Insurance Contracts as Cashflow Models: Premiums, Benefits and Expenses

Understand Insurance Contracts as Cashflow Models

Key rules to remember

Present value of a cashflow

PV = C × v^t, where v = 1 ÷ (1 + i)

C is the amount paid at time t. i is the effective annual interest rate.

Expected present value of a contingent cashflow

EPV = C × v^t × P(event occurs at or by time t as specified)

Use the probability of the exact condition for payment, for example survival to time t.

Survival-contingent payment at time t

EPV = C × v^t × tpx

tpx is the probability that a life now aged x survives t years. Use for annuity payments and premiums.

Death benefit paid at end of year of death

EPV = S × Σ v^(k+1) × kpx × qx+k, summed from k = 0 to n − 1

This is a term assurance with sum assured S and term n years. Payment is at the end of the year of death.

Equivalence principle

EPV(premiums) = EPV(benefits) + EPV(expenses)

Gives the premium when there is no profit loading. Use the same basis for all three terms.

Net cashflow at time t

NCFt = premiums − benefits − expenses (at time t)

Used for projecting cashflows. The expected value allows for the probability the policy is in force.

How to solve Insurance Contracts as Cashflow Models questions

1. 1Identify the contract type: assurance, annuity, endowment or general insurance policy. Note the term, age and sum assured or annuity amount.
2. 2List each cashflow type: premiums, benefits and expenses. Separate in-flows from out-flows.
3. 3For each cashflow, write down the timing: start or end of year, or continuous. Check whether premiums are in advance.
4. 4For each cashflow, write the condition that triggers it: survival, death within a given year, or a claim occurring.
5. 5Find the probability of each condition using the given mortality basis or claim assumptions.
6. 6Multiply amount × discount factor × probability for each time. Sum the terms to get the EPV.
7. 7Apply the equivalence principle or compare EPVs as the question asks. State the basis and assumptions.
8. 8Check that the answer is sensible: premiums should be positive and smaller than the total sum assured for a short term.

Quickest way: Timeline and condition table

Common mistakes in Insurance Contracts as Cashflow Models

• Discounting a death benefit by v^k instead of v^(k+1).

  The index k counts completed years before death, so students forget that payment is at the end of the year of death.

  Fix: Death in year k+1 means payment at time k+1 when paid at year end. Write the time next to every term.

• Using the probability of death at time t when the cashflow needs survival to time t.

  Students mix up premium and annuity payments, which need survival, with death benefits.

  Fix: Ask 'what must be true for this payment to happen?' Use survival probabilities for premiums and annuities.

• Paying a premium in the year of death when premiums are in advance and there was no payment due.

  Timing of premiums and benefits within the year is overlooked.

  Fix: Premiums in advance are paid at the start of the year if the life is alive then. The death benefit is paid later.

• Leaving out expenses or including them at the wrong time.

  Students focus on premiums and benefits only.

  Fix: Read the expense wording. Initial expenses occur at time 0, renewal expenses at each premium date, and claim expenses with the benefit.

• Treating general insurance like life insurance with long-term mortality discounting.

  Students apply the same template without checking the contract.

  Fix: For general insurance, model claim frequency and claim size, and note that the term is short. Discount only if claims settle late.

• Mixing bases, such as using one interest rate for premiums and another for benefits.

  Different parts of the question give different rates.

  Fix: Use one stated basis for the whole equation of value unless the question tells you otherwise.

Worked examples