Page 1 Additional Topics Pricing Umbrella and Excess on Excess The ISO Mixed Exponential Chris Svendsgaard Casualty Exposure Rating CARe Boot Camp 2005

Page 2 Pricing Umbrella and XS on XS Issues – Who wrote the primary contract? - Over Own - Over Other – How does the reinsurance contract handle Over Own/Over Other? – How does the reinsurance contract handle ALAE (esp. Over Own)? – Shares Chris Svendsgaard Casualty Exposure Rating CARe Boot Camp 2005

Page 3 IMPORTANT! You must understand the proposed reinsurance contract wording Chris Svendsgaard Casualty Exposure Rating CARe Boot Camp 2005

Page 4 Example Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp Primary Umbrella Reinsured Layer: 2x1

Page 5 Example Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp Loss Reinsured Layer: 2x1 fgu loss is 2.5

Page 6 Example: Over Other Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp Loss Reinsured Layer: 2x1 Reinsurance pays 0.5

Page 7 Example: Over Own Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp Loss Reinsured Layer: 2x1 Primary and Umbrella losses are combined to define loss Reinsurance pays 1.5

Page 8 ALAE: Some Considerations Large loss ALAE tends to be lower (as a %) than fgu ALAE Umbrella policies are relatively passive on defense so ALAE may be low. When Over Own, how does the client/reinsurance contract allocate ALAE between the primary policy and the umbrella policy? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 9 Shares: Some considerations Need to inflate reinsurance limits, etc. to 100% basis in order to use the severity distribution correctly. Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 10 Further Considerations For treaty, the Subject Premium depends on the reinsurance contract wording. – If the S.P. is the Umbrella premium, then the reinsurance rate will be much higher (because the denominator is smaller) Reinsurance contract wording has many variations. You have to figure out each one on its own. Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 11 The ISO Mixed Exponential Distribution What it is Why ISO uses it Pitfalls Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 12 What it is Distribution function Exponential distribution Mixing Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 13 Distribution function If X is a random variable, then F(x) = Probability (X <= x) is its distribution function – Density function f(x) is derivative with respect to x of F(x) Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 14 A distribution function Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 15 A density function Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 16 Exponential distribution F(x) = 1 – exp(-X/μ) “exp” is the exponential function μ is the mean of the exponential distribution f(x) looks like a slide Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 17 Exponential density function μ = 400 Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 18 NO HUMP! Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 19 What it is: MIXING  Randomly pick a type of claim by flipping a coin that has a certain probabilty for each claim type.  Each type of claim has its own severity distribution. Overall severity distribution is the result of steps 1 and 2—it is a “mixture” of the the claim types Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 20 What it is: MIXING—the math If F and G are distribution functions, then so is wF+(1-w)G for 0<w<1 More generally, so is if all the w’s are between 0 and 1 and they sum to 1. Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 21 ISO-like mixed exponential distribution Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp DistributionMeanWeight 11,50038% 25,00031% 325,00018% 490,0009% 5400,0003% 61,700,0000.9% 710,000,0000.1% Weights and means are fake.

Page 22 Why ISO uses the mixed exponential distribution Lots of parameters Some theory – Can approximate any (sufficiently nice, humpless) distribution with a mixture of exponentials Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 23 Pitfalls A single term by itself does not correspond to any particular claim characteristic – Vs. old “Pareto Soup”, where each term corresponded to the settlement lag of the claim – You can’t tell much about a distribution by looking at a single term (μ and w). - The term might be low-severity but the overall distribution might be high-severity – Can’t compare terms from year to year - At least, not meaningfully - Term 2 from this year does not correspond to term 2 from last year Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 24 More pitfalls Tail is problematic – A finite mixture of exponentials will always be less severe in the tail than any Pareto distribution – ISO caps the largest μ at $10,000,000 - In practice, most ILF tables have a capped largest μ Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp

Page 25 Still more pitfalls Curves do not reflect parameter risk ISO data falls off dramatically at a relatively low level (say, $2,000,000—depends on the line) – Anti-selection picked up? ISO nudges the curves for the highest layers Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp