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Swiss Re Winner’s Curse Chris Svendsgaard 2
Outline
• Basic model
• Implications of the basic model
• Questions that can be explored using this model
• Rational expectations and risk load
Swiss Re Winner’s Curse Chris Svendsgaard 3
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
Swiss Re Winner’s Curse Chris Svendsgaard 4
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
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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
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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)
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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?
Swiss Re Winner’s Curse Chris Svendsgaard 8
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
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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
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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
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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
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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
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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
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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.
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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?
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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
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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
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Best Terms
• Assume three bidders, each willing to take 50%
– Clearing price is median of bid distribution
– No apparent bias
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Best Terms
• Implication: More bias for smaller risks
– Because take 100%
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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