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Conditional Risk Charge. Main principle of conditional risk charge: Each risk receives a charge that represents how much it contributes to undesirable portfolio outcomes. Framework can reflect any dependence structure Beyond correlation matrices.
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Conditional Risk Charge • Main principle of conditional risk charge: Each risk receives a charge that represents how much it contributes to undesirable portfolio outcomes. • Framework can reflect any dependence structure • Beyond correlation matrices
How to Calculate Conditional Risk Charges, in Six Minutes or Less Simple 2x2 example (Exhibit 2): Risk 2 100200Total 100 35% 15% 50% Risk 1 20025%25% 50% Total 60% 40% Correlation: 20%
How to Calculate Conditional Risk Charges, in Six Minutes or Less • Step 1: Select or calculate risk weights for all possible portfolio outcomes using the risk measure of your choice: OutcomeWeight 200 0.50 300 1.00 400 1.25
How to Calculate Conditional Risk Charges, in Six Minutes or Less • Step 2: Normalize the weights (optional, but recommended for arbitrage-free prices): OutcomeProbWeightNorm’d 200 35% 0.50 0.563 300 40% 1.00 1.127 400 25% 1.251.408 Expected Value: 0.89 1.000
How to Calculate Conditional Risk Charges, in Six Minutes or Less • Step 3: Calculate the total portfolio price, as the weighted expected value: OutcomeProbWeightProduct 200 35% 0.563 39 300 40% 1.127 135 400 25% 1.408 141 E[X] = 290 Risk-Loaded Price = 315
How to Calculate Conditional Risk Charges, in Six Minutes or Less • Step 4: Select a component of the portfolio for which you want to calculate the risk charge. Can be a risk, an excess layer of a risk, or any part. • Example: Portfolio = Risk 1 + Risk 2 Calculate the risk charge for Risk 1.
How to Calculate Conditional Risk Charges, in Six Minutes or Less • Step 5: Calculate Price = E[ZR], using conditional distributions of the portfolio: PortfolioRisk 1=100Risk 1=200Z = Weight 200 70% 0% 0.563 300 30% 50% 1.127 400 0% 50% 1.408 E[Z | Risk 1]: 0.732 1.268 P[Risk 1= y]: 50% 50% E[ZR]: 163.4 Risk Charge = 13.4
Advantages of Method • Each risk receives a charge that represents how much it contributes to undesirable portfolio outcomes. • Additive prices. • Extends aggregate risk valuation to any individual risk, including layers of risks. • Handles any underlying dependence structure.
Preview of Mathematical Attractions • This risk charge method can be expressed in a concise formula: Conditional Risk Charge = Cov[Z,R] • The risk charges in CAPM prices are conditional risk charges. • Includes arbitrage-free pricing (e.g., options pricing formulas). • All complete, additive pricing structures implicitly have conditional risk charges in their prices (though they are not, in general, arbitrage-free). They are all represented by the generalized risk pricing formula: Price = W(Cov[Z,R] + E[R])
Example • Can use this method on DFA Model output of a company to develop risk charges by line of business • DFA Insurance Company from 2001 CAS DFA Call Paper Program • An example of a utility-type approach applied to company underwriting result
Example • The valuation formula used to determine the risk-averse outcome weighting is a two-sided utility transform of total underwriting income UITto risk-adjusted underwriting income RUIT via the following formula: • If UIT >= 0 RUIT = UIT * [ 1 + (UIT / 1M )2 ] • Else RUIT = UIT * [ 1 + (-UIT / 100K )0.5 ]
Example • Section (2) on Exhibit 1 shows these parameters and curve forms selected to calibrate to a desired overall implied portfolio risk premium, calculated as follows: • (1) E[UIT] = ($96.9M) • (3) E[RUIT] = ($244.7M) • (4) Implied Portfolio Risk Premium = E[UIT] - E[RUIT] = $147.8M
