[SOLVED] ACTU PS5821 Actuarial Methods – Autumn 2024 Assignment – 3 C/C

ACTU PS5821 Actuarial Methods – Autumn 2024

Assignment – 3

Assigned 9/20/24, Due 9/28/24

Problem 1.

Let F0(t) = 1 − e
−λt, where λ > 0.

(a) Show that ex = (e
λ − 1)−1

(b) Calculate ex when λ = 0.01

Problem 2. Suppose we modify the Makeham’s law to the following force of mortality,

µx = A + Bx + CDx

(a) Derive a formula for tpx

(b) Numerically evaluate the following with

A = 0.00005, B = 0.0000005, C = 0.0003, D = 1.07

(b1) 20|10q30

(b2) e70

(b3) ˚e70 to 7 decimal places, using the Trapezium rule at n = 100 and the Repeated Simpson’s rule at n = 10. The “exact” value is 13.5435472

Problem 3. Mortality follows

Calculate 

Problem 4. Mortality follows qx+k = 0.1(k + 1) for k = 0, 1, 2, …, 9. X = min(K, 3) where K is the curtate future lifetime random variable. Calculate V ar(X).