Insurance Risk

Study guides: Exam 9, Exam GI-201, Exam ILA-201I, Exam ILA-201U

Insurance Risk is the risk that claims, their timing and the cost of settling them differ from what was expected when the business was priced and reserved. For a P&C insurer it has three parts: underwriting (premium) risk on business written or to be written, reserve risk on claims already incurred, and catastrophe risk from single events that strike many policies at once.

 $\begin{aligned} \text{Var}(L) &= E\big[\text{Var}(L \mid \theta)\big] \\ &\quad + \text{Var}\big(E[L \mid \theta]\big) \end{aligned}$

$L$ is aggregate loss and $\theta$ the true, unknown parameters of its distribution. The first term is process risk — randomness given the parameters, which pooling diversifies away (Law of Large Numbers). The second is parameter risk — the chance the parameters are wrong — which does not diversify, because it moves every policy the same way. (Credibility theory splits variance the same way, into the EPV and the VHM.)
Brehm's components of underwriting risk: process risk in frequency and severity (an aggregate loss model); pricing risk from mis-estimation and competitive pressure; parameter risk, subdivided into estimation (too little data), projection (trend and development, often driven by inflation), event (a court ruling, a latent exposure, a new cause of loss) and systematic (drivers that hit all policies together); and catastrophe risk, where the correlation is physical.
Reserve risk applies the same analysis to liabilities already on the balance sheet — the distribution of unpaid claims (Unpaid Claim Distribution, Stochastic Reserving). For long-tailed lines it is often the largest single risk, and A.M. Best's impairment studies have consistently found deficient reserves and inadequate pricing to be the leading cause of P&C insurer impairment.
Analysing it quantitatively means giving each part a distribution and a risk measure — coefficient of variation, VaR, TVaR, PML — and combining it with financial risk in the internal risk model.

Parameter Risk Does Not Diversify Given the true mean $\theta$ , each policy's annual loss has standard deviation $\$3{,}000$ and losses are independent. The mean is estimated at $\$1{,}000$ per policy but is uncertain, with standard deviation $\$50$ . Find the coefficient of variation of total losses for $100$ and $10{,}000$ policies.
Answer For $n$ policies, $\text{Var}(S) = n(3{,}000)^2 + n^2(50)^2$ .
 $\begin{align*} n = 100: \quad \text{Var}(S) &= 9.0 \times 10^8 + 2.5 \times 10^7 \\ &= 9.25 \times 10^8 \\ \text{CV} &= \frac{30{,}414}{100{,}000} \\ &= 30.4\% \\[6pt] n = 10{,}000: \quad \text{Var}(S) &= 9.0 \times 10^{10} + 2.5 \times 10^{11} \\ &= 3.4 \times 10^{11} \\ \text{CV} &= \frac{583{,}095}{10{,}000{,}000} \\ &= 5.83\% \end{align*}$ 
Growing the book a hundredfold cuts the process CV from $30\%$ to $3\%$ , but the CV can never fall below $50/1{,}000 = 5\%$ : at $10{,}000$ policies parameter risk is already $74\%$ of the variance. Volume diversifies process risk, not parameter risk.

Catastrophe Risk: Same Mean, Different Tail An insurer covers $1{,}000$ homes worth $\$250{,}000$ each. In book A the homes face independent fires, each destroying a home with probability $0.2\%$ a year. In book B there is no fire risk, but a hurricane with annual probability $2\%$ would damage every home by $10\%$ . Compare the mean, standard deviation and $99$ th percentile of annual loss.
Answer Book A. $N \sim \text{Bin}(1{,}000,\ 0.002)$ homes lost.
 $\begin{align*} E[L_A] &= 2 \times \$250{,}000 \\ &= \$500{,}000 \\ \text{SD}(L_A) &= \$250{,}000\sqrt{1{,}000(0.002)(0.998)} \\ &= \$353{,}000 \end{align*}$ 
$P(N \leq 5) = 0.984$ and $P(N \leq 6) = 0.996$ , so the $99$ th percentile is $6$ homes, $\$1.5$ million.

Book B. The loss is $\$25$ million with probability $2\%$ , else zero: mean $\$500{,}000$ , standard deviation $\$25\text{M}\sqrt{0.02 \times 0.98} = \$3.5$ million, and $99$ th percentile $\$25$ million.

Same expected loss; the catastrophe book's 1-in-100 loss is about $17$ times larger, and writing more homes in the same region does not help. That is why catastrophe risk is managed through exposure limits, reinsurance and cat bonds rather than volume.