Page 1 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 2 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 3 The Winner’s Curse Quotes incorporate randomness The auction is won by the lowest quote This creates a bias The expected value of the minimum of (say) 5 bids is lower than the expected value of the average bid Chris Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 4 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 parameters Chris Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 5 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 6 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 Re Cas. Actuaries in Reins. 2004

Page 7 Economists do not agree with one another Duh Chris Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 8 Why things tend to turn out worse than expected Your average bid has ample profit built in The bids you win do not Chris Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 9 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 10 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 11 Extreme values in big populations are more extreme Chris Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004

Page 12 The effect of competition Chris Svendsgaard Swiss Re Cas. Actuaries in Reins Simulation of gamma-distributed bids (mean 2, variance as given. iid) Mean winning bid

Page 13 Useful facts Random sample X 1 … X n X (k) = k th 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 Svendsgaard Swiss Re Cas. Actuaries in Reins. 2004