Discussion of Ruhm Arbitrage Paper

1. Discussion of Ruhm Arbitrage Paper

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

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. 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. MMM and MEM for Surplus Process • Start with actual expected claim count l and size g(y) • Minimum martingale measure with 0<s<1 • l* = l/(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. Minimum Entropy Measure • Has parameter c • l* = lEecY – only works if moment exists • g*(y) = g(y)ecy/EecY • Severity probability increases iff y > ln[(EecY)1/c] • For small claims g(y) > g*(y) > g(y)/EecY so probability never decreases more than frequency probability increases • Avoids potential problem Mack noted for many transforms of negative loading of lower layers

12. Hypothetical Example • l=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% • l* = l/(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. 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. Fits • HErrorEErrorFEError • 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

15. Graphs

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