Risk Loads

Study guide: Exam 9

Risk Loads are the amounts added to expected loss to pay an insurer's capital providers for bearing the risk of a contract. Expected loss plus the risk load is the risk-loaded technical premium: the price a pricing model indicates for the risk itself, before expenses and commercial adjustments.

P = E[X] + R(X)

• X is the loss to the contract (after its limits and attachment), R(X) \geq 0 the risk load. In Mildenhall and Major's notation the risk load is the margin M = P - L — see Insurance Margin.
• Classical premium principles, each a Risk Measure applied to X:
  • expected value: P = (1 + \theta)\,E[X] — the load ignores variability, so a remote catastrophe layer and a stable working layer with the same mean pay the same load;
  • standard deviation: P = E[X] + k\,\sigma(X) — the load scales with share, so a 50% share pays half;
  • variance: P = E[X] + k\,\text{Var}(X) — the load scales with the square of share, so a 50% share pays a quarter and two 50% shares together pay half the load on 100%.
• Cost-of-capital loads set capital first, with a capital risk measure (VaR, TVaR or a regulatory standard), then charge the target return on it: R = \iota\,Q = \delta\,(a - L), where a is assets, \iota the target return and \delta = \iota/(1+\iota). See Cost of Capital and Risk Capital.
• Distortion loads R = \rho_g(X) - E[X] weight remote layers more heavily than their probability. The distortion can be calibrated to observed market prices, for example of catastrophe bonds and reinsurance layers.
• Loads are not additive. Pooling reduces risk, so the load on a portfolio is less than the sum of the stand-alone loads. The portfolio load has to be set top-down and allocated to lines or policies — the problem of Risk-Adjusted Pricing. In Exam 5 ratemaking the same idea sits inside the Profit and Contingency Provision.

Five Risk Loads for One Catastrophe Layer Losses ($ millions) to a $10M xs $10M catastrophe layer are 0 with probability 0.90, 5 with probability 0.06 and 10 (a full limit loss) with probability 0.04. Compute the technical premium under (a) the expected value principle with \theta = 0.5, (b) the standard deviation principle with k = 0.25, (c) the variance principle with k = 0.05, (d) a proportional hazard distortion g(s) = s^{0.7}, and (e) a constant 15\% cost of capital with the layer fully collateralised (a = 10).

Answer

\begin{align*}
E[X] &= 0.06(5) + 0.04(10) \\
&= 0.70 \\
\text{Var}(X) &= 0.06(25) + 0.04(100) - 0.70^2 \\
&= 5.01 \\
\sigma(X) &= 2.238
\end{align*}

For (d), S = 0.10 on [0, 5) and 0.04 on [5, 10). For (e), \delta = 0.15/1.15 = 0.1304.

\begin{align*}
P_{(a)} &= 1.5(0.70) \\
&= 1.050 \\
P_{(b)} &= 0.70 + 0.25(2.238) \\
&= 1.260 \\
P_{(c)} &= 0.70 + 0.05(5.01) \\
&= 0.951 \\
P_{(d)} &= 5(0.10^{0.7}) + 5(0.04^{0.7}) \\
&= 0.998 + 0.525 \\
&= 1.523 \\
P_{(e)} &= 0.70 + 0.1304(10 - 0.70) \\
&= 1.913
\end{align*}

The same loss distribution produces risk loads from 0.25 to 1.21 — the choice of measure and parameter, not the data, drives the price. The cost-of-capital price is the highest because the layer ties up $10M of capital against $0.7M of expected loss; its loss ratio is 0.70/1.913 = 37\%, and its return is 1.213/(10 - 1.913) = 15\% as required.

The Diversification Problem Two independent lines each have expected loss 50 and standard deviation 30. The insurer loads premiums at 0.2 standard deviations. Compare the stand-alone loads with the load on the combined portfolio.

Answer

\begin{align*}
\text{Stand-alone} &= 0.2(30) + 0.2(30) \\
&= 12.0 \\
\sigma_{\text{pool}} &= \sqrt{30^2 + 30^2} \\
&= 42.43 \\
\text{Pooled load} &= 0.2(42.43) \\
&= 8.49
\end{align*}

Pooling saves 3.51 of load. If each line is priced stand-alone the insurer over-charges by 41\% of the load it needs, and a competitor that passes the diversification benefit on will win the business. The pooled load has to be allocated — here symmetry gives 4.24 each — and for unequal or correlated lines the allocation method determines each line's price.