Risk Modeling

Study guide: Exam 9

Risk Modeling is the construction of a stochastic model of an organisation's material risks and their dependencies — for an insurer, an internal risk model (IRM), also called a dynamic financial analysis model — that simulates the distribution of its results, so that risk measures can be computed and strategies compared on one basis.

$S_1^{(i)} = S_0 + P - L^{(i)} - X + I^{(i)}$

$\widehat{\text{TVaR}}_{\alpha} = \frac{1}{m}\sum_{j=1}^{m} Y_{(n+1-j)}$

In scenario $i$ of $n$ , year-end surplus $S_1^{(i)}$ is opening surplus $S_0$ plus premium $P$ , less simulated losses $L^{(i)}$ and expenses $X$ , plus the simulated investment result $I^{(i)}$ . $Y_{(1)} \leq \cdots \leq Y_{(n)}$ are the simulated losses (or surplus declines) in order, and TVaR at level $\alpha$ averages the worst $m = n(1 - \alpha)$ of them.
Brehm's components: underwriting risk (frequency and severity, pricing risk, parameter risk, catastrophe risk), reserve risk, asset risk from an economic scenario generator, and — what makes the model realistic — dependencies: common drivers such as inflation acting on claims and asset values together, catastrophes crossing lines, the cycle moving lines together, and tail dependence. Without them the enterprise looks unrealistically stable.
Selecting models for diverse risks. Each category needs a model suited to its nature, data and materiality — see Risk Taxonomies for the mapping and Model Selection for the statistical criteria. Parsimony matters: a model management cannot follow will not be used, and every added parameter adds estimation error and model risk.
The effect on strategic management. An IRM answers strategic questions: how much capital a target rating or ruin probability requires, how to allocate it (Risk Capital), which reinsurance programme or asset mix sits on the efficient frontier of return against risk, and whether the business plan fits appetite. Brehm stresses integration — the model should sit in the planning and reinsurance-buying calendar, not be run once for a report.
The choice of risk measure is itself strategic. VaR reads one quantile; TVaR averages the tail beyond it; they can rank the same strategies differently. Brehm recommends looking at several metrics rather than one.

VaR and TVaR Pick Different Growth Plans An insurer's model gives next year's net loss (in $$$$$M) under two growth plans with the same premium.

Plan A (grow coastal property): $10$ with probability $90\%$ , $40$ with $8\%$ , $200$ with $2\%$ .

Plan B (grow commercial liability): $12$ with probability $90\%$ , $60$ with $9\%$ , $100$ with $1\%$ .

Compare the mean, $\text{VaR}_{95\%}$ and $\text{TVaR}_{95\%}$ .
Answer
 $\begin{align*} E[L_A] &= 9 + 3.2 + 4 \\ &= 16.2 \\ E[L_B] &= 10.8 + 5.4 + 1 \\ &= 17.2 \end{align*}$ 
The $95$ th percentile is where the cumulative probability first reaches $95\%$ : $\text{VaR}_A = 40$ (cumulative $98\%$ ) and $\text{VaR}_B = 60$ (cumulative $99\%$ ). TVaR averages the worst $5\%$ :
 $\begin{align*} \text{TVaR}_A &= \frac{0.02(200) + 0.03(40)}{0.05} \\ &= 104 \\ \text{TVaR}_B &= \frac{0.01(100) + 0.04(60)}{0.05} \\ &= 68 \end{align*}$ 
On the mean and on VaR, plan A looks better; on TVaR, plan B is far safer. VaR does not see A's $2\%$ chance of a $200$ loss at all. A board whose appetite is written in VaR will choose A, one whose appetite is written in TVaR will choose B — the risk measure has made the strategic decision.

Ignoring Dependence Understates Capital Two lines each lose $50$ with probability $90\%$ and $150$ with probability $10\%$ . Find $\text{TVaR}_{95\%}$ and capital $= \text{TVaR}_{95\%} - E[L]$ for the combined book (a) assuming independence, (b) if a common inflation driver makes both lines bad together $5\%$ of the time.
Answer Either way $E[L] = 2(0.9 \times 50 + 0.1 \times 150) = 120$ .

(a) Independent: $100$ with $81\%$ , $200$ with $18\%$ , $300$ with $1\%$ .
 $\begin{align*} \text{TVaR}_{95\%} &= \frac{0.01(300) + 0.04(200)}{0.05} \\ &= 220 \end{align*}$ 
Capital $= 220 - 120 = 100$ .

(b) Common driver: both bad $5\%$ (loss $300$ ), exactly one bad $2(10\% - 5\%) = 10\%$ (loss $200$ ), neither $85\%$ (loss $100$ ). The worst $5\%$ is all $300$ , so $\text{TVaR}_{95\%} = 300$ and capital $= 180$ .

Same marginal distributions, same mean — but the independent model understates capital by $80$ , or $44\%$ . The implied correlation between the lines' bad years is only $(0.05 - 0.01)/(0.1 \times 0.9) = 0.44$ .