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

Guy Carpenter 2

Risk-Adjusted Profit from ERM ModelsERM quantifies risk of company and each business unitManagement 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 ProfitabilityStrategic planning for insurerGrow business units that have higher profit in relationship to riskDe-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 quantileCompute – 1/100 quantile for each business unit and for companyAllocate 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 measure1/3333 quantile impossible to quantify accuratelyProfit not measured relative to marginal cost of riskArbitrary 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-measuresGoal 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

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DefinitionDenoting loss for the total company as Y, and for each line of business as Xi let:

R(Y) = E[ Y | F(Y) > ] . ThenCo-R(Xi) = E[ Xi | 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(Xj) = E[h(Xj)g(Y)| condition on Y]

VaR(Y) = E[Y|F(Y) = ], r(Xj) = E[Xj|F(Y) = ]

Guy Carpenter 8

Improvements Round 2 – Marginal Decompostion

Applies when allocation of capital is based on allocating a risk measureMarginal 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 shareMarginal 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 measureAll 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 riskThen the comparable return for the whole company goes up

Not all allocation does that; marginal decomposition does

Thus useful for strategic planning

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How to Achieve Marginal DecompositionFirst of all, risk measure must be scalable

Proportional increase in business produces a proportional increase in the risk measure

Standard deviation, tail risk measures areVariance isn’t

Also requires that change in business unit is scale increase – homogeneous growthAllocation is a co-measure defined by a derivative of the company risk measureSums up under these conditions: Euler

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Formal DefinitionMarginal r(Xj) = lim0[(Y+Xj) – (Y)]/ .

Take derivative of numerator and denominator wrt . L’Hopital’s rule then gives r(Xj) = ’(Y+Xj)|0 .

Consider (Y) = Std(Y)(Y+Xj) = [Var(Y)+2Cov(Xj,Y)+2Var(Xj)]½ so ’(Y+Xj)|0 = [Var(Y)+2Cov(Xj,Y)+2Var(Xj)]-½ [Cov(Xj,Y) + Var(Xj)]|0

r(Xj) = Cov(Xj,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(Xj) =E[(Xj – EXj)(Y – EY)/Std(Y)] = Cov(Xj,Y)/Std(Y)

So this co-measure gives marginal allocation

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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

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 measure1/3333 quantile impossible to quantify accuratelyProfit not measured relative to marginal cost of riskArbitrary choices required (1/100, etc.)Not clear that growing units with higher returns will actually increase risk adjusted return or firm value

Improvements Round 3Improvements Round 3Risk Measures and Capital Risk Measures and Capital

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Purposes of Risk Measures

Have a consistent way of comparing different risks, including asset risk, results from different businessesComparing profit to risk one key application

For strategic planning – which lines to grow, which to re-organizeMaybe for paying bonuses to managers

Measuring impact of risk-managementAll 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 measureUseful 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 1/3333Includes 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 importantBut the idea seems sound

Losing part of capital can be a big hit to valueEven profit less than target profit can be also

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Classification of Risk MeasuresMoment based measures

Variance, standard deviation, semi-standard deviationGeneralized moments, like E[YecY/EY]

Tail based measuresLook only at the tail of the distribution

Transformed distribution measuresChange the probabilities then take mean or other risk measure with the transformed probabilitiesUses whole distribution but puts more weight in tails by increasing the probabilities of large losses

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Variance and Standard Deviation

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

1.0%

5,000 7,500 10,000 12,500 15,000 17,500 20,000 22,500 25,000

Size of Loss

Rel

ativ

e P

roba

bilit

y

.

Risk A Risk B

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

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Spectral Measures

for nonnegative function . gives TVaRq.

gives blurred VaR

Co-measure isMarginal for step function or smooth .

YFYE

pqqqp

p1,11,0

21

21exp

21

qpp

YFXEr jj

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Tail-Based Measures

Probability of defaultValue at riskTail value at riskExcess tail value at riskExpected policyholder deficitVaR criticized for not being subadditive but not very important with co-VaRTVaR criticized for linear treatment of large loss

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Transformed Probability Measures

Risk measure is the mean (but could be TVaR, etc.) after transforming the loss probabilities to give more weight to adverse outcomesPrices for risky instruments in practice and theory have been found to be approximated this way

Wang transform for bonds and cat bondsEsscher transform for compound Poisson process tested for catastrophe reinsuranceBlack-Scholes and CAPM are of this form as well

More potential to be proportional to the market value of the risk

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Possible Transforms

G*(x) = Qk[-1(G(x)) + ] where Qk is the t-distribution with k dof - Wang transform

= .0453 and k [5,6] fit prices of cat bonds and various grades of commercial bondsk can be non-integer with beta distribution

Compound Poisson martingale transformRequires function (x), with (x) > – 1 for x>0 * = [1+E(X)]g*(x) = g(x)[1+(x)]/[1+ E(X)]

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Reinsurance Pricing Compared to Minimum Entropy and Least Squares

g*(y) = g(y)ecy/EY/EecY/EY * = EecY/EY

Quadratic

Average

Loading Factors for Martingale Pricing of FE

2.4

3.4

4.4

5.4

6.4

7.4

0 0.005 0.01 0.015 0.02Expected Loss on Line

Load

ing MMM Loading

MEM Loading

Mixed Loading

Premium Loading

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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

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Some Criticisms Historically

Quantile is a very limited risk measure1/3333 quantile impossible to quantify accuratelyProfit not measured relative to marginal cost of riskArbitrary 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 AllocationInherently arbitrary

Several risk measures are equally possible

Basically artificialUnits are not limited to their allocations

Alternative methods of risk-adjusting profit may be better

One possibility is capital consumption

Improvements Round 4 – Capital Improvements Round 4 – Capital ConsumptionConsumptionRisk Adjusted Performance Without Risk Adjusted Performance Without Capital AllocationCapital Allocation

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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 approachAvoids arbitrary and artificial notions of allocating capitalBusiness 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 anyPricing of these options can determine economic value added

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Insurance ViewpointCompany 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 unitsSubtracting this value from profit is the value added of the unit

A form of risk adjusted profitabilityRight measure of profit to compare is expected value of profit if positive times probability it is positive

Company gets the profit if it is positiveCompany pays the losses otherwiseComparing value of these options

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Some Approaches to ValuingUnits that have big loss when firm overall does cost more to reinsure, so correlation is an issueLimits on worth of stop loss

Probably worth more than expected valueProbably worth less than market value

Stop-loss pricing includes moral hazardCompany should be able to control this for unit

Or look at impact of unit loss on firm valueNeed to understand relationship of risk and value

Guy Carpenter 32

Capital Consumption Summary

Perhaps more theoretically sound than allocating capitalDoes not provide return on capital by unit Instead shows economic value of unit profits after accounting for riskA few approaches for calculation possibleReally requires market value of risk

Improvements Round 5Improvements Round 5Market Value of RiskMarket Value of Risk

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Market Value of Risk Transfer

Needed for right risk measure for capital allocationNeeded to value options for capital consumptionIf known, could compare directly to profits, so neither of other approaches would be needed

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Two Paradigms

CAPMArbitrage-free pricing

And their generalizations

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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 quantifyCummins-Phillips using full information betas found required returns around 20%

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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 pricedAssume it is not priced

Possible compromise: price co-jump risk

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Arbitrage-Free Pricing

Incomplete market so which transform?Same transform for all business units?

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So …

Marginal decomposition with co-measures improves allocation exerciseChoice of risk measure can make result more meaningfulCapital consumption removes some arbitrary choices and artificial notionsMarket value of risk is really what is needed