Risk Adjusted Profitability by Business Unit: How to Allocate Capital and How Not to

1. Risk Adjusted Profitability by Business Unit: How to Allocate Capital and How Not to

2. Risk-Adjusted Profit from ERM Models • ERM quantifies risk of company and each business unit • Management would like to use that information to identify units that have better and worse profitability compared to risk

3. Uses of Risk Adjusted Profitability • Strategic planning for insurer • Grow business units that have higher profit in relationship to risk • De-emphasize or restructure business that does not give enough profit for the risk

4. Typical Approach • Quantify risk by a percentile of the distribution of profit • Maybe start with capital = – 1/3333 quantile • Compute – 1/100 quantile for each business unit and for company • Allocate capital by ratio of business unit quantile to company quantile • Divide unit profits by capital so allocated

5. Some Criticisms Historically • Quantile is a very limited risk measure • 1/3333 quantile impossible to quantify accurately • Profit not measured relative to marginal cost of risk • Arbitrary choices required (1/100, etc.) • Not clear that growing units with higher returns will actually increase risk adjusted return or firm value

6. Improvements Round 1 – Co-measures • Goal is additive allocation • Capital allocated separately to lines A and B will equal the capital allocated to lines A and B on a combined basis. • Start with a risk measure for the company, for example the average loss in the 1 in 10 and worse years • Then, consider only the cases where the company’s total losses exceed this threshold. In this example it is the worst 10% of possible results for the company. • For these scenarios co-measure is how much each line of business is contributing to the poor results

7. Definition • Denoting loss for the total company as Y, and for each line of business as Xi let: • R(Y) = E[ Y | F(Y) > a ] . Then • Co-R(Xi) = E[ Xi | F(Y) > a] • More generally: • Risk measure r(Y) defined as: E[h(Y)g(Y)| condition on Y], where h is additive, i.e., h(U+V) = h(U) + h(V) • Allocate by r(Xj) = E[h(Xj)g(Y)| condition on Y] • VaRa(Y) = E[Y|F(Y) = a], r(Xj) = E[Xj|F(Y) = a]

8. Improvements Round 2 – Marginal Decompostion • Applies when allocation of capital is based on allocating a risk measure • Marginal impact of a business unit on company risk measure is decrease in overall risk measure from ceding a small increment of the line by a quota share • Marginal allocation assigns this marginal risk to every such increment in the line • Treats every increment as the last one in • If sum of all such allocations over all lines is the overall company risk measure, this is called a marginal decomposition of the risk measure • All co-measures are additive but not all are marginal

9. Advantage of Marginal Decomposition • You would like to have it so that: • If you increase business in a unit that has above average return relative to risk • Then the comparable return for the whole company goes up • Not all allocation does that; marginal decomposition does • Thus useful for strategic planning

10. How to Achieve Marginal Decomposition • First of all, risk measure must be scalable • Proportional increase in business produces a proportional increase in the risk measure • Standard deviation, tail risk measures are • Variance isn’t • Also requires that change in business unit is scale increase – homogeneous growth • Allocation is a co-measure defined by a derivative of the company risk measure • Sums up under these conditions: Euler

11. Formal Definition • Marginal r(Xj) = lime0[r(Y+eXj) – r(Y)]/e . • Take derivative of numerator and denominator wrt e. • L’Hopital’s rule then gives r(Xj) = r’(Y+eXj)|0 . • Consider r(Y) = Std(Y) • r(Y+eXj) = [Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]½ so r’(Y+eXj)|0 = • [Var(Y)+2eCov(Xj,Y)+e2Var(Xj)]-½ [Cov(Xj,Y) + eVar(Xj)]|0 • r(Xj) = Cov(Xj,Y)/Std(Y) • With h(X) = X – EX and g(Y) = (Y – EY)/Std(Y) • r(Y) =E[(Y – EY)(Y – EY)/Std(Y)] = Std(Y) • r(Xj) =E[(Xj – EXj)(Y – EY)/Std(Y)] = Cov(Xj,Y)/Std(Y) • So this co-measure gives marginal allocation

12. Example – Tail Value at Risk, etc. • Co-TVaR, co-Var are marginal decompositions • Increasing Xj by (1+a) increases co-measure and measure by same amount • EPDa = (1 –a)[TVaRa– VaRa] is expected insolvency cost if capital = VaRa • Co – EPD is a[co-TVaR – co-VaR] and is marginal

13. Some Criticisms Historically • Quantile is a very limited risk measure • 1/3333 quantile impossible to quantify accurately • Profit not measured relative to marginal cost of risk • Arbitrary choices required (1/100, etc.) • Not clear that growing units with higher returns will actually increase risk adjusted return or firm value

14. Improvements Round 3 Risk Measures and Capital

15. Purposes of Risk Measures • Have a consistent way of comparing different risks, including asset risk, results from different businesses • Comparing profit to risk one key application • For strategic planning – which lines to grow, which to re-organize • Maybe for paying bonuses to managers • Measuring impact of risk-management • All of these work better if risk measures proportional to economic value of the risk

16. Relating Capital to Risk Measure • Do not have to set capital = risk measure • Useful alternative is capital as a multiple of a risk measure • Capital = 10 times TVaR @ 80% • Average loss in worst 20% of years is 10% of capital • Models can measure this better than 1/3333 • Includes more adverse scenarios

17. Which Risk Measure? • “It has been clearly demonstrated that the possibility of extreme adverse results is not the only risk driver of importance.” • Wish I knew who said it, what literature it refers to, and what other risk is important • But the idea seems sound • Losing part of capital can be a big hit to value • Even profit less than target profit can be also

