Portfolio Optimisation

In this talk we review portfolio optimisation, with a focus on financial applications. Here the problem is to decide the assets (a portfolio) to hold that have desired characteristics. Markowitz mean-variance portfolio optimisation is relatively well known, but has been extended in recent years to encompass cardinality constraints. Less considered in the scientific literature are problems such as: • index tracking – where the objective is to match the return achieved on a benchmark index such as the S&P500 • enhanced indexation-where the objective is to exceed the return achieved on a benchmark index; here we may have a desired specified excess return, or we simply wish to do better than the benchmark • absolute return – where the objective is to achieve a desired return (irrespective of how the market, as represented by the benchmark index, performs) We will outline the mathematical optimisation models that can be adopted for portfolio problems such as these and review the results achieved to date. Markowitz mean-variance portfolio optimisation To proceed with Markowitz mean-variance portfolio optimisation we need some notation, let: N be the number of assets (e.g. stocks) available µ i be the expected (average, mean) return of asset i ρ ij be the correlation between the returns for assets i and j (-1≤ρ ij ≤+1) s i be the standard deviation in return for asset i R be the desired expected return from the portfolio chosen Then the decision variables are: w i the proportion of the total investment associated with (invested in) asset i (0≤w i ≤1) Using the standard Markowitz mean-variance approach we have that the unconstrained portfolio optimisation problem is: