Riskiness Leverage Models

Riskiness Leverage Models

AKA RMK algorithm

Risk/Surplus/Cost of 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

Riskiness be expressed as the mean value of a linear functional in the total times an arbitrary leverage function depending only on the total.

1

N

nn

Y X

Riskiness Leverage Models

Typical example: functional is variable minus constant times the mean of the variable.

The simple allocation of the riskiness to an individual variable is

R E Y L Y

k k kR E x L Y

Riskiness Leverage Models

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

NL Y Y

TVAR as Riskiness Leverage

TVAR has leverage zero below some value corresponding to a percentile , and constant above it:

This can be re-framed as

and individual riskiness as

( )

1qy y

L yq

1| 1R E Y Y F q

1| 1k kR E X Y F q

Riskiness Leverage Examples

x

L

TVaR:L

x

Semi-variance:

VaR:

L

x

EPD as Riskiness Leverage

Expected Policyholder Deficit has (1) functional = variable - some value and (2) leverage zero below the value and 1 above it:

This is

and individual riskiness as

( )L y y b

|R E Y b Y b S b

| |k kkR E X Y b E X Y b S b

Generalization

h(Y) is any linear functional, for example involving the variable and its mean value and its covariance with, say, the market.

E h Y L Y

,h Y aY bE Y cCov Y M

h aX bZ ah X bh Z This holds for all constant a and b and any random variables X and Z

Special case for the Leverage

L is a function of the ratio of Y to its mean value

Y

E Y

The marginal allocation is then

2'1

k k kk E h X L E h Y L E Y X YE XE Y

The first part is the naïve comeasure, and the second is a correction to make it marginal. The second part sums to zero over all the comeasures.

Generic Riskiness Leveragefor 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

not increase for excess significantly exceeding capital – once you are buried it doesn’t matter how much dirt is on top. Note: the regulator’s leverage increases.

Mango capital consumption example

Loss

RentalConsumption

L

Beyond Allocation

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 companyportfolio 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.

Lessons learned

Returns on allocated surplus are treacherous, and must be used circumspectly.

Surplus cannot really be allocated, but the cost of capital can.

For a cedent to evaluate the worth of a reinsurance contract, he needs only to know its effect on his own risk profile.

– If the contract can release capital (to be used in whatever fashion) then it is worth taking on.

– The profitability to the reinsurer is irrelevant.

In the evening…