Skip to content

Actuarial Mathematics for Modelling · Gross random future loss

Loss Variables for Common Contracts: Gross Future Loss

Updated 11 October 2026 · Fact-checked

The gross future loss is the present value of future benefits and expenses minus the present value of future premiums, written as a function of the random future lifetime K or T. For each contract, list the cash flows for each possible death time, discount them, and subtract premiums. Then split into cases where needed.

Understand Loss Variables for Common Contracts

A policy's cash flows depend on when the life dies. So the present value of those cash flows is a random variable. The gross future loss L is that present value, measured from the insurer's side: outgo minus income.

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

A positive L is a loss to the insurer. A negative L is a profit. Expenses are included because this is the gross version. The net version has no expenses.

The random input is the future lifetime. In the discrete case you use the curtate future lifetime K, the whole number of complete years lived. If the benefit is paid at the end of the year of death, the payment is at time K + 1. In the continuous case you use the complete future lifetime T, and the benefit is paid at the moment of death, at time T.

To build L for any contract, ask two questions. When is each payment made, and does it depend on survival or on death? Premiums and annuity payments are made while the life is alive. Death benefits are made at death. Endowment and term contracts also have a time n after which the pattern changes, so you usually write L in two cases: K < n and K ≥ n.

An annuity-certain over m years has the present value ä_m| = (1 − v^m) ÷ d, with v = 1 ÷ (1 + i) and d = i ÷ (1 + i). Premiums paid in advance for K + 1 years are worth P × ä_(K+1)|. Use this to write L in closed form for each case.

Key rules to remember

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.

How to solve Loss Variables for Common Contracts questions

Use this method for any gross future loss question. Work from the timeline, not from memory of a formula.

  1. 1Write down the contract: sum assured or annuity amount, term n, premium pattern, when the benefit is paid, and the interest rate.
  2. 2Choose the random variable: K for annual-in-advance or end-of-year payments, T for benefits at the moment of death or for continuous premiums.
  3. 3Draw a timeline from 0 and mark the benefit, each expense and each premium. Note which ones depend on survival and which on death.
  4. 4Find the cases. Whole life has one case. Term and endowment have two: death before n and survival to n. A deferred annuity has two: death before the deferment and survival beyond it.
  5. 5Write the present value of each benefit, each expense and each premium in that case. Count the number of payments carefully. A death in year K + 1 means premiums at times 0 to K.
  6. 6Combine as outgo minus income. Use annuity-certain functions for premiums and for expense strings.
  7. 7If numbers are given, substitute v and ä_m| and check the sign: a negative result is a profit.
  8. 8Check each case once for reasonableness. For example, a longer life should mean more premiums received.

Quickest way: Case table by death time

When to use it: Use this when you must write L symbolically or evaluate it for a given K under time pressure, especially in the multiple-choice section.

  1. Write the benefit and its payment time first. Death benefit: time K + 1 or T. Survival benefit: time n.
  2. Write the number of premiums: K + 1 if death occurs before premiums stop, otherwise the full count (n for term and endowment).
  3. Put the result in the form benefit − P × ä_(number of payments)|. Add expenses as extra terms in the same shape.
  4. For a given K, put the numbers into v^(K+1) and ä_(K+1)| and compute once. Reuse the powers of v for the premium and expense strings.

Common mistakes in Loss Variables for Common Contracts

  • Using v^K instead of v^(K+1) for a benefit paid at the end of the year of death.

    K is the number of complete years lived, so students discount for K years.

    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)|.

    Premiums are paid in advance, so the premium at time 0 counts but there is none at the time of the death payment.

    Fix: List the premium times 0, 1, ..., K and count them. Do this on the timeline every time.

  • Continuing premiums after the end of the term for a term or endowment contract.

    The whole life form is copied without the cap at n.

    Fix: For K ≥ n use ä_n|, not ä_(K+1)|. Write the two cases explicitly.

  • Leaving out the survival benefit in an endowment when K ≥ n, or paying it at time K + 1.

    The term assurance pattern is copied by mistake.

    Fix: The endowment survival benefit is S v^n, paid at time n regardless of K once K ≥ n.

  • Reversing the sign of L, so that premiums are positive and benefits negative.

    The loss is mixed up with a profit or with the insurer's cash flow.

    Fix: L is outgo minus income. Benefits and expenses are positive, premiums negative. State this at the start of the answer.

  • Mixing discrete and continuous quantities, for example S v^T with ä, or S v^(K+1) with ā.

    The formulas look alike and the payment basis is not read from the question.

    Fix: Match the variable to the payment basis. T goes with v^T and ā_T|. K goes with v^(K+1) and ä_(K+1)|.

Worked examples

Example 1

A 10-year endowment assurance has a sum assured of ₹5,00,000 paid at the end of the year of death, or at the end of year 10 on survival. The annual premium is ₹40,000, paid at the start of each year while the policyholder is alive and within the term. Expenses are ₹1,000 at the start and ₹200 at the start of each later policy year while the contract is in force. Interest is 6% per year. (a) Write the gross future loss in terms of K. (b) Evaluate it if the policyholder dies in policy year 4. (c) Evaluate it if the policyholder survives the full term.

