good-better-best

The emphasis on portfolio structure and asset allocation over security selection and market timing has served investors well and is supported by volumes of academic research. However, aspects of this research have been somewhat overstated.

For example, asset allocation recommendations are often accompanied by illustrations of mean-variance optimal portfolios that sit along an efficient frontier where expected returns are maximized for each level of standard deviation.

Part of the allure of mean-variance optimization and the efficient frontier is the connection with Harry Markowitz's Nobel Prize-winning theory of portfolio selection. But so-called optimizers create only an illusion of scientific accuracy, and these illusory effects are magnified when precise weights in various asset classes are specified to two decimal places.

The output of optimizers is sensitive to the accuracy of the inputs, and even slight errors can result in dramatically different allocations for a supposedly optimal portfolio.

Mistake maximizers

The parameters required to run a mean-variance optimization are the expected returns and standard deviations of each individual asset class, along with the correlation matrix of all asset classes in the portfolio.

The required number of inputs for a portfolio of N asset classes is equal to N x (N+3)/2. With five asset classes, you must estimate twenty parameters, and this number increases dramatically with the addition of asset classes to the portfolio.

If we compute the optimal allocation for a portfolio containing five asset classes—Canadian Equity, U.S. Equity, International Equity, Emerging Markets Equity, and Fixed Income—the estimated expected returns for each asset class (see "Input Error Illustration," below) and historical data are used as the inputs for standard deviations and correlations. The asset classes are represented by the following indexes: TSX Composite Index (Canadian Equity), S&P 500 Index (U.S. Equity), MSCI EAFE Index (International Equity), MSCI Emerging Markets Index (Emerging Markets Equity), and DEX Universe Bond Index (Fixed Income).

Input error illustration

EXPECTED RETURN
Asset Class TRUE ERROR + ERROR -
Canadian Equity 8.0 .50 .50
US Equity 8.0 0 0
International Equity 8.0 0 0
Emerging Markets Equity 9.5 0 0
Fixed Income 5.0 0 0

To illustrate the impact of input errors, we assume 19 of the 20 estimated parameters (four expected returns, five standard deviations, and 10 correlations) are calculated with complete precision and that they represent the true parameter. However, we assume we are unable to accurately estimate the expected return of one asset class, Canadian Equity. We either underestimate or overestimate its expected return by 50 basis points.

This level of precision for expected return estimates is wildly conservative given the amount of noise in return data. For example, the 95% confidence interval for the U.S. equity premium is roughly 3.5% to 12.1%, based on data from 1927-2010. Yet we only make a slight error estimating one parameter while the remaining 19 are assumed to be accurate.