In Defense of Optimization: The Fallacy of 1/N

Abstract

Previous research has shown that equally weighted portfolios outperform optimized portfolios, which suggests that optimization adds no value in the absence of informed inputs. This article argues the opposite. With naive inputs, optimized portfolios usually outperform equally weighted portfolios. The ostensible superiority of the 1/N approach arises not from limitations in optimization but, rather, from reliance on rolling short-term samples for estimating expected returns. This approach often yields implausible expectations. By relying on longer-term samples for estimating expected returns or even naively contrived yet plausible assumptions, optimized portfolios outperform equally weighted portfolios out of sample.

Previous research has shown that equally weighted portfolios outperform optimized portfolios, which suggests that in the absence of informed inputs, optimization adds no value. We argued the opposite. Using naive inputs, we demonstrated that optimized portfolios usually outperform equally weighted portfolios. The ostensible superiority of the 1/N approach arises not from limitations in optimization but, rather, from reliance on rolling short-term samples for estimating expected returns. By relying on longer-term samples for estimating expected returns or even naively contrived yet plausible assumptions, optimized portfolios outperform equally weighted portfolios out of sample.

Our study covered 13 datasets comprising 1,028 data series. We constructed more than 50,000 optimized portfolios and evaluated their out-of-sample performance as compared with the market portfolio and the 1/N portfolio. We grouped portfolios into three categories: asset class, beta, and alpha.

We used three approaches to estimate expected returns for optimized portfolios: (1) We generated the minimum-variance portfolio; (2) for each asset, we estimated a risk premium over a long data sample before the backtest start date and assumed that it remained constant throughout the backtest; and (3) in the spirit of classical statistics, we used a growing sample that included all available out-of-sample data.

Although extremely simple, these expected returns have an important difference from most of the expected returns used in previous studies: They do not rely on rolling samples of realized returns, which often imply implausible expectations. For example, we might forecast that cash will outperform stocks because it did so in the past five years. Why would this particular realization be a good forecast of the next one? We should not use these past data; all we need is a reasonable forecast tied to economic intuition.

Our results show that optimized portfolios significantly outperform the 1/N portfolio, even across beta universes, which are notorious for the exceptional performance of the 1/N portfolio as compared with the market portfolio. We showed that even without any ability to forecast returns, optimization of the covariance matrix by itself adds value. In our view, 1/N is not a viable alternative to thoughtful optimization but is, rather, a capitulation to cynicism.