Tham khảo tài liệu 'review of the economic theory of annuities_5', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 64 Chapter 8 Now let . _ pFi z ri z 1 - p Ỉ2 z r2 z r z pFi z 1 - p F2 z Sri z 1 - S r2 z M z T where ỗ s z 0 s z 1. pFi z 1 - p F2 z The future instantaneous rate of return at any age z M on long-term annuities held at age M is a weighted average of the risk-class rates of return the weights being the fraction of each risk class in the Inserting into the latter becomes p F1 z w z - C1 dz 1 - p F2 z w z - C2 dz Jm Jm Ị- M F z w z - c dz 0. Jo From it is now straightforward to draw the following conclusion The unique solution to and that satisfies with r z given by is c c1 c2 c and R R2 R1 where c R is the First-Best solution and . A separating competitive equilibrium with long-term annuities supports the first-best allocation. Individuals are able to insure themselves against uncertainty with respect to their future risk class by purchasing long-term annuities early in life. In equilibrium these annuities yield at every age a rate of return equal to the population average of risk class rates of return. The returns from these annuities provide an individual with a consumption level that is independent of risk-class realization. The transfers across states of nature necessary for the first-best allocation are obtained through the revaluation of long-term annuities. The stochastically dominant risk class obtains a windfall because the annuities held by individuals in this class are worth more because of the . . A Í Sr1 f1 z Í 1 - f2 z lhe change in r z is r z ---- T- - - ----. - r . 1 - S r2 J f1 z Vr1 1 - S r2 J f2 z The sign of this expression can be negative or positive. The change in the hazard rate is equal to f z . A nondecreasing hazard rate implies that F z F z f z F z J f z f z . f z _ but does not sign . v for the exponential function f z 0 and this f zfi F z f z inequality becomes an equality . Uncertain Future Survival Functions 65 higher life expectancy of the .
