Information and optimisation in investment and risk measurement
The thesis explores applications of optimisation in investment management and risk measurement. In investment management the information issues are largely concerned with generating optimal forecasts. It is difficult to get inputs that have the properties they are supposed to have. Thus optimisation is prone to 'Garbage In, Garbage Out', that leads to substantial biases in portfolio selection, unless forecasts are adjusted suitably for estimation error. We consider three case studies where we investigate the impact of forecast error on portfolio performance and examine ways of adjusting for resulting bias. Treynor and Black (1973) first tried to make the best possible use of the information provided by security analysis based on Markovitz (1952) portfolio selection. They established a relationship between the correlation of forecasts, the number of independent securities available and the Sharpe ratio which can be obtained. Their analysis was based on the assumption that the correlation between the forecasts and outcomes is known precisely. In practice, given the low levels of correlation possible, an investor may believe himself to have a different degree of correlation from what he actually has. Using two different metrics we explore how the portfolio performance depends on both the anticipated and realised correlation when these differ. One measure, the Sharpe ratio, captures the efficiency loss, attributed to the change in reward for risk. The other measure, the Generalised Sharpe Ratio (GSR), introduced by Hodges (1997), quantifies the reduction in the welfare of a particular investor due to adopting an inappropriate risk profile. We show that these two metrics, the Sharpe ratio and GSR, complement each other and in combination provide a fair ranking of existing investment opportunities. Using Bayesian adjustment is a popular way of dealing with estimation error in portfolio selection. In a Bayesian implementation, we study how to use non-sample information to infer optimal scaling of unknown forecasts of asset returns in the presence of uncertainty about the quality of our information, and how the efficient use of information affects portfolio decision. Optimal portfolios, derived under full use of information, differ strikingly from those derived from the sample information only; the latter, unlike the former, are highly affected by estimation error and favour several (up to ten) times larger holdings. The impact of estimation error in a dynamic setting is particularly severe because of the complexity of the setting in which it is necessary to have time varying forecasts. We take Brennan, Schwartz and Lagnado's structure (1997) as a specific illustration of a generic problem and investigate the bias in long-term portfolio selection models that comes from optimisation with (unadjusted) parameters estimated from historical data. Using a Monte Carlo simulation analysis, we quantify the degree of bias in the optimisation approach of Brennan, Schwartz and Lagnado. We find that estimated parameters make an investor believe in investment opportunities five times larger than they actually are. Also a mild real time-variation in opportunities inflates wildly when measured with estimated parameters. In the latter part of the thesis we look at slightly less straightforward optimisation applications in risk measurement, which arise in reporting risk. We ask, what is the most efficient way of complying with the rules? In other words, we investigate how to report the smallest exposure within a rule. For this purpose we develop two optimal efficient algorithms that calculate the minimal amount of the position risk required, to cover a firm's open positions and obligations, as required by respective rules in the FSA (Financial Securities Association) Handbook. Both algorithms lead to interesting generalisations.