Chapter 24 - Profit Testing

Question 1

List the items that we have to consider in the unit fund for a unit-linked life assurance contract.

Answer 1

• premium received
• premium allocated
• cost of allocation
• fund at the start of the year
• fund at the end of the year
• management charge
• fund at the end of the year after deduction of the management charge

Question 2

List the items that we have to consider in the non-unit fund for a unit-linked life assurance contract.

Answer 2

• premium minus cost of allocation
• expenses
• interest
• expected non-unit death cost
• expected not-unit surrender profit
• management charge transferred from the unit fund
• end of year in-force cashflow
• cost of setting up reserves (if required)
• profit vector
• profit signature

Question 3

Explain how to calculate the insurer’s expected death cost for a unit-linked policy.

Answer 3

If the policy provides a guaranteed minimum death benefit and the bid value of the unit fund is less than this minimum at the end of the year of the policyholder’s death, then there will be a non-zero death cost in the non-unit fund. The expected non-unit death cost is determined by:
- calculating the difference between the guaranteed sum assured and the bid value of the unit fund at the end of the year of death
- multiplying this difference by the probability that the policyholder (who is known to be alive at the start of the policy year) dies during that policy year.

Question 4

A unit-linked endowment assurance policy issued to a life aged 60 provides a minimum guaranteed death benefit of 10,000. At the end of the first year, the bid value of the unit fund is 2348.88. Calculate the insurer’s expected death cost at the end of Year 1 for this policy assuming TMNL16 Select mortality and that death is the only decrement operating.

Answer 4

(10000 - 2348.88) * q_[60] = 7651.12 * 0.0001218 = 9.32

Question 5

When there is more than one way in which a policy can be terminated, which type of decrement probability should you use when calculating the expected benefit costs, expected surrender profits and in-force probabilities?

Answer 5

If there is more than one way in which a policy can be terminated, the dependent probabilities of decrement should be used to calculate the expected benefit costs, expected surrender profits and in-force probabilities.

Question 6

A unit-linked endowment assurance policy is issued to a life aged 60. At the end of the first year, the bid value of the unit fund is 2348.88. If the policyholder surrenders the policy at the end of the first year, the insurer will pay out 96% of the bid value of the units. The probability that a policy in force at the end of Year 1 is surrendered is 10%.
Calculate the probability that a policy is still in force at the start of Year 2 assuming that the policyholder is subject to TMNL16 Select mortality.

Answer 6

(ap)60 = (1 - q[60])(1 - 0.1) = (1 - 0.001218) * 0.9 = 0.898904

Question 7

Explain how to calculate the insurer’s expected surrender profit for a unit-linked policy.

Answer 7

When a unit-linked policy is surrendered, the policyholder often faces a surrender penalty. For example, the policyholder may receive 95% of the bid value of the units and the insurer keeps the remaining 5%. To calculate the insurer’s expected surrender profit, the bid value of the funds retained by the insurer is multiplied by the dependent probability of surrender for that policy year.

Question 8

A unit-linked endowment assurance policy is issued to a life aged 60. At the end of the first year, the bid value of the unit fund is 2348.88. If the policyholder surrenders the policy at the end of the first year, the insurer will pay out 96% of the bid value of the units. The probability that a policy in force at the end of Year 1 is surrendered is 10%.
Calculate the insurer’s expected surrender profit at the end of Year 1 for this policy assuming that the policyholder is subject to TMNL16 Select mortality.

Answer 8

Insurer keeps:
0.04 * 2348.88 = 93.96
Expected surrender profit:
93.96 * p_[60] * 0.1 = 93.96 * 0.998782 * 0.1 = 9.38

Question 9

List the items that we have to consider when profit testing a conventional (ie non unit-linked) life assurance policy.

Answer 9

• premium income
• expenses
• interest
• expected benefit payouts (death, surrender, maturity)
• end of year in-force cashflow
• expected cost of increasing reserves
• profit vector
• profit signature

Question 10

Explain how to calculate the expected cost of increasing reserves at the end of policy year t for a conventional life assurance policy.

Answer 10

Suppose we have a reserve of (t-1)V for each policy in force at time t - 1 (ie the start of year t). This earns interest over the course of the year, and by time t it has become (t-1)V * (1 + i). At time t, we need to hold a reserve for t_V for each policy then in force. The probability that a policy in force at time t - 1 is still in force at time t is (ap)(x+t-1), assuming the policyholder is initially aged x (if there is only one decrement operating, this simplifies to p(x+t-1)). So the expected amount of reserve required at time t, per policy in force at time t - 1 is t_V * (ap)(x+t-1), and the expected cost to the company of increasing the reserve at time t is:
t_V * (ap)(x+t-1) - (t-1)_V * (1 + i)

Question 11

For a certain class of endowment assurance, an insurer holds reserves of 1000 per policy in force at time 1 year and 2000 per policy in force at time 2 years. Assuming that the policyholders are aged 50 at the outset, mortality follows TMNL16 Select mortality and interest is 4% pa, calculate the expected cost to the insurer of increasing the reserves for one of these policies at the end of Year 2.

Answer 11

Grows to 1040 by the end of year 2.
Need 2000 per policy still in force. So expected amount needed per policy in force at the start of year 2 is:
2000p_([50] + 1) = 2000 * (1 - 0.000815) = 1998.37
So the expected cost to the insurer of increasing the reserves for one of these policies at the end of year 2 is 1998.37 - 1040 = 958.37

Question 12

Define the terms ‘profit vector’ and ‘profit signature’.

Answer 12

Each entry in a profit vector represents the expected profit at the end of that policy year, per policy in force at the start of the policy year.
Each entry in the profit signature represents the expected profit at the end of that policy year, per policy in force at time 0.

Question 13

You are given the following profit vector for a 4-year endowment assurance policy issued to a life aged 56:
(-56.32, 43.87, 44.90, 18.21).
Calculate the corresponding profit signature assuming that the life is subject to TMNL16 Select mortality.

Answer 13

To calculate the profit signature we multiply each entry in the profit vector by the probability that the policy is in force at the start of the given policy year.
Year | Profit vector | In-force probability | Profit signature
1 | -56.32 | 1| -56.32
2 | 43.87 | p_[56] = 0.999152 | 43.83
3 | 44.90 | 2p[56] = 0.997657 | 44.79
4 | 18.21 | 3p[56] = 0.995752 | 18.13

Question 14

Define:
- the net present value
- the profit margin
- the internal rate of return
of a life assurance contract.

Answer 14

• the present value of the profit signature, discounted using the risk discount rate
• NPV / EPV premiums, where both numerator and denominator are calculated using the risk discount rate
• the interest rate at which the net present value of the contract is equal to zero.

Question 15

You are given the following profit signature for a 4-year endowment assurance policy issued to a life aged 56:
(-56.32, 43.83, 44.79, 18.13).
Calculate the corresponding net present value of the policy using a risk discount rate of 10% pa.

Answer 15

NPV = -56.32/1.1 + 43.83/1.1^2 + 44.79/1.1^3 + 18.13/1.1^4 = 31.06