maximize the expected portfolio return for a specified level of return volatility or conversely minimize the amount of return volatility for a specified expected return. The standard mean variance framework considers a portfolio of N risky assets. Let μ i and σ i denote the mean and standard deviation, respectively, of the return distribution for asset i, andletρ ij denote the correlation between the return of asset i and asset j. Let Q with elements q ij = ρ ij σ i σ j denote the variance/covariance matrix, and let λ denote the risk aversion parameter. Finally, let z denote the optimal objective value to the mean variance optimization problem. That is, z = max μ T x λ 2 xt Qx x T 1 = 1 Let x denote the asset weights that achieve z. Note that the mean variance formulation implicitly assumes asset returns are normally distributed and stationary, that the investor has a single period horizon (no opportunity for recourse), and that there are no transaction costs. Criticism of mean-variance optimization generally focuses on two areas. The first area of criticism revolves around the input. Specifically, the appropriateness of a stationary normal distribution and the ability to accurately assign its parameters for an asset class have been challenged in numerous studies (see, e.g, Fama [3], Mandlebrot [5], and Mandlebrot and Taylor [6]). Empirically, the distribution of financial return data tends to be approximately symmetric, but with fatter tails and 1

4 a more peaked center than a normal distribution (see, e.g., Alexander [1]). Even accepting the normality assumption, the ability to forecast the parameters of the distribution is challenging, as the inability to run controlled experiments has forced the development of other models (e.g., ARCH, GARCH) that attempt to forecast future return distribution parameters based on the single observation provided by historical data. Considerable effort has been expended attempting to improve the ability to accurately forecast future return distribution parameters. The second area of criticism of the mean-variance framework concerns the often extreme output (i.e., several large long and short positions) and the sensitivity of the output to small changes to the input. Numerous approaches have been developed to tame these perceived weaknesses. The most commonly used practical solution is to include user-defined lower and upper bounds for each asset class which forces weights to be well-behaved. Unfortunately, this practice raises the question of whether the solution is being driven by the characteristics of the asset return distributions or by the users perception of what the optimal weights should be. Michaud [8] developed the resampled frontier, in which asset weights are derived from a set of meanvariance solutions that result from asset return distribution parameters from multiple samples. Others have sought more theoretically justifiable approaches to control mean-variance solution sensitivity. In particular, the Black-Litterman approach [2] seeks to extract market return values implied by market capitalizations for asset choices, which generally leads to mean-variance solutions which are less extreme and less sensitive. More recently, robust optimization techniques have been applied to the mean-variance framework to generate weights that maximize performance over a range of possible asset return parameter values (see Pachamanova et. al. [9] for a summary of the application of robust optimization). The criticisms of mean-variance optimization highlight the extreme mismatch in 2

5 the precision of the input and output. Even accepting the stationarity and normality assumptions, the ability to forecast return distributions is an imprecise art. In fact, most strategic asset allocation studies attempt to address this shortcoming by explicitly considering a range of possible expected return and standard deviation values for the return distribution of each asset class. On the other hand, the machinery of quadratic programming (of which mean-variance optimization is an example) can provide arbitrarily precise answers. It is worth noting that while few practitioners or researchers would argue over the first decimal value for the expected return of an asset (e.g., 7.5% vs. 7.6%), the typical commercial optimization program terminates when the solution has converged to within 10 6 of the true optimal objective value. Clearly, the dull level of precision of the inputs is completely mismatched with the sharp level of precision of the output. In this paper, we reconsider the mean-variance framework and attempt to refocus the output from a precise definition of optimal asset weights to a more general guide to assets weights and how they should work together. The true role of mathematical modeling in portfolio management should be to provide insight, not solutions, and the contribution of this paper is a practical approach that increases the insight that can be gained from the powerful mean-variance framework. This approach also has potential application to building a rebalancing strategy consistent with the meanvariance framework. 2 Approach Instead of optimizing the mean-variance objective function, consider the set of asset weights that are in the neighborhood of optimality. Let S(ɛ) be the set of feasible 3

6 asset weights that achieve objective function value within ɛ of optimality: S(ɛ) = {x R N : μ T x λ 2 xt Qx z ɛ, x T 1=1}. Note that, by definition, we know S(0) = {x }, and S(ɛ) =, for all ɛ< Finding Asset Weights in the Neighborhood of Optimality For ɛ>0, first note that the unity constraint can be removed by setting one asset weight equal to one minus the sum of the remaining asset weights and adjusting μ and Q accordingly. That is, S(ɛ) ={x R,x N =1 x i : μ T x λ 2 xt Qx + μ N (1 x i ) λ 2 q NN(1 x i ) 2 λ q jn x j (1 x i )=z ɛ} where = {x R,x N =1 j=1 x i :ˆμ T x λ 2 xt ˆQx = z ˆɛ} ˆμ i = μ i μ N + λσn 2 λρ inσ i σ N ˆq ij = ρ ij σ i σ j + σn 2 ρ inσ i σ N ρ jn σ j σ N ˆɛ = ɛ + μ N λ 2 σ2 N By setting the derivative of the restated objective function to zero and solving for x, the optimal asset weights and solution value are: 4

