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Riskiness Leverage Models. Paper by Rodney Kreps accepted for the 2005 ProceedingsOne criticism of capital allocation" in the past has been that most implementations are actually superadditiveIf Ck is the capital need for line of business k and C is the total capital need, thenThe formulation
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1. Riskiness Leverage Models Paper in submission to the CAS for publicationPaper in submission to the CAS for publication
2. 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
The problem becomes finding a leverage function which accurately reflects the management attitude toward risk, as expressed by different outcomes. The allocation is automatic.The problem becomes finding a leverage function which accurately reflects the management attitude toward risk, as expressed by different outcomes. The allocation is automatic.
3. 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
The problem becomes finding a leverage function which accurately reflects the management attitude toward risk, as expressed by different outcomes. The allocation is automatic.The problem becomes finding a leverage function which accurately reflects the management attitude toward risk, as expressed by different outcomes. The allocation is automatic.
4. 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
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
5. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
6. 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
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
7. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
8. 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 >0
In our situation, ?(x-xq) is the indicator function of the half space where x1+?+xN > xq The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
9. TVaR Here we compute the total capital instead:
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
10. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
11. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
12. 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:
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
13. 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
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
14. Back To VaR We compute the total capital again with x = (x1,x2)
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
15. 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
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
16. 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):
The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
17. 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 ą 1% The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
18. 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) The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
19. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
20. 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 The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.The current formulation as mean plus risk load is used because it is fairly intuitive for actuaries.
From the formulas, any linear combination of variable and mean would create additive co-measures.
Leverage at an outcome can also be thought of as the disutility of the outcome.
21. 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
22. 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
23. 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
24. 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 Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.
25. 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
Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.
26. 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 Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.
27. 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% Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.
28. 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
Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.Riskiness Leverage Models.doc will be published by the CAS this year. This spreadsheet is a modified version of an accompanyment to it.
