Uncertainty is inevitable during portfolio optimisation, and ignoring it can lead to a large gulf between the realised and expected performance of a strategy.

## “Optimal” portfolios

The mathematics of portfolio optimisation is well documented and deceptively simple. If we have a number of assets in a portfolio, then in order to maximise return for a given level of risk, we should weight the assets according to the Markowitz__ __solution where is the covariance matrix and is the vector of expected return for each asset [1].

We demonstrate this with a toy example, in which we have four assets with expected annual returns of 2, 4, 6, and 8%, respectively, and an expected annualised volatility of 10% and correlation to each other of 50%. With this information, we compute the optimal portfolio, and see how it performs.

The optimal portfolio is expected to achieve annualised returns of 8.9%, when geared to 10%. This is compared to an inferior portfolio that simply equally weights the assets and is expected to achieve a return of 6.3%. There will be some variation in the results – especially over shorter time frames, when the equally weighted portfolio could get lucky. But, overall, we expect an average outperformance of 2.6% a year, as seen in Figure 1.

## Figure 1: Randomly simulated performance for 25 "optimal" and equally weighted portfolios