Riskiness Leverage Models

1. Riskiness Leverage Models Rodney Kreps Guy Carpenter Instrat

2. Riskiness Leverage Models • 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.

3. Riskiness Leverage Models • Capital needed is mean plus a risk load. • Risk load is the average of the deviation from the mean times a leverage factor. • Leverage factor is a function ONLY of the sum of random variables and other exogenous quantities. • Other formulations are possible, e.g. linear combinations of means and variables.

4. Risk Load on the Total • x is the total loss, the sum of all random variables; f(x) is its probability density; m is its mean. • g(x) is the dimensionless leverage, which probably will depend upon exogenous variables such as surplus. • Leverage is determined by management disutility of outcomes. • Equal leverage is zero risk load; the mean is OK by itself as capital to support the distribution.

5. Risk Load on the Parts • The risk loads for the individual variables always sum up to R, for any dependency structure. • These are called “co-measures” in parallel with the co-variance terminology.

6. Risk Load Reformulated • The risk loads can be thought of as proportional to expectations on a risk adjusted probability

7. g x g x g m x Riskiness Leverage Examples VaR: TVaR: Semi-variance: