Swiss ReWinner’s Curse Chris Svendsgaard1 THE WINNER’S CURSE.

Swiss Re Winner’s Curse Chris Svendsgaard 1

THE WINNER’S CURSE


Outline

• Basic model

• Implications of the basic model

• Questions that can be explored using this model

• Rational expectations and risk load


Winner’s Curse Basic Model

• You, and (k - 1) competitors, bid to reinsure a risk

• Bids are independent, identically distributed, unbiased estimates of the correct price

• Lowest bid wins the deal


Bias as a function of number of bidders and std. dev. of bid distributionBid distribution is Normal(10, SD)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

1 2 3 4 5 6 7 8 9 10

Number of bidders

Me

an

Win

nin

g B

id

SD = 1

SD = 3


Implications of the basic model

• Winning bid will be biased

• Bias increases as variance of bid distribution increases

– Greater bias for risky lines, high layers

• Bias increases as number of bidders increases

– At a decreasing rate


Questions that can be explored using this model

• The benefit (and cost) of being more accurate

• Different auction mechanisms

– “Best Terms”

• State Farm makes money using those rates--why can’t we?

• Why renewal business is more profitable

• A-priori loss ratios (Murphy’s Law)


Rational Expectations and Risk Load

• “Rational bidders will adjust bids to eliminate bias”

– Not supported by research

– See “The Winner’s Curse” by Thaler

– However, rules-of-thumb may have evolved to fix bias

– Same way poker hands were ordered in terms of rarity before theory of probability developed

– Is risk load such a rule-of-thumb?


Risk Load vs Auction Bias

• Risk Load

– Based on higher moments

– Many measures suggested

– Standard Deviation

– Variance

– Shortfall

– etc.

– Scale factor is subjective

– Some risk diversifies away

– Don’t need for some segments?

• Bias

– Based on expected value

– Measure is expected value

– .

– .

– .

– Scale factor is 1

– Bias does not diversify away

– Need for all segments


Risk Load vs Auction Bias (continued)

• Risk Load

– Does not depend on the number of competitors

– Probably should depend on how good you are at pricing, but not 100% clear how

• Bias

– Depends on the number of competitors

– Depends directly on how accurate your pricing is


Auction Theory and Risk Load

THE END


Appendix 1: Simple Example

Risk Bidder A Bidder B Winning Bid

1. 200 100 100

2. 200 100 100

3. 100 200 100

4. 100 200 100

Total 600 600 400


Appendix 2: More Realistic Examples

• Swiss Re in-house comparison of individual risk cat models

• SR model (“Single SNAP”) and two vendor models

• Standard risk in different locations (165 for EQ, 66 for Wind)

• “Correct Price” is average of three models at location

• Winning bid is lowest of three models at location

• Note that all three models have been changed since this study


Examples: Raw Data (sample)

Comparison of individual risk earthquake pricing tools Method A Method B Method CPOLICYNUM LATITUDE LONGITUDE COUNTYNAME STATE

1 34.037 -118.310 Los Angeles CA 463,047 154,321 294,000 2 34.014 -118.272 Los Angeles CA 655,734 271,280 210,000 3 33.994 -118.231 Los Angeles CA 644,980 281,610 210,000 4 33.971 -118.196 Los Angeles CA 636,757 298,176 210,000 5 33.953 -118.156 Los Angeles CA 630,286 295,634 210,000


Results of winner-takes-all auction based on Single-SNAP study

1 Earthquake Method A Method B Method C Total/Avg23 # Risks Sold 7 68 90 1654 Hit Ratio 4% 41% 55% 100%56 Prem Sold 703,875 9,678,778 10,751,840 21,134,493 7 Correct Prem 822,731 18,503,518 19,579,504 38,905,753 8 Bias In Sold (118,856) (8,824,741) (8,827,664) (17,771,261) 9

10 Sold Bias % -14% -48% -45% -46%1112 Total Prem* 55,296,976 28,618,324 32,801,960 38,905,753 13 Bias in Total 16,391,223 (10,287,429) (6,103,793) 14 Bias % 42% -26% -16% 0%151617 *if no competition


Results of winner-takes-all auction based on Single-SNAP studyWind Coastal Method A Method B Method C Total/Avg

1 # Risks Sold 1 39 26 662 Hit Ratio 2% 59% 39% 100%34 Prem Sold 12,000 3,625,507 6,110,996 9,748,503 5 Correct Prem 17,800 6,478,751 9,453,294 15,949,846 6 Bias In Sold (5,800) (2,853,244) (3,342,298) (6,201,343) 78 Sold Bias % -33% -44% -35% -39%9

10 Total Prem* 23,086,000 12,215,757 12,547,780 15,949,846 11 Bias in Total 7,136,154 (3,734,089) (3,402,066) 12 Bias % 45% -23% -21% 0%

*if no competition Thanks to Yash Gawri for help on this.


Appendix 3: Accuracy

• Being more accurate reduces your bias

• If you are perfectly accurate, you will suffer no bias

– BUT hit ratio goes from 1/k to 1/[2^(k-1)] (assuming symmetric bid distributions)

• Or does it? How do people correct for bias in practice? Would you put some bias back in to get your volume up?


Appendix 4: “Best Terms”

• Bias changes radically depending on form of auction

• Property fac cert per-risk uses “best terms”

– Highest price from among successful bidders is given to all successful bidders


Best Terms Example

Bidder 1 2 3 4

Bid 100 120 130 140

Authorized Share 40% 50% 50% 50%Actual Share 40% 50% 10% 0%

Sold Price 130 130 130 NA


Best Terms

• Assume three bidders, each willing to take 50%

– Clearing price is median of bid distribution

– No apparent bias


Best Terms

• Implication: More bias for smaller risks

– Because take 100%


References

• http://www.economics.harvard.edu/~aroth/alroth.html

– Look for Auction Theory bibliography by Paul Klemperer

• The Winner’s Curse: Paradoxes and Anomalies of Economic Life

– Richard H. Thaler

– Princeton University Press, 1992


Thanks and a tip o’ the hat to

• Shaun Wang for encouragement

• Rob Downs for collaboration

• Isaac Mashitz and Gary Patrik for comments

• Gene Gaydos for original idea

Swiss ReWinner’s Curse Chris Svendsgaard1 THE WINNER’S CURSE.

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