Deflators, used as a valuation technique,

have been the subject of a lot of discussion

in recent months. They can be used to place market-consistent values on uncertain cashflows, in particular the cashflows that result from investment guarantees. They will be invaluable in product pricing, risk management, and the valuation of guarantees for IAS purposes. Given the amount of literature around and the benefits of using deflators, why is it that insurers are not already using these techniques? Many examples have been published of how and why deflators work for simple one-step binomial models, but a lot of actuaries are still either not comfortable with using them or not aware of how they work. This article is aimed at such actuaries, with the objective of demonstrating the use of deflators in valuing stochastic model outputs. It concerns itself only with the use of deflators and not with the specification and calibration of the asset model, nor with the calculation of the deflators.

Answering the question

A question frequently asked is, ‘What is a deflator’? Sometimes the shortest questions are the most difficult to answer! A correct, mathematically rigorous answer to the question is not very helpful, doing nothing to aid the questioner’s understanding. Instead, my reply is to describe a deflator as a stochastic discount factor. The best way to understand what I mean by stochastic discount factors is to see them in action.

I will be demonstrating a valuation example based on a set of five stochastic runs (table 1). They are five randomly generated examples of how £100 of equities could change over a period of five years. It has been assumed that dividends are reinvested and that there is no tax. The associated deflators appear in table 2. Given my definition of a deflator as a stochastic discount factor, it may come as a surprise to see that some of the deflators are greater than one. Close examination of the table will, however, reveal certain underlying features. First, although some individual deflators are greater than one, the average of the deflators in a single column is less than one. Second, deflators are higher when equity prices are lower. The examples given later will provide further insight into these two features.

In practice, an insurer would feed the stochastic investment runs into a model office system to get a set of stochastic cashflow streams. Given a matrix of cashflows Cn,t (where t refers to the timing of the cashflows and n to the particular stochastic run) and a matrix of deflators Dn,t the value of the time t cashflow is the average (over all the stochastic runs n=1,,1,000, say) of SDn,tCn,t. Summing over all the times t would give the value of the insurance contract.

Sensible results

Before looking at the particular example, it is instructive to demonstrate how £100 of equities and £100 of bonds can both be valued at £100 by discounting back their values from time 1. This is (almost) shown in tables 3 and 4, but with only five stochastic runs, some sampling error is inevitable. It is encouraging to see that deflators can at least value securities sensibly. Notice too how the same set of deflators are used in both cases: the same set of deflators can be used to value all cashflows at time 1 that are dependent only on the time 1 equity index. This is in stark contrast to the alternative methodology of discounting back the average cashflow at a single rate if the calculation is to give sensible results, different discount rates would need to be used for equities and fixed interest.

Correct pricing

Looking at how table 3 works, it is clear that the average of the time 1 deflators is a ‘risk-free’ one-year discount factor. This explains why the average of the deflators in individual columns in table 2 is less than 1. In fact, the average of the time-t deflators is 1.05-t, reflecting the flat 5% yield curve that I have assumed in the underlying investment model. Given that the average of a single column of deflators is fixed, the variability of the deflators in that column can be thought of as freedom to place more or less weight on the results of individual simulations. Examination of table 4 will reveal that more weight is effectively put on simulations where equities perform badly and that this neatly results in equities being priced correctly.

The product that I will be valuing involves only a single cashflow: it is a guaranteed equity bond that pays out at time 1 the greater of (i) the value that £100 of equities at time 0 grows to at time 1 and (ii) a fixed amount of £100. Only the maturity amount will be valued: expenses are ignored and mortality and lapses are set to zero. Looking down the time 1 figures in table 1, it can be easily seen that the maturity values that our model office would generate from the stochastic runs are £100, £100, £104.1, £124.7, £156.1.

Table 5 shows how deflators can be applied to the stochastically generated maturity values to value the guaranteed equity bond. The value comes out as 105.7. This could be thought of as made up of 100 of equities plus a put option worth 5.7. Of course, this cannot be expected to be accurate, based on only five stochastic runs. In fact, if enough runs were carried out, the value of the put option would converge towards its actual value of 7.4.

Look at the theory

And that is what I mean when I speak of stochastic discount factors. Of course, having only seen how to use deflators, actuaries will still want to understand why deflators work and how they can be derived. It is hoped that this article has at least helped to ease actuaries’ fears about deflators and will encourage them to take another look at the theory that lies behind them.

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