Gross random future loss: formula sheet

Key formulas

Future loss at time t

L_t = PV of future benefits and expenses − PV of future premiums

All present values are taken at time t, on the stated basis, for a policy still in force at t.

Whole life assurance, sum assured S, death benefit paid at end of year of death, annual premium P in advance, initial expense e0, renewal expense e per year

L_0 = S v^(K_x + 1) + e0 + e ä_(K_x + 1) − P ä_(K_x + 1)

Here K_x is curtate future lifetime. Premiums are paid at times 0, 1, ..., K_x, which is K_x + 1 payments. Renewal expenses are assumed to be paid with each premium, including the first, unless e0 is stated to replace it.

Whole life assurance, benefit paid at moment of death

L_0 = S v^(T_x) + expenses − P ä_(⌈T_x⌉)

Premiums are annual in advance, so the number of payments is the number of policy years started, ⌈T_x⌉ (T_x rounded up). If premiums are payable continuously, use ā_(T_x) instead.

n-year term assurance

L_0 = S v^(K_x + 1) − P ä_(K_x + 1) if K_x < n; L_0 = − P ä_n if K_x ≥ n

Benefit is paid only on death within the term. Premiums stop at death or at time n.

n-year endowment assurance

L_0 = S v^(K_x + 1) − P ä_(K_x + 1) if K_x < n; L_0 = S v^n − P ä_n if K_x ≥ n

The sum assured is paid either on death in the term or on survival to n.

Immediate life annuity-due of 1 pa, premium paid as a single sum π

L_0 = ä_(K_x + 1) − π

A single premium has no premium annuity. It is just π at time 0.

Loss at later time t

L_t uses K_(x+t) or T_(x+t) in place of K_x or T_x, and discounts to time t

Condition: the policy is in force at t, so the life has survived to age x + t.

Gross future loss (general)

L(g) = PV(benefits) + PV(claim expenses) + PV(initial and renewal expenses) + PV(commission) − PV(gross premiums)

Every PV is a random variable that depends on future lifetime. Use this as your starting point.

Whole life, annual premium in advance, benefit at end of year of death

L(g) = (S + C) v^(K+1) + I + αP + (e + βP) (ä(K+1) − 1) − P ä(K+1)

S = sum assured, C = claim expense, I = initial expense, α = initial commission rate, e = renewal expense per year, β = renewal commission rate, P = gross annual premium, K = curtate future lifetime. Renewals fall at times 1, 2, ..., K.

Expected gross loss, whole life

E[L(g)] = (S + C) A(x) + I + αP + (e + βP)(ä(x) − 1) − P ä(x)

Use E[v^(K+1)] = A(x) and E[ä(K+1)] = ä(x). Valid for the stated payment timings only.

Equivalence principle premium

P = [(S + C) A(x) + I + e (ä(x) − 1)] ÷ [ä(x) − α − β (ä(x) − 1)]

Set E[L(g)] = 0 and collect the P terms: P [ä(x) − α − β (ä(x) − 1)] = (S + C) A(x) + I + e (ä(x) − 1). Renewal commission β applies from year 2, so it multiplies ä(x) − 1. It is safer to rebuild this from E[L(g)] = 0 than to memorise it.

Relationship for annuity-due and assurance

A(x) = 1 − d ä(x)

Use it to find A(x) from ä(x) or the reverse, and to form ä(K+1) = (1 − v^(K+1)) ÷ d.

Annuity-due certain, n payments

ä(n) = (1 − v^n) ÷ d

Use this to value premiums and renewals for a given K.

Gross future loss (general)

L = PV(benefits) + PV(expenses) − PV(premiums)

Outgo minus income. A positive value is a loss to the insurer.

Whole life, sum assured S, death benefit at end of year of death, annual premium P in advance

L = S v^(K+1) − P ä_(K+1)| + expenses

Premiums are paid at times 0, 1, ..., K, which is K + 1 payments. Add expenses at their own payment times.

n-year term assurance, annual premium P in advance

L = S v^(K+1) − P ä_(K+1)| for K < n; L = − P ä_n| for K ≥ n

If the life survives n years there is no benefit. Premiums stop at time n − 1, so there are n premiums.

n-year endowment assurance, annual premium P in advance

L = S v^(K+1) − P ä_(K+1)| for K < n; L = S v^n − P ä_n| for K ≥ n

The survival benefit is paid at time n. The death benefit is paid at time K + 1.

Whole life, benefit at moment of death, continuous premium rate P̄

L = S v^T − P̄ ā_T|

ā_T| = (1 − v^T) ÷ δ, where δ is the force of interest.

Whole life annuity-due of R per year, single premium π

L = R ä_(K+1)| + expenses − π

Payments are made at times 0 to K, which is K + 1 payments, if the life survives to each date.

Deferred annuity-due of R per year from time n, single premium π

L = R v^n ä_(K+1−n)| − π for K ≥ n; L = − π for K < n

Pay nothing if death occurs before time n. Add expenses at their own times.

Annuity-certain in advance

ä_m| = (1 − v^m) ÷ d

d = i ÷ (1 + i) is the effective annual rate of discount.

