Gross Premium Reserves from Expected Future Loss

Understand Gross Premium Reserves from Future Loss

Key rules to remember

Definition of gross premium reserve

tV = E[ L(t) | T(x) > t ] = PV of future benefits + PV of future expenses − PV of future gross premiums

All present values are at time t, on the policy value basis, for a life alive at t.

Whole life, annual premiums in advance

tV = S × A(x+t) + (renewal expenses) − G × ä(x+t)

Add claim expenses as e × A(x+t) and renewal expenses as e × ä(x+t) if they apply for as long as premiums are paid.

Net premium reserve (for contrast)

tV(net) = S × A(x+t) − P × ä(x+t)

Uses the net premium P and ignores expenses. Do not mix this with G.

Reserve at issue

0V = 0 when G is found by the equivalence principle on the same basis

A useful check on your premium and basis.

One-year recursion (premium and expense at start, death benefit at end of year)

(tV + G − e) × (1 + i) = q(x+t) × (S + claim expense) + p(x+t) × (t+1)V

e is the expense at the start of year t+1. Use the benefit that applies in year t+1.

Annuity-assurance link

A(x) = 1 − d × ä(x), where d = i ÷ (1 + i)

Lets you get an assurance value from an annuity value at the same age and rate.

How to solve Gross Premium Reserves from Future Loss questions

1. 1Write down the policy details: benefit, term, premium G, payment pattern, and the age x and duration t.
2. 2Write down the policy value basis: interest rate, mortality, and expenses. If it differs from the pricing basis, use the policy value basis.
3. 3Define the future loss at time t and condition on the life being alive at t (age x + t).
4. 4List future outgo at time t: benefits and all expenses still to come, with their timing.
5. 5List future income at time t: gross premiums still to come, with their timing.
6. 6Convert each item to actuarial functions at age x + t, with remaining term n − t where relevant.
7. 7Compute reserve = PV benefits + PV expenses − PV premiums. Check the sign and size.
8. 8If asked for profit or loss, or the next reserve, use the one-year recursion and keep the timing of each cash flow.

Quickest way: Three-line reserve

Common mistakes in Gross Premium Reserves from Future Loss

• Using the net premium P instead of the gross premium G.

  Net premium reserves are taught first, and the formulas look the same.

  Fix: Read the question for the words gross or expenses. If G and expenses are given, use G and include the expenses.

• Leaving out future expenses or treating them as one-off.

  Students focus on benefits and premiums and forget the expense schedule.

  Fix: List every expense with its timing. Renewal expenses continue while the policy is in force; claim expenses are paid at death.

• Using age x and the full term instead of age x + t and the remaining term.

  The policy was written at age x, so the old values feel familiar.

  Fix: At time t, the life is aged x + t, with n − t years left. Rewrite every function accordingly.

• Valuing on the pricing basis when a different policy value basis is given.

  The premium was calculated on the pricing basis, so students assume it stays in force.

  Fix: The premium G stays fixed at its stated value, but all present values use the policy value basis.

• Applying the recursion with the wrong timing, for example including the start-of-year expense after interest.

  The recursion has several terms and the timing is easy to confuse.

  Fix: Anything paid at the start of the year goes inside the bracket before multiplying by (1 + i). Anything paid at the end of the year is on the right.

• Forgetting to condition on survival, so a death-before-t contribution appears.

  The reserve is confused with an unconditional expectation.

  Fix: State that the reserve is for a policy still in force at t. Only cash flows after t, for a life aged x + t, appear.

Worked examples