Riskiness Leverage Models
Rodney Kreps’ Stand In (Stewart Gleason)
CAS Limited Attendance Seminar on Risk and Return
September 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

N
Ck  C
i 1

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
X  i 1 X k
N

Denote by m the mean of X, C the capital to support X and R then
the risk load
C  m R


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
Ck  mk  Rk
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
R   ( x  m )Lx  dF ( x )
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
R  b  x  m  dF ( x )  bVar  X 
2
Riskiness Leverage Models

Riskiness of each line of business is defined analogously and
results in the additive allocation
Rk   ( x k  m k )Lx  dF ( x )
  ( x k  m k )Lx1    xN  f ( x1,..., xN ) dx1 dxN

It follows directly that
R  i 1Rk
N
and
C  i 1Rk  mk  R  m
N
regardless of the joint dependence of the Xk

For example, if L(x) = b(x – m), then
Rk  b  xk  mk x  m  dF ( x )  bCov ( X k , X )  R
Cov ( X k , X )
Var ( X )
Riskiness Leverage Models

Covariance and higher powers have
L( x )  ( x  m )
n

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
1
TVaR( b )  E X X  b 
x f ( x1,..., xN ) dx1 dxN
Pr{ X  b } { x: x1  xN b }

To reproduce TVaR, choose
L(x) 




 ( x  xq )
1- q
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:
C  m  ( x  m )
m

 ( x  xq )
m
Pr{ X  xq
1 q
}
dF ( x )
{ x : x1  x N  x q }
f ( x1 ,..., xN ) dx1 dxN
1
x f ( x1 ,..., xN ) dx1 dxN

Pr{ X  xq } { x : x1  xN  xq }
1
mm
x f ( x1 ,..., xN ) dx1 dxN

{
x
:
x



x

x
}
1
N
q
Pr{ X  xq }
 E [ X | X  xq ]
TVaR

The capital allocation is then:
Ck  m k   ( x k  m k )
 mk  mk 
 ( x  xq )
1 q
dF ( x )
1
x k f ( x1 ,..., xN ) dx1  dxN

{
x
:
x


x

x
}
1
N
q
Pr{ X  xq }
 E [ X k | X  xq ]

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 ( x  xq )
L(x) 
b

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,
D b g   g( b )

Formally, we write
D b g    d ( y  b )g( y )dy  g( b )


– 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:
D b g   g( b1,..., bN )
The Dirac Delta Function

The following was suggested for the leverage function:
L(x) 




d ( x  xq )
b
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)
d ( x  xq )
C  m  ( x  m )
dF ( x )
b
m
 m    d ( x1  ( xq  x 2 ))f ( x1 , x 2 ) dx1 dx 2
b




(x


b
1
m
1

 x 2 )d ( x1  ( xq  x 2 ))f ( x1 , x 2 ) dx1 dx 2
m
1
f
(
x

x
,
x
)
dx

( xq  x 2  x 2 )f ( xq  x 2 , x 2 )dx 2
q
2
2
2


b
b
Back To VaR

Now we see what “f(xq)” actually means – the right choice for b
is
b   f ( xq  x2 , x2 )dx 2

We also recognize that
f* (s ) 
f ( xq  s , s )
f( x
q
 u ,u ) du
is just the conditional probability density above the line
t = xq - s
Back To VaR

With this choice of b we get:
C  m  m  ( xq  x2  x2 )  xq

The comeasure is
Ck  m k   ( x k  m k )


1
x d( x


b
k
1
d ( x  xq )
dF ( x )
b
 x 2  xq )f ( x1 , x 2 ) dx1dx 2
For C2, we integrate with respect to x1 first (and vice versa):
C2 

1
b
  x d( x
2
1
1
x f( x

b
2
q
 ( xq  x 2 ))f ( x1 , x 2 ) dx1dx 2
 x 2 , x 2 ) dx 2
  s f * ( s ) ds
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
L( x )   ( x  b )

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)
R  E [ X | X  b ] Pr{ X  b }
Mean Downside Deviation

Mean downside deviation has
L( x ) 

b( x  m )
Pr{ X  m }
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
L( x )  b ( x  m ) ( x  m )

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
 ( x  S )
Lregulator (x) 
Pr{X  S}
Considerations in Selecting a Leverage
Function

A possibility for a leverage function that satisfies management
criteria is
if x  m
0

Lmanagement (x)   
( x  m )
b
1


 
 if x  m
 


S

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 
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