Premium Rates Under Inflationary Conditions

This short note has as its starting point an interesting article by Taylor (1979) in which he considered the effects of inflation on a risk process. Taylor showed that if the premium density increased at the same rate as the cost of individual claims then, under certain conditions, ultimate ruin was certain. This raises a natural question, viz. “If the cost of individual claims is increasing how should the premiums be increased in order to keep the probability of ruin under control ?†It is this question that we shall be considering in this note.In the next section we define the risk process that we shall be studying for the remainder of this note. Our process is essentially a compound Poisson process except that we allow the distribution function of an individual claim to depend on the time at which the claim occurs. We start the third section by deriving, with the help of a general result of Gerber (1973), a formula for the future premium density that will keep the probability of ruin for our process below a predetermined bound. We then derive a simple approximation to this formula that shows more clearly how we require the premium density to change in relation to the change in claims costs. Finally we show that if we consider the same process with annual premiums then the probability of ultimate ruin will be kept below a predetermined bound if the annual premium is calculated according to the principle of zero utility with an exponential utility function or, as a first approximation, according to the variance principle.