AN OPTION PRICING APPROACH TO BONUS POLICY BY A. D. WILKIE

OF THE DISCUSSION Mr J.M. Maud (opening the discussion): Until a few months ago I was almost entirely ignorant about option pricing theory. Having been introduced to the subject, I was astonished that a theory with so many obvious uses in actuarial work is so little known to many of us. In the Appendix to his paper the author has produced a description of options and an explanation of the Black-Scholes formula which should not present too many difficulties to actuarial readers. Nevertheless, some people may have been put off by the fearsome looking differential equations in section A4. If this applies to you, I would just draw your attention to the remarkable result noted by the author in § A4.3. 3. We would intuitively expect that the higher the expected return on an equity portfolio, the lower would be the value of an option to sell the portfolio at a given fixed price at some future date. In fact the value is quite independent of the expected return. This surprising result follows naturally from the hedging strategy which is used to derive the Boundary condition satisfied by the Black-Scholes formula, and which relies on the option writer’s ability to immunize himself against the effect of fluctuating prices, rather than on the need to offset the guarantee against the expectation that an equity portfolio will out-perform a risk-free investment. Turning now to the detail of the paper, the author examines first a single premium with-profit policy and then an annual premium one. In the latter case he outlines, in § 3.2, just one possible strategy for purchasing guarantees out of a series of annual premiums. This is based on a uniform percentage of the maximum present guarantee which could theoretically be bought with each premium. Another possible approach would be by analogy with the proportionate paid-up policy values. In this example of a 20-year policy with a guaranteed pay-out of £l,000, a guarantee of £50 might be purchased in year 1 and £104, which includes £4 bonus, in year 2. This compares with £76.42 and £163.22 using the author’s method. The lower level of guarantee in the earlier years seems to correspond more closely with common practice in the case of endowment assurances and I see no need for a higher guarantee. In §4.3 the analogy between the author’s example and a real life policy breaks down when bonus cannot be maintained. He assumes that existing bonus would be cancelled if a risk-free investment could not otherwise meet the guaranteed sum at maturity. He is looking at a single policy when he does this. In practice, there might be scope for cross-subsidy between generations in these circumstances. It is also probably the case that some of the guarantees used in Table 3.2 are rather greater than a prudent life office would have offered in practice at that time. Also, of course, the office would not have invested solely in equities, so I do not think that we should read too much into these results. In § 5.1, the author reveals to us the assumptions he has used in the earlier sections: he has allowed for tax on income at appropriate rates but no capita1 gains tax. Although this would have been a distraction in the present paper, a suitable CGT assumption would be needed to apply the method in practice. In § 5.7, the author describes the 7% level rate of interest in Table 5.1 as being intermediate between his original 5% assumption in Table 2.3 and the variable rates he then used in Table 5.2. In fact the variable interest rates are equivalent to a level interest rate of 7% a year over the term for a single premium investment. I do not see why the 7% interest rate is described here as an intermediate one. Certainly the results in Table 5.1 do not generally lie between those in the other two tables. The variable interest rates over the period follow a pattern which is rather like the cross-section of an escarpment: climbing up the steep slope in the first nine years and then going gently down the other side. This presumably accounts for the variation of the results in the three tables and demonstrates how volatile the results are as the interest rate assumption is varied. Similarly, the results are highly variable as the bonus rate and policy date assumptions are changed, in sections 6 and 7. I think this demonstrates what we should have known all along: if they are set at a moderate level, guarantees can be financially insignificant but, beyond a certain point, their effect can rapidly become overwhelming. If the paper helps us to assess when that point is reached, it is certainly valuable. An Option Pricing Approach to Bonus Policy 79 I was puzzled by § 9.6. In the examples given, on average the guarantee costs the policyholder a positive amount. Remembering the hedging strategy and the derivation of Black-&holes, why should this be so? Is the option writer getting a free lunch? I came to the conclusion that he is making a profit only if he invests wholly in equities rather than in the hedging portfolio, which will probably put some money in risk-free interest-bearing deposits. In all cases I think the expected equity return assumed by the author exceeds that on risk-free investments. This is an entirely reasonable assumption, but if the two rates of return were brought into line I suspect the profits would disappear. I would have welcomed a mention of this in the paper. I have not commented, so far, on the assumptions which underlie the author’s results. The absence of initial expenses and mortality are just details which are unlikely to alter the pattern of the results. The absence of transaction costs and the ability to deal in large amounts of securities in an infinitesimal time are necessary approximations. In practice one would aim to include a margin to allow for them. The assumptions which concern me most are those which may be good enough for most traded options, whose lifetime is usually a matter of months, but may break down if we are looking at a much longer time scale. For example, I suspect that the nature of the equity market may change over the years so that the standard deviation of the return in successive periods could vary. More seriously the assumption that risk-free investments will continue to be available at a given fixed interest rate looks unrealistic. It could produce substantial distortions if we are looking at a period of up to 20 years. We have only to compare the results assuming 5% and 7% interest rates to imagine how critical this assumption might be. The author is aware of these aspects and in § A6.3 of the Appendix he has noted alternative models which have been developed taking into account the variability of risk-free interest rates and the changing volatility of equity markets. I have not followed up these references so I cannot tell whether they lead to workable methods. I would certainly feel more confident of the results from a model which allowed for the stochastic nature of interest rates. With this proviso I would like to see life office actuaries using and developing the methods which the author has outlined for us. Certainly any development which helps us to assess the cost of guarantees and bonus expectations must be of interest to the profession, particularly at a time when lower inflation and interest rates combined with sharper competition have put more pressure on appointed actuaries to set bonus rates at the highest level they can justify. Professor S. Benjamin: I am a little surprised that the author managed to avoid producing stochastic bundles in order to illustrate his paper since it might have shown in a more diagrammatic way what happens in these stochastic processes. The author uses the Black-Scholes formula for pricing yet practitioners tell me that the market does not necessarily conform to this formula. The Maturity Guarantees Working Party did not find that the random walk model, which underlies the Black-Scholes formula, was a good representation over the longer term; the author had a great part to play in coming to that conclusion. Options, as I understand it, are available in the market for only up to about nine months. Yet the author is using option pricing over 20 years, so perhaps the model is not so important for his purpose. This appears to show how the ideas behind option-pricing can be used to price the values of guaranteed bonuses, as compared with non-guaranteed bonuses, from the point of view of equity between policyholders. I wonder if this degree of sophistication is suitable for this particular purpose since there are some important loose ends left. For example, we do not know the policyholders’ utility function and also, as the author has shown, there are many different investment strategies which the policyholder could have adopted. If the purpose is an in-house exercise in equity then the model may not matter too much, but once we start trying to use it to produce ad hoc numerical rules of thumb, as the author does in § 9.9, then the model and the values of the parameters become much more important. The rule he gives there, in my words, is: do not declare reversionary bonuses if the value of the liabilities on a market basis and excluding future bonuses is more than 90% of the market value of the assets. I think we would all agree with that, but our agreement would be in spite of the calculations via options, rather than because of them. The author could say that his production of a result which is acceptable does not disprove what he is doing. It is very difficult to see a simplifying pattern in the many tables and examples which the author gives. I attempted a simplification myself trying to stay within the underlying model used by the author. There seemed to be some odd consequences of assuming that there is a risk-free rate of 80 An Option Pricing Approach to Bonus Policy interest. Firstly, if it is risk-free then I would say that by definition it must be known. It must be a contradiction in terms to say that