Insurance Margin

Study guide: Exam 9

Insurance Margin $M$ is the part of the premium above expected loss — in Mildenhall and Major's notation $M = P - L$ . It is what policyholders pay for the use of the capital that makes their insurer's promise credible, and equally the owners' expected profit for providing that capital.

$M = P - L$

$M = \iota\, Q = \delta\,(a - L)$

$P$ is premium, $L$ expected loss ( $E[X \wedge a]$ strictly), $Q = a - P$ capital, $a$ assets, $\iota = M/Q$ the return and $\delta = \iota/(1+\iota)$ . In the one-period model with no expenses or investment income, the margin is the entire expected underwriting profit and plays the role of the risk load.
Why $\delta(a - L)$ . Assets above expected loss, $a - L$ , are the "unfunded" part of the promise. Owners pay in $Q$ at inception and expect $a - L = M + Q$ back, so premium funds the expected loss plus a fraction $\delta$ of $a - L$ and investors fund the remaining $\nu = 1 - \delta$ .
Margin ratio $M/P = 1 - L/P$ is the complement of the loss ratio. Multiplied by leverage $P/Q$ it gives the return — see Insurance Profitability.
Gross versus net of expenses. Expenses enlarge the premium without changing the margin, so the same margin is a smaller percentage of an expense-loaded premium. Quote margins on a stated basis.
Where the margin sits. Pricing layer by layer, a distortion $g$ charges margin $g(S(x)) - S(x)$ per unit of the layer at $x$ . A constant cost of capital ( $g(s) = \nu s + \delta$ ) charges almost $\delta$ per unit of capacity whatever the layer's attachment probability, which loads remote layers heavily; other distortions spread the margin more evenly. Exam 5's counterpart is the Profit and Contingency Provision.

The Margin a Target Return Requires A book has expected loss $60$ and is supported by assets of $100$ . Owners require $10\%$ . Find the margin, premium and capital, then express the margin as a percentage of premium before and after grossing up for a $25\%$ expense ratio.
Answer
 $\begin{align*} \delta &= \frac{0.10}{1.10} \\ &= 0.09091 \\ M &= 0.09091(100 - 60) \\ &= 3.64 \\ P &= 60 + 3.64 \\ &= 63.64 \\ Q &= 100 - 63.64 \\ &= 36.36 \end{align*}$ 
Check: $3.64/36.36 = 10\%$ . The margin is $3.64/63.64 = 5.7\%$ of the risk premium. Grossed up, the premium is $63.64/0.75 = 84.85$ and the same margin is $3.64/84.85 = 4.3\%$ of it — a lower-looking figure with no change in profitability.

Where a Constant Cost of Capital Puts the Margin A catastrophe portfolio's loss $X$ ($ millions) is $0$ , $10$ , $50$ or $100$ with probabilities $0.80$ , $0.12$ , $0.06$ and $0.02$ ; $E[X] = 6.2$ and assets are $a = 100$ . At $\iota = 15\%$ ( $\nu = 0.8696$ , $\delta = 0.1304$ ), allocate the margin to the layers $0$ – $10$ , $10$ – $50$ and $50$ – $100$ .
Answer Each layer's attachment probability is $S = 0.20$ , $0.08$ and $0.02$ . The priced probability is $g(S) = 0.8696\,S + 0.1304$ , and the margin per unit of layer is $g(S) - S$ :
 $\begin{align*} M_{0\text{--}10} &= 10(0.3043 - 0.20) \\ &= 1.04 \\ M_{10\text{--}50} &= 40(0.2000 - 0.08) \\ &= 4.80 \\ M_{50\text{--}100} &= 50(0.1478 - 0.02) \\ &= 6.39 \end{align*}$ 
The total, $12.23$ , matches $\delta(a - L) = 0.1304(93.8)$ . The top layer carries $16\%$ of the expected loss ( $1.0$ of $6.2$ ) but $52\%$ of the margin: a constant cost of capital charges for capacity, not for the chance of using it.