Bayesian Statistics
In this chapter, we describe modern Bayesian parameter estimation and show how the method is applied to the RSLN model for stock returns. The major advantage of this method is that it gives us a scientific but straightforward method for quantifying the effects of parameter uncertainty on our projections. Unlike the maximum likelihood method, the information on parameter uncertainty does not require asymptotic arguments. Although we give a brief example of how to include allowance for parameter...
BlackScholes Formula for the GMMB
The GMMB is a straightforward put option on the segregated fund. Assume a fund value at the valuation date t 0 of Fo. Let G denote the guarantee, and assume first that the guarantee is fixed. The insurer's liability under the GMMB at maturity in, say, T years is G - FT . This is identical to the put option, with strike price G and underlying asset Ft. Under standard Canadian contract terms for policies of this type, G is typically 75 percent or 100 percent of the initial single premium for the...
BlackScholes Formula for the GMAB
The GMAB is a more complicated option with curious put- and call-type features. Ignoring exits for the moment, we will derive the hedge portfolio at time t 0 for a GMAB with renewals at t1 gt 0 and at t2 gt t1, maturing at t3 gt t2. The starting guarantee is G, and the starting separate fund is F0. At t1, if the fund value is more than G, then the guarantee is reset to the fund level. On the other hand, if the guarantee is greater than the fund at t1, then the insurer pays the difference into...
Using the RSLN2 Model
Although the regime-switching model has more parameters than the ARCH and GARCH models, the structure is very simple and analytic results are readily available. In this section, we will derive the distribution function for the accumulated proceeds at some time n of a unit investment at time t 0. Let Sn denote the proceeds, so that The key technique is to condition on the time spent in each regime. Let R denote the number of months spent in regime 1, so that n - R is the number of months spent...
The Wilkie Model Structure 1
The Wilkie model Wilkie 1986, 1995 was developed over a number of years, with an early version applied to GMMBs in the MGWP Report 1980 and the full version first applied to insurance company solvency by the Faculty of Actuaries Solvency Working Party 1986 . The Wilkie model differs in several fundamental ways from the models covered so far It is a multivariate model, meaning that several related economic series are projected together. This is very useful for applications that require...
The Actuarial Approach 1
In the mid 1970s the ground-breaking work of Black, Scholes, and Mer-ton was relatively unknown in actuarial circles. In the United Kingdom, however, maturity guarantees of 100 percent of premium were a common feature of the unit-linked contracts, which were then proving very popular with consumers. The prolonged low stock market of 1973 to 1974 had awakened the actuaries to the possibility that this benefit, which had been treated as a relatively unimportant policy tweak with very little value...
Quantile Matching
A p-quantile of a distribution with distribution function F y is the value Zp such that We can determine parameters for a model by matching the model and empirical quantiles. For example, to fit the lognormal distribution we need any two quantiles of the empirical distribution. Say we decide to use the 10th and 25th percentiles of the empirical and lognormal distributions. The 10th percentile of the log-return for the TSE 300 monthly data from 1956 to 2001 is - 0.04682 and the 25th percentile...
The Wilkie Model Structure
The Wilkie model Wilkie 1986, 1995 was developed over a number of years, with an early version applied to GMMBs in the MGWP Report 1980 and the full version first applied to insurance company solvency by the Faculty of Actuaries Solvency Working Party 1986 . The Wilkie model differs in several fundamental ways from the models covered so far It is a multivariate model, meaning that several related economic series are projected together. This is very useful for applications that require...
Guaranteed Minimum Maturity Benefit
In this section, we show how to generate the distribution of the present value of the guarantee liability for a simple GMMB policy held by a life aged x with remaining duration n years. We assume a monthly discrete time model for equity returns and management charges. Withdrawals and deaths are assumed to occur at month ends. As discussed, exits are treated deterministically, so the only random process simulated is the equity price process. Clearly other assumptions and approaches are possible...
The Stable Distribution Family
Stable distributions appear in some econometrics literature, for example, McCulloch 1996 . Panneton 1999 and Finkelstein 1995 both used stable distributions for valuing maturity guarantees. One reason for their popularity is that stable distributions can be very fat-tailed, and are also easy to combine, as the sum of stable distributions is always another stable distribution. Stable distributions are related to Levy processes if Yt t gt o is a Levy process, then at any fixed time Yt has a...
The Likelihood Ratio Test
The likelihood ratio test see, for example, Klugman, Panjer, and Willmot 1998 compares embedded models, where a model with k1 parameters is a special case of a more complex model with k2 gt k1 parameters. Let l1 be the log-likelihood of the simpler model, and l2 be the log-likelihood of the more complex model. The test statistic is 2 l2 - l1 . The null hypothesis is H0 No significant improvement in Model 2 Under the null hypothesis, the test statistic has a distribution, with degrees of freedom...
