# Portfolio Optimization by Means of Resampled Efficient Frontiers Global Asset Allocation and Stock Selection Estimation Error and Portfolio Optimization Campbell R. Harvey Duke University, Durham, NC USA National Bureau of Economic Research, Cambridge MA USA [email protected] +1 919.660.7768 office || +1 919.271.8156 mobile http://www.duke.edu/~charvey Motivation The Markowitz Mean-Variance Efficiency is the standard optimization framework for modern asset management. Given the expected returns, standard deviations and correlations of assets (along with constraints), the optimization procedure solves for the set of portfolio weights that has the lowest risk for a given level of portfolio expected returns. Standard algorithms (linear programming/quadratic programming) are available to compute the efficient frontier with or without short-selling/borrowing constraints.

Motivation A number of objections to MV Efficiency have been raised: 1. 2. 3. 4. Investor Utility: Utility might involve preferences for more than means and variances and might be a complex function. Multi-period framework: The one-period nature of static optimization does not take dynamic factors into account. Liabilities: Little attention is given to the liability side. Instability and Ambiguity: Small changes in input assumptions often imply large changes in the optimized portfolio. The MVE procedure overuses statistically estimated information and magnifies estimation errors.

Jorion (1992, Financial Analyst Journal) addresses Portfolio Optimization in Practice and proposes the first resampling method. Motivation Richard Michaud (1998) has built a business around resampling. Implemented in Northfield optimizers. Ibbotson Associates also uses a resampling technique The Procedure The procedure described below has U.S. Patent #6,003,018 by Michaud et al., December 19, 1999. Estimate the expected returns and the variance-covariance matrix () [say, the MLE (average) or using Bayesian techniques]. Suppose there are m assets.

Solve for the minimum-variance portfolio. Call the expected return of this portfolio L. Solve for the maximum return portfolio. Call the expected return of this portfolio H. Choose the number of discrete increments, in returns, for characterizing the frontier. That is, for the purpose of the Monte Carlo analysis, one can think of looking at a series of points on the frontier - rather than every point on the frontier. The Procedure Suppose for L=.05 and H=.20 and we choose the number of increments, K=16. This means that we evaluate the frontier at expected return={.05, .06, ..., .19, .20}, that is 16 different points. We will represent the frontier as ak, where a

represents weights. So for m assets, ak is Kxm (rows represent the number of points on the frontier and columns are the assets). The pair (ak, ) then represents the efficient frontier. In our example, we would have 16 rows (discrete increments) and m columns (number of assets). The Procedure E[r] 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 H L 0 0.2 0.4 0.6 0.8

1 1.2 The Procedure Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16 E[r] 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Asset 1 Columns are Asset Weights

Asset 2 Asset 3 Asset 4 Matrix = ak m=5 columns Asset 5 k=16 rows The Procedure Now begin the Monte Carlo analysis. Generate i from the likelihood function L(). Hence i and are statistically equivalent. For example, suppose we have m=5 assets and T=200 observations. Give a random number generator, (means, and variancecovariance matrix), assume a multivariate normal (not necessary

but simplest), and draw five returns 200 times. With the generated data, calculate the simulated means and variance-covariance matrix, i. Note that i and are statistically equivalent. Using i, calculate the minimum variance portfolio (expected return Li) and the maximum expected return portfolio (expected return Hi). Use these to determine the size of the expected return increments. The Procedure Following our example of K=16, suppose using i, Li=.03 and Hi=.25. Hence, the discrete points would be expected returns of {.030, .044, .058, ..., .236, .250}. Calculate the efficient portfolio weights at each of these K points. (Solve for the minimum variance weights given expected returns of .030, ...).

With this information, we now have ak,i. This is the same dimension, Kxm. Repeat the simulations, so that we have 1,000 ak,is. The Procedure Average the 1000 ak,is. For each increment, this gives us average portfolio weights. Call this ak* We can redraw the efficient frontier, by using the original means and variances, i.e. combined with the new weights, ak* The Procedure

Note, the new efficient frontier is inside the original frontier. Why? If we look at any particular Monte Carlo draw, say i, we could draw a frontier (which could be to the right or left of the original frontier). However, we are keeping track of the weights at the discrete increments. We average the weights (not the frontiers) - and then apply these average weights to the original . We know that the efficient weights for , are ak. If we apply, ak*, then we must be to the right of the original frontier. In other words, if ak is the best, then ak* cannot be the best. However, importantly, ak* is taking estimation error into account. Variations

Instead of using K increments for the indexation based on the high and low returns, assume a quadratic utility function is used. A function of the form = 2 is minimized. Each value of ranging from zero to infinity defines a portfolio on the mean-variance and simulated frontiers. Decide on the values of the s and then use for the indexation. Does the resampled frontier outperform? Based on simulations, yes. Here is the evaluation. Given the first draw of i, do another Monte Carlo exercise to determine the resampled frontier defined by ak,i* (notice difference in notation).

Do this extra Monte Carlo on each i - so we will have 1,000 different ak,i* Compare the average of ak,i* to the average of the ak,i,(that is, compare the average of the simulated portfolios based on the true value of i to the average of the portfolios that do not incorporated any allowance for estimation error). Simply draw frontiers based on . Is Resampling What We Want to Do? Resampling provides an improvement over traditional methods However, there are issues:

Average of maximums is not the maximum Hence, allocation will be suboptimal and we should be able to improve on this work New research on the horizon that provides an alternative. See Harvey, Liechty, Liechty and Muller (2010) article in Quantitative Finance.