7 x = x N = 1 ˆμT ˆQ 1 λ z = ˆμT ˆQ 1 ˆμ 2λ Hence, we can restate S(ɛ) as follows: S(ɛ) ={x R,x N =1 x i + μ N λ 2 σ2 N x i :ˆμ T x λ 2 xt ˆQx = ˆμ T ˆQ 1ˆμ 2λ ɛ} Let W with elements w ij denote the inverse matrix of the modified variance-covariance matrix ˆQ. Then, for ɛ>0, by applying the quadratic formula, we get that all asset weights that achieve an objective value within ɛ of optimality are bounded as follows: ( ) 0.5 ( ) 0.5 x i 2wii ɛ x i x i λ + 2wii ɛ for all x S(ɛ),i N 1. (1) λ That is, the set of feasible asset weight vectors within ɛ of optimality in terms of objective value are centered at the optimal solution x and the range of possible weights around x grows as the optimality tolerance parameter ɛ increases and as the risk aversion parameter λ decreases. Note that the numbering of the assets is arbitrary, so the range of near optimal weights on asset N can be found simply by renumbering the assets and reapplying the method. 2.2 Building the solution landscape Further, the solution landscape around the optimal solution x can be derived by fixing an individual asset weight within the range defined by (1) and then solving for the range of possible asset weights for the remaining assets which together achieve an objective value within ɛ of optimality. 5

8 Let x be a fixed weight for asset N 1, and let S(ɛ x )denotethesetof asset weights that achieve an objective function value within ɛ of the optimal value z, given the weight on asset N 1isfixedat x. Then, by definition, S(ɛ x )={x S(ɛ) :x = x } where = {x R N 2,x = x,x N =1 x N 2 ˆμ T x λ 2 xt ˆQx λ = {x R N 2,x = x,x N =1 x N 2 x i : ˆq i, x i x = z ˆɛ (ˆμ x λ 2 ˆq, x 2 )} N 2 x i : ˆμ T x λ 2 xt ˆQx = z ˆɛ (ˆμ x λ 2 ˆq, x 2 )} ˆμ i = ˆμ i λˆq i, x By setting the derivative of the modified objective function to zero and solving for x, the conditionally optimal solution and objective value are given by: x x = ˆμ T ˆQ 1 λ x x = x x N x = 1 x i x z x = ˆμ T ˆQ 1 ˆμ 2λ +ˆμ x + μ N λ 2 (ˆq, x 2 + σ2 N ) For x in the range defined by (1) for a given ɛ, by construction we have: z ɛ z x z. Let ɛ x z x z + ɛ. Note that ɛ represents the amount of allowable deviation from optimality and ɛ x represents the amount of allowable deviation 6

9 that remains after the weight on asset N 1 has been fixed at x. By construction, for x on the boundaries defined by (1), ɛ x the boundaries defined by (1), ɛ x is positive. By substitution, S(ɛ x ) can then be restated as: is zero and for x strictly within S(ɛ x ) = {x R N 2,x = x,x N =1 x ˆμ T x λ 2 xt ˆQx = ˆμ T ˆQ 1 ˆμ 2λ ɛ x } N 2 Similar to before, by applying the quadratic formula, we get that all asset weights that achieve an objective value within ɛ of optimality, given the weight on asset N 1 is x, are bounded as follows: For all x S(ɛ x ),i N 2, x i : x i x ( 2wii ɛ x λ ) 0.5 x i x i x + ( 2wii ɛ x λ ) 0.5 (2) By fixing the weight on asset N 2 in (2), we could similarly then bound the asset weights on assets 1,...,N 3 so that the resulting complete allocation achieves an objective value within ɛ of optimality. 3 Computational Example To see the insight provided by a broader consideration of the solution landscape, consider the 5-asset class problem with return distribution parameters defined in Table 1. Figure 1 shows the efficient frontier defined by mean-variance optimization solutions for different risk aversion values, and the corresponding portfolio risk versus return 7

