Riskiness Leverage Models

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Riskiness Leverage Models
Rodney Kreps Stand In (Stewart Gleason)CAS
Limited Attendance Seminar on Risk and
ReturnSeptember 26, 2005
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Riskiness Leverage Models

• Paper by Rodney Kreps accepted for the 2005
  Proceedings
• One criticism of capital allocation in the past
  has been that most implementations are actually
  superadditive
• If Ck is the capital need for line of business k
  and C is the total capital need, then
• The formulation presented by Kreps provides a
  natural way to allocate capital to components of
  the business in a completely additive fashion

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Riskiness Leverage Models

• Capital can be allocated to any level of detail
• Line of business
• State
• Contract
• Contract clauses
• Understanding profitability of a business unit is
  the primary goal of allocation, not necessarily
  for creating pricing risk loads
• Riskiness only needs to be defined on the total,
  and can be done so intuitively
• Many functional forms of risk aversion are
  possible
• All the usual forms can be expressed, allowing
  comparisons on a common basis
• Simple to do in simulation situation

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Riskiness Leverage Models

• Start with N random variables Xk (think of unpaid
  losses by line of business at the end of a policy
  year) and their total X
• Denote by m the mean of X, C the capital to
  support X and R then the risk load
• With analogy to the balance sheet, m is the
  carried reserve, R is surplus and C is the total
  assets
• Denote by mk the mean of Xk, Ck the capital to
  support Xk and Rk the risk load for the line of
  business is

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Riskiness Leverage Models

• Riskiness be expressed as the mean value of a
  linear function of the total times an arbitrary
  function depending only on the total
• where dF(x) f(x1,...,xN) dx1...dxN and
  f(x1,...,xN) is the joint density function of all
  of the variables
• Key to the formulation is that the leverage
  function L depends only on the sum of the
  individual random variables
• For example, if L(x) b(x m), then

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Riskiness Leverage Models

• Riskiness of each line of business is defined
  analogously and results in the additive
  allocation
• It follows directly that
• regardless of the joint dependence of the Xk
• For example, if L(x) b(x m), then

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Riskiness Leverage Models

• Covariance and higher powers have
• Riskiness models for a general function L(x) are
  referred to as co-measures, in analogy with the
  simple examples of covariance, co-skewness, and
  so on.
• What remains is to find appropriate forms for the
  riskiness leverage L(x)
• A number of familiar concepts can be recreated by
  choosing the appropriate leverage function

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TVaR

• TVaR or Tail Value at Risk is defined for the
  random variable X as the expected value given
  that it is greater than some value b
• To reproduce TVaR, choose
• q is a management chosen percentage, e.g. 99
• xq is the corresponding percentile of the
  distribution of X
• ?(y) is the step function, i.e., ?(y) 0 if y
  0 and
• ?(y) 1 if y gt0
• In our situation, ?(x-xq) is the indicator
  function of the half space where x1?xN gt xq

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TVaR

• Here we compute the total capital instead

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TVaR

• The capital allocation is then
• Ck is the average contribution that Xk makes to
  the total loss X when the total is at least xq
• In simulation, you need to keep track of the
  total and the component losses by line
• Throw out the trials where the total loss is too
  small
• For the remaining trials, average the losses
  within each line

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VaR

• VaR or Value at Risk is simply a given quantile
  xq of the distribution
• The math is much harder to recover VaR than for
  TVar!
• To reproduce VaR, choose
• d(y-y0) is the Dirac delta function (which is not
  a function at all!)
• d(y-y0) is really defined by how it acts on
  other functions
• It picks out the value of the function at y0
• May be familiar with it when referred to as a
  point mass in probability readings
• b is a constant to be determined as we progress

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The Dirac Delta Function

• The Dirac delta function is actually an operator,
  that is a function whose argument is actually
  other functions
• If g is such a function and Db is the Dirac delta
  operator with a mass at y b,
• Formally, we write
• As a Riemann integral, this statement has no
  meaning
• Manipulating d(?) as if it was a function often
  leads to the right result
• When g is a function of several variables and b
  is a point in N space, the same thing still
  applies

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The Dirac Delta Function

• The following was suggested for the leverage
  function
• We know what d(x-xq) means if both x and xq are
  points in RN, but xq is a scalar!
• In this case, d(x-xq) is actually not a point
  mass but a hyperplane mass living on the plane
  x1?xN xq
• One more thing in the paper, the constant b is
  given as f(xq)
• f(x) is a function of several variables and xq is
  a scalar!
• We will walk through the calculation in two
  variables to see how to interpret these
  quantities

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Back To VaR

• We compute the total capital again with x
  (x1,x2)

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Back To VaR

• Now we see what f(xq) actually means the
  right choice for b is
• We also recognize that
• is just the conditional probability density above
  the line
• t xq - s

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Back To VaR

• With this choice of b we get
• The comeasure is
• For C2, we integrate with respect to x1 first
  (and vice versa)

