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Riskiness Leverage Models
Riskiness Leverage Models

AKA RMK algorithm

Capital can be allocated to any level of detail in a completely
additive fashion.

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 (think underwriting and other cash
flows) and their total
N
Y   Xn
n 1
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Riskiness be expressed as the mean value of a linear functional in
the total times an arbitrary function depending only on the total.

Simple example: functional is variable minus constant times the
mean of the variable.
R  E Y    L Y  
Riskiness Leverage Models
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The allocation of the riskiness to an individual variable is
Rk  E  xk  k  L Y  
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Surplus, risk load or whatever can be allocated proportionally and
everything will add no matter what the dependency structure.
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These are referred to a co-measures, in analogy with the simple
examples of covariance, co-skewness, and so on.
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Covariance and higher powers have  = 1 and
L  Y   Y   
N
TVAR as Riskiness Leverage

TVAR has (1) functional = variable and (2) leverage zero below some
value corresponding to a percentile , and constant above it:
L( y ) 

  y  yq 
1 q
This can be re-framed as
1

R  E Y | Y  F 1  q  

and individual riskiness as
1

Rk  E  X k | Y  F 1  q  
EPD as Riskiness Leverage
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Expected Policyholder Deficit has (1) functional = variable - some
value and (2) leverage zero below the value and 1 above it:
L( y)    y  b 

This is
R  E Y  b | Y  b S  b 

and individual riskiness as


Rk  E  X k | Y  b   E  X k | Y  b  S  b 
Riskiness Leverage Examples
VaR:
L
x
TVaR:
L
x
Semivariance:
L

x
Generic Riskiness Leverage
for management should
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be a down side measure (the accountant’s point of view);
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be more or less constant for excess that is small compared to
capital (risk of not making plan, but also not a disaster);
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become much larger for excess significantly impacting capital; and
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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.
How to choose measures?
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Try out various measures on simulation to see how different they
are.
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Try out various measures on past history to see what would have
guided you well.
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Try out various measures on different levels of management to see
what kind of buy-in you can get.
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Run candidates in parallel with current processes for a while to see
what they suggest.
A miniature company
portfolio example
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ABC Mini-DFA.xls is a spreadsheet representation of a company
with two lines of business.
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How do we as company management look at the business?
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“For the x percent of possibilities of net income that are less
than $BAD we want the surplus to be a prudent multiple of the
average value so that we can go on in business.”
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Looking at the numbers quantifies x% as 2% and prudent as 1.5.
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Two lessons from the model:
– Returns on allocated surplus can be VERY misleading and
need careful interpretation.
– that we do not need to know what the reinsurer’s rate of return
is on a contract to know how good or bad it is for us.