FALLACY: EQUALLY WEIGHTED PORTFOLIOS ARE SUPERIOR TO OPTIMIZED PORTFOLIOS

Optimized portfolios are designed to maximize expected return for a chosen level of risk by accounting for differences in expected returns, standard deviations, and correlations. Yet it has been argued that equally weighted portfolios outperform optimized portfolios out of sample. The “1/N” approach to investing assigns asset weights based purely on the number of asset classes, N, and ignores all other information. Those who favor this naïve allocation method believe that estimation error is so insurmountable that incorporating any views about expected return and risk damages portfolios more than it helps them.

Should we really conclude that expectations about return, risk, and correlation, derived from some combination of theory and data, are useless for portfolio construction? We argue not. The notion that 1/N is superior to optimization is based on tests that blindly extrapolate small‐sample historical means as estimates of expected return. It is this naïve approach to estimating expected return, and not the process of optimization, that is flawed. Thoughtful practitioners know not to rely on recent return outcomes as expected returns. Successful portfolio optimization does not require perfect estimates of return and risk—merely estimates that are plausible. If we use reasonable intuition along with long samples of historical data to estimate optimization inputs, optimization regularly outperforms 1/N out of sample.

THE CASE FOR 1/N

The case for 1/N is appealing. It is simple. It avoids concentrated positions. It always outperforms the worst‐performing asset class. And it always allocates some amount to the best‐performing asset class.

One of the most influential studies supporting 1/N is by DeMiguel, Garlappi, and Uppal (2009). They performed out‐of‐sample backtests on seven different data sets and used 14 different methods to estimate inputs, including Bayesian shrinkage for managing estimation error. They found that, on average, 1/N portfolios generated Sharpe ratios 50 percent higher than optimized portfolios.

SETTING THE RECORD STRAIGHT

We argue that the perceived failure of optimization arises from overreliance on short return samples to estimate expected returns. For example, DeMiguel, Garlappi, and Uppal used rolling 60‐ and 120‐month return samples to estimate expected returns. These estimates are prone to small‐sample error, and in many cases are utterly implausible. For example, what should we expect about the future return of the equity market if it experienced a significant downturn over the past five years? Should we expect another five‐year loss contrary to capital market theory and long‐term evidence? Most informed investors would expect the future return of the equity market to bear some positive relationship to its riskiness and not to its recent performance. In fact, those who believe in long‐term valuation cycles might believe that equity prices are more likely to rise following a downturn. In any event, most thoughtful investors rely on information beyond short‐term historical returns to form expectations about the future.

As an analogy, think of optimization as a sophisticated navigation system. 1/N is the navigational equivalent of wandering aimlessly in search of our destination, ignoring not only the GPS instructions but also any posted road signs. We argue that it is better to use the GPS; we just need to specify the correct destination. If our goal is to drive to the beach on Saturday but we mistakenly instruct the GPS to take us to the office, we should not fault the GPS for directing us to the office. But those who favor 1/N think of optimization this way.

EMPIRICAL EVIDENCE IN DEFENSE OF OPTIMIZATION

In a 2010 article, Kritzman, Page, and Turkington conducted backtests to compare optimization to 1/N. They tested a wide range of applications, including allocation across broad asset classes, equity industries, factors, individual stocks, commodities, hedge fund styles, and actively managed funds. In each case, they constructed long‐only portfolios with weights that sum to 1 but were otherwise unconstrained. They used three different methods to estimate expected returns, each of which is simple yet avoids the drawbacks of relying on short‐sample means. The first method assumes constant expected returns across assets, which yields the minimum‐variance portfolio. The second method derives expected returns from risk premiums spanning multiple decades prior to the start of the backtest. The third method estimates expected returns from long historical samples to reflect all monthly return data available up to that point in time. They calculated covariance matrices as sample covariances from rolling 5‐, 10‐, and 20‐year windows, as well as from all available historical data.¹

