# MA312-6/7-SP: Consider Two Lives (x) and (y) Currently aged x and y Exact with an age Difference: Contingencies II Assignment, UOE, UK

University |
university of Essex (UOE) |

Subject |
MA312-6/7-SP: Contingencies II |

This is the question sheet for the Mock lab test. Download and open the Excel file named “Mock lab workbook”, which is where you should show your work and answers. The Excel file contains a few sheets (see instructions below), including “Base” which contains the probability of dying for each age.

1. Consider two lives (x) and (y) currently aged x and y exact with an age difference y − x = −7, where both lives follow the mortality given in sheet “Base”. Suppose an annual interest rate of i = 2.6%.

(a) Calculate the joint survival probabilities pxy for x = 7, 8, . . . , 101. [5 marks]

(b) Use the backward recursive method and ¨a101:94 = 1 to calculate ¨axy for x = 7, . . . , 100. [Hint: Recall the formula ¨axy = 1 + vpxya¨x+1:y+1.]

(c) Use the conversion relationship to find Axy for x = 7, . . . , 101.

(d) A life office issues a special policy to the two lives above which pays a lump sum of £50,000 immediately on the death of the first to die. Level premiums of Px are payable annually in advance until the first death, where initial expenses are £250 and renewal expenses are 5% of all annual premiums. Calculate the gross annual premium Px for x = 45, . . . , 65.

(e) How does Px vary with x? Explain why.

2. An insured population is subject to only 2 modes of decrement, death (d) and surrender (s). Suppose also, that surrenders are only allowed at the end of each policy year (mode s operates only at k = 1), while a uniform distribution of decrement (UDD) is assumed for mode d. The independent rates of dying are in accordance with those in sheet “Base”. The independent annual rates of surrender for each age between 30-64 are given as follows:

(a) Input the values of q d x and q s x for x = 30, 31, . . . , 64.

(b) Calculate the dependent probabilities of exits due to death and surrender,

(c) Construct a multiple decrement table for ages x = 30, 31, . . . , 64 using a radix of (al)30 = 100, 000. Also include the expected number of insured people alive at age 65.

(d) A life office issues a 35-year term assurance policy to a live currently aged 30 exact using the multiple decrement table in (c). This policy pays a lump sum of £250, 000 at the end of the year of death. Level premiums are payable annually in advance for at most 35 years, or until death/withdrawal, whichever earlier. Additionally, the policy returns all premiums paid (without interest) at the end of the year of surrender within the policy term. Suppose an annual interest rate of 5.5%.

i. Calculate the total EPV of premiums assuming unit premium, i.e. P34 t=0 v t t(ap)30.

ii. Calculate the EPV of a death benefit for each policy year, and hence, the EPV of the total death benefit. Recall the EPV of death benefit between t and t + 1 is Sum Assured×v t+1

iii. Similarly, calculate the total EPV of surrender benefit per unit premium.

iv. Hence, find the annual premium payable for this life.

3. A life office issues a 35-year nonprofit endowment assurance policy to a life currently aged 30 exact for a sum assured of £250, 000 payable on survival to the end of the term or at the end of the year of death if earlier. A premium of £2838.39 is payable annually in advance throughout the term of the policy. There is a surrender benefit of £1000 payable at the end of the year of surrender. The projected expenses in the first year are £500, and renewal expenses of £70 at the time of payment of the second premium and

subsequent payments.

In addition, the office assumes it earns 5.5% per annum on funds. The life office wishes to carry out profit testing using the multiple decrement table constructed in Q2(c).

(a) Import the values of (aq) d x and (aq) s x for x = 30, · · ·, 64, from sheet “Q2” and complete the rest of the multiple decrement table in sheet “Q3”. [5 marks]

(b) You are given the reserve at the beginning of each policy year. Calculate, using a risk discount rate of 8% per annum, the expected profit margin on this contract.