1. 1 Capital Consumption Don Mango American Re-Insurance 2003 CARe Seminar

2. 2 Goals for Today 1. Get you to admit this is a valid alternative framework (albeit orthogonal) to capital allocation / release / IRR 2. Demonstrate how it can be practically implemented as a means of pricing reinsurance 3. Demonstrate connections to leading edge thinking in financial science

3. 3 Problem Statements Capital allocation is a de facto paradigm  a requirement or necessity Therefore we force-fit our business into a manufacturing-based capital investment framework

4. 4 Problem Statements But insurance capital usage is fundamentally different than it is for manufacturing, being in fact the mirror-image in time For these decision evaluation processes, capital allocation is sufficient but not necessary

5. 5 Problem Statements Even worse, the resulting insurance IRR framework is now completely fictional (“imputed”), since no capital is actually transferred or returned However, insurance capital is actually consumed when results are worse than planned

6. 6 Actually…  This IS capital allocation for insurance, done right  But I needed new terminology to shake loose the old thought processes

7. 7 Two Bets Bet #1 You pay me $10 now I might pay you $50 later Bet #2 I pay you $10 now You might have to pay me $50 later

8. 8 Payoff Diagrams

9. 9 Bet #1 Spend then Maybe Receive You spend now, hope to receive later You spend NOW, voluntarily With the odds I give you, you can compute an expected value and decide if you want to make the bet

10. 10 Bet #2 Receive then Maybe Spend You receive now, hope you don’t have to spend later You MAY spend LATER, involuntarily With the odds I give you, you can compute an expected value and decide if you want to make the bet

11. 11 Capital? Bet #1 = $10 You spend $10 capital NOW no matter what The capital investment is current and certain – i.e., not contingent Allocated = spent Natural capacity constraint = your budget

12. 12 Capital? Bet #2 = $??? I should be sure you have $40 available LATER, but you don’t spend anything NOW If Bet #2 hits, you spend $40 capital LATER Capital expenditure (= allocation) is contingent and in the future Capacity constraints = ??? Perception

13. 13 Two Bets? Bet #1 = the manufacturing investment decision Spend then receive Bet #2 = the insurance investment decision Receive then spend

14. 14 Allocation vs Consumption Two different but equally valid frameworks for Treating capital Evaluating insurance business segments Developing indicated prices for reinsurance Nearly orthogonal

15. 15 Allocation vs Consumption Four questions: 1. What do you do with the total capital? 2. How do you evaluate business segments? 3. What does it mean to be in a portfolio? 4. How is relative risk contribution reflected?

16. 16 Allocation vs Consumption

17. 17 Allocation vs Consumption

18. 18 Allocation vs Consumption The difference between having your own kiddie pool and joining a swim club This is THE CRITICAL SLIDE!

19. 19 Allocation vs Consumption

20. 20 Details of the Framework Scenario analysis Default-free discounting Scenario-level capital consumption Evaluation of capital consumption using a “quasi~utility” approach

21. 21 Default-Free Discounting Conditional on its occurrence, a given scenario’s outcome is certain  discount at the default-free rate Risk-adjusted discounting is too clumsy Overloaded operator Try splitting out default probability from price of risk in risky debt spreads Reflect uncertainty between scenarios, not within What is uncertainty within a scenario anyway? Do you believe the scenario is possible or not?

22. 22 Scenario Capital Consumption Experience fund From Finite Reinsurance Fund into which goes all revenue, from which comes all payments Bakes in investment income When it drops below zero, and further payments need to be made, gotta “call the parents” for some capital That capital is spent  CONSUMED

23. 23 Experience Fund Long-Tailed LOB

24. 24 Experience Fund Short-Tailed LOB

25. 25

26. 26 Scenario Capital Consumption This is more realistic than imputed capital flows. (Imputed = fictional) The capital does flow, but in the future. When a segment’s results deteriorate, the company’s capital is consumed as it is turned into additional reserves. This is what actually happens, so why don’t we model it? Why pretend?

27. 27 Property Cat Example

28. 28 Property Cat Example How would you do this with capital allocation? Allocate a percentage of the limit – say 5% -- based on marginal portfolio capital requirements? What does that mean? What happens if the event occurs? Where does the money to pay the claim come from? Does the sum of the marginals add up to the company’s total capital? If not, what does it mean?