18. Classification of Risk Measures • Moment based measures • Variance, standard deviation, semi-standard deviation • Generalized moments, like E[YecY/EY] • Tail based measures • Look only at the tail of the distribution • Transformed distribution measures • Change the probabilities then take mean or other risk measure with the transformed probabilities • Uses whole distribution but puts more weight in tails by increasing the probabilities of large losses

19. Variance and Standard Deviation • Do not differentiate between good and poor deviations. • Two distributions with same mean and standard deviation but Risk B has a much higher loss potential. It will produce losses in excess of 20,000 while Risk A will not. • Semi-variance does

20. Spectral Measures • for nonnegative function h. • gives TVaRq. • gives blurred VaR • Co-measure is • Marginal for step function or smooth h.

21. Tail-Based Measures • Probability of default • Value at risk • Tail value at risk • Excess tail value at risk • Expected policyholder deficit • VaR criticized for not being subadditive but not very important with co-VaR • TVaR criticized for linear treatment of large loss

22. Transformed Probability Measures • Risk measure is the mean (but could be TVaR, etc.) after transforming the loss probabilities to give more weight to adverse outcomes • Prices for risky instruments in practice and theory have been found to be approximated this way • Wang transform for bonds and cat bonds • Esscher transform for compound Poisson process tested for catastrophe reinsurance • Black-Scholes and CAPM are of this form as well • More potential to be proportional to the market value of the risk

23. Possible Transforms • G*(x) = Qk[F-1(G(x)) + l] where Qk is the t-distribution with k dof - Wang transform • l = .0453 and k  [5,6] fit prices of cat bonds and various grades of commercial bonds • k can be non-integer with beta distribution • Compound Poisson martingale transform • Requires function f(x), with f(x) > – 1 for x>0 • l* = l[1+Ef(X)] • g*(x) = g(x)[1+f(x)]/[1+ Ef(X)]

24. Reinsurance Pricing Compared to Minimum Entropy and Least Squares • g*(y) = g(y)ecy/EY/EecY/EY • l* = lEecY/EY • Quadratic • Average

25. Which Risk Measures? • Useful to be proportional to value of risk being measured • Favors transformed probability measures • Tail measures are popular but ignore some of the risk

26. Some Criticisms Historically • Quantile is a very limited risk measure • 1/3333 quantile impossible to quantify accurately • Profit not measured relative to marginal cost of risk • Arbitrary choices required (1/100, etc.) • Not clear that growing units with higher returns will actually increase risk adjusted return or firm value

27. Problems with Capital Allocation • Inherently arbitrary • Several risk measures are equally possible • Basically artificial • Units are not limited to their allocations • Alternative methods of risk-adjusting profit may be better • One possibility is capital consumption

28. Improvements Round 4 – Capital Consumption Risk Adjusted Performance Without Capital Allocation

29. Alternative to Capital Allocation(for measuring risk-adjusted profit) • Charge each business unit for its right to access the capital of the company • Profit should exceed value of this right • Essentially an economic value added approach • Avoids arbitrary and artificial notions of allocating capital • Business unit has option to use capital when premiums plus investment income on premiums run out (company provides stop-loss reinsurance at break-even) • Company has option on profits of unit if there are any • Pricing of these options can determine economic value added

30. Insurance Viewpoint • Company implicitly provides stop-loss reinsurance to each business unit • Any unit losses above premium and investment income on premium are covered • Value of this reinsurance is an implicit cost of the business unit • Higher for higher risk units • Subtracting this value from profit is the value added of the unit • A form of risk adjusted profitability • Right measure of profit to compare is expected value of profit if positive times probability it is positive • Company gets the profit if it is positive • Company pays the losses otherwise • Comparing value of these options

31. Some Approaches to Valuing • Units that have big loss when firm overall does cost more to reinsure, so correlation is an issue • Limits on worth of stop loss • Probably worth more than expected value • Probably worth less than market value • Stop-loss pricing includes moral hazard • Company should be able to control this for unit • Or look at impact of unit loss on firm value • Need to understand relationship of risk and value

32. Capital Consumption Summary • Perhaps more theoretically sound than allocating capital • Does not provide return on capital by unit • Instead shows economic value of unit profits after accounting for risk • A few approaches for calculation possible • Really requires market value of risk

33. Improvements Round 5 Market Value of Risk

34. Market Value of Risk Transfer • Needed for right risk measure for capital allocation • Needed to value options for capital consumption • If known, could compare directly to profits, so neither of other approaches would be needed

35. Two Paradigms • CAPM • Arbitrage-free pricing • And their generalizations

36. CAPM and Insurance Risk • Insurance risk is zero beta so should get risk-free rate? • But insurance companies lose money on premiums but make it up with investment income on float • Really leveraged investment trust, high beta? • Hard to quantify • Cummins-Phillips using full information betas found required returns around 20%

37. Problems with CAPM • How to interpret Fama-French? • Proxies for higher co-moments? • Could co-moment generating function work? • What about pricing of jump risk? • Earthquakes, hurricanes , … • Two standard approaches to jump risk: • Assume it is priced • Assume it is not priced • Possible compromise: price co-jump risk

38. Arbitrage-Free Pricing • Incomplete market so which transform? • Same transform for all business units?

39. So … • Marginal decomposition with co-measures improves allocation exercise • Choice of risk measure can make result more meaningful • Capital consumption removes some arbitrary choices and artificial notions • Market value of risk is really what is needed