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VaR In Simulations

• When running simulations, calculating the
  contributions becomes problematic
• Ideally, we would select all of the trials for
  which X is exactly xq and then average the
  component losses to get the co VaR
• In practice, we are likely to have exactly one
  trial in which X xq
• The solution is to take all of the trials for
  which X is in a small range around xq, e.g. xq
  1

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Expected Policyholder Deficit

• Expected Policyholder Deficit (EPD) has
• This is very similar to TVaR but without the
  normalizing constant
• It becomes expected loss given that loss exceeds
  b times the probability of exceeding b
• The riskiness functional becomes (R, not C)

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Mean Downside Deviation

• Mean downside deviation has
• This is actually a special case of TVaR with xq
  m
• It assigns capital to outcomes that are worse
  than the mean in proportion to how much greater
  than the mean they are
• Until this point we have been thinking in terms
  of calibrating our leverage function so that
  total capital equals actual capital and
  performing an allocation
• What is the right total capital?
• Interesting argument in the paper suggests b ? 2
  for this (very simplistic) leverage function

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Semi Variance

• Semi Variance has
• Similar to the variance leverage function but
  only includes outcomes that are greater than the
  mean
• Similar to mean downside deviation but increases
  quadratically instead of linearly with the
  severity of the outcome

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Considerations in Selecting a Leverage Function

• Should be a down side measure (the accountants
  point of view)
• Should be more or less constant for excess that
  is small compared to capital (risk of not making
  plan, but also not a disaster)
• Should become much larger for excess
  significantly impacting capital and
• Should go to zero (or at least not increase) for
  excess significantly exceeding capital
• once you are buried it doesnt matter how much
  dirt is on top

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Considerations in Selecting a Leverage Function

• Regulators criteria for instance might be
• Riskiness leverage is zero until capital is
  seriously impacted
• Leverage should not decrease for large outcomes
  due to risk to the guaranty fund
• TVaR could be used as the regulators choice with
  the quantile chosen as an appropriate multiple of
  surplus

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Considerations in Selecting a Leverage Function

• A possibility for a leverage function that
  satisfies management criteria is
• This function
• Recognizes downside risk only
• Is close to constant when x is close to m, i.e.,
  when x m is small
• Takes on more linear characteristics as the loss
  deviates from the mean
• Fails to flatten out or diminish for extreme
  outcomes much greater than capital
• Testing shows that allocations are almost
  independent of a

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Implementation Example

• ABC Mini-DFA.xls is a spreadsheet representation
  of a company with two lines of business
• X1 Net Underwriting Income for Line of Business
  A
• X2 Net Underwriting Income for Line of Business
  B
• X3 Investment Income on beginning Surplus
• Lines of Business A and B are simulated in
  aggregate and are correlated
• B is much more volatile than A
• The first goal is to test the adequacy of capital
  in total

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Implementation Example

• We want our surplus to be a prudent multiple of
  the average net loss for those losses that are
  worse than the 98th percentile.
• Prudent multiple in this case is 1.5
• Even in the worst 2 of outcomes, you would
  expect to retain 1/3 of your surplus
• Prudent multiple might mean having enough surplus
  remaining to service renewal book
• Summary of results from simulation
• 98th percentile of net income is a loss of 4.7
  million
• TVaR at the 98th percentile is 6.2 million
• Beginning surplus is 9.0 million almost (but
  not quite) the prudent multiple required

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Implementation Example

• Allocation Line B is a capital hog
• Line A 13.6
• Line B 84.3
• Investment Risk 2.1
• Returns on allocated capital
• Line A 40.9
• Line B 5.3
• Investments 190.6
• Overall 14.0
• Misleading perhaps Line B needs so much capital,
  other returns are inflated

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Implementation Example

• Could shift mix of business away from Line B but
  also could buy reinsurance on Line B
• X4 Net Ceded Premium and Recoveries for a Stop
  Loss contract on Line of Business B
• Summary of results from simulation with
  reinsurance
• 98th percentile of net income is a loss of 2.9
  million
• TVaR at the 98th percentile is reduced to 3.6
  million
• Capital could be released and still satisfy the
  prudent multiple rule
• Allocation
• Line A 36.3
• Line B 73.9
• Investment Risk 14.2
• Reinsurance -24.4

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Implementation Example

• Reinsurance is a supplier of capital
• In the worst 2 of outcomes, Line B contributes
  significant loss
• In those scenarios, there is a net benefit from
  reinsurance
• The values of X4 averaged to compute the co
  measure have the opposite sign of the values for
  Line B (X2)
• Returns on allocated capital including
  reinsurance
• Line A 15.3
• Line B 6.0 (5.1 if Line B and Reinsurance are
  combined)
• Investments 28.3
• Reinsurance 7.9
• Overall 12.1
• Overall return reduced because of the expected
  cost of reinsurance
• Releasing 1.2 million in capital would restore
  overall return to 14 and still leave surplus at
  more than 2 times TVaR