Smoothness Improvement Of Smith-Wilson Model

The Smith-Wilson（SW）model is a risk-free term structure method specified by the European Insurance and Occupational Pension Administration in the SolvencyⅡsystem mainly used to construct the basic interest rate curve for life insurance and pension actuarial calculation.The SW model has both interpolation and extrapolation and the calculation is simple.And it also can make the long-term interest rate converge to the ultimate forward rate（UFR）.Thus,the method is suitable for insurance companies and pension funds since by the very nature of their business their liabilities stretch out much further into the future than those of the average financial institution.The non-parametric model fit fixed-income asset prices with free error but ignore the noise in the price of bonds.Therefore,the SW model has three drawbacks.Firstly,the influence of price noise will make the interest rate curve of the model lack smoothness.Second,the error-free fitting within the sample will make the model’s out-of-sample prediction ability is poor.Third,when the number of interest-bearing bonds is large,the model will useless because the equation can’t be solved.Even if fewer interest-bearing bonds are used,the cash flow matrix may be rank deficient due to many cash flow dates.This makes the optimization problem ill-posed.In order to solve the above problems,this paper adds the l₂ norm penalty term to the SW model convert to the interpolation model to the ridge regression.And then we design a reasonable calculation for the novel model.In the empirical part,this paper uses the transaction data of Chinese government bonds,British STRIPS bonds,British Phnom Penh bonds,and US Treasury bonds to test the performance of the improved model.The empirical results show that l₂ norm penalty can reduce the model’s sensitivity to noise so improve smoothness and stable of interest rate curve.And it also can increase the model’s out-of-sample prediction ability.The smoothness of the curve of the SW model and the out-of-sample predictability indicators will vary significantly depending on the type of bonds and the bond market,but the improved model can perform a good effect in all bond types and bond markets tested in this paper.At last,the improved model always constructs a reasonable interest rate curve in large bonds or ill-posed.The novel model does not change other properties of the SW model.The long term interest rates of the improved model can still converge to UFR.The risk-free interest rate term structure curve is a basic tool in the field of life insurance and pension actuarial,which has an important influence on the solvency assessment of these two industries.How to construct a basic interest rate curve has become an urgent project to be solved because the next step in the reform of the C-ROSS system is enhanced risk measurement,and China’s social pension insurance is also required to follow the principle of actuarial balance.The research in this article has certain reference significance for the selection of the basic interest rate curve model in the actuarial calculation of China’s life insurance and basic old-age insurance.