1 The Cost of Conditional Risk Financing CAS Ratemaking Seminar March 11-12, 2004 Frank Schnapp National Crop Insurance Services, Inc. 2 Standard Approaches for Insurance Risk Pricing Economic methods Adopts Risk-takers perspective Expected Utility Theory Key concept: Preferences Shape of utility function is unknown Ignores the insurers ability to reduce risk through diversification Financial methods Takes Investors 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 Model Basic Concepts Adopts the Risk takers 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 insurers risk May (or may not) affect the price paid by policyholder Insurer operates under a Capital Preservation Objective 4 Risk Takers 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 Insurers 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) Method 1 st Example: Premium = Expected Damages OutcomePremiumDamagesProbabilityReturnDeficit A B C Expected 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 OutcomePremiumDamagesProbabilityReturnDeficit A B C Expected 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 insurers 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 The Optimal Retroactive Price Policyholder pays for potential use of the insurers 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 Insurers risk diversification Can be interpreted as the Self-insurance price P = E(X) + xi>P (x i P)s i p i 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 Overview & Outlook for the Property/Casualty Insurance Industry. Dr. Robert P. Hartwig. July, Insurance Information Institute.Insurance Market Pricing Cycle 14 Overview & Outlook for the Property/Casualty Insurance Industry. Dr. Robert P. Hartwig. July, Insurance Information Institute.Hard Markets Follow Years of Deteriorating Results 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 insurers average risk per exposure: V() = 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 insurers price (mostly) Reduces the insurers 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 s i, the cost of borrowed funds Helps satisfy the Insurers Capital Preservation Objective 17 Illustration of Risk Diversification without Price Reduction 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 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 insurers diversification Consistent with pricing formulas: P(c) = c P(X + c) = P(X) + c P(aX) = aP(X) for a >= 0 P(X + Y)