1. 1 The Cost of Conditional Risk Financing CAS Ratemaking Seminar March 11-12, 2004 Frank Schnapp National Crop Insurance Services, Inc.

2. 2 Standard Approaches for Insurance Risk Pricing Economic methods –Adopts Risk-taker’s perspective –Expected Utility Theory Key concept: Preferences Shape of utility function is unknown Ignores the insurer’s ability to reduce risk through diversification Financial methods –Takes Investor’s perspective –Net Present Value model Key concept: Cost of capital Capital is invested at the time an insurance policy is issued Focus is on timing of cash flows Ignores the uncertainty of the cash flows

3. 3 Proposed Risk Pricing Model Basic Concepts Adopts the Risk taker’s perspective No capital is needed to issue an additional policy Considers both the uncertainty and timing of cash flows Based on real costs –Expected Utility Theory is preference based –Cost of capital represents a competitive return, not an actual cost Risk diversification (pooling) –Reduces insurer’s risk –May (or may not) affect the price paid by policyholder Insurer operates under a Capital Preservation Objective

4. 4 Risk Taker’s Perspective Return to the insurer  Return to the investor Company actions to provide an adequate return to investor: –Increase or decrease expenses Commissions, salaries, bonuses –Portfolio selection Pursue markets where Insurer has a competitive advantage Higher risk markets producer higher returns –Increase or decrease amount of insurance or investment risk

5. 5 Two Varieties of Pricing Model Based on Actual costs  Retroactive pricing Based on Expected costs  Prospective pricing

6. 6 No Capital is needed to issue an additional policy Capital is used when a claim is paid –And only if the Damages exceed the Premium If Damages < Premium  Insurer earns a profit If Damages > Premium  Insurer contributes capital –Capital contribution = max(Damages – Premium, 0) –Takes into account the uncertainty of the outcomes Analysis is similar if expenses are included

7. 7 Insurance as a Risk Financing Mechanism Self-insurance –Self-insurer borrows funds to pay any deficit on policy –Repays the loan over time Purchase of an insurance policy –Insurer provides funds as needed to pay any deficit on policy Insurer functions as the “Bank” Risk financing is treated as a loan, not as an investment –Insurer’s Capital Preservation Objective Insurer needs to recover the borrowed funds –Loan can be repaid by: Policyholder, or All policyholders in the market segment Policyholders in all market segments

8. 8 Retroactive Pricing (Payback) Method 1 st Example: Premium = Expected Damages OutcomePremiumDamagesProbabilityReturnDeficit A10000.25010000 B 500.5005000 C10003000.250-20002000 Expected1000 1.0000500 Return = Profit Deficit = Amount of Capital consumed Outcomes A & B – Insurer earns a profit –Premium in year 2 is $1000 = Expected damages Outcome C – Insurer makes a capital contribution (loan) of $2000 –Loan must be repaid by next expected occurrence in 4 years = 1/.250 –Annual payment on loan = $500 –Premium in Years 2-5 = Expected Damages + Annual Payment on Loan = $1500 ·Premium may change again if outcome C occurs in years 2, 3, or 4 ·Long term average premium > $1000

9. 9 OutcomePremiumDamagesProbabilityReturnDeficit A14000.25014000 B 500.5009000 C14003000.250-16001600 Expected140010001.000400 Assume outcome C occurs every 4 th year –Insurer makes $1600 capital contribution every 4 th year –Policyholder contributes $400 expected profit every year –Total of $1600 over four years –Policyholder pays for the insurer’s capital contributions (over the long term) Result represents the “optimal” retroactive premium –Premium will vary depending on the actual outcome Retroactive Pricing (Payback) Method 2 nd Example: Premium = $1400

10. 10 The “Optimal” Retroactive Price Policyholder pays for potential use of the insurer’s capital –Pays the long term average cost Premium = Expected Damages + Average cost of loan Cost based surcharge s i on loan of x i – P –Term of loan is 1/p i –s i represents the interest charged on the loan –s i is reduced for the time value of money –s i >= 1 Equivalent to the Prospective price for the exposure –Insurer charges for its expected, not actual, capital contribution –No recognition of the effect of Insurer’s risk diversification –Can be interpreted as the Self-insurance price P = E(X) +  xi>P (x i – P)s i p i

11. 11 Retroactive vs. Prospective Pricing Prospective method –Useful for small exposures –Used if Insurer is not permitted to recoup losses from policyholder –Cost of loan may be spread across all exposures in market segment Retroactive pricing –For exposures large enough to be self-rated –Reinsurance and large accounts

