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. 1.Allocate by Risk Measure Take one from each column and mix carefully 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

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 X i each with mean m i  Then co-measure for X i is: –CoR(X i ) = E[(X i – am i )g(x)|condition] –Note that CoR(X 1 )+CoR(X 2 ) = CoR(X 1 +X 2 ) and so the sum of the CoR’s of the n X i ’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  TVAR q = E[X|X>q]  Co-TVaR q (X i ) = E[X i |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  XTVAR q = E[X – m|X>q]  Co- XTVAR q = E[X i – m i |X>q]  Allocates average loss excess of mean when total losses are above the target value  Allocates nothing to a constant X i

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