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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
100
35%
100
Risk 1
200 25%
Total 60%
200
15%
Total
50%
25%
50%
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:
Outcome
200
300
400
Weight
0.50
1.00
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):
Outcome
Prob
Weight Norm’d
200
35%
0.50
0.563
300
40%
1.00
1.127
400
25%
1.25
1.408
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:
Outcome
200
300
400
E[X] = 290
Prob
Weight Product
35%
0.563
39
40%
1.127
135
25%
1.408
141
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:
Portfolio
Risk 1=100
Risk 1=200
Z = 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, arbitragefree). 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
Exhibit 1 - Conditional Risk Charge Demo using DFAIC
(1) Expected U/W Income
(96,952)
(2) Risk Adjustment Curve Parameters
Upside Scale
Upside Shape
Downside Scale
Downside Shape
(3) Risk-Adjusted
Expected U/W Income
(244,714)
(4) Portfolio Risk Premium
= (1) - (3)
(5)
LOB
CA
CMP
HO
PPA
WC
TOTAL
1,000,000
200.00%
100,000
50.00%
147,762
(6)
(7)
Expected Loss
Expected U/W
Income
(8)
Expected Riskadjusted U/W
Income
(10,946)
(7,910)
(19,460)
(54,963)
(3,673)
(96,952)
(23,014)
(23,152)
(67,474)
(117,554)
(13,520)
(244,714)
115,995
221,025
220,787
437,352
145,131
1,140,291
(9) = (7) - (8)
Allocated Risk
Premium
12,068
15,242
48,013
62,591
9,847
147,762
(10)
Risk
Premium
as % of E[L]
10.4%
6.9%
21.7%
14.3%
6.8%
13.0%
Example
• The valuation formula used to determine the riskaverse outcome weighting is a two-sided utility
transform of total underwriting income UIT to
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