1. Guy Carpenter Instrat ® One Madison Avenue New York New York 10010 Discussion of Ruhm Arbitrage Paper

2. Guy Carpenter 2 Arbitrage-Free Pricing (No chance of profits unless losses possible ) General idea is that means under transformed probabilities give arbitrage-free prices But there are details: Probabilities transformed are of events  “States of nature” Probability zero events same before and after Transforming probabilities of outcomes of deals does not does not always lead to arbitrage-free pricing – Ruhm’s example shows this for stock options Transforming probabilities of stock prices does work

3. Guy Carpenter 3 States of Nature For stock option prices these are the stock prices For insurance they are frequency and severity distributions These are probabilities that have to be transformed Transforming aggregate loss probabilities will not necessarily give arbitrage-free prices Seen already with Wang’s 1998 transform paper

4. Guy Carpenter 4 Requirement on Transform Equivalent martingale transform Equivalent requires same impossible states Martingale requires that mean at any future point is the current value For insurance prices such a transform would weight adverse scenarios more heavily in order to compensate for profit loadings (so no transformed expected profit on present value basis) Then all layers priced as transformed mean

5. Guy Carpenter 5 Why Arbitrage-Free? In complete markets arbitrage opportunities are quickly competed away In incomplete markets it may be impossible to realize theoretical arbitrage opportunities But competitive pressures could still penalize pricing that violates arbitrage principles Pricing that would create arbitrage profits would be too good to last for long In complete markets principle of no-arbitrage actually determines unique prices

6. Guy Carpenter 6 Arbitrage-free Pricing in Incomplete Markets Not impossible – in fact problem is a proliferation of choices Requires a probability transform which makes prices the expected losses under transformed probabilities In complete markets the transform can be uniquely determined and has a no-risk hedging strategy Incomplete markets have many possible transforms but all hedges are imperfect

7. Guy Carpenter 7 Transforms for Compound Poisson Process Møller (2003 ASTIN Colloquium) shows how to create such transforms Co-ordinated transforms of frequency and severity Starts with  (y) function that is > –1 Frequency parameter is transformed to [1+E  (Y)]. Severity g(y) transformed to g(y)[1+  (y)]/[1+ E  (Y)]. Scaled so that transformed mean total loss is price of ground-up coverage

8. Guy Carpenter 8 Two Popular Transforms in Finance Minimum martingale transform Corresponds to hedge with minimum variance Minimum entropy martingale transform Minimizes a more abstract information distance Recognizes that markets are sensitive to risk beyond quadratic

9. Guy Carpenter 9 Application to Insurance Surplus Process Process is premium flow less loss flow Transformed probabilities make this a martingale Makes expected transformed losses = premium Møller paper 2003 ASTIN demonstrated what the minimum martingale and minimum entropy martingale transforms would be for this process Frequency and severity get linked transforms

10. Guy Carpenter 10 MMM and MEM for Surplus Process Start with actual expected claim count and size g(y) Minimum martingale measure with 0<s<1 * = /(1 – s) g*(y) = [1 – s + sy/EY]g(y) Claim sizes above the mean get increased probability and below the mean get decreased No claim size probability decreases more than the frequency increases Thus no layers have prices below expected losses s selected to give desired ground-up profit load

11. Guy Carpenter 11 Minimum Entropy Measure Has parameter c * = Ee cY – only works if moment exists g * (y) = g(y)e cy /Ee cY Severity probability increases iff y > ln[(Ee cY ) 1/c ] For small claims g(y) > g * (y) > g(y)/Ee cY so probability never decreases more than frequency probability increases Avoids potential problem Mack noted for many transforms of negative loading of lower layers

12. Guy Carpenter 12 Hypothetical Example =2500, g(y) = 0.00012/(1+y/10,000) 2.2, policy limit 10M To get a load of 20%, take the MMM s = 0.45% * = /(1 – s) = 2511 g*(y) = [1–s+sy/EY]g(y) = (.9955+y/187,215)g(y) Probability at 10M goes to 0.055% from 0.025% 4M x 1M gets load of 62.3%, 5M x 5M gets 112.8% For MEM these are 50.8% and 209.1%, as more weight is in the far tail 89% of the risk load is above $1M for MEM; 73% for MMM

13. Guy Carpenter 13 Testing with Pricing Data Had prices and cat model losses for a group of reinsurance treaties Fit MMM, MEM and a mixture of them to this data with transforms based on industry loss distribution = distribution of sum across companies Had separate treaties and modeled losses for three perils: H, E, and FE Mixture always fit best, but not usually much better than MEM alone, which was better than MMM Fit by minimizing expected squared relative errors

14. Guy Carpenter 14 Fits H ErrorE Error FE Error MMM s.017.381.021.470.036.160 MEM ln c: -28.2.308-26.3.311 -26.6.082 Mixed.011.298 0.220.116.064 … -27.0 -25.5 -26.7 Quadratic effects not enough to predict prices Especially problematic in high layers

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19. Guy Carpenter 19 Beyond No Arbitrage Principle of no good deals Good deal defined as risk everyone would want to buy, no one would want to sell Involves a cutoff point Expanding literature on how to define cutoff Restricts prices more than does no arbitrage Some similarities to risk transfer testing