Portfolio Optimization When Expected Returns Are Uncertain
Summary
The document asks how to choose portfolio weights when estimated mean returns are too uncertain to distinguish reliably. It frames the problem with three assets that share the same correlations and variances, where some pairwise tests fail to reject equal means while another does reject it. This illustrates how noisy estimates can make mean–variance optimization sensitive to the choice of expected returns.
Two responses suggest ways to handle that uncertainty. One points to Black–Litterman as a model developed for practical portfolio optimization. The other describes risk parity, which sets equal risk contributions or equal marginal risk contributions and does not use alpha estimates to determine weights. Risk parity can reduce reliance on uncertain expected returns, but it may not produce the highest Sharpe ratio. The discussion also notes that covariance estimates can be difficult to make accurately; it offers no empirical comparison or specific implementation guidance.
Key ideas
- Noisy mean-return estimates can make mean–variance portfolio weights difficult to choose reliably.
- Black–Litterman is suggested as an approach to practical portfolio optimization.
- Risk parity can set equal asset risk contributions without using expected returns as an optimization input.
- Risk parity may be a useful alternative when alpha estimates are uncertain, though it may not maximize the Sharpe ratio.
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Full text
# portfolio optimization with uncertain returns # portfolio optimization with uncertain returns What is the usual method of dealing with many uncertain mean returns in portfolio optimization? For example say you have a 3 asset portfolio with assets A, B and C. All the correlations and variances are the same. You can't reject the null that the assets have the same mean return for (A and B) and (B and C), but you can reject the null for (A and C). It seems there is no nonbiased way to choose the means? ## Answer by jaamor (score 2) https://quant.stackexchange.com/a/16831 Your question accurately addresses of the practical problems of applying Modern Portfolio Theory (i.e. mean-variance optimization) in practice. Generally the correlations are considered much more difficult to accurately estimate in this context. I am not sure I understand the question. Isn't $E(r)$ is an unbiased estimator of $r$? You may want to look at the Black-Litterman model. It is an optimization model developed to address the practical problems in applying portfolio optimization. ## Answer by jeonw (score 1) https://quant.stackexchange.com/a/54649 A simple way that Ray Dalio suggested is using Risk-Parity portfolio. Even though Modern Portfolio Theory with mean-variance optimization gives great framework for mathematically designing a tradeoff between alpha, risk and costs, some of the drawback of this framework is that one needs to come up with 1) correct alpha and 2) correct estimation of covariance matrix. Assuming our estimate of covariance matrix is relatively stable, then the only problem that we'd need to deal with is Alpha. You can have a look at risk-parity optimization where you can set 1) risk contribution of each asset is equal (equal risk contribution optimization) 2) marginal risk contribution of each asset is equal. In these framework, it removes alpha from your utility function and let the optimizer to choose the optimal portfolio only using historical available data (historical realized volatility). This may not be optimal in realizing the best sharpe, however, it could be a good alternative to the approach that you are taking if you are unsure of your alpha estimation.
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