29. 29 Capital Calls (Philbrick/Painter) The entire surplus is available to every policy to pay losses in excess of the aggregate loss component. We can envision an insurance company instituting a charge for the access to the surplus. This charge should depend, not just on the likelihood that surplus might be needed, but on the amount of such a surplus call.

30. 30 Capital Calls (Philbrick/Painter) We can think of a capital allocation method as determining a charge to each line of business that is dependant on the need to access the surplus account. Conceptually, we might want to allocate a specific cost to each line for the right to access the surplus account. In practice though, we tend to express it by allocating a portion of surplus to the line, and then requiring that the line earn (on average) an adequate return on surplus.

31. 31 Capital Call Cost Function Risk-based overhead expense loading Pricing decision variable Application of utility theory Borch (1961): To introduce a utility function which the company seeks to maximize, means only that such consistency requirements (in the various subjective judgments made by an insurance company) are put into mathematical form.

32. 32 Capital Call Cost Function Make the implicit explicit Express your preferences explicitly, in mathematical form, and apply them via a utility function The mythical Risk Appetite Enforce consistency in the many judgments being made

33. 33 Implicit Preferences Preferences buried in Kreps’ “Marginal Standard Deviation” risk load approach: The marginal impact on the portfolio standard deviation is our chosen functional form for transforming a given distribution of outcomes to a single risk measure. Risk is completely reflected, properly measured and valued by this transform. Upward deviations are treated the same as downward deviations.

34. 34 This links up with:  Utility theory in actuarial pricing – from Longley-Cook, Halliwell, Heyer and Schnapp  Probability measure change – from financial mathematics The Wang Transform – from Shaun Wang Additive Co-Measures – from Rodney Kreps Conditional Risk Charges – from David Ruhm and Don Mango, 2003 Bowles Symposium

35. 35 Risk Charge Both Expected Utility and Distorted Probability determine a risk charge by: Risk Charge = Expected Value – Modified Expected Value How do we calculate the Modified Expected Value?

36. 36 Expected Utility Valuex1x2…xn Probp1p2…pn ValueU(x1)U(x2)…U(xn) Probp1p2…pn Modified Expected Value = Sumproduct of Modified Values and Probabilities Utility function is the modifier

37. 37 Distorted Probability Valuex1x2…xn Probp1p2…pn Valuex1x2…xn Probq1q2…qn Modified Expected Value = Sumproduct of Values and Modified Probabilities Probability Distortion Function is the modifier (changes p  q; impress your friends by discussing the “q-measure”)

38. 38 Distorted Probability A.k.a. “Measure Change” (change in the probability measure) In the Black-Scholes world… Constant interest rate, complete market, no transaction costs, instantaneous perfect hedging, … …the q-measure is unique. As soon as a few of those constraints are relaxed, there are infinite q- measures, all of which work.

39. 39 Every value is standard deviations worse If the asset return R has a normal distribution F(x), transformed F*(x) is also normal with E*[R] = E[R] – [R] = r (risk-free rate) = { E[R] – r }/  [R] = the “market price of risk”, also called the Sharpe ratio It recovers CAPM for assets, and Black-Scholes formula for Options Wang Transform

40. 40 Risk load R(X) is a probability-weighted average of “riskiness” r(x) over outcomes of the total net loss g(x) can be thought of as the “riskiness leverage ratio” that multiplies the actual dollar excess that an outcome would entail to get the riskiness. It reflects that not all dollars are equal, especially dollars that trigger analyst or regulatory tests. Kreps’ Co-Measures

41. 41 Conditional Risk Charge David Ruhm and Don Mango, 2003 Bowles Symposium paper www.casact.org/coneduc/specsem/sp2003/pa pers/ruhm-mango.doc www.casact.org/coneduc/specsem/sp2003/pa pers/ruhm-mango.doc Main principle of conditional risk charge: Each risk receives a charge that represents how much it contributes to undesirable portfolio outcomes. Generalization of Appendix B of my paper

42. 42 Advantages of Method Additive prices. Extends aggregate risk valuation to any individual risk, including layers of risks. Handles any underlying dependence structure. Really works well for Property Cat.

43. 43 Goals for Today 1. Get you to admit this is a valid alternative framework (albeit orthogonal) to capital allocation / release / IRR 2. Demonstrate how it can be practically implemented as a means of pricing reinsurance 3. Demonstrate connections to leading edge thinking in financial science