Sensitivity Of Dynamic Optimal Portfolio Investments By Simulation

Optimal decisions on portfolio investments in stocks are closely related to problems of statistical modelling and forecasting the rates of return on stocks and risk of investments. Forecasting prices of stocks and rates of return is difficult as a separate problem, some scientists say it is even impossible to give satisfactory solutions. In Taylor(1986), Azoff (1994), Campbell (1997) we can find various techniques of modelling and forecasting the behaviour of capital markets, among them statistical and econometrical models, neural networks techniques, methods having the origin in the theory of chaos and fractal statistics. To obtain the solution of the Markowitz model of the optimal portfolio the following parameters must be estimated: a) expected rates of return on individual securities in the investment period, b) standard deviations of rates of return on individual securities in the investment period, c) the elements of variance-covariance matrix for securities in the investment period. Some significant problems are connected with estimation of the above mentioned parameters. The estimates are obtained on the base of historical data. Each of them is evaluated with estimation error. The rules of construction confidence ranges under some assumptions about the rates of return probability distributions are known. As we observe in practice, there is not too much importance attached to verification of assumptions. Values of estimates are very sensitive on different time samples. Consequently, they influence the solutions of the optimal portfolio model, its optimal composition and risk/return characteristics. The real aim of the optimal portfolio model is to construct in the moment (period) t an investment portfolio; profit from the investment is expected in the moment (period) t+k. Constructed portfolio should be effective (dominant over any other portfolios given value of minimal return) in market conditions which will be present after k periods since the moment of investment. Finding precise forecasts of model parameters for k periods ahead becomes the most important thing. The choice of proper forecasting method depends on examination the parameters changes in time, finding the model or specifying the causal relation. The fact is that regardless of the chosen method with calculated forecasts are always associated prediction errors. It also applies to models which combine risky (f.i. stocks) and not risky components (f.i. bonds with constant value of rate of return) in portfolio. The magnitude of prediction error depends closely on time horizon of investment and on properties of financial market, degree of its efficiency. In our work we try to analyse the sensitivity of optimal portfolio model solution on errors of forecasts of return, risk, elements of covariance matrix. We will use time series forecasting methods (ARIMA, GARCH ), Monte Carlo simulation techniques, among them simulation of returns or time series model residuals from empirical probability distributions, and propose some analytical sensitivity measures. Every proposition is verified on the data coming from Warsaw Stock Exchange.