Actuarial Science exam

AS 3429/9429 Long Term Actuarial Math II

Final Practice Questions

1. A fully discrete 10-year endowment insurance policy with sum insured of $75,000 is issued to a life aged 40. You are given the following information:

• Mortality follows the Standard Ultimate Life Table with interest rate i = 5%. • Death benefit is payable at the end of year of death. • Expenses are as follows:

– taxes are 4% of premiums – Commissions are 30% of the first premium, and 5% of all premiums paid thereafter – The annual policy maintenance fee is $25 in year one and $5 thereafter – There is a termination expense of $100 that is to be paid at the time the policy benefit is paid

(a) Find the gross premium P under Equivalence Principle. (b) Write an expression for Lg0 and calculate the probability that the policy makes a loss. (c) Find the gross premium policy value, net premium policy value and the FPT policy value at

time 5.

2. An insurance company issues identical whole life insurance policies to 10, 000 individuals aged 50. You are given the following;

• Each policy has sum insured of $150,000 payable at end of year of death • Issue expenses are 25%of the first year premiumand renewal expenses are 5%of each subsequent annual premium

• Mortality follows the Standard Ultimate Life Table with interest rate i = 5%. • Annual premium is calculated using portfolio percentile principle such that the probability of a loss on the portfolio is 5%

Calculate the annual premium per policy.

3. A special fully discrete whole life insurance is issued to (40). The death benefit payable at the end of the year of death and the premium paid by the insured at the beginning of the year (if alive) are respectively

bK+1 =

{ 1000,K = 0, 1, ..., 24 500,K = 25, 26, ... and Pk =

 3π, k = 0, 1, ..., 14 2π, k = 15, 16, ...29 0, k = 30, 31, ...

.

Given that k A40+k A

1 40+k:25−k| ä40+k:15−k| ä40+k:30−k|

0 0.202 0.114 10.005 13.414

10 0.316 0.149 4.386 10.806

20 0.458 0.112 − 6.929 40 0.744 − − −

,

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(a) define the prospective loss rv Lj for all possible values of j (b) compute π under the equivalence principle and the policy values at times 10, 20 and 40.

4. Consider a 20-year endowment insurance on (60) with a sum insured of $100,000 payable at the end of the year of death or on survival to age 80, whichever occurs first. An annual premium of $5,200 is payable for at most 10 years. The insurer uses the following basis for the calculation of policy values:

Survival model: Standard Select Survival Model Interest: 5% per year Expenses:10% of the first premium, 5% of subsequent premiums, and $200 on payment of the sum insured

An insurer issued a large number of policies identical to the policy above. Ten years after they were issued, a total of 200 of these policies were still in force. In the following year,

• expenses of 4% of each premium paid were incurred • interest was earned at 4% on all assets • two policyholders died, and • a $250 expenses was incurred on payment of sum insured for the policyholder who died

(a) Calculate profit or loss on this group of policies for this year. (b) Determine how much of this profit/loss is attributable to the sources from expenses, interest,

and mortality (in this order).

5. Suppose you are given the following estimated parameters of the CBD model

K (1) 2017 = −3.2 K

(2) 2017 = 0.01 c

(1) = −0.02 c(2) = 0.0006 x̄ = 70

σk1 = 0.03 σk2 = 0.005 ρ = 0.2

(a) Calculate the mean and standard deviation of logit(q(65, 2018)). (b) Calculate the median and 95th percentile of p(65, 2018).

6. You are given the following data set. Observations marked with an asterisk (*) were censored at that value.

1 2 3* 4 4 4* 4* 5 7* 8 8 8 9 9 9 9 10* 12 12 15*

(a) Determine the numbers at risk for the data. (b) (i) Construct the Kaplan–Meier estimate of S(y). Apply all three tail correction methods to the

data. Assume γ = 22. (ii) Indicate how the answer in (i) would change if s7 = 3 and no censoring after 12.

(c) Construct the Nelson–Aalen estimate of S(y). Apply all three tail correction methods to the data. Assume γ = 22.

(d) Estimate the variances of the Kaplan–Meier estimator and Nelson–Aalen estimator of S(2). (e) Construct a 95% log-transformed confidence interval for S(2) based on the Kaplan–Meier

estimator. (f) Construct a 95% log-transformed confidence interval for S(2) based on the Nelson–Aalen

estimator.

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