Insurance Companies and Pension Plans: formula sheet

Key formulas

Balance sheet identity

Assets = Reserves (liabilities) + Capital (equity)

Capital is the buffer. A fall in asset value or a rise in reserves reduces capital one for one.

Underwriting result

Underwriting profit = Premiums earned − Claims incurred − Expenses

A negative result can still be offset by investment income.

Overall profit

Total profit = Underwriting profit + Investment income

Explains why insurers can write at an underwriting loss when float is invested.

Reserve for a life policy

Reserve = PV(future benefits) − PV(future premiums)

Discount rate matters. A lower discount rate raises reserves.

Net retained risk with proportional reinsurance

Retained loss = (1 − cession share) × Gross loss

Applies to quota-share reinsurance only.

One-year survival probability

p(x) = 1 − q(x)

q(x) is the table probability that a person aged x dies within one year.

Survival to the end of year t

S(t) = (1 − q₀) × (1 − q₁) × ... × (1 − q(t−1))

Multiply the one-year survival probabilities. Here q₀ is for the starting age, q₁ for the next age, and so on.

Probability of dying in year t (t = 1, 2, ...)

P(death in year t) = S(t−1) × q(t−1)

You must survive to the start of the year, then die during it.

Expected PV of a death benefit (benefit paid at end of year of death)

EPV = Σ B × P(death in year t) ÷ (1 + r)^t

This is the net single premium for a term policy, ignoring costs and profit.

Level annual premium, paid at start of each year while alive

P = EPV of benefits ÷ Σ [S(t−1) ÷ (1 + r)^(t−1)]

The denominator is the expected PV of $1 paid at the start of each year, starting with 1 at time 0.

Expected PV of an annuity-immediate

EPV = Σ C × S(t) ÷ (1 + r)^t

Payment C is made at the end of year t only if the annuitant is alive then.

Loss ratio

Loss ratio = (Incurred losses + Loss adjustment expenses) ÷ Net premiums earned

Some questions use losses only. Use the definition given in the question.

Expense ratio

Expense ratio = Underwriting expenses ÷ Premiums

The denominator can be premiums written or earned. Use the one the question states. Check the wording.

Combined ratio

Combined ratio = Loss ratio + Expense ratio

Below 100% = underwriting profit. Above 100% = underwriting loss.

Operating ratio

Operating ratio = Combined ratio − Investment income ÷ Premiums

Shows overall profitability including investment income. Below 100% = profit overall.

Underwriting profit

Underwriting profit = Premiums × (1 − Combined ratio)

Negative result means an underwriting loss.

Claim payment with deductible

Payment = max(Loss − Deductible, 0)

Subject to the policy limit. The policyholder bears the first part of the loss.

Claim payment with coinsurance

Payment = Coinsurance share × Loss

If the insurer pays 80%, the policyholder bears 20%.

Combined deductible, coinsurance and limit

Payment = min(Limit, Insurer share × max(Loss − Deductible, 0))

Apply in this order unless the policy states otherwise: deductible, then share, then cap.

Solvency II SCR calibration

SCR ≈ 99.5% one-year VaR of own funds

Equivalent to a failure probability of about 1 in 200 years.

DB annual pension (final salary)

Annual pension = accrual rate × years of service × final salary

Accrual rate is a percentage per year of service, for example 1.5% or 2%. Final salary may be an average of the last few years if the question says so.

DB replacement ratio

Replacement ratio = annual pension ÷ final salary

With a flat accrual rate this equals accrual rate × years of service.

DC account value at retirement

FV = Σ contribution_t × (1 + r)^(N − t)

For level end-of-year contributions C: FV = C × [(1 + r)^N − 1] ÷ r.

Risk allocation rule

DB: sponsor bears investment and longevity risk. DC: employee bears them.

Remember the exceptions: DB employees still face sponsor default risk, and DC sponsors face none of the funding risk.

Funded status

Funded status = Plan assets − PV of pension liabilities

Negative means underfunded (a deficit). Positive means overfunded (a surplus).

