26 September 2005

Stephen Lowe

Survey Results / Overview of Methods

CAS Limited Attendance Seminar on Risk and Return in Reinsurance

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Sixteen survey participants

ACE Tempest Re

AWAC

Chubb Re

GE

GMAC Re

Hannover Re

Max Re

Montpelier Re

Odyssey Re

Partner Re

Platinum Re

QBE Re

Scor Re

Signet Star

Toa Re

Transatlantic Re

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Traditional approaches to pricing

Measure

Target Combined Ratio

Target Return

Approach

Return on Sales

Return on Capital

Variations

Nominal versus Discounted

Fixed versus Variable Target

ROE based on NPV of Internal Cash Flows versus IRR of Free Cash Flows

Fixed Versus Variable Target

Rating Agency Capital versus Economic Capital

These methods are usually applied to deterministic (i.e., expected) cash flows

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Stochastic pricing methods

Description

Required capital a fn of contract outcome distribution Tail VaR

Required capital a fn of marginal impact on portfolio outcome distribution Tail VaR

Calculate R2R from contract outcome distribution

Price is expected outcome using modified probabilities

Price is expected outcome using modified amounts

Approach

Standalone Tail VaR

Marginal Tail VaR

R2R

Wang Transform

Capital Consumption

Measure

Target Return

Target Return

Target R2R

Adjusted Expected Value

Adjusted Expected Value

Thanks, Don

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Typical descriptions of method

Target ROE, comparing NPV of contract cash flows to equity based on leverage ratios

Target underwriting profit by class of business

Target ROE, using NPV model that balances to capital requirements

Target IRR, based on free cash flows (capital and profits in/out)

Target ROE, reflecting corporate cost of capital, based on NPV of contract cash flows and internal RBC factors

Variety of methods that look at downside risk and utility metrics; game theory considered

Metrics relating to simulated contract results distribution used to determine leverage required, then target ROE

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How are profit margins set in pricing?

Nominal NPV IRR

Return on Capital

Return on Sales

One company responded that they used “a variety of methods”

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Do pricing methods vary by line?

Most companies indicated that they use the same general method for all lines

Exceptions: Property catastrophe, where pricing reflects the marginal

impact of the contract on the portfolio Clash covers, where a bank approach is taken Property business, where volatility of individual contract and

portfolio concentration is taken into account Contracts with loss sensitive features treated differently

One company responded that they used “a variety of methods” that vary by line

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How is risk reflected?

At the class of business level

Fixed ROC, but RBC allocates more capital to volatile classes

Variable ROC, and RBC allocates more capital to volatile classes

Profit margins vary with volatility of class

At the individual contract level

Risk loads determined by individual contract simulation

Contracts with unusually high risk have target set higher than the standard target for the class

Underwriters make judgmental adjustments

Volatility of contract is benchmarked to other contracts in class

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How is capital allocated?

Rating agency RBC factors

Leverage ratios

Management allocation

Internal capital model (Economic Capital)

Volatility of class

Individual contract simulation distribution

Individual contract downside risk

Contract characteristics

Not allocated

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How are pricing targets reconciled with corporate financial goals?

They are the same; they are consistent

No reconciliation is made

Reconciliation assures that aggregate pricing return is greater than overall financial target

They are expected to be similar

Differences reflect actual versus rating agency capital

IRR versus ROE make them different

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What enhancements are being developed or considered?

Allocation of capital to contract is being tested

Researching RAROC

Researching greater use of marginal portfolio impact in the allocation of capital

Need to understand correlations between lines to implement marginal impact

Refinements to marginal capital allocation

Looking at game theoretical constructs

Researching internal risk models

Implementing rating agency capital formula into capital allocation

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Additional Materials on Stochastic Pricing

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Pricing to a Target R2R = 8.00Pricing to a Target R2R = 8.00a) b) c) d) e) f) g)

Probability NPV Amount (a-b-d)Premium Expenses of Loss of Loss Loss or Profit Downside Upside$1,000,000 $175,000 20% $0 $825,000 $825,000

15% $100,000 $725,000 $725,00015% $250,000 $575,000 $575,00015% $350,000 $475,000 $475,00010% $500,000 $325,000 $325,00010% $750,000 $75,000 $75,00010% $1,000,000 -$175,000 -$175,0002.5% $1,500,000 -$675,000 -$675,0002.5% $2,500,000 -$1,675,000 -$1,675,000

Exp Value $395,000 -$508,333 $554,412D/U Ratio 0.92 Prob of … 15.0% 85.0%

Product 76,250$ 471,250$ R2R 6.18

a) b) c) d) e) f) g)Probability NPV Amount (a-b-d)

Premium Expenses of Loss of Loss Loss or Profit Downside Upside$1,067,660 $175,000 20% $0 $892,660 $892,660

15% $100,000 $792,660 $792,66015% $250,000 $642,660 $642,66015% $350,000 $542,660 $542,66010% $500,000 $392,660 $392,66010% $750,000 $142,660 $142,66010% $1,000,000 -$107,340 -$107,3402.5% $1,500,000 -$607,340 -$607,3402.5% $2,500,000 -$1,607,340 -$1,607,340

Exp Value $462,660 -$440,674 $622,071D/U Ratio 0.71 Prob of … 15.0% 85.0%

Product 66,101$ 528,761$ R2R 8.00

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Transform the distribution amounts or probabilities?

