Return on Capital

Study guide: Exam 9

Return on Capital — Mildenhall and Major's $\iota$ — is the expected margin earned per unit of capital. Seen from the investor's side it is the expected return on the capital they put up; from the policyholder's side it is the cost of capital built into the premium.

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

$\nu = \frac{1}{1+\iota}$

$\delta = \frac{\iota}{1+\iota}$

$M = P - L$ is the margin and $Q = a - P$ the capital. $\nu$ and $\delta$ are the risk discount factor and risk discount rate, the analogues of $v$ and $d$ in interest theory (Discount Factor, Discount Rate): capital is paid in at the start and the margin, like interest, is earned at the end. The same identities hold — $\delta = \iota\nu$ and $\nu + \delta = 1$ .
Capital is the present value, at the risk discount factor, of what investors expect back: $Q = \nu(a - L)$ . Premium follows as $P = L + \iota Q = \nu L + \delta a$ — see Insurance Pricing.
Return is margin times leverage: $\iota = (M/P)(P/Q)$ . A return target can be met by pricing (margin) or by capital management (leverage).
The target comes from the cost of capital — the return investors need to support the firm's market value — often blended with the cheaper, fixed costs of debt and reinsurance into a weighted average (Capital Structure).
Expected is not reported. Pricing uses the expected return on allocated capital. Reported return on equity is net income over book surplus, including investment income, realised gains and reserve development on prior years; the two answer different questions.
Constant or varying? A constant cost of capital charges the same $\iota$ on every layer of capital. Pricing with a distortion $g$ implies a return of $(g(s) - s)/(1 - g(s))$ on the layer of assets whose attachment probability is $s$ ; for most distortions (the constant-cost-of-capital one excepted) it falls as layers become more remote — as in bond markets, where safer layers of capital earn lower spreads. See Risk-Adjusted Performance for return by line.

Return, Risk Discount Factor and Rate A portfolio is written at premium $105$ with expected loss $95$ and is supported by assets of $180$ . Find $\iota$ , $\nu$ and $\delta$ , and confirm $P = \nu L + \delta a$ .
Answer
 $\begin{align*} M &= 105 - 95 \\ &= 10 \\ Q &= 180 - 105 \\ &= 75 \\ \iota &= \frac{10}{75} \\ &= 13.33\% \\ \nu &= \frac{1}{1.1333} \\ &= 0.88235 \\ \delta &= \frac{0.1333}{1.1333} \\ &= 0.11765 \end{align*}$ 
 $\begin{align*} \nu L + \delta a &= 0.88235(95) + 0.11765(180) \\ &= 83.82 + 21.18 \\ &= 105.00 \end{align*}$ 
Equivalently $\delta = M/(a - L) = 10/85 = 0.11765$ : premium funds $11.8\%$ of the $85$ of assets above expected loss, and investors fund the other $88.2\%$ .

Layer Returns Under a Distortion A pricing model uses the proportional hazard distortion $g(s) = s^{0.6}$ . Find the loss ratio and the return on capital for layers of assets with attachment probabilities $s = 0.20$ , $0.05$ and $0.01$ .
Answer Per unit of layer, expected loss is $s$ , premium $g(s)$ and capital $1 - g(s)$ .
 $\begin{align*} g(0.20) &= 0.3807 \\ \iota(0.20) &= \frac{0.3807 - 0.20}{1 - 0.3807} \\ &= 29.2\% \\ g(0.05) &= 0.1657 \\ \iota(0.05) &= \frac{0.1657 - 0.05}{1 - 0.1657} \\ &= 13.9\% \\ g(0.01) &= 0.0631 \\ \iota(0.01) &= \frac{0.0631 - 0.01}{1 - 0.0631} \\ &= 5.7\% \end{align*}$ 
Loss ratios $s/g(s)$ are $52.5\%$ , $30.2\%$ and $15.8\%$ . The remote layer has the lowest loss ratio yet earns the lowest return, because it is almost all capital. A constant $15\%$ cost of capital would instead price the $s = 0.01$ layer at $g = 0.8696(0.01) + 0.1304 = 0.139$ , more than double the distortion's $0.063$ , because it demands the full $15\%$ on capital that is very unlikely to be used.