Risk-Adjusted Pricing

Study guide: Exam 9

Risk-Adjusted Pricing sets an insurance premium that covers expected losses and expenses plus a margin paying the Cost of Capital on the Risk Capital the business consumes — with that capital, and so the margin, measured by the risk's contribution to the insurer's total risk rather than by its standalone volatility. It is the capital-based derivation of the profit and risk provision in a rate.

$P = E[X \wedge a] + \iota\,(a - P)$

$P = \nu\,E[X \wedge a] + \delta\,a$

In Mildenhall and Major's notation: $X$ is the loss, $a$ the assets held (set by a Risk Measure such as VaR or TVaR), $E[X \wedge a]$ the expected loss the insurer can actually pay, $Q = a - P$ the capital investors supply, $M = P - E[X \wedge a]$ the margin, and $\iota = M/Q$ the cost of capital. Solving the first line for $P$ gives the second, with $\nu = 1/(1+\iota)$ and $\delta = \iota/(1+\iota)$ , so $\nu + \delta = 1$ . This is constant cost of capital (CCoC) pricing; expenses and the time value of money are added on top.
From portfolio to policy. The formula prices the whole portfolio; each line or policy then needs its share of the margin. Allocating capital (see Risk Capital) and charging every unit the same $\iota$ is one choice. Mildenhall and Major instead allocate the margin directly, which lets units earn different implied returns on capital.
RAROC-target form. With allocated capital $C_i$ held for one period and invested at yield $y$ , the premium that earns hurdle $k$ is $P_i = \text{PV}(E[L_i]) + \text{PV}(\text{expenses}_i) + (k - y)\,C_i$ — the capital earns $y$ while held, so pricing supplies only the spread.
Price to the margin, not the standalone risk. A policy that diversifies the book needs less capital and so less margin. Pricing on standalone capital overprices diversifying risks, which competitors pricing on the portfolio then take away.
Relation to classic loads. Risk Loads based on variance or standard deviation, and the Profit and Contingency Provision of ratemaking, are the same idea expressed without an explicit capital model; Insurance Pricing with a capital model makes the load traceable to a cost.

Constant Cost of Capital Price for a Portfolio A portfolio has $E[X \wedge a] = \$1{,}000$ M. The insurer holds assets $a = \$1{,}600$ M, set by its risk measure, and targets a cost of capital $\iota = 12.5\%$ . Find the premium, margin, capital, loss ratio and premium-to-capital leverage.
Answer $\nu = 1/1.125 = 0.8889$ and $\delta = 0.125/1.125 = 0.1111$ .
 $\begin{align*} P &= 0.8889(1{,}000) + 0.1111(1{,}600) \\ &= \$1{,}066.7\text{M} \\ M &= 1{,}066.7 - 1{,}000 \\ &= \$66.7\text{M} \\ Q &= 1{,}600 - 1{,}066.7 \\ &= \$533.3\text{M} \end{align*}$ 
Check: $M/Q = 66.7/533.3 = 12.5\%$ . The loss ratio is $1{,}000/1{,}066.7 = 93.75\%$ and premium-to-capital leverage is $1{,}066.7/533.3 = 2.0$ . Premium funds part of the assets, so investors supply only $\$533$ M of the $\$600$ M above expected loss.

Pricing on Allocated versus Standalone Capital A large commercial account has present value of expected loss $\$800$ K and expenses $\$150$ K. Its marginal (Euler) allocated capital is $\$1{,}000$ K; standalone it would need $\$2{,}500$ K. The hurdle is $12\%$ , capital is invested at $4\%$ , and capital is held for one year. Ignore taxes. Price the account both ways.
Answer
 $\begin{align*} P_{\text{allocated}} &= 800 + 150 + (0.12 - 0.04)(1{,}000) \\ &= \$1{,}030\text{K} \\ P_{\text{standalone}} &= 800 + 150 + (0.08)(2{,}500) \\ &= \$1{,}150\text{K} \end{align*}$ 
The standalone price is $\$120$ K ( $11.7\%$ ) higher. The account diversifies the book — it adds only $\$1.0$ M to the firm's capital need — so the standalone quote charges it for risk the firm never holds. A competitor pricing on its portfolio would write the account at about $\$1{,}030$ K and still earn $12\%$ on the capital it actually uses.