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The Winner’s Curse in Reinsurance
“. . . I have always believed an exception would be made in my case."
William Saroyan (on his deathbed)
The winner’s curse applies to reinsurance– Easier to understand in fac
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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The Winner’s Curse in Reinsurance: Outline
What it is
Useful applications– Why things tend to turn out worse than
expected (HELLOOOO RED SOX FANS!)– Why underwriters whine about the actuaries
so much– The value of accuracy—is it worth hiring
actuaries?– How competition affects profit– A (—the?—) source of risk aversion– How to measure risk
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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The Winner’s Curse
Quotes incorporate randomnessThe auction is won by the lowest quoteThis creates a bias
The expected value of the minimum of (say) 5 bids is lower than the expected value of
the average bid
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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Sources of randomness
Variations in judgment
Selection of data to use, cleaning the data– Also sample error sometimes
Selection of method(s) to use– Getting loss costs– Allocating expenses– Setting profit provision– Reflecting potential investment income
Selection of parametersChris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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Lowest Quote Wins Auction
This is true for small certs
For larger certs and treaties, reinsurers take shares and pricing is often on a “best terms” basis
– Auction theory can be modified to handle this
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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The “Bias”
Do people adjust their bids to counteract winner’s curse bias?
"It is important to keep in mind that rationality is an assumption in economics, not a demonstrated fact." Richard H. Thaler, The Winner's Curse
"…these paradoxes are of relatively little significance for economics." Hirshleifer and Riley, The Analytics of Uncertainty and Information (discussing departures of decision-makers from rationality).Chris Svendsgaard
Swiss ReCas. Actuaries in Reins. 2004
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Economists do not agree with one another
Duh
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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Why things tend to turn out worse than expected
Your average bid has ample profit built in
The bids you win do not
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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Why underwriters complain
“We’re higher than the market 80% of the time.”Well, if there are 5 bidders, …
The average bid tends to be more accurate than individual bids and gets more accurate as you add bidders
The winning bid (= “the market”) is biased downwards and the bias gets worse as you add bidders
The market is your stupdiest competitor
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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The Value of Accuracy
Without adjustment: More accurate lower variance of bid Less WC bias (BUT hit less often)
Result from admittedly made-up bid distribution simulations:
Being smarter than everybody else is nice
Being stupider than everybody else is horrible
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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Extreme values in big populations are more extreme
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
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The effect of competition
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004
Variance# of bids 2.0 3.0 4.0 5.0 6.0 7.0 8.0
1 2.00 2.00 2.00 2.00 2.00 2.00 2.002 1.25 1.11 1.00 0.91 0.84 0.78 0.733 0.96 0.79 0.67 0.57 0.50 0.44 0.384 0.81 0.62 0.50 0.41 0.34 0.29 0.245 0.70 0.52 0.40 0.32 0.25 0.21 0.176 0.63 0.45 0.34 0.26 0.20 0.16 0.127 0.58 0.39 0.29 0.21 0.16 0.12 0.098 0.53 0.36 0.25 0.18 0.13 0.10 0.079 0.50 0.33 0.22 0.16 0.11 0.08 0.06
10 0.47 0.30 0.20 0.14 0.10 0.07 0.05
Simulation of gamma-distributed bids(mean 2, variance as given. iid)Mean winning bid
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Useful facts
Random sample X1 … Xn
X(k) = kth largest (”order statistic”)
Distribution function of X(1) is 1 – [1 - F(x)]n.– Example: Min of n expontials is also
exponential with [new mean] = [old mean]/n
F(X(k) ) is Beta(k, n – k + 1) – Mean = k/(n + 1)
Chris SvendsgaardSwiss ReCas. Actuaries in Reins. 2004