12. 12 Retroactive Pricing and the Insurance Market Pricing Cycle Insurers raise prices to recoup underwriting losses High prices would continue even if coverage is amended –Terrorism, toxic mold coverage Enables insurers to “recoup” capital losses in subsequent years Enhances long term solvency of the industry –Supports Capital Preservation Objective

13. 13 Overview & Outlook for the Property/Casualty Insurance Industry. Dr. Robert P. Hartwig. July, 2003. Insurance Information Institute. http://www.iii.org/media/presentations/industryoutlook/ Insurance Market Pricing Cycle

14. 14 Overview & Outlook for the Property/Casualty Insurance Industry. Dr. Robert P. Hartwig. July, 2003. Insurance Information Institute. http://www.iii.org/media/presentations/industryoutlook/ Hard Markets Follow Years of Deteriorating Results

15. 15 The Effect of Risk Diversification (Pooling) on Price Evaluate Risk from a portfolio perspective Effect of Risk diversification within a market segment –Reduces an insurer’s average risk per exposure: V(Ž) = V(Z) / n –Permits the insurer to reduce its risk margin on each exposure –Competition may prevent Insurer from pricing an exposure for its own risk Price each market segment for its own risk instead Effect of Risk diversification across market segments –No reduction in the insurer’s price (mostly) –Reduces the insurer’s risk instead

16. 16 Example of Risk Diversification Across Market Segments (Without Price Reduction) Assume Insurer prices each market segment for its own risk –Distribution A: Single market segment Insurer frequently uses its own capital –Distribution B: Portfolio consisting of 5 market segments Insurer occasionally uses its own capital –Distribution C: Portfolio consisting of 12 market segments Insurer rarely, if ever, uses its own capital A loss in one market segment is paid by the policyholders in other market segments Affects s i, the cost of borrowed funds Helps satisfy the Insurer’s Capital Preservation Objective

17. 17 Illustration of Risk Diversification without Price Reduction

18. 18 Prospective Pricing Model with Risk Diversification Across Market Segments Price achieves a balance between Risk and Return –For outcome x i, define Return as P – x i –Define Risk as (x i – P)s i or (x i – P)  for x i > P, else 0 –The premium P is the unique solution to: Expected Risk = Expected Return Risk Pricing Model –Assumes insurer rarely, if ever, uses its own capital –Allows insurer to use a uniform surcharge for s i of  >= 1 –For a very well-diversified insurer,  = 1 –Select  to satisfy Capital Preservation Objective P – E(X) =  x>P (x – P) dF(x)

19. 19 Comparison to Expected Utility Theory Risk Pricing Model is consistent with Utility pricing –Model is applied to market segments, not to individual exposures –Shape of Utility function: Two rays with positive slope meeting at 0 Concave downward –“Utility” is independent of wealth –Risk aversion parameter is a function of insurer’s diversification Consistent with pricing formulas: –P(c) = c –P(X + c) = P(X) + c –P(aX) = aP(X) for a >= 0 –P(X + Y) <= P(X) + P(Y) (diversification property) –For X ~ N( ,  2 ), P(X) =  +  (with a constant) Income taxes have little or no effect on price

20. 20 The Mutually Acceptable Price Insurer’s price declines as number of exposures increases – Enables Insurer to compete with self-insurance – Even if the Insurer is more “risk averse” than the Self-insurer – No mutually acceptable price exists if insurer’s expenses are too high

21. 21 Competitive Market Pricing Construct Supply and Demand curves for insurance Limits on the Insurer’s ability to insure additional policies: –Quality of the Insurer’s book declines during rapid expansion –Staffing is insufficient to handle the work load –But: the amount of Capital held by Insurer is not a limitation Intersection of Supply & Demand determines the market price –Low cost Insurers earn more than a “normal” profit –High cost Insurers earn less than a “normal” profit Will continue to write insurance as long as variable costs are met Decision to participate in market is unrelated to Cost of Capital

22. 22 Is the Capital Preservation Objective Realistic? U.S. P&C insurance industry is consistently profitable –Only one exception 2001 (9/11 terrorist attack) –Sharp increase in insurance prices immediately afterward Helped industry to recoup losses from event Stability: Insolvency rate remarkably low –Ten year average = 0.72% –Most insolvencies are small, low rated companies Industry structure –Unconcentrated, with a large number of competitors –Survival & profitability much better than auto, steel, & airlines

