Standard Approaches for Insurance Risk Pricing

1. The Cost of Conditional Risk FinancingCAS Ratemaking SeminarMarch 11-12, 2004Frank SchnappNational Crop Insurance Services, Inc.

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. Proposed Risk Pricing ModelBasic 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. 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. Two Varieties of Pricing Model • Based on Actual costs  Retroactive pricing • Based on Expected costs  Prospective pricing

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. 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. Retroactive Pricing (Payback) Method1st Example: Premium = Expected Damages • 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. Retroactive Pricing (Payback) Method2nd Example: Premium = $1400 • Assume outcome C occurs every 4th year • Insurer makes $1600 capital contribution every 4th 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

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 si on loan of xi – P • Term of loan is 1/pi • si represents the interest charged on the loan • si is reduced for the time value of money • si >= 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(xi – P)sipi

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

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

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. 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 si, the cost of borrowed funds • Helps satisfy the Insurer’s Capital Preservation Objective

17. Illustration of Risk Diversification without Price Reduction

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

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(m,s2), P(X) = m + ls (with l a constant) • Income taxes have little or no effect on price

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. 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. 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. World Trade Center (9/11) Andrew Northridge $28 Billion full year 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/

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

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. Additional Topics

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 X1, … Xn be exposures in market segment W = Xi • Assume prices are additive: P(Xi+Xk;W) = P(Xi;W) + P(Xk;W) • Assume the price PW for the market segment W is known • Let Price be based on Xi’s contribution to systematic risk W • Xi can be uniquely decomposed as b W + Ui • With b = cov(Xi,W) / V(W) • Systematic risk component is b W • Diversifiable risk component is Ui (since Ui = 0) • Ui is uncorrelated with W • Systematic Risk Pricing Model: P(Xi;W) = E(Xi) + b (PW - E(W))

28. Observations on Systematic Risk Pricing Model • Market segment premium PW can be selected arbitrarily • It need not be determined using the Risk Pricing Model • Restriction: E(W) <= PW <= P(Xi) • 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. 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. 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. Pricing of Uncertain Future Damages • Assume stable risk aversion parameters over time: a1 = a0 • Justification: a = 1 for a well-diversified insurer • Given X0 and X1 with identical damage distributions • Damages are paid at times 0 and 1, respectively • Since a is constant, P1(X1) = P0(X0) • What is the price at time 0 for future damages X1? • Let v be the discount factor corresponding to the risk-free rate • Discount the future price for X1 to time 0 = vP1(X1) • Or, discount each outcome to time 0 = P0(vX0) • Both methods give the same price: • P0(vX0) = vP1(X1)

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. End of Presentation