Optimisation techniques have found extensive use in the world of finance. For example, they have been used to select the cheapest portfolio of bonds to match a stream of liabilities, or to determine the best portfolio rebalancing strategy under transaction costs to maximise final wealth.
But today optimisation in a financial context is almost synonymous with the selection of a set of holdings that best meets an investor’s requirements from a universe of risky assets. Consistent with the definitions given above, optimisation seeks to arrive at the best compromise between the conflicting objectives of return maximisation and risk minimisation.
Dr Harry Markowitz first mapped out the way in 1952. His model used the statistical concepts of variance and covariance of returns as proxies for asset risk and the co-movement of assets. He later commented: Since there are two criteria - expected return and risk - the natural approach for an economics student was to imagine the investor selected a point from the set of Pareto optimal expected return, variance of return combinations, now known as the efficient frontier”. His ideas were taken up by, amongst a host of others, Sharpe, Lintner, Mossin, Rosenberg … the list is long. Their work spawned a large part of the body of knowledge that today forms modern portfolio theory.
Nowadays, optimisation is widely used in search of the efficient frontier. Almost all funds use some form of mean/variance analysis in order to find the best strategic mix of assets. Many managers recognise that although investment experts (or models) are usually not reticent in providing forecasts of returns, they are not good at measuring and managing risks.
Thus optimisers are used in conjunction with forecasts to focus bets and control overall risk. Even the most non-quantitative investment manager is going to be asked what level of risk he or she is taking for the client.
So – even after the portfolio has been chosen – mean/variance analysis is likely to be used to measure risk.
Today sophisticated mean/variance optimiser computer programs are widely available to the investment management community and can be bought ‘off the shelf’. So it is important to be aware of some of their weaknesses. The first thing to bear in mind is that the output is very sensitive to the return forecasts. Thus the “optimal” portfolios turn out to be unstable: if expected returns are altered a little, then the “optimal” portfolio can change dramatically.
An example: the German and French equity markets are highly correlated. An optimisation which has forecast French returns below German returns can lead to a dramatic shift in the portfolio out of France. For this reason some refer to optimisers as “error maximisers”. Studies show that optimisers are orders of magnitude more sensitive to return forecasts than to volatility (risk) estimates.
Another weakness is that different forecasts have different payoff horizons. These problems can be overcome to various extents by judicious use of constraints on the optimiser. A more fundamental problem is that mean/variance optimisers assume that asset returns adhere to symmetric probability distributions. Thus they are not equipped to handle asymmetric payoffs such as those of options.
Humans, however, think of risk as risk of loss in a multi-period setting. Surprises on the upside are welcome, whereas those on the downside are not. To address this issue requires much more complex forms of optimisation.
Despite these weaknesses, portfolio optimisation, used cautiously, is a very useful tool for both analysing and implementing investment strategies.