The Winner

1
The Winners Curse in Reinsurance

• . . . I have always believed an exception would
  be made in my case."    
• William Saroyan (on his deathbed)
• The winners curse applies to reinsurance
• Easier to understand in fac

Chris Svendsgaard Swiss Re Cas. Actuaries in
Reins. 2004
2
The Winners 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 accuracyis 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
3
The Winners 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
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
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
6
The Bias

• Do people adjust their bids to counteract
  winners 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
7

• Economists do not agree with one another
• Duh

Chris Svendsgaard Swiss Re Cas. Actuaries in
Reins. 2004
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
9
Why underwriters complain

• Were 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
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
11
Extreme values in big populations are more extreme
Chris Svendsgaard Swiss Re Cas. Actuaries in
Reins. 2004
12
The effect of competition
Simulation of gamma-distributed bids (mean 2,
variance as given. iid) Mean winning bid
Chris Svendsgaard Swiss Re Cas. Actuaries in
Reins. 2004
13
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 Svendsgaard Swiss Re Cas. Actuaries in
Reins. 2004