Riskiness Leverage Models

1. Riskiness Leverage Models

2. Riskiness Leverage Models • AKA RMK algorithm • Risk/Surplus/Cost of Capital can be allocated to any level of detail in a completely additive fashion. • Riskiness only needs to be defined on the total, and can be done so intuitively. • Many functional forms of risk aversion are possible. • All the usual forms can be expressed, allowing comparisons on a common basis. • Simple to do in simulation situation.

3. Riskiness Leverage Models • Start with N random variables (think underwriting and other cash flows) and their total • Riskiness be expressed as the mean value of a linear functional in the total times an arbitrary leverage function depending only on the total.

4. Riskiness Leverage Models • Typical example: functional is variable minus constant times the mean of the variable. • The simple allocation of the riskiness to an individual variable is

5. Riskiness Leverage Models • Surplus, risk load or whatever can be allocated proportionally and everything will add no matter what the dependency structure. • These are referred to a co-measures, in analogy with the simple examples of covariance, co-skewness, and so on. • Covariance and higher powers have a = 1 and

6. TVAR as Riskiness Leverage • TVAR has leverage zero below some value corresponding to a percentile , and constant above it: • This can be re-framed as • and individual riskiness as

7. L x L x L m x Riskiness Leverage Examples VaR: TVaR: Semi-variance:

8. EPD as Riskiness Leverage • Expected Policyholder Deficit has (1) functional = variable - some value and (2) leverage zero below the value and 1 above it: • This is • and individual riskiness as

9. Generalization • h(Y) is any linear functional, for example involving the variable and its mean value and its covariance with, say, the market. This holds for all constant a and b and any random variables X and Z

10. Special case for the Leverage L is a function of the ratio of Y to its mean value The marginal allocation is then The first part is the naïve comeasure, and the second is a correction to make it marginal. The second part sums to zero over all the comeasures.

11. Generic Riskiness Leveragefor management should • be a down side measure (the accountant’s point of view); • be more or less constant for excess that is small compared to capital (risk of not making plan, but also not a disaster); • become much larger for excess significantly impacting capital; and • not increase for excess significantly exceeding capital – once you are buried it doesn’t matter how much dirt is on top. Note: the regulator’s leverage increases.

12. Mango capital consumption example L Loss Rental Consumption Beyond Allocation

13. How to choose measures? • Try out various measures on simulation to see how different they are. • Try out various measures on past history to see what would have guided you well. • Try out various measures on different levels of management to see what kind of buy-in you can get. • Run candidates in parallel with current processes for a while to see what they suggest.

14. A miniature companyportfolio example • ABC Mini-DFA.xls is a spreadsheet representation of a company with two lines of business. • How do we as company management look at the business? • “For the X PERCENT of possibilities of net income that are less than $BAD we want the surplus to be a PRUDENT MULTIPLE of the average value so that we can go on in business.” • Looking at the numbers quantifies x% as 2% and prudent as 1.5.

15. Lessons learned • Returns on allocated surplus are treacherous, and must be used circumspectly. • Surplus cannot really be allocated, but the cost of capital can. • For a cedent to evaluate the worth of a reinsurance contract, he needs only to know its effect on his own risk profile. • If the contract can release capital (to be used in whatever fashion) then it is worth taking on. • The profitability to the reinsurer is irrelevant.

16. In the evening…