The main purpose of the “Optimal Portfolio Project” was twofold. Namely, the project focused on determining the optimal mix of classes of business at Lloyd’s level and determining the optimal mix of classes of business at Syndicate level.

The approach

Mean – Risk models are tools for modelling choice under risk and widely used in the practice of portfolio construction. In these models, distributions are described and compared using 2 scalars: the mean and some suitable risk measure. In the mean – risk approach a random variable is preferred if it has greater expected value and less risk.

The project used variance as a risk measure and proposed two mean – variance models in order to determine the optimal mix of classes of business. An optimal mix portfolio is defined as a portfolio that achieves:

• minimum level of risk (variance of underwriting profit) at a given level of underwriting profit (mean); or

• maximum level of underwriting profit (mean) at a given level of risk (variance of underwriting profit)

and by plotting the optimal portfolios, an Efficient Frontier is constructed. Figure 1 is an example of an Efficient Frontier. All the portfolios on the frontier are optimal. The portfolio at (μ1, σ1) is not optimal; because at the same level of return (μ1), a lower level of risk (σ2) can be achieved and at the same level of risk (σ1), a higher level of return (μ2) can be achieved.

The two models proposed differ in terms of the approach followed. In the first model Lloyd’s and the syndicates were considered individually, whereas in the second model Lloyd’s and the syndicates were considered together. We would expect to achieve a lower risk in the first model compared to the second model, since we deal with Lloyd’s and the Syndicates on a one to one basis. The data used was for the period 1993 – 2007. Premiums and paid claims were used to calculate the ultimate loss ratios and the underwriting profits for each year of account. The models were implemented in AMPL (A Mathematical Programming Language) and solved by using the FortMP solver. The outcome of the models enabled Lloyd’s to gain an appreciable understanding of the risk and the return relationship as a whole, as well as the individual syndicates.