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Deflators: there’s no getting away from them is there?

In the January/February 2002 issue of The Actuary I demonstrated how deflators could be used to value insurance products with investment guarantees. Deflators were seen to be a neat and tidy way to value such contracts, but the volume of calculations involved might give cause for concern. A contract with cashflows at m different points in time that is to be valued using n stochastic runs requires a huge m*n matrix of deflators to be calculated, stored and accessed. Some people might rightly ask whether there are less calculation-intensive techniques around. This article sets out to describe the alternative methodology of risk-neutral valuation.

Methodology
Using the deflator methodology, the value of a random cashflow C at time t is the average over all the stochastic runs i=1,,n of Dt,i*Ct,i. Dt,i is the time-t deflator applicable to the ith stochastic run. I demonstrated last month that the average (over i) of the Dt,i values needs to be equal to a risk-free discount factor vt if deflators are to be able to price zero coupon bonds correctly. An alternative presentation can be derived by substituting Dt,i=vt*Wt,i where the Wt,i can be interpreted as ‘weighting factors’. The average of these weighting factors must be equal to one and I described in the last article how the weightings tend to be heavier in runs where equities perform poorly.
The value of the time-t cashflow can then be rearranged to give:
SDt,i*Ct,i/n = Svt*Wt,i*Ct,i/n = vt*SCt,i*Wt,i/n
This new expression looks like the present value of an expected cashflow but where, rather than assigning identical probabilities of 1/n to each run, the ith run and tth period is assigned a ‘risk-neutral’ probability of Wt,i/n. In return for this extra complication we are rewarded with being able to discount using a single vt factor rather than a set of n deflators. This suggests the possibility of an alternative valuation technique, where the value of a random cashflow is equal to the present value of its expectation but where the expectation is based on artificial risk-neutral probabilities rather than our best estimates. This technique is known as risk-neutral valuation.

Indirect adjustment
The way that risk-neutral valuation is used in practice is slightly different. Rather than being adjusted directly, the probabilities are adjusted indirectly by adjusting the parameters in the underlying investment model. The adjustment made is one that results in all assets earning a risk-free rate of return. Under the reparameterised model, it can be shown that the Wt,is are all equal to one, the deflators all equal to vt and (most importantly) the risk-neutral probabilities all equal to 1/n. The valuation formula simplifies down to vt*SCt,i/n. Now we have not only a single vt discount factor but also equal 1/n probabilities for all the runs.
At this stage, an example would help illustrate risk-neutral valuation in action. Table 1 left shows five random values for an equity index at time 1 from two asset models. The first set was based on a Black-Scholes model with an expected rate of return of 12.9%. These runs were used in the illustration in the last article. The second set are from a risk-neutral version of the same model one that was reparameterised to make the expected return on equities equal to the 5% risk-free rate that has been assumed all along. Both sets of runs are based on the same five random number feeds, so are directly comparable with each other.
Table 2 shows how values can be determined from the risk-neutral runs. In each case the first step is to determine the average cashflow at time 1 and the second step is to discount this payment by dividing by 1/1.05. Just as with the deflator methodology, it is useful to check that £100 of equities and £100 of bonds are being valued correctly. Table 2 shows that this is trivial for bonds but that there is (unsurprisingly) some sampling error that means equities are mispriced. A guaranteed equity bond paying out the greater of £100 and the equity index is valued at £106.8, compared to a theoretical value of £107.4. Increasing the number of stochastic runs will make the result converge towards this figure and valuation using deflators will converge towards that same number.

One disadvantage
So far this article has had nothing bad to say about risk-neutral pricing. If enough stochastic runs are performed, values will converge to the same figure as those calculated using deflators. The volume of calculation and computer power required is smaller. Some people may find that the tables showing the valuation calculation are more intuitive looking. There is, though, one disadvantage associated with risk-neutral valuation that should be highlighted.
Under the deflator methodology, we are free to parameterise our stochastic model using our best estimates of future investment returns. This means that we can look through the stochastic output and calculate expectations, probabilities, etc knowing that these will reflect what we really expect to happen. Such supplementary statistics are not available from risk-neutral valuation runs because they are based on artificial assumptions chosen to make the calculations easier. For example, table 2 suggests (albeit only based on five runs) that there is a 60% chance of the maturity guarantee on the guaranteed equity bond biting, whereas the corresponding realistic results in the January/February article tell us that a better estimate is 40%. Risk-neutral stochastic runs would be dangerous in the wrong hands because they can give a misleading picture of the business.
However, provided the results of the stochastic runs are regarded with care, risk-neutral valuation represents a viable alternative to deflators for a life office wishing to value financial guarantees.

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