Use jensens inequality to show that aoline 1ex ge aolinex

Question

Assume that interest rates are positive, that is, the investment discount function v(t) is decreasing with t.

(a) Show that a‾ (1_t; v) is a concave function of t.

(b) Use Jensen's inequality to show that a‾ (1e_x^? ) ≥ a‾_x

(c) Now assume constant interest. A 1-unit whole life policy on (x) has premiums payable continuously at the annual rate of P. Show that

P E[Val_T(x)(1_T(x); v)] ≥ 1.

That is, the expected amount of premiums accumulated at death by a policyholder is greater than or equal to the benefit payment at that time.

(d) Explain the inequality in (c) by general reasoning.

(e) Show that the inequalities in (b) and (c) are equalities at 0 interest.