A Time Series Framework for Pricing Guaranteed Lifelong Withdrawal Benefit

Abstract

In this work, the pricing problem of a variable annuity (VA) contract embedded with a guaranteed lifelong withdrawal benefit (GLWB) rider has been considered. VAs are annuities whose value is linked with a sub-account fund consisting of bonds and equities. The GLWB rider provides a series of regular payments to the policyholder during the policy duration when he is alive irrespective of the portfolio performance. Also, the remaining fund value is given to his nominee, at the time of death of the policyholder. The appropriate modelling of fund plays a crucial role in the pricing of VA products. In the literature, several authors model the fund value in a VA contract using a geometric Brownian motion (GBM) model with a constant variance. However, in real life, the financial assets returns are not Normal distributed. The returns have non-zero skewness, high kurtosis, and leverage effect. This paper proposes a discrete-time model for annuity pricing using generalized autoregressive conditional heteroscedastic (GARCH) models, which overcome the limitations of the GBM model. The proposed model is analyzed with numerical illustration along with sensitivity analysis.

Appendices

Appendix 1: Statistical Analysis of Partitioned Datasets

Tables 6, 7 and 8 displays the value of some statistics such as mean, variance and kurtosis for the partitioned datasets. These tables also show the results of some important tests for all the partitions of the three datasets. The high kurtosis values of the partitioned series shown in the tables signifies that the distribution of all partitioned datasets is leptokurtic. Further, from these tables, it is evident that the null hypothesis of ADF Test is rejected at $99\%$ level of significance for all the partitions of the three datasets. The normality hypothesis got rejected by the results of the JB test. The ARCH test p values are less than 0.01 for all considered lags rejecting the null hypothesis of “no ARCH effect” at $99\%$ level of significance for all the partitions of the three datasets. Hence, ARCH effect is present in the partitioned data series.

Table 6 Statistical analysis of partitioned datasets of S&P 500 index

Table 7 Statistical analysis of partitioned data of Nikkei 225 index

Table 8 Statistical analysis of partitioned data of MSCI world index

Appendix 2: Asymmetric GARCH Model Selection

To obtain the best-fitted model, we compared the AIC, BIC and Log-Likelihood values. The difference between AIC and BIC is that the later penalizes free parameters more strongly compared to the former. The best-fitted model can be chosen by minimizing AIC, BIC, and by maximizing the log-likelihood function values. If two or more models have the same AIC, BIC, and log-likelihood values, then the model having the least number of parameters is chosen. Also, if there exists no single model with highest AIC, BIC, and log-likelihood values, then a comparison between the models is based only on BIC and log-likelihood values.

We have considered the possible values for parameters p and q to be $\{1,2\}$. From Tables 9, 10 and 11, the AIC, BIC, and log-likelihood values shows that the GJR-GARCH, E-GARCH and T-GARCH models provide better fit to all the series compared to a GARCH model. The bold values in Tables 9, 10, and 11 corresponds to the best-fitted models for the partitioned datasets based on the AIC, BIC and log-likelihood values. Note that NC denotes not convergent models in Tables 9, 10, and 11. The final fitted models to all the partitioned series based on above-mentioned criteria are given in Table 4 in Sect. 5. The parameters corresponding to the fitted models are specified in Tables 12, 13 and 14. For the goodness of fit check of the fitted models, we applied the Engle’s ARCH test to residuals, and the Ljung–Box Portmanteau test to the squared residuals. The results are mentioned in Tables 15, 16, and 17. Based on the p values of the Engle’s ARCH test mentioned in Tables 15, 16, and 17, the null hypothesis of having “no ARCH effect” in the residual series is not rejected in most of the cases, except for a few cases with lag 120. The results for the Ljung–Box Portmanteau test to the squared residuals series are mentioned in Tables 15, 16, and 17. The null hypothesis of the test is “the observations are random and independent”. Based on p values for the Ljung–Box Portmanteau test observed from Tables 15, 16, and 17, the null hypothesis is not rejected in most of the cases, showing that the squared residuals are also independent. The resulting p values are significantly greater than the level of significance, $\alpha$ for $\alpha =1\%,~5\%,~10\%$, hence, supporting the efficacy of the estimated models in both the cases. Hence, the residuals do not have any ARCH effect.

Table 9 AIC, BIC and LLK of fitted Assymetric model to S&P 500 index

Table 10 AIC, BIC and LLK of fitted Assymetric model to Nikkei 225 index

Table 11 AIC, BIC and LLK of fitted Assymetric model to MSCI world index

Table 12 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the S&P 500 index

Table 13 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the Nikkei 225 index

Table 14 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the MSCI world index

Table 15 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of S&P 500 index

Table 16 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of Nikkei 225 index

Table 17 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of MSCI world index

Appendix 3: Standarrdized GARCH Model

For comparing the effect of not taking leverage effect in the modelling of return series, we considered modelling with a standardized GARCH model. The parameters (p, q) are decided based on the AIC, BIC, and Log-Likelihood values mentioned in the last column of first block in Tables 9, 10 and 11. The AIC, BIC, and Log-Likelihood values are almost the same for all the fitted GARCH(p, q) models for $p,~q=1,~2$. Therefore, in order to minimize the number of parameters, GARCH(1, 1) model for all the three datasets is chosen. The parameters of the fitted model are mentioned in Table 18.

Table 18 Parameters of fitted GARCH(1, 1) model to the three datasets

Appendix 4: Geometric Brownian Motion (GBM) Model

To show the significance of varying volatility, we considered returns modelling with GBM model also. The parsimonious GBM model considers the return volatility to be a constant over time. The stock price dynamics for a GBM under risk-neutral measure is given by

$$\begin{aligned} S_t=S_0 e^{\left( r-\frac{1}{2}\sigma ^2\right) t+\sigma B_t} \end{aligned}$$

(27)

here r is the continuously compounded risk-free rate, $\sigma ^2$ is the variance and $B_t$ is a Wiener process. The estimated value of $\sigma$ for the S&P 500, Nikkei 225 and MSCI world index is given by 0.010385, 0.012573 and 0.008291 respectively.