14.2 Estimation error: why sample mean–variance fails

14.2.1Error in means and weights#

The size of the error in means With $T$ years of data, the standard error of an asset's annual mean return is about $\sigma/\sqrt T$. For the Nifty ($\sigma \approx 16\%$) over 5 years that is $16\%/\sqrt5 = 7.2\%$; over 25 years, 3.2%. The expected-return inputs to (14.3) are uncertain by amounts comparable to the risk premia themselves. Covariances are estimated far more precisely (their error falls with the number of observations, not the number of years), which is why the robust methods below lean on $\Sigma$ and avoid $\mu$.

Sensitivity of the tangency weights Unnormalised tangency weights are $x = \Sigma^{-1}e$, so $\partial x/\partial\mu = \Sigma^{-1}$. Errors in $\mu$ are amplified by $\Sigma^{-1}$, whose largest eigenvalue is $1/\lambda_{\min}(\Sigma)$. Highly correlated assets make $\lambda_{\min}$ small. For two assets with volatilities $\sigma_1, \sigma_2$ and correlation $\rho$,

so the amplification factor is $1/(1-\rho^2)$: 5.3 at $\rho = 0.9$. Example (one of the module's tests): two stocks with $\sigma$ = 18% and 20%, $\rho = 0.9$, $r_f$ = 6.5%. With expected returns (13%, 14%) the tangency weights are (0.34, 0.66). Raise the second expected return by one point to 15%, well within estimation error, and the weights become (−0.31, 1.31): a swing from long to short in the first stock.

14.2.2Simulation and evidence#

Simulation Take the four-asset example of 14.1 as the truth. Draw 60 months of returns, estimate $\hat\mu$ and $\hat\Sigma$, compute the sample tangency portfolio, and evaluate its true Sharpe ratio. Over 2,000 repetitions:

Portfolio	True Sharpe
True tangency (infeasible)	0.507
Equal weight $1/N$ (no estimation)	0.500
Sample tangency, median over 2,000 samples	0.26
Sample tangency, 10th percentile	−0.25
Fraction of sample tangency portfolios with negative true Sharpe	27%

The "optimal" portfolio built from five years of data delivers about half the Sharpe ratio of naive equal weighting, and more than a quarter of the time it has negative expected excess return.

Evidence DeMiguel, Garlappi & Uppal ("Optimal Versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?", Review of Financial Studies 22(5), 2009, 1915–1953) evaluated 14 optimisation models on seven empirical datasets and found none consistently better than $1/N$ out of sample on Sharpe ratio, certainty-equivalent return or turnover. Calibrated to US equity data, the estimation window needed for sample-based mean–variance to beat $1/N$ is around 3,000 months for 25 assets (and about 6,000 months for 50).