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

CCN

i k 1

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 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 Xk, Ck the capital to support Xk and Rk the risk load for the line of business is

N

i kXX1

RC

kkk RC

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) = (x – ), then

)( )( xdFxLxR

XVarxdFxR )( 2

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) = (x – ), then

NNNkk

kkk

dxdxxxfxxLx

xdFxLxR

111 ),...,( )(

)( )(

RRCRR k

N

i k

N

i k 11 and

)(

),(),()(

XVar

XXCovRXXCovxdFxxR k

kkkk

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

nxxL )()(

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

}bxx :x{ NNN

dxdx )x,...,x(f x}bXPr{

bXXE)b(TVaR

1

11

1

q-1

)xx( L(x) q

TVaR

Here we compute the total capital instead:

]xX|X[E

dxdx )x,...,x(f x}xXPr{

dxdx )x,...,x(f x}xXPr{

dxdx )x,...,x(f}xXPr{

)x(dFq

)xx()x(C

q

}xxx :x{ NNq

}xxx :x{ NNq

}xxx :x{ NNq

q

qN

qN

qN

1

1

1

11

11

11

1

1

1

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

]xX|X[E

dxdx )x,...,x(f x}xXPr{

)x(dFq

)xx()x(C

qk

}xxx :x{ NNkq

kk

qkkkk

qN

111

1

1

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

(y-y0) is the Dirac delta function (which is not a function at all!)

– (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

is a constant to be determined as we progress

)xx(

L(x) q

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:

)b(ggb

)b(gdy)y(g)by(gb

)b,...,b(gg Nb 1

The Dirac Delta Function

The following was suggested for the leverage function:

We know what (x-xq) means if both x and xq are points in RN, but xq is a scalar!

In this case, (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 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

)xx(

L(x) q

Back To VaR

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

22222222

21212121

212121

1

1

dx)x,xx(f)xxx(dx)x,xx(f

dxdx )x,x(f))xx(x()xx(

dxdx )x,x(f))xx(x(

)x(dF)xx(

)x(C

qqq

q

q

q

Back To VaR

Now we see what “f(xq)” actually means – the right choice for is

We also recognize that

is just the conditional probability density above the line

t = xq - s

222 dx)x,xx(f q

du )u,ux(f

)s,sx(f)s(*f

q

q

Back To VaR

With this choice of we get:

The comeasure is

For C2, we integrate with respect to x1 first (and vice versa):

qq x)xxx(C 22

212121

1dxdx )x,x(f)xxx(x

)x(dF)xx(

)x(C

qk

qkkkk

ds )s(*f s

dx )x,xx(fx

dxdx )x,x(f))xx(x(xC

q

q

2222

21212122

1

1

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)

)bx()x(L

}bXPr{ ]bX|X[ER

Mean Downside Deviation

Mean downside deviation has

This is actually a special case of TVaR with xq = 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

}XPr{

)x()x(L

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

)x()x()x(L

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

S}Pr{X

)Sx( (x)Lregulator

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

x if S

)x(

x if

(x)Lmanagement 1

0

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