28 February 2014 - 3 minute read

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

Randomly simulated performance for 25 optimal (red lines) and equally weighted portfolios (green). Average outperformance of the optimal portfolio is shown in blue.

Theory meets reality

In practice, you do not know the exact returns that the assets in your portfolio will deliver. We investigate the effect that uncertainty can have on the optimisation process, by replacing the true values of the expected returns, volatility, and correlations with estimates and recomputing the “optimal” asset weights. The input estimates are sample values computed from limited amounts of data. The less data we use, the noisier the estimates will be.

We find the outperformance of the “optimal” portfolio versus the equally weighted portfolio is significantly reduced. And in the case of using five years or fewer to determine the inputs, it underperforms the equally weighted strategy, as in Figure 2.

Figure 2: Outperformance of “optimal” portfolio relative to the equally weighted portfolio

Each colour represents a given number of years of data to determine the inputs.

The result is a double-edged sword, though, because had you computed the outperformance of your portfolio on the same data you used to derive the asset weights, then you would also have been very optimistic about how it would perform. This is especially true when using short periods for the evaluation – noise in the data would present opportunities that won’t necessarily exist in the future.

The results to the left of zero in Figure 3 are purely hypothetical as they come from a strategy with the benefit of hindsight. To the right of zero are results more akin to those that could be achieved in live trading – and these are always less favourable.

Figure 3: Outperformance of “optimal” portfolio relative to the equally weighted portfolio

Results are shown on the data used to determine assets weights and then on subsequent data.

Conclusion

Financial forecasts come with large amounts of uncertainty. When making investment decisions based on quantitative information, it is imperative that this uncertainty is properly handled. A variety of statistical tools can help, and these are essential in the field of robust portfolio optimisation.


References

[1] Harry Markowitz is a founding father of modern portfolio theory, publishing seminal work during the 1950s, and receiving the Nobel prize for economics in 1990.

This article contains simulated or hypothetical performance results that have certain inherent limitations. Unlike the results shown in an actual performance record, these results do not represent actual trading.  Also, because these trades have not actually been executed, these results may have under- or over-compensated for the impact, if any, of certain market factors, such as lack of liquidity and cannot completely account for the impact of financial risk in actual trading.  There are numerous other factors related to the markets in general or to the implementation of any specific trading program which cannot be fully accounted for in the preparation of hypothetical performance results and all of which can adversely affect actual trading results. Simulated or hypothetical trading programs in general are also subject to the fact that they are designed with the benefit of hindsight. No representation is being made that any investment will or is likely to achieve profits or losses similar to those being shown.

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