Value x1 x2 … xn

Prob p1 p2 … pn:X

Value U(x1) U(x2) … U(xn)

Prob p1 p2 … pn:)(XU

Value x1 x2 … xn

Prob p1 p2 … pn:X

Value x1 x2 … xn

Prob q1 q2 … qnqXE ][

Probability Transform or

“Measure Change”

modifies the Probabilities

Downside penalty function

modifies the amounts

Either approach uses

SUMPRODUCT of amounts and probabilities

PEN

ALTY

FU

NC

TIO

NW

AN

G

TR

AN

SFO

RM

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Downside Penaltya.k.a. Capital Consumption

Risk Load = E[X*] expected value of adjusted amounts

Adjustment happens by modifying the amounts using a capital consumption penalty:

— Zero if positive NPV outcome

— Multiple of outcome if negative NPV outcome

Expected value = SUMPRODUCT of Amounts and Probabilities

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Capital Consumption

NPV Distribution

-8,000,000

-6,000,000

-4,000,000

-2,000,000

0

2,000,000

4,000,000

6,000,000

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Once NPV Falls Below Zero, Penalties Assessed to Offset

Consumption ofAdditional Capital

NPV Above Zero – No Penalties

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a) b) c) d) e) f)Probability NPV Amount (a-b-d) Adjusted

Premium Expenses of Loss of Loss Loss or Profit Amounts$1,000,000 $175,000 20% $0 $825,000

15% $100,000 $725,00015% $250,000 $575,00015% $350,000 $475,00010% $500,000 $325,00010% $750,000 $75,00010% $1,000,000 -$175,000 -$350,0002.5% $1,500,000 -$675,000 -$1,350,0002.5% $2,500,000 -$1,675,000 -$3,350,000

$395,000 -$1,016,667

Penalty Charge 200.0%Mean $395,000

Risk-Adjusted Mean $318,750

Capital Consumption Pricing Example

Downside(Capital

Consumed) Amounts Increased

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Wang TransformModifies the Probabilities

))(()(* 1 xFxF

In Excel: F* = normsdist( normsinv(F) - lambda )

Makes severe outcomes appear more likely by reducing their implied percentile

For example, if lambda = 0.5, a 3 std deviation outcome becomes a 2.5 std deviation outcome

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0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

500 1,000 1,500 2,000X

F(X

)

Input

Transformed

The Wang Transform shifts the NPV distribution, giving more weight to the tail of the distribution.

Unlike TVaR and VaR, WT considers the entire distribution

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Probability NPV Amount (a-b-d)Premium Expenses of Loss of Loss Loss or Profit Downside Upside$1,000,000 $175,000 20% $0 $825,000 $825,000

15% $100,000 $725,000 $725,00015% $250,000 $575,000 $575,00015% $350,000 $475,000 $475,00010% $500,000 $325,000 $325,00010% $750,000 $75,000 $75,00010% $1,000,000 -$175,000 -$175,0002.5% $1,500,000 -$675,000 -$675,0002.5% $2,500,000 -$1,675,000 -$1,675,000

$395,000 -$508,333 $554,412

Target adjusted ENPV

a) b) c) d) e) f) g) h) i) j)

LambdaProbability Cumulative 0.75 Adjusted Implied NPV Amount

Premium Expenses of Loss Probability NORMSINV Transform Probability Prob of Loss Loss or Profit$1,000,000 $175,000 20.0% 20.0% (0.84) (1.59) 5.6% 5.6% $0 $825,000

15.0% 35.0% (0.39) (1.14) 12.8% 7.2% $100,000 $725,00015.0% 50.0% - (0.75) 22.7% 9.9% $250,000 $575,00015.0% 65.0% 0.39 (0.36) 35.8% 13.1% $350,000 $475,00010.0% 75.0% 0.67 (0.08) 47.0% 11.2% $500,000 $325,00010.0% 85.0% 1.04 0.29 61.3% 14.3% $750,000 $75,00010.0% 95.0% 1.64 0.89 81.5% 20.2% $1,000,000 -$175,0002.5% 97.5% 1.96 1.21 88.7% 7.2% $1,500,000 -$675,0002.5% 100.0% 100.0% 11.3% $2,500,000 -$1,675,000

Exp Value -- Unadjusted $395,000Exp Value -- Adjusted -$9,100

Wang Pricing Transform Modifies the Probabilities

Applies a Greater Weight to Downside …. By Modifying Probabilities