Funded ratio

Funded ratio = Plan assets ÷ PV of liabilities

Below 1 means underfunded. Above 1 means overfunded.

Present value of liabilities

PV = Σ Bₜ ÷ (1 + r)ᵗ

Bₜ is the expected benefit paid at time t. r is the discount rate, usually a high-quality corporate bond yield.

Liability sensitivity to rates

ΔL ≈ −D × L × Δr

D is the modified duration of the liabilities. A fall in rates raises L.

Change in funded status

ΔSurplus = ΔA − ΔL

Compare asset change with liability change. Use each side's own duration.

Quick revision

• Life insurers cover death and longevity; P&C insurers cover property and liability losses.
• Mutual insurers are owned by policyholders; stock insurers are owned by shareholders.
• Reinsurance lets an insurer transfer part of its risk to another insurer.
• Loss ratio = losses ÷ premiums earned; expense ratio = expenses ÷ premiums.
• Combined ratio = loss ratio + expense ratio; below 100% signals an underwriting profit.
• Insurers can still profit with a combined ratio above 100% if investment income is large enough.
• Mortality tables give death and survival probabilities by age.
• Moral hazard: behaviour changes after cover is bought. Adverse selection: riskier people seek cover before the contract.
• Deductibles, co-payments and risk-based pricing reduce both moral hazard and adverse selection.
• DB plan: sponsor bears investment and longevity risk. DC plan: employee bears them.
• Funded status = plan assets − present value of liabilities.
• A lower discount rate raises the present value of pension liabilities.

Common mistakes

• Saying P&C insurers have long-term liabilities like life insurers. Fix: P&C contracts are usually annual. Some liability claims settle slowly, but the contract itself is short.
• Treating reserves as assets. Fix: Reserves are liabilities, the amount owed to policyholders. Assets are the investments backing them.
• Using q for year 2 or 3 directly as the probability of dying in that year. Fix: Always use P(death in year t) = S(t−1) × q(t−1).
• Discounting a benefit paid at the end of the year of death by the wrong number of years. Fix: Write the timing next to each cash flow. A death benefit in year t is discounted t years. A premium paid at the start of year t is discounted (t − 1) years.
• Treating a combined ratio below 100% as meaning overall profit is certain, or above 100% as meaning overall loss. Fix: Combined ratio speaks only to underwriting. Use the operating ratio for overall profitability.
• Adding investment income to the combined ratio instead of subtracting it. Fix: Investment income lowers the ratio because it offsets costs. Operating ratio = combined − investment income ratio.
• Swapping moral hazard and adverse selection. Fix: Use timing: adverse selection is before the contract, moral hazard is after.
• Applying coinsurance before the deductible. Fix: Subtract the deductible first, then apply the insurer's share, then the limit, unless the question says otherwise.
• Saying the employee bears investment risk in a DB plan. Fix: In DB, the promise is fixed. The sponsor must top up the fund if assets fall short, so the sponsor bears investment risk.
• Forgetting longevity risk in DC plans. Fix: In DC, the employee may outlive the account balance. Longevity risk sits with the employee unless an annuity is bought.

Exam tips

• Expect conceptual questions linking insurer type to risks, asset mix and contract horizon. Learn the comparison cold.
• For balance sheet numbers, always compute capital as assets minus reserves, and watch for changes to only one side.
• Read for hidden details such as reinsurance, mutual structure or investment income before choosing an answer.
• Be careful with classification of health insurance. Use the product description in the question, not assumptions.
• Read the timing words closely: start of year or end of year, and benefit on death or on survival. They change the discount exponent and the probability used.
• Most questions give q values, not survival values. Convert them yourself, and watch for questions that give cumulative survival instead.
• For product questions, link each product to who bears investment risk: insurer (whole life, fixed annuity) or policyholder (variable life, variable annuity).
• Sanity check numbers: a death-benefit premium must be below the benefit, and an annuity value below payment times the number of payments.