Capital Allocation using the Ruhm-Mango-Kreps Algorithm

David L. Ruhm, FCAS

Enterprise Risk Management Symposium

Session CS-13: Risk-Adjusted Capital Allocation

July 30, 2003

Washington, DC

The Capital Allocation Problem

• How can total-company capital, and its costs, be allocated to all sources of risk in a way that:– Is internally consistent (summing within parts),– Allocates in proportion to risk contributed,– Attributes diversification benefits to sources,– Can be used with any specified risk measure,– Is consistent with established financial theory.

The RMK algorithm

RMK meets the above requirements, plus:

• Is simple to use,

• Can be explained,

• Has solid underlying mathematical theory,

• Evaluates all risk in terms of total-company, “top-down” view (instead of evaluating each part as if stand-alone)

The RMK Algorithm

Central principle

Each component is evaluated, to measure its contribution to total-company risk.

The RMK Algorithm

• The algorithm, in short:– Simulate possible outcomes, by component and total

company. Calculate unweighted average outcome (expected value) for each.

– Select a risk measure on total company outcomes, expressed as higher weights on adverse outcomes.

– Apply the risk-weights to the individual components, and calculate risk-weighted averages.

– Allocate capital in proportion to risk, as measured by the difference between the risk-weighted average and the unweighted average:

Risk ~ Risk-Weighted Expected Value – Unweighted Expected Value

Selecting a risk measure

• Any standard risk measure (e.g., value at risk, tail-value at risk, default rate of surplus) can be expressed in the form of weights.

• Simplest: Net loss outcomes > 1 , net gain outcomes = 1.– Measures tail of distribution, all loss outcomes equally weighted.– Risk Measure ~ Frequency of Loss x Average Severity of Loss.– Is a good risk measure, similar to TVaR.– Weights could be refined to distinguish among loss, gain levels.

• In general, risk measure weights are:– Non-negative,– Higher for worse (“riskier”) outcomes,– Lower for better outcomes.

Some examplesContext Curve Weights Result

Stock Mkt Normal Wang transform

CAPM

Derivatives Lognormal Wang transform

Black-Scholes

Insurance Any TVaR (wtd) Default Rate of Surplus

Insurance Any P(ruin) Myers-Read

RMK useful properties

• Any additive allocation method can be replicated using the RMK framework.

• Works with user’s choice of risk measure.• Allocates total company risk to parts, down to any

desired level of detail.• Consistent with financial theory, and arbitrage-

free: risk-weighted averages are equivalent to “risk-neutral” valuations.

• Simple, transparent.

Selected References

• Ruhm / Mango, “A Risk Charge Calculation Based on Conditional Probability,” Bowles Symposium, Atlanta, April 2003 (submitted to NAAJ).

• Kreps, “Riskiness Leverage Ratios,” Instrat working paper

• Ruhm / Mango, “A Method of Implementing Myers-Read Capital Allocation in Simulation,” CAS Spring Meeting, May 2003 (submitted to CAS Forum).