Mean and Variance of Gross Future Loss

Understand Mean and Variance of Future Loss

Key rules to remember

Gross future loss

L = PV of benefits + PV of expenses − PV of gross premiums

All present values are at policy issue and are random because they depend on K (or T).

Mean of the loss

E[L] = EPV of benefits + EPV of expenses − EPV of premiums

This is a signed number. A negative value means the premium is more than enough on average.

Equivalence principle

E[L] = 0, so EPV of premiums = EPV of benefits + EPV of expenses

Solve this equation for the premium P. Use the policy's own expense and premium assumptions.

Whole life loss, sum assured S paid at end of year of death

L = S v^(K+1) + I + (f − P) ä_{K+1|}

P is the annual gross premium in advance. I is an initial expense at time 0. f is an expense at the start of every policy year including the first.

Whole life loss in simplified form

L = (S + (P − f)/d) v^(K+1) + I − (P − f)/d

Uses ä_{K+1|} = (1 − v^(K+1)) ÷ d. The term I − (P − f)/d is a constant.

Mean for whole life

E[L] = S A_x + I − (P − f) ä_x

Use ä_x = (1 − A_x) ÷ d when only A_x is given.

Variance for whole life

Var(L) = (S + (P − f)/d)² × [²A_x − (A_x)²]

The initial expense I and other constants do not affect the variance. ²A_x is A_x calculated at force of interest 2δ, that is at rate of interest 2i + i².

Net premium special case (no expenses, equivalence premium)

Var(L) = (S ÷ (1 − A_x))² × [²A_x − (A_x)²]

Applies only when I = 0, f = 0 and P = S A_x ÷ ä_x, so that S + P/d = S ÷ (1 − A_x).

How to solve Mean and Variance of Future Loss questions

1. 1Write down the contract: benefit amount, when it is paid, premium term, and premium timing (in advance or arrears).
2. 2List the expenses and when they occur. Separate the initial expense at time 0 from the regular expense at each premium date.
3. 3Write L in words, then in symbols: PV of benefits + PV of expenses − PV of premiums, using K+1 or T as the random time.
4. 4For the mean, replace each random present value by its expected value, for example v^(K+1) by A_x and ä_{K+1|} by ä_x. If the question asks for the premium, set the mean to zero and solve.
5. 5For the variance, rewrite L as (constant) × v^(K+1) + (another constant). For a whole life policy use ä_{K+1|} = (1 − v^(K+1)) ÷ d.
6. 6Compute Var(L) = (coefficient of v^(K+1))² × [²A_x − (A_x)²]. Take ²A_x at the doubled force of interest, that is the rate 2i + i², unless the question gives it.
7. 7Check the result: the variance must be positive, the units are rupees squared, and the standard deviation should be a sensible size compared with S. State the standard deviation if asked.
8. 8Write the notation, the working and the final answer clearly, with any assumptions stated.

Quickest way: Coefficient method for whole life variance

Common mistakes in Mean and Variance of Future Loss

• Using (A_x)² in place of ²A_x, or forgetting to subtract (A_x)².

  The notations look alike and students rush to the formula.

  Fix: Remember that ²A_x is the EPV of v^(2(K+1)), a number you are given or compute at the rate 2i + i². Always write ²A_x − (A_x)² as the variance of v^(K+1).

• Forgetting to square the coefficient of v^(K+1).

  Students remember that variance is linked to the multiplier but treat it like the mean.

  Fix: Var(c X + k) = c² Var(X). Write c² explicitly before you multiply.

• Putting the initial expense or other constants into the coefficient.

  Students include every term in the loss when they build the coefficient.

  Fix: Only terms that multiply v^(K+1) belong in the coefficient. The initial expense at time 0 is a constant and drops out of the variance, though it stays in the mean.

• Using ä_x in the variance in place of the rewritten form, or mixing up the sign of P − f.

  Students try to apply a variance rule directly to ä_{K+1|}, which is itself random and linked to v^(K+1).

  Fix: Always substitute ä_{K+1|} = (1 − v^(K+1)) ÷ d first. The premium net of expense, P − f, adds to S in the coefficient for a loss written as outgo minus income.

• Treating a non-zero mean as an error when the premium is not the equivalence premium.

  Students expect E[L] to always be zero.

  Fix: E[L] = 0 only when P is the equivalence premium. If P is given, E[L] can be positive or negative.

• Using the wrong interest rate for ²A_x, for example doubling i to 2i.

  Students confuse doubling the force of interest with doubling the rate.

  Fix: Doubling δ gives v² = e^(−2δ), so the rate is 2i + i² (because (1 + i)² = 1 + 2i + i²). Use that when ²A_x must be recomputed.

Worked examples