Capital Allocation Discussion - Maximizing Efficiency and Risk Management

1. Myers Read Capital Allocation DiscussionCAS Marco Island 2003Gary G Venter

2. Mathematical Setup • c iLi = cL, • c iLi is the capital for the ith policy with expected losses Li , and cL is total capital • D/L is value of default put related to losses • M-R require thatD/Li = D/L • D/L is held constant by adjusting capital • So ci’s are sought that will change the allocated capital ciLi by the amount needed to keep D/L constant when there is a small change in Li

3. The Answer (assuming assets and losses are uncorrelated) • c i = c + (bi – 1)Z, where Z does not vary with i andbi is a correlation measure for Li with L • Several examples in written discussion • For instance, a constant risk generally gets a negative capital charge • It does not add risk but both adds stability and accepts risk of non-payment

4. Main Problems • Some losses pay in the first year, others take many years to pay • So what is the time period for the option? • Don’t know the extreme tail of the aggregate distribution so don’t know what distribution to use for evaluating the option • Seems equally valid with other methods that also are completely additive (co-measures) • Aimed at allocating frictional costs of holding capital, but tends to be misused as denominator of return on capital calculation by line

5. 1. Time Period • Some losses on six month policies pay within a few months • Others take years to pay • Current default would affect unpaid losses from many prior years • Could do a sum of three-month increment default options over 20 years • Formula adapted for volatility smile?

6. 2. Measuring Default Probabilities • Usually fit distributions to loss frequency and severity, combine into an aggregate distribution, and extended way out into the tail well beyond the data where the distribution is not known • Actual causes of default are more like court changes in the meaning of contracts, changes in the probabilities of catastrophes, employee fraud, regulatory underpricing, reserve changes showing years of underpricing, asset market collapse, jumps in inflation, etc. • Much of that is difficult to tie to given losses or policies • Risk measurements based on default are problematic • Partial loss of surplus more relevant due to value of renewal book as well as easier to calculate

7. 3. Definition of Co-Measures • Suppose a risk measure for risk X with mean m can be defined as: • R(X) = E[(X– am)g(x)|condition] for some value a and function g • X is the sum of n portfolios Xi each with mean mi • Then co-measure for Xi is: • CoR(Xi) = E[(Xi– ami)g(x)|condition] • Note that CoR(X1)+CoR(X2) = CoR(X1+X2) and so the sum of the CoR’s of the n Xi’s is R(X) • A risk measure could have equivalent definitions with different a’s and g’s so alternative co-measures

8. Example: TVaR • TVARq = E[X|X>q] • Co-TVaRq(Xi) = E[Xi |X>q] • Charges each sub-portfolio for its part of total losses in those cases where total losses exceed threshold value • In simulation, cases where condition is met are selected, and losses of sub-portfolio measured in those cases • Allocates Xito a constant Xi

9. Excess TVAR • XTVARq = E[X – m|X>q] • Co- XTVARq = E[Xi– mi|X>q] • Allocates average loss excess of mean when total losses are above the target value • Allocates nothing to a constant Xi

10. M-R vs. Co-Measures • M-R based on marginal method, which has economic precedent • Co-measures can be adapted to any capital standard • Pretty much of a toss-up

11. 4. Frictional Costs and Return • Taxation of investment income • Investing conservatively to be more secure • Agency and liquidity costs of letting someone else control the capital • All these accrue even if you don’t write any business – they are from the intention to write • They are proportional to capital held, so make sense to relate to capital

12. Risk Bearing Costs • Cost for actually bearing risk, not just from intention to write • Not proportional to capital • E.g., suppose BIG insurance company wants its reinsurers to sell it put options on its stock • Value of put option is not dependent on capital needed to support it – except for a reduction in value for credit risk of seller • Reinsurer should charge for value of risk transfer plus allocation of frictional costs of capital • Return on allocated capital should vary across lines, depending on risk bearing costs

13. Estimating Market Price of Bearing Risk • CAPM might be starting point • Company-specific risk needs to be reflected • Froot-Stein, Mayers-Smith • The estimation of beta itself is not an easy matter • Full information betas • Other factors besides beta are needed in actual pricing • Fama & French Multifactor Explanations of Asset Pricing Anomalies • Heavy tail beyond variance and covariance • Wang A Universal Framework For Pricing Financial And Insurance Risks • Kozik and Larson The N-Moment Insurance CAPM PCAS 2001 • Impact of jump risk

14. Alternative Profitability Measure: Charge Capital Cost against Profits • Instead of return rate, subtract cost of capital from unit profitability • Use true marginal capital costs of business being evaluated, instead of an allocation of entire firm capital • If evaluating growing the business 10%, charge the cost of the capital needed for that much growth • If evaluating stopping writing in a line, use the capital that the company would save by eliminating that line • This maintains financial principle of comparing profits to marginal costs

15. Calculating Marginal Capital Costs • Could use change in overall risk measure of firm that results from the marginal business – but requires selection of the overall risk measure • Or could set capital cost of a business segment as the value of the financial guarantee the firm provides to the clients of the business segment • This could be called capital consumption

16. Value of Financial Guarantee • Cost of capital for subsidiary is a difference between two put options: • 1. The cost of the guarantee provided by the corporation to cover any losses of the subsidiary • 2. The cost to the clients of the subsidiary in the event of the bankruptcy of the corporation • Economic value added of the subsidiary is value of profit less cost of capital • Value of profit is contingent value of profit stream if positive • A pricing method for heavy-tailed contingent claims would be needed • The individual put-option for the business unit may be more heavy-tailed than for the company as a whole