HELSINKI UNIVERSITY OF TECHNOLOGY Department of Industrial Engineering and Management Taras Beletski FORECASTING THE TERM STRUCTURE OF INTEREST RATES WITH STOCHASTIC MODELS Master’s thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Technology

The objective of this thesis is to provide a life insurance company with a stochastic model for the term structure of interest rates. The term structure of interest rates affects the structure of an optimal portfolio and the calculation of risk measures. Therefore, the term structure of interest rates for the DEM/Euro market is analyzed. On the basis of the theory of affine term structure models, the Kalman filter is applied for estimating multi-factor models including both Vasicek (1977) models and square-root models of Cox, Ingersoll and Ross (1985). This thesis reviews mathematical preliminaries including the theory of continuous time stochastic processes and the Kalman filter. Next the concept of affine term structure models in continuous time is briefly introduced based on the work of Duffie and Kan (1996). On this basis specific multi-factor models are proposed including the state space representation for estimating the model parameters with the Kalman filter. Two key results are presented in this thesis. First, the two-factor and three-factor square-root models produce estimates that are on the boundary of the restricted parameter set. Thus, the multi-factor square-root models of Chen and Scott (1992) should be used with care. Second, the estimation procedure for the three-factor models is numerically demanding. The result of the estimation is highly dependent on finding good starting values that initiate the maximization of the log-likelihood function and further the estimation results are highly sensitive to minor modifications of the data set. This difficulty increases with the number of factors and hence a restriction to two-factor models seems reasonable. We also discuss the mean, variance and correlation structure of the yields that is implied by the estimated parameters and that is further compared to the empirical structure of the yields. This analysis is carried out for each single-factor model and two-factor model. The main drawback of the single-factor models is that they imply a perfect correlation of all yields. Thus, they cannot reflect a possible diversification when holding a portfolio with bonds of different maturities. For the two-factor models, the striking feature is that the models with independent factors cannot describe the variance and correlation structure that is observed empirically in the market data. The two-factor generalized Vasicek model of Babbs and Nowman (1998) allows for two correlated Gaussian factors and produces mean, variance and correlation almost the same as it is observed empirically. For the DEM/Euro term structure, the results indicate …