1. Riskiness Leverage Models Rodney Kreps’ Stand In (Stewart Gleason) CAS Limited Attendance Seminar on Risk and Return September 26, 2005

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 C k 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

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

4. Riskiness Leverage Models Start with N random variables X k (think of unpaid losses by line of business at the end of a policy year) and their total X Denote by  the mean of X, C the capital to support X and R then the risk load With analogy to the balance sheet,  is the carried reserve, R is surplus and C is the total assets Denote by  k the mean of X k, C k the capital to support X k and R k the risk load for the line of business is

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(x 1,...,x N ) dx 1...dx N and f(x 1,...,x N ) 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) =  (x –  ), then

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 X k For example, if L(x) =  (x –  ), then

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

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% x q 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-x q ) is the indicator function of the half space where x 1 +  +x N > x q

9. TVaR Here we compute the total capital instead:

10. TVaR The capital allocation is then: C k is the average contribution that X k makes to the total loss X when the total is at least x q 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

11. VaR VaR or Value at Risk is simply a given quantile x q of the distribution – The math is much harder to recover VaR than for TVar! To reproduce VaR, choose  (y-y 0 ) is the Dirac delta function (which is not a function at all!) –  (y-y 0 ) is really defined by how it acts on other functions – It “picks out” the value of the function at y 0 – May be familiar with it when referred to as a “point mass” in probability readings  is a constant to be determined as we progress

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  b is the Dirac delta operator with a “mass” at y = b, Formally, we write – As a Riemann integral, this statement has no meaning – Manipulating  (  ) 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:

13. The Dirac Delta Function The following was suggested for the leverage function: We know what  (x-x q ) means if both x and x q are points in R N, but x q is a scalar! In this case,  (x-x q ) is actually not a point mass but a “hyperplane mass” living on the plane x 1 +  +x N = x q One more thing: in the paper, the constant  is given as “f(x q )” – f(x) is a function of several variables and x q is a scalar! We will walk through the calculation in two variables to see how to interpret these quantities

14. Back To VaR We compute the total capital again with x = (x 1,x 2 )

15. Back To VaR Now we see what “f(x q )” actually means – the right choice for  is We also recognize that is just the conditional probability density above the line t = x q - s

16. Back To VaR With this choice of  we get: The comeasure is For C 2, we integrate with respect to x 1 first (and vice versa):

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 x q and then average the component losses to get the “co VaR” In practice, we are likely to have exactly one trial in which X = x q The solution is to take all of the trials for which X is in a small range around x q, e.g. x q ± 1%

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)

19. Mean Downside Deviation Mean downside deviation has This is actually a special case of TVaR with x q =  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   2 for this (very simplistic) leverage function

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

21. Considerations in Selecting a Leverage Function Should be a down side measure (the accountant’s 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 doesn’t matter how much dirt is on top”

22. Considerations in Selecting a Leverage Function Regulator’s 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 regulator’s 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 , i.e., when x –  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 

24. Implementation Example ABC Mini-DFA.xls is a spreadsheet representation of a company with two lines of business – X 1 : Net Underwriting Income for Line of Business A – X 2 : Net Underwriting Income for Line of Business B – X 3 : 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

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 98 th 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 – 98 th percentile of net income is a loss of $4.7 million – TVaR at the 98 th percentile is $6.2 million – Beginning surplus is $9.0 million – almost (but not quite) the prudent multiple required

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

27. Implementation Example Could shift mix of business away from Line B but also could buy reinsurance on Line B – X 4 : Net Ceded Premium and Recoveries for a Stop Loss contract on Line of Business B Summary of results from simulation with reinsurance – 98 th percentile of net income is a loss of $2.9 million – TVaR at the 98 th 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%

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 X 4 averaged to compute the co measure have the opposite sign of the values for Line B (X 2 ) 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