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).

Future loss

L = PV of outgo − PV of income

Define it for one policy at the start date. Include expenses if the question is a gross premium one.

Probability of loss

Pr(loss) = Pr(L > 0)

A loss means L is strictly positive. L = 0 is not a loss.

Percentile premium principle

smallest P such that Pr(L > 0) ≤ α, equivalently Pr(L ≤ 0) ≥ 1 − α

α is given in the question, for example 0.05 for a 5% chance of loss.

Cut-off lifetime (L decreasing in T)

Pr(L > 0) = Pr(T < t₀), where L(t₀) = 0

Solve L = 0 for t₀. Then use Pr(T < t₀) = 1 − ₜ₀pₓ.

Constant force of mortality

Pr(T < t) = 1 − e^(−μt)

Use only when μ is constant.

Normal approximation for L

Pr(L > 0) ≈ 1 − Φ((0 − E[L]) ÷ √Var(L))

Use only when told to, or for a large portfolio where the sum of losses is close to normal.

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.

Quick revision

• L = PV future benefits + PV future expenses − PV future gross premiums.
• Premium by the equivalence principle: set E[L at time 0] = 0.
• Gross premium reserve at time t = E[L at t | alive at t].
• Use T for continuous and K for curtate future lifetime, and match the formula to the payment timing.
• Premiums are paid in advance as an annuity-due while the policy is in force, so stop them at death or the end of the premium term.
• For term contracts, L has separate expressions for death in the term and survival past it.
• For a level contract, L is a linear function of v^T (or v^(K+1) for discrete payments). For a whole life assurance with annual premiums in advance, in the net-premium, no-expense case with unit sum assured, Var(L) = (1 + P/d)² [²A − (A)²], where ²A is evaluated at double the force of interest.
• For gross loss with expenses, the coefficient changes. The next formula applies only to a whole life assurance with level annual premiums payable in advance for life, where the sum assured and the claim expense are both paid at the end of the year of death, and d is the annual discount rate. With sum assured S, claim expense e_c, gross premium P_g and no renewal expenses, Var(L) = (S + e_c + P_g/d)² [²A − (A)²]. If there is a renewal expense e_r with each premium, use P_g − e_r in place of P_g. Initial expenses are a constant and do not affect the variance. For term or endowment contracts this formula does not hold. Derive Var(L) from the piecewise form of L instead.
• Percentile premium: choose P so that Pr(L > 0) equals the stated probability of loss.
• Because L usually falls as death time increases, Pr(L > 0) becomes a probability of death before a critical time.
• Include expenses at the right dates: initial at time 0, renewal with each premium, claim expenses with the benefit.
• State assumptions on mortality, interest and expenses clearly in written answers.

Common mistakes

• Writing the loss as premiums minus benefits. Fix: Remember loss = outgo − income. Write "PV benefits + PV expenses − PV premiums" every time.
• Using the wrong number of premiums, such as ä_(K_x) instead of ä_(K_x + 1). Fix: If death occurs in year K_x + 1, premiums are paid at times 0 to K_x. That is K_x + 1 payments.
• Applying renewal expenses at time 0 as well, or using ä instead of ä − 1. Fix: Renewal items start at time 1. Use (ä − 1) for the first-year-excluded annuity. Check which year initial items cover.
• Discounting the claim expense separately or leaving it out of the benefit discounting. Fix: Claim expenses are paid at the same time as the benefit. Combine them as (S + C) before applying v^(K+1) or A(x).
• Using v^K instead of v^(K+1) for a benefit paid at the end of the year of death. Fix: If death occurs in year K + 1, the end-of-year payment is at time K + 1. Use v^(K+1), and use K + 1 premiums in advance.
• Using too many or too few premiums, for example ä_K| instead of ä_(K+1)|. Fix: List the premium times 0, 1, ..., K and count them. Do this on the timeline every time.
• Using (A_x)² in place of ²A_x, or forgetting to subtract (A_x)². 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). Fix: Var(c X + k) = c² Var(X). Write c² explicitly before you multiply.
• Using the equivalence premium (E[L] = 0) when the question asks for a percentile premium. Fix: Read the wording. If it mentions a probability of loss or a percentage, set Pr(L > 0) ≤ α and ignore E[L].
• Reversing the inequality, using T > t₀ for a loss. Fix: Test a very small T. If L is positive there, loss happens for T below the cut-off.

Exam tips

• Define K_x or T_x in your first line. Marks are often given for correct definitions.
• Always write the loss in cases when the contract has a term. Examiners expect the death case and the survival case.
• State the time of valuation and 'given alive at time t' when t > 0.
• In MCQs, check the number of premiums first. Wrong counts are the commonest source of wrong options.
• In computer-based questions, show the formula in notation before the code or spreadsheet formula, then the numerical result.
• Draw a timeline before writing anything. Mark time 0, premium dates, and the benefit date. Most lost marks come from timing errors.
• In written answers, write the loss random variable in full before taking expectations. Examiners award marks for the setup, even if the arithmetic goes wrong.
• State your assumptions: when the benefit is paid, how commission is defined, and which basis you use. Do this in one line.