ST301 – ACTUARIAL MATHEMATICS 2022
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January 2022 Exam
ST301 – ACTUARIAL MATHEMATICS (LIFE)
You will need the following select life table in order to answer question 2 .
x |
l[x] |
l[x]+1 |
lx+2 |
x +2 |
75 |
1593 |
1567 |
1529 |
77 |
76 |
1551 |
1522 |
1482 |
78 |
77 |
1505 |
1474 |
1431 |
79 |
. . . |
. . . |
. . . |
. . . |
. . . |
80 |
|
|
1258 |
82 |
81 |
|
|
1193 |
83 |
82 |
|
|
1125 |
84 |
83 |
|
|
1054 |
85 |
84 |
|
|
981 |
86 |
85 |
|
|
906 |
87 |
You may need the following information for questions 6 and 8 .
10 p30 = 0.9785
30 p30 = 0.8452
You may also need the values of some of the temporary assurances calculated at various forces of interest r.
r |
A(¯)1 30:30 |
|
A(¯)1 40:20 |
|
0.01 |
0.1271 |
0.1206 |
||
0.02 |
0.1051 |
0.1071 |
||
0.03 |
0.0874 |
0.0953 |
||
. . . |
. . . |
. . . |
||
0.10 |
0.0296 |
0.0464 |
||
0.11 |
0.0261 |
0.0425 |
||
0.12 |
0.0232 |
0.0390 |
1. Explain with words why
Axy = 1 − iaxy
[3]
2. Using the select life table provided at the beginning of the paper, calculate 4 | 2 q[77]+1 [3]
3. Two independent lives x and y experience the same Gompertz law of mortality µx = bcx where b > 0 and c > 0 . Show that
A(¯)x(1)y
A(¯)xy
is independent of the interest rate used for calculations.
[4]
4. The benefits of a life policy issued by an office are an amount S1 at time n if the policy holder is alive at the time or an amount S2 on earlier death. The premium is paid up front and is calculated using a first order basis consisting of a force of interest r and a force of mortality µx .
(a) Show that if S2 ≥ S1, the office should use a first basis consisting of a lower force of interest and a higher force of mortality than expected.
[3]
(b) Comment on the case S2 < S1 . What first basis should the office use? [4]
5. Consider two independent lives aged x andy.
(a) Let K1 represent the present value of an amount of 1 payable on the first death and K2
the present value of an amount of 1 payable on the second death. Show that
Cov(K1, K2 ) = (A(¯)x − A(¯)xy)(A(¯)y − A(¯)xy)
[4]
(b) Let L1 represent the present value of an amount of 1 payable 5 years after the death of the life now aged x provided the life aged y is alive at the time of payment and L2 represent the present value of an amount of 1 payable 5 years after the death of the life now aged x provided the life aged y is dead at the time of payment. Express Cov(L1, L2 ) in terms of assurance functions and survival probabilities.
[9]
6. An office issues 30-year unit-linked endowment assurance policies to lives aged 30 . The policies are financed by a continuous premium payable at rate ⇡ . In order to calculate the premium the office makes the following assumptions. At any time t(< n) a proportion γt of the premium s.t.
γt⇡ = µx+t(1000000 − Ut)++0 .1⇡
is allocated to a cash fund Ut that grows at a constant force of interest 0 per annum . A proportion 1 − γt of the premium is invested in a unit fund that is assumed to grow at a force of interest of 0.03 per annum . µx denotes the force of mortality. At time 30 or on earlier death the policyholder will receive the accumulated amount of the fund or a guaranteed sum 1000000, whichever is larger. There is also the possibility of surrenders which occur independently of deaths and with rate 0.02 − µx+t (You may assume this is positive) . Surrenders are entitled to their share of the unit fund (without guarantees) but not to any of the cash fund. Calculate ⇡ .
[16]
7. Consider a health-sickness Markov model. The sickness rate is σx, the recovery rate is ⇢x , the mortality rate for a healthy life is µx for a female life aged x and the mortality rate for a sick life is ⌫x . An office is issuing a policy under which a continuous sickness benefit at a rate of b per annum payable while the life is sick is provided. The life is now 35 years old. The policy is financed by a continuous premium at a rate ⇡ per annum payable while the life is healthy. The policy duration is 25 years. After that period has elapsed, no more premiums are paid. If the life is healthy, no more benefits are paid but if at time 25 the life is sick, she continues receiving the benefit for as long as this period of sickness lasts. Upon recovery the policy expires.
(a) What problem is the extra benefit while sick at time 25 addressing? [3]
(b) Explain by writing down appropriate diferential equations and their terminal conditions how the office can calculate ⇡ using a force of interest r.
[10]
(c) Explain how the company will calculate reserves at all times and states. [10]
8. Employees of a company start working in a clerical position and are eventually promoted to a managerial position. The rate of transition from clerical to managerial is 0.1 per annum and once moving to managerial the employee can not go back to clerical. Employees leave the company with a force 0.01 per annum while in clerical and 0.02 per annum while in managerial. They are also subject to a force of mortality µx+t at time t regardless of position. An employee aged 30 is initially at a clerical position. Let p1 (t) denote the probability that she is working in a clerical position at time t and p2 (t) denote the probability that she is in a managerial position at time t.
(a) Write down the forward equations.
[3]
(b) Hence or otherwise derive p1 (t) and p2 (t) in terms of
tpx = exp ✓ − Z0 t µx+sds◆ .
[7]
(c) The company is offering her a lump sum of 100000 on death while in a clerical position and 200000 while in a managerial position. The benefit will stop when the employee reaches the age of 60 or if she leaves the company. This is financed by a single premium payable upfront. Calculate the premium using a force of interest of 0.01 per annum . [7]
(d) The employee is also entitled to any profits generated by the policy. Using a second basis with a force of interest of 0.03 per annum and a force of mortality of µx+t−0 .001, calculate the rate that profit emerges at time t = 0.
[4]
(e) Repeat part (d) for t = 10
[10]
2023-08-17