I would like to thank Taylor and Francis, publishers of the Scandinavian Actuarial Journal, for permission to reproduce material. I learned a lot from my fellow members of the Canadian Institute of Actuaries' great Task Force on Segregated Funds.
APPENDIX A
APPENDIX C
In the bulk of the book, the contracts used to illustrate the methods are single premium, separate account products. In Chapter 3 we discuss parameter estimation for some of the models using maximum likelihood estimation (MLE).
INTRODUCTION
The guaranteed minimum maturity benefit (GMMB) guarantees the policyholder a specific monetary amount at the maturity of the contract. Guaranteed Minimum Death Benefit (GMDB) guarantees the policyholder a specified cash sum upon death during the policy term.
EQUITY-LINKED INSURANCE AND OPTIONS
Let denote the value to of a European call option on a unit of stock and the value of a European put option on a unit of the same stock. The payout at maturity of the portfolio of the pure discount bond plus call option will be.
PROVISION FOR EQUITY-LINKED LIABILITIES
Most of the academic literature related to equity-linked insurance assumes a dynamic hedge management strategy. The present value of the quantile is held in risk-free bonds so that the office can be 99 percent sure that the obligation will be met.
PRICING AND CAPITAL REQUIREMENTS
DETERMINISTIC OR STOCHASTIC?
A stochastic analysis of the warranty obligations requires a credible long-term model of the underlying inventory return process. The lognormal model is the discrete-time version of the geometric Brownian motion of stock prices, which is an assumption underlying the Black-Scholes theory.
ECONOMICAL THEORY OR STATISTICAL METHOD?
By knowing the price-volatility relationship in the market, the volatility implied by market prices can be calculated from the quoted prices. Figure 2.5 shows the net return of a 10-year single premium investment in the S&P 500 index.
THE LOGNORMAL MODEL
The maximum likelihood estimates of the parameters and are the mean and variance of the log returns (i.e., the mean and variance of log. From Table 2.1, the one-month autocorrelation is small but potentially significant in the tail of the distribution of accumulation factors.
AUTOREGRESSIVE MODELS
ARCH(1)
The GARCH model is more flexible and has been found to fit many econometric applications significantly better than the ARCH model. The parameter estimation method does not automatically match the means, and apparently the estimated ARCH and GARCH models have higher means and variances than LN.
REGIME-SWITCHING LOGNORMAL MODEL (RSLN)
1] is the probability that the last unit of time is not spent in regime 1, given that the. Using the probability function for, the distribution of the total return index at a given point in time can be calculated analytically.
THE EMPIRICAL MODEL
THE STABLE DISTRIBUTION FAMILY
GENERAL STOCHASTIC VOLATILITY MODELS
The term denotes an AR parameter; is a (conditional) variance parameter; and is a weighting applied to the force of inflation within the other processes. For example, the stock dividend yield process contains a term ( ), which indicates how the current force of inflation ( ( )) affects the current logarithm of the dividend yield (see Equation 2.36).
VECTOR AUTOREGRESSION
The advantage of this model is that much of the correlation between the series is explained by correlations with inflation. By removing inflation from the formula, many of the covariance terms in v can be set to zero.
冱 ⫺ ⫺
This is found by maximizing , which is just the joint probability function of the data expressed as a function of the parameters. Since the observations are independent, the likelihood function, which is the joint probability density function (pdf) for the data, is simply the product of the individual density functions. Maximum likelihood estimators can be found in terms of the sample and are random variables.
PROPERTIES OF MAXIMUM LIKELIHOOD ESTIMATORS
Given the parameters and (the mean and variance of the associated normal distribution), the mean of the lognormal distribution is First, we briefly discuss the case of stochastic models and some interesting features of stock return data. In fact, the maximum likelihood estimate (MLE) is where the variance of the log returns is.
This version of the model allows for volatility pooling and autocorrelations in the data. This is the balance of added complexity and improving the fit of the model to the data.
SOME LIMITATIONS OF MAXIMUM LIKELIHOOD ESTIMATION
So the MLE for the mean of the log-returns is the mean of the log data. Maximum likelihood estimators for the United States and Canadian total return indices are given in Table 3.2. For the ARCH(1) and GARCH(1,1) models, we adopt a similar approach to that used for AR(1) estimation.
LIKELIHOOD-BASED MODEL SELECTION60
AR-ARCH—the ARCH model with an additional autoregressive component for the mean, described in the section on ARCH in Chapter 2. STABLE—the stable distribution described in the section on the stable family of distributions in Chapter 2. AR-GARCH—the GARCH model with an additional autoregressive component for the mean, described in the section on GARCH(1,1) in chapter 2.
MOMENT MATCHING
For a satisfactory overall fit, it is better to use more of the distribution than the first two moments. In the next chapter we will see how to adjust the estimates if we are interested in other parts of the distribution. In this chapter we first look at the methodology of the Canadian Institute of Actuaries (CIA) report (SFTF 2000) and consider some of the empirical evidence.
QUANTILE MATCHING66
THE CANADIAN CALIBRATION TABLE
QUANTILES FOR ACCUMULATION FACTORS
THE EMPIRICAL EVIDENCE68
However, we can use the bootstrap method of statistics to infer some information about the tails of the distribution. So if we have 528 monthly observations of the log return (representing the monthly data from 1956 to 1999), we can, with replacement, sample 120 values to get a new “observation” of the 10-year accumulation factor. We repeat this a number of times to construct a new 'sample' of hypothetical observations of the 10-year accumulation factor.
