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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 7 Definition Denoting loss for the total company as Y, and for each line of business as X i let: R(Y) = E[ Y | F(Y) >  ]. Then Co-R(X i ) = E[ X i | F(Y) >  ] More generally: Risk measure  (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(X j ) = E[h(X j )g(Y)| condition on Y] VaR  (Y) = E[Y|F(Y) =  ], r(X j ) = E[X j |F(Y) =  ]

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 11 Formal Definition Marginal r(X j ) = lim  0 [  (Y+  X j ) –  (Y)]/ . Take derivative of numerator and denominator wrt . L’Hopital’s rule then gives r(X j ) =  ’ (Y+  X j )| 0. Consider  (Y) = Std(Y)  (Y+  X j ) = [Var(Y)+2  Cov(X j,Y)+  2 Var(X j )] ½ so  ’ (Y+  X j )| 0 = [Var(Y)+2  Cov(X j,Y)+  2 Var(X j )] -½ [Cov(X j,Y) +  Var(X j )]| 0 r(X j ) = Cov(X j,Y)/Std(Y) With h(X) = X – EX and g(Y) = (Y – EY)/Std(Y)  (Y) =E[(Y – EY)(Y – EY)/Std(Y)] = Std(Y) r(X j ) =E[(X j – EX j )(Y – EY)/Std(Y)] = Cov(X j,Y)/Std(Y) So this co-measure gives marginal allocation

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

Guy Carpenter 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

Improvements Round 3 Risk Measures and Capital

Guy Carpenter 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

Guy Carpenter 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 80% Average loss in worst 20% of years is 10% of capital Models can measure this better than 1/3333 Includes more adverse scenarios

Guy Carpenter 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

Guy Carpenter 18 Classification of Risk Measures Moment based measures Variance, standard deviation, semi-standard deviation Generalized moments, like E[Ye cY/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

Guy Carpenter 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

Guy Carpenter 20 Spectral Measures for nonnegative function . gives TVaR q. gives blurred VaR Co-measure is Marginal for step function or smooth .

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 23 Possible Transforms G*(x) = Q k [  -1 (G(x)) + ] where Q k is the t-distribution with k dof - Wang transform =.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  (x), with  (x) > – 1 for x>0 * = [1+E  (X)] g*(x) = g(x)[1+  (x)]/[1+ E  (X)]

Guy Carpenter 24 Reinsurance Pricing Compared to Minimum Entropy and Least Squares g*(y) = g(y)e cy/EY /Ee cY/EY * = Ee cY/EY Quadratic Average

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Guy Carpenter 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

Improvements Round 5 Market Value of Risk

Guy Carpenter 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

Guy Carpenter 35 Two Paradigms CAPM Arbitrage-free pricing And their generalizations

Guy Carpenter 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%

Guy Carpenter 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

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

Guy Carpenter 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