In total, Kritzman, Page, and Turkington evaluated more than 50,000 optimized portfolios. As a measure of overall performance they reported average Sharpe ratios for each category of tests. They evaluated asset allocation in the context of asset/liability management with an investable universe of U.S. equity, foreign equity, U.S. government bonds, U.S. corporate bonds, real estate investment trusts (REITs), commodities, and cash equivalents. Their test period spanned 1978 to 2008, with portfolios reoptimized yearly and held until the following year. In their analysis of asset allocation, the capitalization‐weighted market portfolio and the 1/N portfolio delivered similar Sharpe ratios of around 0.7. The minimum variance and long‐term risk premium approaches fared better, with Sharpe ratios above 1.0. To prove a point, the authors also tested a set of completely contrived yet plausible expected returns (which we happen to know were chosen by one of the coauthors while on vacation at the beach and communicated by phone to the office), which resulted in an even higher Sharpe ratio. Though less directly related to the asset allocation focus of this book, the tests for industry and factor allocation using data beginning in 1926 supported the same conclusions, as did out‐of‐sample tests on commodities (since 1971), active funds (since 1987), hedge fund styles (since 1996), and individual U.S. stocks (since 1998). In these experiments the authors intentionally used simple models to demonstrate the efficacy of optimization. One could easily make a stronger case for optimization by building portfolios in line with best practices.

PRACTICAL PROBLEMS WITH 1/N

While 1/N is provocative in its simplicity and appealing when simplistically evaluated, it is difficult to defend for reasons other than the fact that it doesn't work. First, equal weighting is not sensitive to return and risk estimates; instead, it is entirely dependent on the choice of the asset class universe. Each asset class is assigned equal importance regardless of how many there are. Splitting an asset class into two subcomponents would, in effect, nearly double that asset class's allocation. The 1/N approach essentially transfers the risk of input estimation error to the risk of selecting the right asset classes.

Second, the 1/N heuristic offers investors only one portfolio no matter their attitude toward risk. The efficient frontier allows investors the choice of a wide variety of portfolios, each with different combinations of expected return and risk.

Third, equal weighting ignores the capacity of each asset class as well as a variety of other considerations, beyond expected return and risk, which might favor one asset class over another.

BROKEN CLOCK

As the adage goes, even a broken clock is right twice a day. It is possible, given certain conditions, that mean‐variance analysis will direct investors toward an equally weighted portfolio. We reverse engineer the optimization process to determine what equal weighting implies for expected returns and covariances. In the absence of constraints, the optimal portfolio is given by the following expression, where is the covariance matrix and is a vector of expected returns. (See Chapter 18 for details.)

(7.1)

For any level of risk aversion, the optimal weights are proportionately identical, so for convenience we ignore this multiple and express this relationship as a proportionality:

(7.2)

Equation (7.2) reveals that for the optimal allocation to equal 1/N, the expected returns, , must be proportional to the sum of the rows of the covariance matrix.

(7.3)

It is worth noting that the Capital Asset Pricing Model (CAPM) implies that expected returns are proportional to the systematic risk of each asset class, estimated as its regression beta (or in proportionality terms, its covariance) with the market portfolio, :

(7.4)

To the extent market capitalization weights differ from equal weights, 1/N and CAPM imply different expected returns. Ultimately, these relationships show that it is possible, though not necessarily likely, that optimal weights will equal 1/N. To justify 1/N in terms of optimality, one needs to believe that prices are adequately efficient to reflect their covariances with other assets accurately, and yet at the same time do not converge to the CAPM market portfolio equilibrium.

THE BOTTOM LINE

The bottom line is that 1/N makes sense only for investors who believe they have no insight whatsoever about differences in the expected returns and risk of asset classes. We encourage investors who have access to historical data and who are capable of sound judgment to apply optimization to identify the portfolio that best suits their investment goals.

REFERENCES

1. V. DeMiguel, L. Garlappi, and R. Uppal. 2009. “Optimal versus Naïve Diversification: How Inefficient Is the 1/N Portfolio Strategy?” Review of Financial Studies, Vol. 22, No. 5 (May).
2. M. Kritzman, S. Page, and D. Turkington. 2010. “In Defense of Optimization: The Fallacy of 1/N,” Financial Analysts Journal, Vol. 66, No. 2 (March/April).