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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
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
The allocation of the riskiness to an individual variable is
Rk E xk k L Y
Surplus, risk load or whatever can be allocated proportionally and
everything will add no matter what the dependency structure.
These are referred to a co-measures, in analogy with the simple
examples of covariance, co-skewness, and so on.
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
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
be a down side measure (the accountant’s point of view);
be more or less constant for excess that is small compared to
capital (risk of not making plan, but also not a disaster);
become much larger for excess significantly impacting capital; and
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?
Try out various measures on simulation to see how different they
are.
Try out various measures on past history to see what would have
guided you well.
Try out various measures on different levels of management to see
what kind of buy-in you can get.
Run candidates in parallel with current processes for a while to see
what they suggest.
A miniature company
portfolio example
ABC Mini-DFA.xls is a spreadsheet representation of a company
with two lines of business.
How do we as company management look at the business?
“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.”
Looking at the numbers quantifies x% as 2% and prudent as 1.5.
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.