RMK and Covariance Seminar on Risk and Return in Reinsurance September 26, 2005 Dave Clark American Re-Insurance Company This material is being provided to you for information only, and is not permitted to be further distributed without the express written permission of American Re. This material is not intended to be legal, underwriting, financial or any other type of professional advice. Examples given are for illustrative purposes only. © Copyright 2005 American Re-Insurance Company. All rights reserved.

RMK Framework Introduction: A Confusing List of Terms Capital Allocation Capital Allocation Cost of Capital Cost of Capital Capital Consumption Capital Consumption Risk Load Risk Load Risk / Reward trade-offs Risk / Reward trade-offs RMK ??? RMK ??? 1234

RMK Framework Introduction: The RMK Framework sources RMK = Ruhm, Mango, Kreps Mango:Capital Consumption: An Alternative Methodology for Pricing Reinsurance CAS Forum, Winter 2003 Kreps:Riskiness Leverage Models PCAS 2005 – Originally circulated in bootleg version as “A Risk Class with Additive Co-Measures” “A Risk Class with Additive Co-Measures” 1234

RMK Framework Agenda: Examples from Reinsurance Pricing Examples from Reinsurance Pricing Allocation of stop-loss premium Allocation of stop-loss premium Multi-year profit commission Multi-year profit commission Example of “Capital Consumption” Example of “Capital Consumption” Properties Properties The Mathematics The Mathematics Problems & Challenges Problems & Challenges 1234

RMK Framework Resource divided between individuals 1234 Shared Resource:

RMK Framework Instead of the “capital allocation” problem we will start with two other examples in which dollar amounts need to be allocated in reinsurance applications. Example #1: A ceding company purchases an aggregate “stop-loss” cover that applies to all lines of business combined. How should the cost of this reinsurance be allocated to the individual lines of business? 1234

RMK Framework Example #2: A reinsurer has sold an Excess WC treaty with a profit commission that applies on a 3-year block. It is now the end of the second year and we want to evaluate the “expected” profit commission on the prospective third year. How do we estimate the expected profit commission by year? 1234

RMK Framework Example #3: Risk Measures and “Capital Consumption” Given an overall profit target, what is the fairest method for setting corresponding profit targets for individual products? 1234

RMK Framework Review Excel Examples 1234

RMK Framework Strengths of the RMK Framework: Results are additive: business segments can be defined any way you want and it will not affect the answer. Results are additive: business segments can be defined any way you want and it will not affect the answer. Risk measures by business segment are logically connected to the risk measure for the company in total.* Risk measures by business segment are logically connected to the risk measure for the company in total.* Theory works for any correlation or dependence structure in the losses: If you can simulate it, RMK will work! Theory works for any correlation or dependence structure in the losses: If you can simulate it, RMK will work! *This is why Kreps calls RMK “additive co-measures” 1234

RMK Framework Mathematics We will follow Kreps’ notation for describing formulas for allocating the risk measure Assume that we have a portfolio “Y”, made up of the sum of three business segments: X 1, X 2, and X 3. These three segments do not have to be independent or identically distributed.

RMK Framework Mathematics Kreps’ notation continued: Define a risk measure R, which is based on the total losses to the portfolio Amount by which an actual loss exceeds the average. The “Leverage” or pain associated with the loss amount.

RMK Framework Mathematics Kreps’ notation continued: We then introduce a simplified notation where

RMK Framework Mathematics Kreps’ notation continued: The overall risk-load is allocated as follows Changes to…

RMK Framework Mathematics Kreps’ notation continued: Note that ADDITIVITY is preserved regardless of the dependence between business segments or the way in which we define “segment” (line of business, SBU, etc). 1234

RMK Framework Mathematics 1234 The Leverage Ratio L(y) is a “pain” function: If (y- μ y ) is negative: If (y- μ y ) is negative: no capital used, L(y)=0; there is no “pain” If (y- μ y ) is small but positive: If (y- μ y ) is small but positive: some capital is consumed, L(y)>0 If (y- μ y ) is large: If (y- μ y ) is large: capital is consumed, solvency is imperiled If (y- μ y ) is huge (a multiple of capital): If (y- μ y ) is huge (a multiple of capital): L(y) stops growing, since we no longer care if we are dead many times over

RMK Framework Mathematics 1234 Kreps shows that virtually all of the common proposals for risk measures can be formulated in terms of a leverage ratio L(y) : Variance Variance Standard Deviation Standard Deviation Value at Risk (VaR) Value at Risk (VaR) Conditional Downside (TVaR or “tail value at risk”) Conditional Downside (TVaR or “tail value at risk”)

RMK Framework Mathematics Kreps’ notation and covariance: Alternative formulation (from Ruhm & Mango): 1234 This implies that if Then a covariance allocation results:

RMK Framework Mathematics Conditions for RMK to reproduce covariance: 1234 (1) If the portfolio risk measure is variance (2) If there is a linear relationship between expected losses, then for any leverage ratio: Or …

RMK Framework Mathematics Conditions for RMK to reproduce covariance: 1234 This second condition is EXACTLY met when 1.All business segments write identical policies (even if they have different commission percents) 2.All business segments take different shares of a pool 3.All business segments are drawn from certain multivariate distributions We can also find cases when it is APPROXIMATELY met…

RMK Framework Mathematics Loss Distributions for which RMK = covariance: 1234 Examples: Elliptical Distributions Elliptical Distributions Normal, Student-t, Logistic, etc Normal, Student-t, Logistic, etc Additive Form of Exponential Family Additive Form of Exponential Family Normal, Poisson, Gamma, Inverse Gaussian Normal, Poisson, Gamma, Inverse Gaussian Note “Additive” = closed-under-convolution Note “Additive” = closed-under-convolution Other… Other…

RMK Framework Mathematics Generalized Pareto (a.k.a. Beta of Second Kind) 1234 Bivariate Generalized Pareto (common  and  )

RMK Framework Mathematics General Principle, informally stated: 1234 Covariance allocation is a linear approximation to any arbitrary risk co-measure. Note: subject to conditions such as all variances existing, and the risk measure being on a “central” basis (x- μ).

RMK Framework Mathematics Condition for RMK to approximate covariance: 1234

RMK Framework Mathematics 1234 Condition when RMK does NOT approximate covariance:

RMK Framework Mathematics Condition when RMK does NOT approximate covariance: 1234

RMK Framework Problems & Challenges: Two hurdles: Calibration down to individual contract level will probably always be an approximation. Calibration down to individual contract level will probably always be an approximation. More work needed on theory for including the time value of money. More work needed on theory for including the time value of money. 1234

RMK Framework Questions & Discussion David R. Clark “Reinsurance Applications of the RMK Framework”; Spring 2005 CAS Forum