Insurance Risk & Premium Pricing

Insurance works through risk pooling: many people each face a small probability of a large loss. By combining their risks, the insurer can predict total claims far more accurately than any individual can predict their own loss.

The Law of Large Numbers guarantees that as the pool of $n$ independent policyholders grows, the average claim per person converges to the expected loss per person $\mu$. The uncertainty (standard deviation of the mean) shrinks as $\sigma/\sqrt{n}$, making the aggregate predictable even though individual outcomes remain random.

This predictability lets the insurer charge a premium that covers expected losses and still earn a stable profit.

$\text{Premium} = \underbrace{p \times L}_{\text{expected loss}} + \underbrace{\text{loading}}_{\text{costs + safety buffer}}$

where $p$ = probability of a claim event and $L$ = average claim size.

Standard deviation of the mean claim across a pool of $n$ policyholders:

$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$

A health insurer covers 10,000 people. Each person has a 5% annual chance of a ₹50,000 hospital claim.

$\text{Expected loss per person} = 0.05 \times 50{,}000 = ₹2{,}500$

With 20% loading for operating costs and reserves:

$\text{Premium} = ₹2{,}500 \times 1.20 = ₹3{,}000 \text{ per year}$

Total premiums: $10{,}000 \times ₹3{,}000 = ₹3{,}00{,}00{,}000$. Expected total claims: ₹2,50,00,000. The ₹50,00,000 surplus absorbs years with above-average claims.

Car insurance pools millions of drivers — only a small fraction crash each year, but the insurer predicts the aggregate precisely. Reinsurance companies pool risks across primary insurers and entire countries, reducing catastrophe exposure. The 2008 financial crisis arose partly from assuming mortgage defaults were independent — in a nationwide housing crash they were highly correlated, breaking the pooling assumption and causing massive underestimation of risk.

1. A fire insurer covers 5,000 shops, each with a 2% chance of a ₹1,00,000 claim. What is the expected loss per shop, and what premium should be charged with 25% loading?

2. Why does the Law of Large Numbers help an insurer with 10,000 policyholders but not a single individual facing the same risk?