and policyholders. At t = 0 all amounts are fixed. At t = 1 the random loss outcome X is revealed. The model can include stochastic asset returns.
Table 2.1 Investor and insured transactions with Ins Co. at t=0,1. In the last row X∧a′=min(X,a′)
| View | Total | Loss | Margin | Capital | |||
|---|---|---|---|---|---|---|---|
| At issue, t = 0 | |||||||
| Insured | Premium | = | Expected loss cost | + | Margin | ||
| Investor | Capital | = | Capital | ||||
| Ins Co. Total | Assets | = | Expected loss cost | + | Margin | + | Capital |
| expected | a | = | E[X] | + | M | + | Q |
| At expiration, t = 1 | |||||||
| Ins Co. Total | Assets | = | Loss outcome | + | Residual value | ||
| random | a′=a(1+r) | = | X∧a′ | + | (a′−X)+ |
In an unregulated insurance market, insured relative risk aversion interacts with investor opportunity costs of capital to determine the amount of assets supporting the risk and the split of those assets into premium and capital. In a regulated market, the amount of assets is determined, or at least heavily influenced by a regulator (or quasi-regulatory rating agency), but the market still determines the split of funding between premium and equity.
2.3 Model vs. Reality
The reader should bear in mind:
Man plans, and God laughs (old Yiddish saying)
Anything that can go wrong, will go wrong (Edward Murphy)
The map is not the territory (Alfred Korzybski)
All models are wrong but some are useful (George E.P. Box)
All the analysis presented in this book presumes we know the objective probability distribution underlying the phenomenon being studied. Cue laughter. This conceit is particularly pernicious when coupled with management optimism and its attempts at optimization.
To make the results useful, sensitivity analysis must be part of the effort. Perturb the assumptions in a realistic manner and repeat the risk measure calibration, capital calculation, margin allocation, etc. Then do it again. Coherent risk measures are all about taking the worst case among a set of alternatives. The alternatives reflect our uncertainty about event probabilities by positing different values. Lift this strategy to the entire engagement. Generate a set of assumptions and look across the set of answers. Only then will you know whether the best estimate is robust or fragile. With that caveat, we proceed, assuming that we have a tin opener as the joke about economists has it.
2.4 Examples and Case Studies
In this section we introduce a Simple Discrete Example and three more realistic Case Studies. These are used throughout the book to illustrate the various methods presented in the theory chapters.
We include simple examples to familiarize practitioners with the computations required to implement the methods we cover. The Cases aim to help practitioners develop an intuition for how each method prices business, informing their selection of an appropriate method for an intended purpose without resorting to trial and error.
The Cases share several common characteristics.
Each includes two units, one lower risk and one higher.
Reinsurance is applied to the riskier unit.
Total unlimited losses are calibrated to ¤ 100. (The symbol ¤ denotes a generic currency.)
Losses are in ¤ millions, although the actual unit is irrelevant.
For each Case Study:
Figures 2.2, 2.4, and 2.6 show densities. The horizontal axis shows loss amount. In each figure the top row shows gross losses and the bottom row shows net. The left column uses a linear scale and the right a log scale.
Figures 2.3, 2.5, and 2.7 show bivariate plots of one unit against the other. The three plots show the gross and net logdensities, and the right plot shows a scatter plot of a sample. The two units are assumed independent. These plots show where the bivariate distribution is concentrated.
Figure 2.2 Tame Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.
Figure 2.3 Tame Case Study, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.