Show the solution
  1. (a) For K < 10: L = 5,00,000 v^(K+1) + 1,000 + 200 (v + ... + v^K) − 40,000 ä_(K+1)|. Expenses are at times 0, 1, ..., K, which is 1,000 at time 0 and 200 at times 1 to K.
  2. For K ≥ 10: L = 5,00,000 v^10 + 1,000 + 200 (v + ... + v^9) − 40,000 ä_10|.
  3. (b) Death in year 4 means K = 3. Use v = 1 ÷ 1.06. Then v = 0.943396, v² = 0.889996, v³ = 0.839619, v⁴ = 0.792094.
  4. ä_4| = 1 + 0.943396 + 0.889996 + 0.839619 = 3.673011.
  5. Benefit: 5,00,000 × 0.792094 = ₹3,96,047.
  6. Expenses: 1,000 + 200 × (0.943396 + 0.889996 + 0.839619) = 1,000 + 200 × 2.673011 = ₹1,534.60.
  7. Premiums: 40,000 × 3.673011 = ₹1,46,920.40.
  8. L = 3,96,047 + 1,534.60 − 1,46,920.40 = ₹2,50,661 (rounded).
  9. (c) Survival: v^10 = 0.558395. d = 0.06 ÷ 1.06 = 0.0566038. ä_10| = (1 − 0.558395) ÷ 0.0566038 = 7.801692.
  10. Benefit: 5,00,000 × 0.558395 = ₹2,79,197.50.
  11. Expenses: 1,000 + 200 × (7.801692 − 1) = 1,000 + 1,360.34 = ₹2,360.34.
  12. Premiums: 40,000 × 7.801692 = ₹3,12,067.70.
  13. L = 2,79,197.50 + 2,360.34 − 3,12,067.70 = −₹30,509.86.

Answer: (a) L = 5,00,000 v^(K+1) + 1,000 + 200 (v + ... + v^K) − 40,000 ä_(K+1)| for K < 10, and L = 5,00,000 v^10 + 1,000 + 200 (v + ... + v^9) − 40,000 ä_10| for K ≥ 10. (b) For death in year 4, L ≈ ₹2,50,661, a loss. (c) For survival to 10 years, L ≈ −₹30,510, a profit.

Example 2

A whole life annuity-due of ₹60,000 per year is bought by a single premium of ₹8,00,000. The insurer incurs an initial expense of ₹16,000 at the start. There are no other expenses. Interest is 5% per year. (a) Write the gross future loss in terms of K. (b) Evaluate it if the annuitant dies in policy year 15, so K = 14.

Show the solution
  1. (a) Payments of ₹60,000 are made at times 0, 1, ..., K, while the annuitant is alive. That is K + 1 payments.
  2. PV of benefits = 60,000 ä_(K+1)|. The expense is ₹16,000 at time 0. The single premium of ₹8,00,000 is received at time 0.
  3. L = 60,000 ä_(K+1)| + 16,000 − 8,00,000.
  4. (b) K = 14 gives 15 payments. v = 1 ÷ 1.05, so v^15 = 0.481017. d = 0.05 ÷ 1.05 = 0.047619.
  5. ä_15| = (1 − 0.481017) ÷ 0.047619 = 0.518983 ÷ 0.047619 = 10.89864.
  6. PV of benefits = 60,000 × 10.89864 = ₹6,53,918.
  7. L = 6,53,918 + 16,000 − 8,00,000 = −₹1,30,082.

Answer: (a) L = 60,000 ä_(K+1)| + 16,000 − 8,00,000. (b) For K = 14, L ≈ −₹1,30,082, a profit to the insurer, because the annuitant died early.

Exam tips

  • Always state the random variable and the payment timing first, for example: K is the curtate future lifetime and the death benefit is paid at the end of the year of death. This earns method marks even if the arithmetic fails.
  • Write term and endowment answers as two cases. Examiners expect K < n and K ≥ n shown separately.
  • In multiple-choice questions, count the premiums before calculating. Many wrong options differ only by one premium or one year of discounting.
  • State the sign convention: L is outgo minus income. Say so in one line.
  • For written and computer-based questions, show the formula in standard notation, the working with v and ä, and the final rupee value with its sign. Check that a loss and a profit are described correctly.

Practice questions from Gross random future loss

Loss Variables for Common Contracts: frequently asked questions

What is the future loss random variable in actuarial maths?

It is the present value at the policy start of the insurer's future outgo minus income, treated as a function of the policyholder's random future lifetime. Positive means a loss to the insurer. Negative means a profit.

What is the difference between gross and net future loss?

The gross future loss includes expenses, and it uses the actual (office) premium. The net future loss includes only benefits and premiums, and is used with the net premium. The structure of the two is otherwise the same.

Why do we use v^(K+1) and not v^K for the death benefit?

K is the number of complete years lived. If the benefit is paid at the end of the year of death, that is at time K + 1. So the discount factor is v^(K+1). If the benefit is paid at the moment of death, you use T and v^T instead.

How do I write the future loss for a term assurance with level premiums?

For K < n, L = S v^(K+1) − P ä_(K+1)|. For K ≥ n, L = − P ä_n|, since there is no benefit and n premiums are paid. Add any expenses at their payment times in each case.

How does the future loss for an annuity differ from an assurance?

For an annuity the outgo is a stream of payments while the annuitant lives, so the benefit is R ä_(K+1)| for an annuity-due. The premium is usually a single amount at time 0, so the loss is the annuity value plus expenses minus the premium.