Issues and challenges in each main practice area: formula sheet

Key formulas

Persistency rate

Persistency = Policies in force at end of period ÷ Policies in force at start of period

Define the cohort and period clearly. Insurers often track persistency by policy year (for example 13th month, 25th month).

Lapse rate

Lapse rate = Policies lapsed in period ÷ Policies exposed at start of period

For a single period with no other exits, lapse rate = 1 − persistency rate. Deaths and maturities also remove policies, so exclude them or adjust.

Profit margin on premium

Profit margin = Present value of profits ÷ Present value of premiums

Both are discounted at the same rate. Used to compare product designs under different assumptions.

Value of a guarantee (intrinsic)

Intrinsic value at exercise = max(Guaranteed benefit − Market-based benefit, 0)

This ignores time value. The full option cost includes time value and needs a stochastic or option-pricing model.

Solvency position

Excess of assets over liabilities = Available capital − Required capital

Insurer is solvent on this test if the result is not negative. Use the regulator's valuation basis, not your own.

Ultimate claims

Ultimate claims = Paid claims + Case reserves + IBNR

IBNR here includes both claims not yet reported and, in many usages, further development on reported claims (IBNER). State which you mean.

Claims ratio (loss ratio)

Claims ratio = Incurred claims ÷ Earned premium

Use earned, not written, premium.

Combined ratio

Combined ratio = Claims ratio + Expense ratio

Above 100% means an underwriting loss before investment income.

Excess of loss recovery

Recovery = min(max(X − R, 0), L)

X is the loss, R the retention and L the limit of the layer.

Quota share recovery

Reinsurer's claim share = q × X

q is the share ceded. Premium is shared in the same proportion, normally with a ceding commission.

Claims inflation effect

Future cost = Current cost × (1 + i)^n

i is the annual claims inflation rate and n the number of years.

Incurred claims ratio

Incurred claims ratio = Incurred claims ÷ Earned premium

A key monitoring measure. Use incurred claims, not only paid claims, and earned premium, not written premium.

Gross premium (expected-cost approach)

Gross premium = (Expected claims cost + Expenses + Profit and contingency loading) per policy, with each loading allowed for explicitly

A simple structure for explaining pricing. Always list the assumptions behind each part.

Claim cost per policy

Expected claim cost = Claim frequency × Average claim size

Useful to separate the effect of utilisation from the effect of medical inflation.

Effect of cost-sharing on insurer payment

Insurer payment = (Claim − Deductible) × (1 − Co-payment %), if Claim > Deductible, otherwise 0

Applies when the deductible is applied first and the co-payment is a percentage of the remaining amount. Check the policy order in the question.

Compound cost growth

Future cost = Current cost × (1 + i)^n

Use for projecting claim costs under a constant annual medical inflation rate i over n years.

Present value of a liability

PV = Σ B(t) × v(t), where v(t) = 1 ÷ (1 + i)^t

B(t) is the expected benefit payment at time t. A higher i gives a lower PV.

Funding level

Funding level = Assets ÷ Value of liabilities

Below 100% means a deficit on the chosen basis. The result depends on the basis used.

Surplus or deficit

Surplus (deficit) = Assets − Liabilities

A negative answer is a deficit. Always state the valuation basis.

Final salary DB benefit (typical form)

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

For example, 1/60 × 30 × final salary. Scheme rules decide the exact formula.

DC fund at retirement (level contribution at end of each year)

Fund = C × s(n at i) = C × ((1 + i)^n − 1) ÷ i

Assumes a constant return i and a fixed contribution C. Real returns vary, which is why DC members bear investment risk.

Present value of liabilities

PV = Σ Cₜ ÷ (1 + i)ᵗ

Cₜ is the liability cash flow at time t and i is the valuation rate. A lower i gives a higher PV, and the effect is larger for longer liabilities.

Surplus

Surplus = Value of assets − Value of liabilities

Both sides must be valued on a consistent basis. ALM aims to keep surplus stable when conditions change.

Macaulay duration

D = Σ t × Cₜ ÷ (1 + i)ᵗ ÷ PV

The weighted average term of the cash flows, with weights equal to their present values.

Modified duration (approximate price change)

ΔPV ≈ −D_mod × PV × Δi, where D_mod = D ÷ (1 + i)

Valid for small changes in yield. For large changes, convexity matters.

Redington immunisation conditions

(1) PV assets = PV liabilities; (2) Duration of assets = Duration of liabilities; (3) Convexity of assets > convexity of liabilities

Protects against small parallel shifts in the yield curve only. It needs regular rebalancing.

