Assume that death benefits are nonincreasing show that the

Question

For a certain whole life policy on (x), given by a death benefit vector b and premium pattern vector ρ;, reserves are calculated according to two mortality tables, with the 1-year probabilities of dying denoted by q_x+k and q∗_x+k, respectively. The same positive interest rates are used in both calculations, and in each case the premiums are the net premiums as determined by the particular table being used. Suppose the two ‘curves' cross at one point. That is

q_x+k ≥ q∗_x+k,         k=0,1,.......,n,

q_x+k ≤ q∗_x+k,         k=n,n+1,.......,ω-x-1.

Assume that death benefits are nonincreasing. Show that the reserve at time n is higher for the starred rates (the steeper curve).