A Pure Endowment is, in some sense, the opposite of term insurance. All insuranc

Question

A Pure Endowment is, in some sense, the opposite of term insurance. All insurance companies sell them. A $1 n-year pure endowment pays $1 at time nyear if the insured is alive. A $1 n-year Endowment (distinct from “pure” endowment) is as follows: if the insured is alive at time n-year, they receive $1; if they die before time n-year, their beneficiary receives $1. Thus, it is a combination of a pure endowment and n-year term insurance. Derive a formula for the actuarial present value of a $1 n-year endowment. Then apply the formula to the case of a life age x=55, n=10, r=.05, and the GoMa parameters are m=86.34, b=9.5 and lambda=0.

Explanation / Answer

IFM(t)=+EXP((Age-m)/b)/b m 86.34 b 9.5 0 r 5% Ps=1-IFM(t) Exp Pay PV of Probability Cum Probability if early death Exp Pay Age Year(t) IFM(t) of survival(Ps) of survival(CPs(t)=CPs(t-1)*Ps(t)) (A=IFM(t)*1*CPs(t-1) if early death=A/(1+r)^t 56 1 0.43% 99.57% 99.57% 0.004318 0.004112 57 2 0.48% 99.52% 99.091% 0.004777 0.004332 58 3 0.53% 99.47% 98.562% 0.005281 0.004562 59 4 0.59% 99.41% 97.979% 0.005836 0.004801 60 5 0.66% 99.34% 97.334% 0.006446 0.00505 61 6 0.73% 99.27% 96.623% 0.007114 0.005309 62 7 0.81% 99.19% 95.838% 0.007846 0.005576 63 8 0.90% 99.10% 94.974% 0.008646 0.005852 64 9 1.00% 99.00% 94.022% 0.009519 0.006136 65 10 1.11% 98.89% 92.975% 0.01047 0.006428 Total Present value of Expected pay if early death 0.052159 (=sum of above PVs) Present value of Expected pay if early death 0.052159 Expected pay at age 65=E=PV*Cumulative prob. Of survival till age 65 0.929747 Present value of Expected pay at age 65 if alive (=E/(1.05)^10 0.570784 Total Present value of the $1 n-year endowment. $       0.62

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