Solvency ratio

Solvency ratio = Available capital ÷ Required capital

A ratio of 1 (100%) means capital equals the requirement. Regulators set the minimum level; check the current IRDAI rule.

Surplus over requirement

Excess capital = Available capital − Required capital

Positive means the firm meets the requirement. Available capital is assets less liabilities, after any admissibility rules.

Value at Risk (VaR)

P(Loss > VaR) = 1 − α

α is the confidence level, such as 99.5%, over a stated time horizon. VaR is the loss threshold, not the average loss beyond it.

Tail VaR (expected shortfall)

TVaR(α) = E[Loss | Loss ≥ VaR(α)]

Average loss in the worst cases. For continuous losses it is at least as large as VaR at the same level.

Economic capital (loss-based view)

Economic capital = VaR(α) − Expected loss

One common definition: capital covers unexpected loss, while expected loss is covered by pricing and reserves. State the definition you use.

Risk response options

Avoid | Reduce | Transfer | Accept

Choose by comparing the risk with appetite and the cost of the response.

Quick revision

• Use a fixed pattern for each issue: challenge, cause, effect, action.
• Life: long-term guarantees, persistency, mis-selling risk, interest rate risk and customer outcomes.
• General insurance: claims uncertainty, reserving, pricing cycles, catastrophe exposure and reinsurance.
• Health: medical cost inflation, fraud, moral hazard, adverse selection and limited reliable data.
• Pensions: longevity, funding shortfalls, sponsor strength, investment risk and clear member communication.
• ALM: mismatch of asset and liability cash flows, duration, liquidity and interest rate movement.
• ERM: identify, measure, manage and report risks, and hold enough capital for them.
• Capital issues include cost of capital, regulatory requirements and the trade-off between safety and returns.
• Emerging areas bring new data, new models and limited history, so state assumptions and uncertainty clearly.
• Always consider stakeholders: customers, regulator, shareholders, employees and the public.
• Link every recommendation to the specific case facts, not general statements.
• Tie in professionalism: act with integrity, communicate clearly and be open about limits of your work.

Common mistakes

• Listing risks without linking them to the product in the question. Fix: Start from the product features. For example, say a guaranteed annuity option creates longevity and interest rate risk because the rate is fixed in advance.
• Saying lapses are always bad for the insurer. Fix: Compare the surrender value paid with the reserve released and with unrecovered acquisition costs. Lapses can create a loss or a gain depending on that comparison.
• Treating IBNR as only claims not yet reported. Fix: State your definition. Say whether you include further development on already reported claims.
• Saying a soft market means insurers are making high profits. Fix: Say that soft markets mean low premiums and weak margins, so profits fall and reserve shortfalls may show later.
• Mixing up anti-selection and moral hazard. Fix: Anti-selection happens before or at purchase. Moral hazard happens after cover starts. Use that timing test.
• Listing issues without explaining effects or solutions. Fix: For every issue, write cause, effect on the insurer and at least one control.
• Saying DC schemes have no risk for the employer or no cost to anyone. Fix: Say the employer's cost is certain, but members bear investment, longevity and conversion risk. Add that poor outcomes can create pressure on employers.
• Thinking a higher discount rate raises the liability. Fix: Remember that discounting reduces present values. Higher rate means lower liability, so check direction every time.
• Saying rising interest rates are always bad for insurers. Fix: Check the liabilities. Rising yields lower the PV of liabilities too. Mismatch decides the net effect. Also note surrender risk when rates rise and reinvestment risk when rates fall.
• Looking at assets alone. Fix: Always compare with liabilities. A risk is only a problem relative to what must be paid.

Exam tips

• Always tie your points to the product and details in the scenario. Generic lists score poorly.
• Use the structure risk, effect, action. It keeps answers short and complete.
• State the direction of adverse experience: for example, fewer deaths is bad for annuities, more deaths is bad for term cover.
• In MCQs on lapses, check whether surrender value is above or below the reserve before choosing whether lapses help or hurt.
• Add one line on customer fairness or regulatory capital in longer answers. Examiners look for this wider view.
• Structure answers as issue, cause, effect, response. Examiners reward reasoning, not lists.
• Use general insurance terms precisely: IBNR, retention, quota share, excess of loss, earned premium.
• In comparison questions, set out life and general points side by side in prose: contract length, claim uncertainty, key assumptions and reserving.