Insurance Leverage

Study guide: Exam 9

Insurance Leverage is the volume of business written per unit of capital supporting it — in Mildenhall and Major, the premium-to-capital ratio $P/Q$ . It is the lever that turns a thin margin into an adequate return, paid for with a thinner cushion against bad outcomes.

$\text{Leverage} = \frac{P}{Q}$

$\iota = \frac{M}{P} \cdot \frac{P}{Q}$

$P$ is premium, $Q = a - P$ capital, $a$ assets, $M$ margin and $\iota$ return. Leverage can equally be stated as premium to assets, $P/a$ ; since $Q = a - P$ , the two are linked by $P/Q = (P/a)/(1 - P/a)$ .
Return versus safety. For a given margin ratio, doubling leverage doubles the return — but halves the capital behind each dollar of premium, so a smaller adverse deviation exhausts it. Regulators and rating agencies watch leverage for exactly that reason; statutory screens include premium-to-surplus tests (IRIS Ratios), alongside risk-based capital requirements (Risk-Based Capital).
Leverage follows risk. Diversified, low-volatility business such as personal auto supports high leverage at thin margins. Business whose losses do not diversify — catastrophe layers above all — supports little, and needs a high margin ratio to earn the same return. Under distortion pricing a layer whose attachment probability is $s$ has leverage $g(s)/(1 - g(s))$ : remote layers are funded almost entirely by capital.
Other capital raises equity leverage. Debt and reinsurance are forms of risk capital with a fixed cost; replacing part of the equity with them raises premium-to-equity leverage and, if they cost less than equity, the return on what equity remains (Capital Structure).
The industry measures leverage from the statutory balance sheet — written premium, and reserves, relative to policyholders' surplus (Capital and Surplus). Pricing uses capital allocated to the business being priced. See Insurance Profitability.

Same Margin, Different Leverage Two insurers write premium of $100$ with expected loss $94$ , a $6\%$ margin ratio. Insurer A holds capital of $66.67$ ; insurer B holds $33.33$ . Compare their leverage, return, and how far losses can exceed expectation before capital is gone.
Answer
 $\begin{align*} \iota_A &= 0.06 \times \frac{100}{66.67} \\ &= 0.06 \times 1.5 \\ &= 9\% \\ \iota_B &= 0.06 \times \frac{100}{33.33} \\ &= 0.06 \times 3.0 \\ &= 18\% \end{align*}$ 
Assets are $166.67$ for A and $133.33$ for B, so losses would have to exceed expectation by $166.67 - 94 = 72.67$ to exhaust A but by only $39.33$ to exhaust B. B's doubled return is bought with a much higher chance of default — which is why a return comparison means nothing without the leverage behind it.

Buying Security Lowers Leverage and Raises the Price A portfolio has expected loss $80$ and investors require $12\%$ . Compare the premium, capital and leverage if the capital standard sets assets at $120$ and at $150$ .
Answer With $\delta = 0.12/1.12 = 0.10714$ and $P = L + \delta(a - L)$ :
 $\begin{align*} P_{120} &= 80 + 0.10714(40) \\ &= 84.29 \\ Q_{120} &= 120 - 84.29 \\ &= 35.71 \\ P_{150} &= 80 + 0.10714(70) \\ &= 87.50 \\ Q_{150} &= 150 - 87.50 \\ &= 62.50 \end{align*}$ 
Leverage falls from $84.29/35.71 = 2.36$ to $87.50/62.50 = 1.40$ and the loss ratio from $94.9\%$ to $91.4\%$ . Policyholders pay $3.21$ more for an insurer that can absorb $30$ more of adverse loss; both prices earn exactly $12\%$ .