23. 23 P&C Industry Profitability Based on “Overview & Outlook for the Property/Casualty Insurance Industry.” Dr. Robert P. Hartwig. July, 2003. Insurance Information Institute. http://www.iii.org/media/presentations/industryoutlook/ $28 Billion full year AndrewNorthridge World Trade Center (9/11)

24. 24 Overview & Outlook for the Property/Casualty Insurance Industry. Dr. Robert P. Hartwig. July, 2003. Insurance Information Institute. http://www.iii.org/media/presentations/industryoutlook/ P&C Industry Insolvency Rates

25. 25 Summary Retroactive Pricing –Reinsurance, large accounts, Insurance Market Pricing Cycle Prospective Pricing –Without risk diversification Each exposure is priced for its own risk (e.g., self-insurance) –With risk diversification Insurer determines its price for each market segment, not each exposure Diversification across market segments minimizes use of insurer’s capital Price is determined from Risk-taker perspective –Cost of capital is not relevant to the model –Model accounts for: Risk/Return tradeoff, Expenses, Taxes, Time value of money Competition, Self-insurance, Heterogeneity of exposures Investment Income on insurance cash flows Unified treatment of insurance and investment pricing

26. 26 Additional Topics

27. 27 Pricing for Systematic Risk ( Once the Insurer determines the premium it needs for a market segment, how does it determine its price for each policy?) Let X 1, … X n be exposures in market segment W =  X i –Assume prices are additive: P(X i +X k ;W) = P(X i ;W) + P(X k ;W) –Assume the price P W for the market segment W is known Let Price be based on X i ’s contribution to systematic risk W –X i can be uniquely decomposed as  W + U i With  = cov(X i,W) / V(W) Systematic risk component is  W Diversifiable risk component is U i (since  U i = 0) U i is uncorrelated with W Systematic Risk Pricing Model: P(X i ;W) = E(X i ) +  (P W - E(W))

28. 28 Observations on Systematic Risk Pricing Model Market segment premium P W can be selected arbitrarily –It need not be determined using the Risk Pricing Model –Restriction: E(W) <= P W <=  P(X i ) Application to Insurance pricing –Accounts for systematic risk –Price is unrelated to security market returns –Formula can be converted to a rate of return on price Formula does not involve capital –Finding: Rate of return formula is identical to the CAPM formula

29. 29 Application to Security Market Pricing Model can be applied to determine security prices –Not rates of return –Price is tied to the underlying earnings of the business Consistent with Dividend Discount Model But, it recognizes the uncertainty of the dividends Relationship to Rate of Return –Model determines rate of return on price, not on capital –Finding: The CAPM does not apply to security market pricing Reason: A security is tied to the earnings of a business But, a business need not have a fixed risk exposure over time A company can enter or leave markets, change its pricing, etc. –CAPM is consistent with the model under narrow restrictions

30. 30 Comparison of the Role of Capital Risk Pricing Model –Capital expenditure is no different from any other cash flow Purchase of productive goods: Capital investment = Up-front Expense (fixed) Purchase of a security: Capital expenditure = Price (may be negotiable) –Analysis of an investment depends on the responsibility for losses A business pays for operating losses out of its capital A security holder has no obligation to use its capital to pay losses Insurance: Other policyholders provide the capital needed to settle claims Systematic Risk Pricing Model and the Revised CAPM –Capital has no bearing on price –Rate of return is defined in relation to price, not capital Actuarial Pricing Models (per Standards of Practice) –Cost of Capital is fundamental

31. 31 Pricing of Uncertain Future Damages Assume stable risk aversion parameters over time:  1 =  0 –Justification:  = 1 for a well-diversified insurer Given X 0 and X 1 with identical damage distributions –Damages are paid at times 0 and 1, respectively –Since  is constant, P 1 (X 1 ) = P 0 (X 0 ) What is the price at time 0 for future damages X 1 ? –Let v be the discount factor corresponding to the risk-free rate –Discount the future price for X 1 to time 0 = vP 1 (X 1 ) –Or, discount each outcome to time 0 = P 0 (vX 0 ) –Both methods give the same price: P 0 (vX 0 ) = vP 1 (X 1 )

32. 32 The Time Value of Money Risk-free rate –Assume Lender’s objective is to maintain purchasing power Purchasing power is affected by future inflation Future inflation is uncertain Define the risk-free rate as the price needed to offset the risk of future inflation Apply risk vs. return analysis to future purchasing power –Example: model inflation as a Markov chain Shape of yield curve: –Short term rate is similar to expected inflation rate –Increases as payment horizon increases –Long term rate stabilizes after several periods

33. 33 End of Presentation