Maximum Likelihood Estimation of ARCH and GARCH Models
For the ARCH 1 and GARCH 1,1 models, we adopt a similar approach to that used for the AR 1 estimation. Conditional on the previous value or values of the series, each value is normally distributed with fixed volatility, leaving only the first term of the series, for the probability density for Y1, to be determined. That is, for the ARCH 1 model where at2 ao a1 Yt-1 - 2 3.29 Yt Yt-1 N L,ao a1 Yt-1 - 2 t 2,3, ,n 3.30 1 All the likelihoods in this chapter were maximized using solver in Excel. As...
Regimeswitching Lognormal Model Rsln 1
Regime-switching models assume that a discrete process switches between, say, K regimes randomly. Each regime is characterized by a different parameter set. The process describing which regime the price process is in at any time is assumed here to be Markov that is, the probability of changing regime depends only on the current regime, not on the history of the process. One of the simplest regime-switching models is the regime-switching LN model RSLN , where the process switches randomly at...
Mcmc For The Rsln Model
The candidate variance is chosen to give an appropriate probability of acceptance. The acceptance probabilities for and depend on the distributions used for the other parameters using those described below, we have acceptance probabilities of around 40 percent for both variables. It is conventional to work with the inverse variance, t a-2, known as the precision. The prior distribution for t1 is the gamma distribution with prior mean 865 and variance 8492 the prior distribution for t2 is gamma...
The Lognormal Model Xrs
For the lognormal model, with Yj factor is S12 exp Y1 Y2 Y12 3 logS12 X So, the one-year accumulation factor has a lognormal distribution with parameters 12 and a 12 a. It is possible to use any two of the table values to solve for the two unknown parameters and a , but this tends to give values that lie outside the acceptable range for the mean. So the recommended method from Appendix A of SFTF 2000 is to keep the mean constant and equal to the empirical mean of 1.116122 the data set is TSE...
Maximum Likelihood Estimation for the RSLN2 Model
The RSLN-2 model is the two-regime LN model, introduced in the section on the RSLN model in Chapter 2. The log-returns Yt are assumed to depend on an underlying two-state Markov process, where the state in the interval t to t 1 is denoted by pt 1,2, and within each regime the log-returns are normally distributed, with parameters specific to the regime. The six parameters of the RSLN-2 distribution are the values of j and o for either regime, denoted 1, o1, 2, o2, and the two transition...
Some Comments on the Wilkie Model
The Wilkie model has been subject to a unique level of scrutiny. Many companies employ their own models, but few issue sufficient detail for independent validation and testing. The most vigorous criticism of the Wilkie model has come from Huber 1997 . Huber's work is concerned with Evidence of a permanent change in the nature of economic time series in Western nations around the second world war is not allowed for. This criticism applies to all stationary time-series models of investment, but...
Using the RSLN2 Model 1
Although the regime-switching model has more parameters than the ARCH and GARCH models, the structure is very simple and analytic results are readily available. In this section, we will derive the distribution function for the accumulated proceeds at some time n of a unit investment at time t 0. Let Sn denote the proceeds, so that The key technique is to condition on the time spent in each regime. Let R denote the number of months spent in regime 1, so that n - R is the number of months spent...
The Actuarial Approach
In the mid 1970s the ground-breaking work of Black, Scholes, and Mer-ton was relatively unknown in actuarial circles. In the United Kingdom, however, maturity guarantees of 100 percent of premium were a common feature of the unit-linked contracts, which were then proving very popular with consumers. The prolonged low stock market of 1973 to 1974 had awakened the actuaries to the possibility that this benefit, which had been treated as a relatively unimportant policy tweak with very little value...
Probability Function for Total Sojourn in Regime 1
Let Rn be the total number of months spent in regime 1 for a process S 0, then Rn 0,1, ,n . We want to derive the probability function Pr Rn r pn r . Let Rn t be the total sojourn in regime 1 in the interval t, n , and consider for r 0,1, ,n - t and t 1, ,n -1. Clearly PrR t r pt-1 0 for r gt n - t or r lt 0. For example, Pr Rn n - 1 0 pt-1 1 is the probability that the last time unit is not spent in regime 1, given that the process is in regime 1 in the previous period, that is, for t n - 2,n...
Probability Functions for Sn
Using the probability function for Rn, the distribution of the total return index at time n can be calculated analytically. Let Sn represent the total return index at n, assume S0 1, then Sn Rn LN Rn , a Rn where X Rn Rn n - Rn M2 2.27 and a R jR a2 n - R of 2.28 Then, if pn r is the probability function for Rn Fs x Pr Sn lt x XPr Sn x Rn r pn r 2.29 where is the standard normal probability distribution function. Similarly, the probability density function for Sn is fn x XX 4r x-p r Pn r 2.31...
Introduction
This book is designed for all practitioners working in equity-linked insurance, whether in product design, marketing, pricing and valuation, or risk management. It is written with actuaries in mind, but it should also be interesting to other investment professionals. The material in this book forms the basis of a one-semester graduate course for students of actuarial science, insurance, and finance. The aim is to provide a comprehensive and self-contained introduction to modeling and risk...