the winner’s curse in reinsurance

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

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


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


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

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


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

Economists do not agree with one another

Duh


Why things tend to turn out worse than expected

Your average bid has ample profit built in

The bids you win do not


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


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


Extreme values in big populations are more extreme


The effect of competition


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

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)


the winner’s curse in reinsurance

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