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

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

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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

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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

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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

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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 companys
  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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Definition

• Denoting loss for the total company as Y, and for
  each line of business as Xi let
• R(Y) E Y F(Y) gt a . Then
• Co-R(Xi) E Xi F(Y) gt a
• More generally
• Risk measure r(Y) defined as Eh(Y)g(Y)
  condition on Y, where h is additive, i.e.,
  h(UV) h(U) h(V)
• Allocate by r(Xj) Eh(Xj)g(Y) condition on Y
• VaRa(Y) EYF(Y) a, r(Xj) EXjF(Y) a

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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

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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

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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 isnt
• 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

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Formal Definition

• Marginal r(Xj) lime?0r(YeXj) r(Y)/e .
• Take derivative of numerator and denominator wrt
  e.
• LHopitals rule then gives r(Xj) r(YeXj)0
  .
• Consider r(Y) Std(Y)
• r(YeXj) Var(Y)2eCov(Xj,Y)e2Var(Xj)½ so
  r(YeXj)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

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Example Tail Value at Risk, etc.

• Co-TVaR, co-Var are marginal decompositions
• Increasing Xj by (1a) increases co-measure and
  measure by same amount
• EPDa (1 a)TVaRa VaRa is expected
  insolvency cost if capital VaRa
• Co EPD is aco-TVaR co-VaR and is marginal

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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

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Improvements Round 3 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 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

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

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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

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Classification of Risk Measures

• Moment based measures
• Variance, standard deviation, semi-standard
  deviation
• Generalized moments, like EYecY/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

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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

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Spectral Measures

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

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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

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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 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

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Possible Transforms

• G(x) QkF-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) gt 1 for xgt0
• l l1Ef(X)
• g(x) g(x)1f(x)/1 Ef(X)

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Reinsurance Pricing Compared to Minimum Entropy
and Least Squares

• g(y) g(y)ecy/EY/EecY/EY
• l lEecY/EY
• Quadratic
• Average

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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 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

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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

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Improvements Round 4 Capital Consumption Risk
Adjusted Performance Without Capital 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 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

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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

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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

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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

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Improvements Round 5 Market Value of Risk
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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

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Two Paradigms

• CAPM
• Arbitrage-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 quantify
• Cummins-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 priced
• Assume 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 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