Question

A single premium whole life annuity-due is issued to (65) that pays a monthly benefitof $1,000. The first 10 years of payments are guaranteed. The only expenses are $700at issue and $45 per year in renewal years.(Assume that the renewal expenses areincurred only while the policyholder is alive.) Assume that mortality is given by theILT; use the UDD fractional age assumption where necessary. Also assume an annualeffective interest rate of 6%.(a) Calculate the net single premium for this policy. [123,570](b) Calculate the gross single premium for this policy. [124,670](c) Calculate8Vn,8Vg,and8Vefor this policy, assuming the insured is alive at thattime. [88,420, 88,770, 350](d) Calculate12Vnfor this policy. [74,610](e) Calculate0.05p76.95for this person. [0.997](f) Calculate11.95Vnfor this policy. [74,170]

Answer 1

Below is the solution for the given problems.

Step-by-step explanation:

(a) Calculate the net single premium for this policy.

P = 12, 000 [ a¨
`(12)/(10)` + 10E65 a¨
`(12)/(75)` ]

P = 12, 000[ a¨
`(12)/(10)`+ 10E65 (α(12) ¨a75 − β(12)}]

P = 12, 000 [(7.59716) + (0.39994) ((1.00028) (7.217) − (0.46812))]

P = 123, 565.38

(b) Calculate the gross single premium for this policy.

G = 12, 000[ a¨
`(12)/(10)` + 10E65 a¨
`(12)/(75)` ]+ 655 + 45 ¨a65

G = 123, 565.38 + 655 + 45 (9.8969)

G = 124, 665.74

(c) Calculate 8Vn, 8Vg, and 8Ve for this policy, assuming the insured is alive at that time.

Prospectively compute the net and gross premium reserve, then the expense reserve

is the difference:

8V n= 12, 000[ a¨
`(12)/(2)` + 2E73 a¨
`(12)/(75)` ]

= 12, 000 (1.892455 + 0.811178((1.00028) (7.217) − (0.46812)))

= 88, 423.64

8Vg = 88, 423.64 + 45 ¨a73

= 88, 772.70

8Ve = 349.06

(d) Calculate 12Vn for this policy.

12V n = 12, 000 ¨a (12) 77

12V n = 12, 000 (α(12) ¨a77 − β(12))

12V n = 12, 000 ((1.00028) (6.6836) − (0.46812))

12V n =74, 608.22

(e) Calculate 0.05p76.95 for this person.

p76 = 0.95p76 · 0.05p76.95

0.94353 UDD= (1 − (0.95) · q76) · 0.05p76.95

0.94353 UDD= (1 − (0.95)(0.05647)) · 0.05p76.95

0.05p76.95 UDD= 0.997

(f) Calculate11.95Vn for this policy.

11.95Vn =
`v^(0.05)` 0.05p76.95 12Vn

11.95Vn =
`(0.997)/(1.06^(0.05) )` · (74, 608.22)

11.95V n = 74, 169.22