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Actuarial Mathematics for Modelling · Gross random future loss

Gross Future Loss Random Variable with Expenses

Updated 11 October 2026 · Fact-checked

The gross future loss is the present value of future benefits, expenses and commission minus the present value of gross premiums, all depending on future lifetime. It differs from the net future loss because it includes expenses. Fix the timings, discount each cash flow, then take the expectation if asked.

Understand Gross Premium Loss Including Expenses

The net future loss looks only at benefits and net premiums. It ignores the cost of running the policy. A real insurer pays for selling the policy, collecting premiums and settling claims. The gross future loss adds these costs.

The gross future loss at policy start is a random variable. It is the present value of all outgo minus the present value of all premium income. Outgo now has four parts: the benefit, initial expenses, renewal expenses and claim expenses. Commission is also outgo. It is usually a percentage of premium, higher in the first year and lower afterwards.

Each part has its own timing. Initial expenses and initial commission are paid at time 0, with the first premium. Renewal expenses and renewal commission are paid with each later premium. Claim expenses are paid when the benefit is paid, so they are discounted like the benefit. The timing of every item depends on the future lifetime, so each item is a random variable.

The gross premium is the premium charged to the policyholder, so it is the P in this loss. Under the equivalence principle you set E[L] = 0 and solve for P. The result is always higher than the net premium, because expenses must be recovered.

A useful check: the expected gross loss at time 0 equals the expected outgo minus the expected premiums. If expenses are zero, it reduces to the net future loss. Use this to test your setup.

Key rules to remember

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.

How to solve Gross Premium Loss Including Expenses questions

This method works for any gross future loss question: whole life, term, endowment, level or single premium.

  1. 1Write down the contract: benefit, term, premium pattern, and when the benefit is paid (end of year of death or moment of death).
  2. 2List every cost: initial expense, renewal expense, claim expense, initial commission and renewal commission. Note whether each is a fixed amount or a percentage of premium.
  3. 3Fix the timing of each item. Initial items go at time 0. Renewal items go at each premium date after the first. Claim expense goes with the benefit.
  4. 4Write the loss L(g) as a present value in terms of K (or T). Count payments carefully: if death occurs in year K+1, premiums are paid at times 0 to K, so there are K+1 of them and K renewals.
  5. 5Group the terms. Collect all the P terms together, so that the loss reads: outgo not linked to P, plus P times a factor.
  6. 6If asked for a specific outcome, put in the value of K and compute the numbers using v^(K+1) and ä(K+1).
  7. 7If asked for the expected value, replace v^(K+1) by A(x) and ä(K+1) by ä(x). If asked for the premium, set the expected value to 0 and solve.
  8. 8Check: with expenses set to zero, you should recover the net future loss. Also check that the gross premium exceeds the net premium.

Quickest way: Build the loss in three blocks

When to use it: Use this when time is short, for expected loss or premium questions with a standard whole life or term contract.

  1. Block 1: benefit and claim expense. Write (S + C) times the assurance factor.
  2. Block 2: expenses and commission. Write the time 0 items, then (renewal items) times (ä − 1).
  3. Block 3: premium income. Write P times ä.
  4. Add blocks 1 and 2, subtract block 3. This is E[L].
  5. For the premium, put E[L] = 0, move all P terms to one side, and divide.
  6. Check the sign: gross premium must be above the net premium for the same basis.

Common mistakes in Gross Premium Loss Including Expenses

  • Applying renewal expenses at time 0 as well, or using ä instead of ä − 1.

    Renewal expenses are linked mentally to every premium, including the first.

    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.

    It is listed as an expense, so students treat it with the other expenses.

    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).

  • Treating commission as a fixed rupee amount when it is a percentage of premium.

    Students copy the expense pattern for fixed amounts.

    Fix: Write commission as αP and βP. These terms go in the P factor when you solve for P.

  • Using the net premium in the gross loss.

    The net loss is learned first and the premium symbol carries over.

    Fix: The premium in the gross loss is the gross premium. If the question gives a net premium, it is for a different calculation.

  • Miscounting the number of premiums when K is given.

    Confusing K (completed years) with K+1 (year of death).

    Fix: Death in year K+1 means premiums at times 0, 1, ..., K, which is K+1 payments, ä(K+1). Draw a short timeline.

  • Using the net premium basis for the gross loss without checking the question's assumptions.

    Students reuse earlier mortality or interest values.

    Fix: Always state the assumptions and use the basis given for this question. Expenses and the interest rate for pricing can differ from the reserving basis.

