The Cost of Conditional Risk Financing
CAS Ratemaking Seminar
March 11-12, 2004
Frank Schnapp
National Crop Insurance Services, Inc.
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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
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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
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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
4
Two Varieties of Pricing Model
• Based on Actual costs
 Retroactive pricing
• Based on Expected costs  Prospective pricing
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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
6
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
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Retroactive Pricing (Payback) Method
1st Example: Premium = Expected Damages
Outcome
Premium
Damages
Probability
Return
Deficit
A
1000
0
.250
1000
0
B
1000
500
.500
500
0
C
1000
3000
.250
-2000
2000
Expected
1000
1000
1.000
0
500
• 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
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Retroactive Pricing (Payback) Method
2nd Example: Premium = $1400
Outcome
Premium
Damages
Probability
Return
Deficit
A
1400
0
.250
1400
0
B
1400
500
.500
900
0
C
1400
3000
.250
-1600
1600
Expected
1400
1000
1.000
400
400
• 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
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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
P = E(X) + xi>P (xi – P)sipi
• 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
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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
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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
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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/
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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/
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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
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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
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Illustration of Risk Diversification without Price Reduction
Figure 3
Distribution of the Insurer's Returns
Probability
C
B
A
Insurer's Actual Return (Premium - Damages) as Percent of Premium
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Prospective Pricing Model with Risk Diversification
Across 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
P – E(X) = a x>P (x – P) dF(x)
• 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:
Expected Risk = Expected Return
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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)
For X ~ N(m,s2), P(X) = m + ls
(diversification property)
(with l a constant)
• Income taxes have little or no effect on price
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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
Figure 4
Comparison of Certainty Equivalent Prices
Price
Self-insurance price
Insurer's price
Number of Exposures Insured
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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
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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
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P&C Industry Profitability
World Trade
Center (9/11)
Andrew
Northridge
$28 Billion
full year
Based on “Overview & Outlook for the Property/Casualty Insurance Industry.” Dr. Robert P. Hartwig. July, 2003.
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Insurance Information Institute. http://www.iii.org/media/presentations/industryoutlook/
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/
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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
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Additional Topics
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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))
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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
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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
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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
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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)
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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
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End of Presentation
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