Stop playing with your optimizer

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.

Even with this unlikely level of precision, the composition of the optimal portfolio at a standard deviation of 12.5% varies dramatically (see “Optimal allocations”). When the expected return on Canadian Equity is overestimated by 50 basis points, the optimal portfolio contains a 45% allocation to this asset class.

But when it is underestimated by the same amount, it drops to merely 2% of the allocation. Consequently, optimizers function as mistake maximizers when more weight is given to the asset class with an overestimated expected return, and vice versa.

Faulty premise

Even if you could estimate inputs with enough accuracy to truly identify the efficient frontier, the whole concept of optimization is still based on the faulty premise that portfolio variance is a complete measure of risk.

The empirical analysis of the relation between risk and return, in the context of market equilibrium, is known as asset pricing. The capital asset pricing model (CAPM) was the first asset pricing model, and it assumes the only risk investors are compensated for is volatility relative to the market, as measured by beta.

Optimal allocations for an annual standard deviation of 12.5%

The model is simple, elegant, and intuitive. Unfortunately, it doesn’t work very well. All models are, by definition, false, because they are abstractions from reality and simplifications of a complex world. The relevant question is how good the model is at describing reality or, in this case, the relation between risk and return.

Many subsequent models that built on the CAPM—and resolved some of its deficiencies—suggest that risk is multi-dimensional and cannot be distilled into a single statistic. For example, in “The Cross Section of Expected Stock Returns” (Journal of Finance 47, no. 2 [June 1992]: 427–465), Eugene F. Fama and Kenneth R. French provide compelling evidence that average stock returns can largely be explained by exposure to three dimensions: market, size, and relative price.

An alternative framework

The final chapter on asset pricing may never be written, but the efficient frontier is only observable in hindsight, and building portfolios solely focused on the trade-off between expected return and standard deviation is not robust. When building lifetime portfolios for clients, advisors must apply a much broader perspective by considering a host of planning and portfolio management issues.

A good starting point for the asset allocation decision is the market, which is always an efficient portfolio. For every investor overweighted in an asset class, there must be another who is underweighted. Asking how an investor differs from others can help guide any decision to allocate assets in a way that deviates from the market portfolio. Risk preferences, tracking error, home bias, human capital, non-financial assets, consumption patterns, and other variables are some considerations that can help answer this question (see “Three key portfolio construction decisions,” below).

Three key portfolio construction decisions

If you encounter the proverbial mean-variance investor, go ahead and run your optimizer. For everyone else with multi-dimensional goals, forget optimal and focus on reasonable when designing a portfolio.

Brad Steiman is director, head of Canadian Financial Advisor Services, and vice president of Dimensional Fund Advisors Canada ULC. brad.steiman@dimensional.com.