Gross Premium Policy Value Calculations

Understand Gross Premium Policy Value Calculations

Key rules to remember

Prospective gross premium policy value

tV = EPV of future benefits + EPV of future expenses − EPV of future gross premiums

Use the policy value basis for mortality, interest and expenses. Value at time t for a policy still in force at t.

Whole life, annual premiums in advance, sum assured S

tV = (S + E) × A(x+t) + e × ä(x+t) − (1 − f) × P × ä(x+t)

Here E is the claim expense paid with the sum assured, e is the fixed renewal expense each year at the start of the year, and f is the percentage-of-premium expense rate. The percentage expense is taken off the premium once, through the factor (1 − f). Do not also add it to the outgo.

Retrospective policy value

tV = [PV at time 0 of past premiums less expenses, less past death benefits and claim expenses] ÷ (v^t × tp_x), or equivalently [accumulated value at time t of the same cashflows] ÷ tp_x

Use one form or the other. If you accumulate with interest to time t, divide by tp_x only. If you use present values at time 0, divide by v^t × tp_x. Do not divide by v^t after accumulating, as that counts interest twice. All parts are on the same basis. The retrospective value equals the prospective value only when both use the same basis and the premium satisfies the equivalence principle on that basis.

Recursive relationship (premium and expense at start of year, death benefit at end of year)

(tV + P_t − e_t)(1 + i) = q(x+t) × (S_{t+1} + E_{t+1}) + p(x+t) × (t+1)V

P_t is the gross premium at time t. e_t is the expense at time t. S_{t+1} is the death benefit paid at the end of the year. E_{t+1} is any claim expense. Rearrange to find tV from (t+1)V, or the reverse.

Terminal condition

nV = maturity benefit for an n-year endowment (including any survival benefit); nV = 0 for a term assurance

Start a backward recursion from here. No premium is due at time n, so there is no before or after premium distinction at that point.

Policy values just before and after a premium

(tV + P_t − e_t) is the amount available just after the premium and expense at time t

Be clear about whether your tV is before or after the premium. The standard convention is that tV is the value just before the premium due at time t.

How to solve Gross Premium Policy Value Calculations questions

1. 1Read the policy value basis. Note the mortality table, interest rate, expense assumptions and whether future bonuses are allowed for. Do not use the pricing basis unless the question says so.
2. 2Identify the policy duration t and the remaining term. Write down the benefits still to come, the premiums still to come and the expenses still to come.
3. 3Write the policy value equation in words first: EPV of benefits + EPV of expenses − EPV of premiums.
4. 4Convert each part into actuarial functions: A, ä and so on at age x+t, with the right interest rate. Split percentage-of-premium expenses from fixed expenses.
5. 5Use the actual gross premium given, not a recomputed net premium. If the premium has to be found, set the EPV of all cashflows at time 0 to zero on the pricing basis.
6. 6Compute the value. Check the sign: premiums are income, so they are subtracted. Check that the answer is sensible: a policy value for an endowment should rise towards the maturity benefit.
7. 7For a recursion, place the cashflows on the time line for the year. Apply (tV + P − e)(1 + i) = expected cost at year end, then solve for the unknown value. State the terminal condition clearly.
8. 8If asked for a retrospective value, accumulate past premiums net of expenses, deduct the cost of past claims allowing for survivorship, and divide by the survival probability. State the basis.

Quickest way: Write the cashflow line, then use the recursion

Common mistakes in Gross Premium Policy Value Calculations

• Using the net premium instead of the gross premium in the policy value

  Students recall net premium reserve formulas, where the premium is the net premium from the equivalence principle.

  Fix: Use the actual gross premium given. For net premium reserves, ignore expenses and use the net premium. For gross premium reserves, include expenses and use the gross premium.

• Leaving out expenses or claim expenses

  Expense items are listed in the question text and are easily skipped, especially claim expenses paid on death.

  Fix: Tick off each expense item from the question. Make a list: initial, renewal, claim and any percentage-of-premium expenses. Include each in the right EPV.

• Treating percentage-of-premium expenses as if they were an extra cost without reducing the premium income

  Students add them to the benefit EPV but use the full premium. This is acceptable only if done consistently.

  Fix: Either deduct the expense from each premium or add it to the outgo, but do not do both. Net it from the premium first, then multiply by the annuity.

• Mixing up values just before and just after a premium

  The recursion includes the premium at time t, and some students include it in tV as well.

  Fix: Define tV as the value just before the premium due at time t. Then add P and subtract expenses explicitly in the recursion.

• Assuming prospective and retrospective values are always equal

  Textbooks show the equality for the pricing basis, and the condition is forgotten.

  Fix: Remember the condition: same basis, and the premium satisfies the equivalence principle on that basis. If the bases differ, the values differ.

• Ignoring bonuses already declared on a with-profits policy

  Students value only the basic sum assured.

  Fix: Include the sum assured plus declared bonuses in the benefit. Allow for future bonuses only if the policy value basis says so.

Worked examples