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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.