Riskiness Leverage Models

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

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

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

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

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

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

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

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

TVaR • Here we compute the total capital instead:

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

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 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 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

Back To VaR • We compute the total capital again with x = (x1,x2)

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

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):

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%

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)

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

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

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”

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

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

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

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

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

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%

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