Capital Allocation Survey

1. Capital Allocation Survey

2. Purpose of Allocating Capital • Not a goal in itself • Used to make further calculations, like adequacy of business unit profits, incentive compensation • Can also be used for strategic planning: • Would growing this business unit by 10% generate enough profits to make up for the cost of the capital it would need? • Best methods to some extent depend on purpose of allocation

3. Approaches Used for Evaluating Profitability of Business Units • Divide return by allocated capital • Allocate capital by some risk measure and divide • Compare return to price of bearing risk • Use a theory of market risk pricing to set targets • Charge actual marginal capital costs against profits • Compare the profits expected from a strategic plan to the cost of the extra total capital the firm needs • Direct marginal cost – not an allocation • Compare value of float generated by the business to a leveraged investment fund with the same risk • Do for whole firm, then look at marginal impact of business unit or growth plan

4. Risk Measures VaR EPD Tail VaR X TVaR Standard Deviation Variance Semi-Variance Cost of Default Option Mean of Transformed Loss Allocation Methods Proportional Spread Marginal Analysis By whole business unit Increment of business unit Game Theory Equalize Relative Risk Apply Co-Measure Allocate by Risk MeasureTake one from each column and mix carefully

5. 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

6. 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

7. 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

8. Myers-Read Capital Allocation • Overall capital: target default put cost as % of expected losses • Limited capital of an insurer gives it an implicit option to put losses that exceed capital to the policyholders • Allocation method is incremental marginal • Last dollar of expected loss in a business unit is charged with the capital needed to keep the company put cost ratio constant • The whole business unit (or policy) gets charged at that ratio to expected losses – a pure marginal method • But all capital is allocated, as the sum of the marginal capital charges equals the whole capital of the firm • From the additivity of option prices • A constant risk generally gets a negative capital charge • It does not add risk but both adds stability and accepts risk of non-payment

9. Allocation by Risk Measure • Myers-Read and Co-XTVaR both additive and reasonable • But pricing to equalize returns on capital so allocated may not tie in to risk pricing standards • Myers-Read do not advocate pricing purely in proportion to allocated capital – a risk charge is also added e.g. for covariance with market • Makes most sense for allocation of frictional costs • Costs from holding capital even if no risk taken • Cost is proportional to capital

10. 2. Target to 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 to account for actual risk pricing • Fama and 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

11. 3. 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

12. 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

13. 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

14. 4. Compare to Closed-end Mutual Fund • Insurer can be viewed as a tax-disadvantaged leveraged mutual fund • Combined ratio less 100% is cost of funds • Measure mean and risk for insurer’s post-tax return • Then find parameters for mutual fund that give same after-tax distribution of return • Find amount to be borrowed, investment mix, and borrowing rate • Evaluate financial worth of writing the insurance by the risk-equivalent borrowing rate • A high rate easy to obtain says writing insurance is not adding value • A low risk-equivalent rate indicates value is added • Business units can be evaluated based on their marginal impact on the equivalent borrowing rate

15. Allocation Summary and Evaluation • Allocating by risk measure straightforward but arbitrary and might be allocating fixed capital costs; works for allocating frictional costs • Using risk pricing appropriate for profitability comparison but requires a good theory of pricing • Actual marginal surplus most useful for determining economic contributions of business units. This is not the same as allocation in proportion to marginal risk. • Leveraged mutual fund comparison a basically qualitative method for evaluating return on total capital and the marginal contribution of each business unit to that