Transcript THE WINNER’S CURSE Swiss Re 1

THE WINNER’S CURSE
Swiss Re
Winner’s Curse
Chris Svendsgaard
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Outline
• Basic model
• Implications of the basic model
• Questions that can be explored using this model
• Rational expectations and risk load
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Chris Svendsgaard
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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
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Bias as a function of number of bidders and std. dev. of bid distribution
Bid distribution is Normal(10, SD)
12.0
10.0
SD = 1
Mean Winning Bid
8.0
SD = 3
6.0
4.0
2.0
0.0
1
2
3
4
5
6
7
8
9
10
Number of bidders
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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?
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Risk Load vs Auction Bias
• Risk Load
Swiss Re
• Bias
– Based on higher moments
– Based on expected value
– Many measures suggested
– Measure is expected value
–
Standard Deviation
–
.
–
Variance
–
.
–
Shortfall
–
.
–
etc.
– Scale factor is subjective
– Scale factor is 1
– Some risk diversifies away
– Bias does not diversify away
– Don’t need for some segments?
– Need for all segments
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Risk Load vs Auction Bias (continued)
• Risk Load
• Bias
– Does not depend on the number of
competitors
– Probably should depend on how
good you are at pricing, but not
100% clear how
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Winner’s Curse
– Depends on the number of
competitors
– Depends directly on how accurate
your pricing is
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Auction Theory and Risk Load
THE END
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Appendix 1: Simple Example
Risk
Swiss Re
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 C
POLICY
NUM
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
2
3
# Risks Sold
4
Hit Ratio
5
6
Prem Sold
7
Correct Prem
8
Bias In Sold
9
10
Sold Bias %
11
12
Total Prem*
13
Bias in Total
14
Bias %
15
16
17 *if no competition
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Method A
Method B
Method C
Total/Avg
7
4%
68
41%
90
55%
165
100%
703,875
822,731
(118,856)
9,678,778
18,503,518
(8,824,741)
10,751,840
19,579,504
(8,827,664)
21,134,493
38,905,753
(17,771,261)
-14%
-48%
-45%
-46%
55,296,976
16,391,223
42%
28,618,324
(10,287,429)
-26%
32,801,960
(6,103,793)
-16%
38,905,753
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Results of winner-takes-all auction based on Single-SNAP study
Wind Coastal Method A
1
2
3
4
5
6
7
8
9
10
11
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Method C
Total/Avg
# Risks Sold
Hit Ratio
1
2%
39
59%
26
39%
66
100%
Prem Sold
Correct Prem
Bias In Sold
12,000
17,800
(5,800)
3,625,507
6,478,751
(2,853,244)
6,110,996
9,453,294
(3,342,298)
9,748,503
15,949,846
(6,201,343)
Sold Bias %
-33%
-44%
-35%
-39%
Total Prem*
Bias in Total
Bias %
23,086,000
7,136,154
45%
12,215,757
(3,734,089)
-23%
12,547,780
(3,402,066)
-21%
15,949,846
*if no competition
Swiss Re
Method B
0%
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
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Bidder
1
2
3
4
Bid
Authorized
Share
Actual
Share
100
120
130
140
40%
50%
50%
50%
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
–
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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
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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
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