MTH 434/214 Life Insurance Mathematics 2021 - 2022 S2
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MTH 434/214
Life Insurance Mathematics
2021 - 2022 S2
Universal Life Examples
1. Example 14.1 (Step 1: account value projection) A UL policy is sold to a 45-year- old man. The initial premium is $2250 and the ADB is $100 000. The policy charges are:
. Cost of Insurance: 120% of the Standard Select Survival Model, iq = 5% per year
interest.
. Expense Charges: $48 + 1% of premium at the start of each year.
Surrender penalties at each year end are the lesser of the full account value and the following surrender penalty schedule:
8-10 > 10
Penalty $4500 $4100 $3500 $2500 $1200 $0
Assume
(a) the policy remains in force for 20 years,
(b) interest is credited to the account at 5% per year,
(c) all cash flows occur at policy anniversaries, and
(d) there is no corridor factor requirement for the policy.
Project the account value and the cash value at each year end for the 20-year projected term, given that the policyholder pays the full premium of $2250 for six years, and then pays no further premiums.
2. (Step 2: the profit test) For the scenario described below, calculate the profit signa- ture, the discounted payback period and the net present value, using a hurdle interest rate of 10% per year effective, for the UL policy described in previous example.
Assume
❼ Policies remain in force for a maximum of 20 years.
❼ Premiums of $2250 are payable for six years, and no premiums are payable thereafter.
❼ The insurer does not change the CoI rates or expense charges from the values given
in Example 1.
❼ Interest is credited to the policyholder’s account value in the tth year using a 2%
interest spread, with a minimum credited interest rate of 2%. In other words, if the insurer earns more than 4%, the credited interest will be the earned interest rate less 2%. If the insurer earns less than 4%, the credited interest rate will be 2%.
❼ The ADB remains at $100 000 throughout.
❼ Interest earned on all insurer’s funds at 7% per year.
❼ Mortality experience is 100% of the Standard Select Survival Model.
❼ Incurred expenses are $2000 at inception, $45 plus 1% of premium at renewal, $50
on surrender (even if no cash value is paid), $100 on death.
❼ Surrenders occur at year ends. The surrender rate given in the following table is the
proportion of in force policyholders surrendering at each year end.
|
Duration at year end |
Surrender rate |
q45(w)+t − 1 |
|
1 |
5% |
|
|
2-5 |
2% |
|
|
6-10 |
3% |
|
|
11 |
10% |
|
|
12-19 |
15% |
|
|
20 |
100%. |
|
❼ The insurer holds the full account value as reserve for this contract.
3. (Type A UL profit test) Consider the following UL policy issued to a life aged 45:
❼ Face amount $100,000.
❼ Type A death benefit with corridor factors (γt ) applying to benefits payable in respect
of deaths in the tth year, as follows:
❼ CoI based on: 120% of mortality rates from the Standard Select Survival Model,
and 4% interest; the CoI is calculated assuming the fund earns 4% interest during
the year.
❼ Expense charges: 20% of the first premium plus $200, 3% of subsequent premiums. ❼ Initial premium: $3500.
❼ Surrender penalties:
Year of surrender 1 2 3-4 5-7 ≥ 8
Penalty $2500 $2100 $1200 $600 $0
(a) Project the account and cash values for this policy assuming level premiums of $3500
are paid annually in advance, that the policyholder surrenders the contract after 20 years, and that the credited interest rate is 4% per year.
(b) Profit test the contract using the basis below. Use annual steps, and determine the
NPV and DPP using a risk discount rate of 10% per year.
Assume
❼ Level premiums of $3500 paid annually in advance.
❼ Insurer’s funds earn 6% per year.
❼ Policyholders’ accounts are credited at 4% per year.
❼ Surrender rates are as in Example 2 above. All surviving policyholders surrender
after 20 years.
❼ Mortality follows the Standard Select Survival Model.
❼ Incurred expenses are:
– pre-contract expenses of 60% of the premium due immediately before the issue date,
– maintenance expenses of 2% of premium at each premium date including the first,
– $50 on surrender,
– $100 on death.
❼ The insurer holds reserves equal to the policyholder’s account value.
2022-08-08