Worked examples

Example 1

A whole life policy is issued to a life aged x. Sum assured ₹10,00,000, paid at the end of the year of death. Gross annual premium ₹30,000 payable in advance throughout life. Initial expenses ₹5,000 at issue. Initial commission is 40% of the first premium. Renewal expenses are 3% of each premium from the second year onwards. Claim expense is ₹2,000, paid with the benefit. Assume d = 0.04, A(x) = 0.40 and ä(x) = 15. (a) Find the expected gross future loss at issue. (b) Find the gross future loss if death occurs in the third policy year.

Show the solution
  1. (a) Write the expected loss: E[L] = (S + C) A(x) + I + αP + eP' (ä(x) − 1) − P ä(x), where the renewal expense is 0.03 × 30,000 = ₹900 a year.
  2. Benefit and claim expense: (10,00,000 + 2,000) × 0.40 = ₹4,00,800.
  3. Initial items: 5,000 + 0.40 × 30,000 = 5,000 + 12,000 = ₹17,000.
  4. Renewal expenses: 900 × (15 − 1) = ₹12,600.
  5. Premium income: 30,000 × 15 = ₹4,50,000.
  6. E[L] = 4,00,800 + 17,000 + 12,600 − 4,50,000 = −₹19,600.
  7. (b) Death in the third year means K = 2. Premiums are paid at times 0, 1, 2. Renewal expenses are at times 1 and 2. The benefit is paid at time 3.
  8. v = 1 − d = 0.96, so v^3 = 0.884736. ä(3) = (1 − 0.884736) ÷ 0.04 = 2.8816.
  9. Benefit and claim expense: 10,02,000 × 0.884736 = ₹8,86,505.47.
  10. Initial items: ₹17,000.
  11. Renewal expenses: 900 × (2.8816 − 1) = 900 × 1.8816 = ₹1,693.44.
  12. Premium income: 30,000 × 2.8816 = ₹86,448.
  13. L = 8,86,505.47 + 17,000 + 1,693.44 − 86,448 = ₹8,18,750.91.

Answer: (a) Expected gross future loss = −₹19,600, so the policy is expected to make a small gain on this basis. (b) If death occurs in the third year, the loss is about ₹8,18,751.

Example 2

Using the same policy and basis as above (sum assured ₹10,00,000, initial expenses ₹5,000, initial commission 40% of the first premium, renewal expenses 3% of premium from year 2, claim expense ₹2,000, A(x) = 0.40, ä(x) = 15), find the gross annual premium P using the equivalence principle.

Show the solution
  1. Set E[L] = 0 with P unknown.
  2. Benefit and claim expense: 10,02,000 × 0.40 = ₹4,00,800.
  3. Initial expense: ₹5,000.
  4. Initial commission: 0.40P.
  5. Renewal expenses: 0.03P × (15 − 1) = 0.42P.
  6. Premium income: 15P.
  7. Equation: 4,00,800 + 5,000 + 0.40P + 0.42P − 15P = 0.
  8. Collect terms: 4,05,800 = (15 − 0.82)P = 14.18P.
  9. P = 4,05,800 ÷ 14.18 = ₹28,617.77 (to the nearest paisa).
  10. Check: the net premium is 10,00,000 × 0.40 ÷ 15 = ₹26,666.67. The gross premium is higher, as expected.

Answer: The gross annual premium is about ₹28,618.

Exam tips

  • 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.
  • When solving for P, collect all P terms, including commission and percentage expenses, before dividing.
  • In multiple-choice questions, check the sign and size first. The gross premium must exceed the net premium, and expected gross loss at the equivalence premium is zero.

Practice questions from Gross random future loss

Gross Premium Loss Including Expenses: frequently asked questions

What is the difference between net and gross future loss?

The net future loss includes only the benefit and the net premium. The gross future loss also includes initial, renewal and claim expenses and commission, and uses the gross premium. Set expenses to zero and the gross loss becomes the net loss.

Where does commission go in the gross future loss?

Commission is outgo, so it is added to the present value of costs. It is usually a percentage of premium, paid at the same time as each premium. Initial commission is paid at time 0 and renewal commission at later premium dates.

Are claim expenses discounted like the benefit?

Yes. Claim expenses are paid when the benefit is paid, so they use the same discount factor. It is simplest to add them to the sum assured and discount the total.

How do I find the gross premium from the future loss?

Write the expected gross loss in terms of P and set it to zero (the equivalence principle). Collect all the P terms, including percentage commission and expenses, and divide. Always check that the answer is above the net premium.