Modeling Long-Term Stock Returns


It has been stated firmly in the previous chapter that this book will use stochastic methods to analyze and manage risks from investment guarantees. To model the investment guarantee risks, we need to model the underlying equity process upon which the guarantee depends. There are many stochastic models in common use for equity returns. The objective of this chapter is to introduce some of these and discuss their different characteristics. This should assist in the choice of an appropriate model for a given contract. First, we discuss briefly the case for stochastic models, and some of the interesting features of stock return data.We also demonstrate how often the guaranteed minimum maturity benefit (GMMB) under a 10 year contract would have ended up greater than the fund using the historical returns. The rest of this chapter introduces the various models. These include the lognormal model, the autoregressive model, the ARCH-type models, the regime-switching lognormal model, the empirical model (where returns are drawn from historic experience), and the Wilkie model. Where it is sufficiently straightforward, we have derived probability functions for the models, but in many cases this is not possible.


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 (This mandate has since been 16 superseded by the recommendations of the Task Force on Segregated Funds (SFTF) in 2000.) However, there are problems with this approach: It is likely that any single path used to model the sort of extreme behavior relevant to theGMMBwill 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 rather sceptical about the need to reserve against such an unlikely
outcome. 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? 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.


Some models are derived from economic theory. For example, the efficient market hypothesis of economics states that if markets are efficient, then all information is equally available to all investors, and it should be impossible to make systematic profits relative to other investors. This is different from
the no-arbitrage assumption, which states that it should be impossible to make risk-free profits. The efficient market hypothesis is consistent with the
theory that prices follow a random walk, which is consistent with assuming
returns on stocks are lognormally distributed. The hypothesis is inconsistent with any process involving, for example, autoregression (a tendency for returns to move toward the mean). In an autoregressive market, it should be possible to make systematic profits by following a countercyclical investment strategy—that is, invest more when recent returns have been poor and disinvest when returns have been high, since the model assumes that returns will eventually move back toward the mean. The statistical approach to fitting time series data does not consider exogenous theories, but instead finds the model that “best fits” the data, in some statistical sense. In practice, we tend to use an implicit mixture of the economic and statistical approaches. Theories that are contradicted by the historic data are not necessarily adhered to, rather practitioners prefer models that make sense in terms of their market experience and intuition, and that are also tractable to work with.

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