## Deterministic Or Stochastic

Traditional actuarial techniques assume a deterministic, usually constant path for returns on assets. There has been some effort to adapt this technique for equity-linked liabilities; for example, the Office of the Superintendent of Financial Institutions (OSFI) in Canada mandated a deterministic test for the GMMB under segregated fund contracts. (This mandate has since been superseded by the recommendations of the Task Force on Segregated Funds (SFTF) in 2000.) However, there are problems with this approach:

1. It is likely that any single path used to model the sort of extreme behavior relevant to the GMMB will lack credibility. The Canadian OSFI scenario for a diversified equity mutual fund involved an immediate fall in asset values of 60 percent followed by returns of 5.75 percent per year for 10 years. The worst (monthly) return of this century in the S&P total return index was around - 35 percent. Insurers are, not surprisingly, rather sceptical about the need to reserve against such an unlikely outcome.

2. It is difficult to interpret the results; what does it mean to hold enough capital to satisfy that particular path? It will not be enough to pay the guarantee with certainty (unless the full discounted maximum guarantee amount is held in risk-free bonds). How extreme must circumstances be before the required deterministic amount is not enough?

3. A single path may not capture the risk appropriately for all contracts, particularly if the guarantee may be ratcheted upward from time to time. The one-time drop and steady rise may be less damaging than a sharp rise followed by a period of poor returns, for contracts with guarantees that depend on the stock index path rather than just the final value. The guaranteed minimum accumulation benefit (GMAB) is an example of this type of path-dependent benefit.

Deterministic testing is easy but does not provide the essential qualitative or quantitative information. A true understanding of the nature and sources of risk under equity-linked contracts requires a stochastic analysis of the liabilities.

A stochastic analysis of the guarantee liabilities requires a credible long-term model of the underlying stock return process. Actuaries have no general agreement on the form of such a model. Financial engineers traditionally used the lognormal model, although nowadays a wide variety of models are applied to the financial economics theory. 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. The model has the advantage of tractability, but it does not provide a satisfactory fit to the data. In particular, the model fails to capture extreme market movements, such as the October 1987 crash. There are also autocorrelations in the data that make a difference over the longer term but are not incorporated in the lognormal model, under which returns in different (nonoverlapping) time intervals are independent. The difference between the lognormal distribution and the true, fatter-tailed underlying distribution may not have very severe consequences for short-term contracts, but for longer terms the financial implications can be very substantial. Nevertheless, many insurers in the Canadian segregated fund market use the lognormal model to assess their liabilities. The report of the Canadian Institute of Actuaries Task Force on Segregated Funds (SFTF (2000)) gives specific guidance on the use of the lognormal model, on the grounds that this has been a very popular choice in the industry.

A model of stock and bond returns for long-term applications was developed by Wilkie (1986, 1995) in relation to the U.K. market, and subsequently fitted to data from other markets, including both the United States and Canada. The model is described in more detail below. It has been applied to segregated fund liabilities by a number of Canadian companies. A problem with the direct application of the Wilkie model is that it is designed and fitted as an annual model. For some contracts, the monthly nature of the cash flows means that an annual model may be an unsatisfactory approximation. This is important where there are reset opportunities for the policyholder to increase the guarantee mid-policy year. Annual intervals are also too infrequent to use for the exploration of dynamic-hedging strategies for insurers who wish to reduce the risk by holding a replicating portfolio for the embedded option. An early version of the Wilkie model was used in the 1980 Maturity Guarantees Working Party (MGWP) report, which adopted the actuarial approach to maturity guarantee provision.

Both of these models, along with a number of others from the econometric literature, are described in more detail in this chapter. First though, we will look at the features of the data.

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