THE LOGNORMAL MODEL70
We can then use any of the nine entries in Table 4.1 as the other equation. For the lognormal model, the strictest test is actually the 2.5 percentile of the one-year accumulation factor. This means that, given the parameters calculated using the 2.5 percentile for , the probability of the five-year accumulation factor falling below 0.75 is just over 3.6 percent, which is greater than the required 2.5 percent, indicating that the test was successful.
ANALYTIC CALIBRATION OF OTHER MODELS72
The are independent, identically distributed (0 1), which gives the result in equation 4.18, so it is possible to calculate probabilities analytically for the accumulation factors. The distribution function of the accumulation factor for the RSLN-2 model is derived in equation 2.30 in the section on RSLN in Chapter 2. Using this equation, it is straightforward to calculate the probabilities for the various maximum quantile points in Table 4.1.
CALIBRATION BY SIMULATION
The maximum likelihood estimates of the conditional heteroscedastic generalized autoregressive (GARCH) model are given in Table 3.4 in Chapter 3. Increasing any of the other parameters will increase the standard deviation for the process and, therefore, increase the part of the distribution in the left tail. Here, fitness is measured in terms of overall fitness at each duration for accumulation factors.
BAYESIAN STATISTICS
If we have good information, we can choose a prior distribution with a small variance, indicating little uncertainty about the parameter. The mean of the prior distribution represents the best estimate of the parameter before observing the data. We can relate all of this in terms of the probability density functions involved, considering the sample and the parameter as random variables.
MARKOV CHAIN MONTE CARLO AN INTRODUCTION
The hanging sample from the posterior distribution, which enables estimation of the joint moments of the posterior distribution. One of the reasons the MCMC method is so efficient is that we can update the parameter vector one parameter at a time. The problem is then reduced to simulating from the posterior distributions for each of the parameters, assuming known values for all the remaining parameters.
THE METROPOLIS-HASTINGS ALGORITHM (MHA)
That is, the candidate value for the parameter (1) value is a random number generated from the ( ) distribution for some , chosen to ensure that the acceptance probability is in an efficient range. Again, there are advantages to centering the candidate distribution on the prior value of the series. The value from the candidate distribution is accepted as the new value for with probability.
MCMC FOR THE RSLN MODEL
The results given here are from 10,000 simulations of the parameters, separately for the TSE and S&P data. It also shows that higher values for the regime 1 to regime 2 transition probability appear. In Figure 5.4 we show the sample paths for the MCMC estimation for the six parameters of the TSE data.
SIMULATING THE PREDICTIVE DISTRIBUTION90
We will illustrate the ideas of the last section using simulated values for the 10-year accumulation factor using the TSE parameters. For each accumulation factor simulation, a new vector was sampled from the set of parameters generated using MCMC. Managing the risk of equity-linked collateral requires a full understanding of the nature of the liability.
THE STOCHASTIC PROCESSES96
SIMULATING THE STOCK RETURN PROCESS
NOTATION98
We assume that the management cost or management expense ratio (MER) is deducted from the fund at the beginning of each month; also for the guaranteed accumulation benefit, the fund can be increased at the end of some months. Sometimes it is convenient to distinguish between the fund immediately before these transactions at the end of the month and the fund immediately after. Let enter the month-end fund before these transactions, and let enter the month-end fund after the transactions.
GUARANTEED MINIMUM MATURITY BENEFIT100
冱 ⫺
GUARANTEED MINIMUM DEATH BENEFIT
GUARANTEED MINIMUM ACCUMULATION BENEFIT102
The warranty in effect at the beginning of the projection period is from the last reset before the projection. In Table 6.3, we show the fund at the beginning of the month, before management fees are deducted, ; the income from the risk premium,. Therefore, at the beginning of the 145th month the fund has increased to the guarantee value of $158.99.
STOCHASTIC SIMULATION OF LIABILITY CASH FLOWS108
The simulated density function for the 10,000 simulations of the GMAB NPV of the commitment is presented in the first plot of Figure 6.2; in the diagram on the right we show a smoothed version. As the guarantee moves to the fund level, both the frequency and the severity of the death liability increase. The risk to the two-year maturity advantage is essentially a catastrophic stock return in the early part of the projection.
THE VOLUNTARY RESET112
This table shows that the effect of the reset option is not very large, although the difference in the right tail is significant enough to be considered. In addition, the reset will limit the risk management of the contract for two main reasons. For a more technical discussion of the financial engineering approach to risk management for the reset option, see Windcliff et al.
THE GUARANTEE LIABILITY AS A DERIVATIVE SECURITY
REPLICATION AND NO-ARBITRAGE PRICING116
If the insurer buys one unit of the risky asset now, it will have enough to accurately meet the due obligation in one month. The stock worth 100 to 0 follows one of the paths in the upper diagram of Figure 7.2. At time 1 we know whether we are in the up or down state.
THE BLACK-SCHOLES-MERTON ASSUMPTIONS
THE BLACK-SCHOLES-MERTON RESULTS124
We derive the risk-neutral distribution using the same requirements as used in the binomial model, described in the Replication and No-Arbitrage Pricing section. The risk-neutral distribution must correspond to the target, and the expected annual return below the risk-neutral distribution must be at the risk-free rate (continuously compounded). For a given risk-free interest rate per time unit is the risk-neutral distribution generated by the GBM, another GBM, with drift parameter 2 and with variance parameter.
THE EUROPEAN PUT OPTION126
However, it is important to remember that these are functions of the variables , , time to expiration, , and .
THE EUROPEAN CALL OPTION
PUT-CALL PARITY128