10 Table 1: Return distribution parameters for 5-asset class example problem. Exp. Return Volatilty Correlations Asset Class (μ) (σ) US Large US Small Intl. Eq. Em. Mkt. US Bond US Large Cap US Small Cap Intl. Equity Emerging Mkt US Bond points defined by asset weights in S(ɛ) for increasing values of ɛ. As shown, the increasing values of ɛ draw a similar frontier but with a wider brush. Table 2 lists the range of asset weights in S(ɛ) for increasing values of ɛ for fixed risk aversion values λ =0.198 and λ = Note that from Fig. 1 that the the points in S(0.05) define a frontier almost identical to the efficient frontier, but from Table2weseethatwithλ =0.198, the asset weights vary by as much as 6.7% from the optimal weights; with λ =0.037, the asset weights vary by as much as 15.3%. Further, the points in S(0.25) define a rougher but essentially similar frontier to the efficient frontier but with λ =0.198, the asset weights vary by as much as 14.8% from the optimal weights; with λ =0.037, the asset weights vary by as much as 34.3%. Given the uncertainty that surrounds the asset return parameters, having these ranges would proven benefical to an investment profession who could then overlay their expertise and judgement to set the final asset weights. Fig. 2 shows the set of risk-reward points defined by weights in S(ɛ) for increasing values of ɛ and for different risk aversion parameter λ values. For both risk aversion values shown and for each optimality tolerance parameter ɛ shown, the near optimal risk-reward points appear to define a plane parallel to the tangent plane to the 8

11 Figure 1: The efficient frontier and (risk, return) points within ɛ =0.05, ɛ =0.15 and ɛ =0.25 of the efficient frontier. efficient frontier at the optimal risk-reward point,and this plane moves further away from the optimal point as ɛ increases. Lastly, Fig. 3 shows the solution landscape for pairs of asset classes for increasing values of ɛ. These charts show graphically that a near optimal solution can be achieved by increasing (or decreasing) the allocation to US Large Cap and simultaneouly decreasing (or increasing) the allocations to both US Small Cap and US Bonds and keeping the allocations to International Equity and Emerging Markets constant. This representation provides greater insight into how asset weights must be coordinated and the amount of flexibility one has in setting an asset allocation that is in the neigborhood of optimality. 9

12 Table 2: Asset weight ranges for points within ɛ =0.05, ɛ =0.15 and ɛ =0.25 of optimality. ɛ =0.05 ɛ =0.15 ɛ =0.25 Optimal Weight (%) Min Weight (%) Max Weight (%) Min Weight (%) Max Weight (%) Min Weight (%) Max Weight (%) λ = US Large Cap US Small Cap Intl. Eq Emrg. Mkt US Bond λ = US Large Cap US Small Cap Intl. Eq Emrg. Mkt US Bond Conclusion Kritzman [4] argues through two examples that much of the discussion concerning the hyper-sensitivity of mean-variance solutions is simply hype. He demonstrates that for portfolios with assets having distinct distribution parameters, the solutions are relatively stable as parameter values change, and that for portfolios with assets having similar distribution parameters, the solutions may vary widely as parameter values change but portfolio performance characteristics are relatively stable. An alternative explanation is that for portfolios with assets having similar dis- 10

13 Figure 2: The efficient frontier and (risk, return) points in S(ɛ =0.05), S(ɛ =0.15) and S(ɛ = 0.25) for different risk aversion parameter (λ) values. tribution parameters, the mean-variance objective function is relatively flat near the optimal asset allocation. That is, although only one point will provide the truly optimal asset allocation (subject to the assumptions of the mean-variance framework), a potentially wide range of asset allocations will provide near-optimal solutions. This paper defines those ranges for each asset and defines how the ranges of pairs of assets are coordinated to stay close to optimal. Experienced investment managers could then overlay their judgement and constraints to make final asset allocation decisions. Further, upon implementation of an asset allocation, the landscape of near-optimal solutions could define no-trade regions which investment managers could use to guide rebalancing decisions. 11

14 Figure 3: Solution landscapes for Asset Class 1 within ɛ =0.05, 0.15 and

15 References [1] Alexander, S. S., Price Movements in speculative markets: trends or random walks?, Industrial Management Review, Vol. 2, 1961, [2] Black, F. and Litterman, R., Global Portfolio Optimization, Financial Analysts Journal, Sept/Oct. 1992, [3] Fama, E., The Behavior of Stock Market Prices, Journal of Business, Vol. 38, 1965, [4] Kritzman, M., Are Optmizers Error Maximizers, Journal of Portfolio Management, Summer 2006, [5] Mandlebrot, B. B., The Variation of Certain Speculative Prices, Journal of Business, Vol. 36, 1963, [6] Mandlebrot, B. B. and Taylor, H. W., On the Distribution of Stock Price Differences, Operations Research, Vol. 15, 1967, [7] Markowitz, H., Portfolio Selection, Journal of Finance, Vol. 7, No. 1, 1952, [8] Michaud, R., The Markowitz Optimization Enigma: Is Optimized Optimal?, Financial Analysts Journal, Jan-Feb 1989, [9] Pachamanova, D. A., Fabozzi, F., Kolm, P., and Focardi, S., Robust Portfolio Optimization: Recent Trends and New Directions, Journal of Portfolio Management, Vol. 33(3), 2007,