Insurance Profitability

Study guide: Exam 9

Insurance Profitability is how well premium rewards the capital that stands behind it. Exam 9 assesses it through three linked ratios — margin (profit relative to premium), return (profit relative to capital) and leverage (premium relative to capital) — because none of them alone says whether a line is profitable.

$\frac{M}{Q} = \frac{M}{P} \times \frac{P}{Q}$

$\frac{M}{P} = 1 - \frac{L}{P}$

Return equals margin ratio times premium-to-capital leverage, and the margin ratio is one minus the loss ratio. A thin margin on a highly leveraged book and a fat margin on a lightly leveraged one can earn the same return.
The insurance pentagon (Mildenhall and Major): five amounts — expected loss $L$ , premium $P$ , margin $M$ , capital $Q$ and assets $a$ — tied by $P = L + M$ and $a = P + Q$ , and described by three ratios: loss ratio $L/P$ , leverage and return $\iota = M/Q$ . That leaves three degrees of freedom, so three well-chosen quantities (say $L$ , $a$ and $\iota$ ) usually determine the rest. The three ratios alone do not: they fix proportions, not volume.
The loss ratio alone misleads. A line with a low loss ratio can be unprofitable if it consumes a great deal of capital, and one with a high loss ratio can be very profitable if it needs little. Comparing lines requires capital allocated to each — see Risk-Adjusted Performance and Risk Capital.
Diversification sets the mix. Where diversification works, capital can be highly leveraged and margins thin; where it fails, as for catastrophe risk, leverage falls and the margin ratio must rise to deliver the same return.
Accounting versus pricing. Reported measures — the combined ratio, the operating ratio, net income over surplus (Key Financial Measures) — mix accident years, reserve development and realised gains. Pricing profitability uses expected values, allocated capital and discounted cash flows (Economic Value).

Completing the Pentagon and Testing Against a Target A book has expected loss $70$ , is supported by assets of $150$ and is written at a premium of $78$ . Investors require $12\%$ . Compute margin, capital, loss ratio, leverage and return, and the shortfall against the target.
Answer
 $\begin{align*} M &= 78 - 70 \\ &= 8 \\ Q &= 150 - 78 \\ &= 72 \\ \iota &= \frac{8}{72} \\ &= 11.1\% \end{align*}$ 
The loss ratio is $70/78 = 89.7\%$ , so the margin ratio is $10.3\%$ ; leverage is $78/72 = 1.083$ ; and $10.3\% \times 1.083 = 11.1\%$ agrees.

The premium that earns $12\%$ , with $\delta = 0.12/1.12 = 0.10714$ :
 $\begin{align*} P_{\text{req}} &= 70 + 0.10714(150 - 70) \\ &= 78.57 \end{align*}$ 
The book is $0.57$ short of its required margin: $11.1\%$ is below the cost of capital even though an $89.7\%$ loss ratio might look acceptable on its own.

A 51% and a 96% Loss Ratio Earning the Same Return Two lines are each supported by assets of $100$ and priced at a $12\%$ return. Personal auto has expected loss $70$ ; a catastrophe-exposed property line has expected loss $10$ . Compare their loss ratios and leverage.
Answer With $\delta = 0.10714$ and $P = L + \delta(a - L)$ :
 $\begin{align*} P_{\text{auto}} &= 70 + 0.10714(30) \\ &= 73.21 \\ P_{\text{cat}} &= 10 + 0.10714(90) \\ &= 19.64 \end{align*}$ 
Auto: loss ratio $70/73.21 = 95.6\%$ , capital $26.79$ , leverage $2.73$ , margin ratio $4.4\%$ .

Catastrophe: loss ratio $10/19.64 = 50.9\%$ , capital $80.36$ , leverage $0.24$ , margin ratio $49.1\%$ .

In both, $\text{margin ratio} \times \text{leverage} = 12\%$ . The catastrophe line's low loss ratio is not excess profit — it is the price of the capital its tail consumes.