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CAS 2004 Spring Meeting Presentation of Proceedings Paper

This presentation provides a discussion of the Option Pricing Theory (OPT) approach in ratemaking, highlighting its strengths and shortcomings. The focus is on correcting and extending the examples presented in the original paper.

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CAS 2004 Spring Meeting Presentation of Proceedings Paper

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  1. CAS 2004 Spring MeetingPresentation of Proceedings Paper Ratemaking: A Financial Economics Approach Stephen P. D’Arcy and Michael A. Dyer DISCUSSION BY MICHAEL G. WACEK

  2. Scope of Wacek Discussion • D’Arcy-Dyer is an important paper* that surveys a range of ratemaking approaches related to modern financial economics • This discussion focuses exclusively on their Section 8, which describes a method based on Option Pricing Theory (OPT)* Paper is on syllabus for CAS Exam 9 (Sections 4,6, 8 directly tested)

  3. Origin of OPT Ratemaking Approach • D’Arcy–Dyer summary of OPT approach based on work of Doherty & Garven (1986) and Garven (1988) • Doherty & Garven (D&G) assumed an insurer’s assets consist of tradable assets on which it is possible to price option

  4. Summary of OPT Ratemaking Approach • Pretax value of an insurer can be seen as a call option on its assets • Strike price of call equal to aggregate amount of claims (variable strike price) • Government tax claim also a call option (under asymmetrical taxation, i.e., no loss carry forwards / carry backs) • Appropriate rate level indicated by equality between beginning surplus and value of the call option, net of tax

  5. Simplifying Assumptions in D’Arcy–Dyer Illustration • Policies written for one-year term at common date • Claims totaling L are paid exactly one year from policy inception • Premium funds (net of expenses) are received at policy inception • Premium receipts, P0, and initial surplus, S0, are invested solely in taxable assets initially valued at Y0,

  6. Summary of Wacek Discussion • Point out shortcomings of D’Arcy–Dyer • Purpose is to correct, clarify and extend their paper • Rework and extend examples • Expand exposition to allow for (a) symmetrical taxation and (b) stochastic claims • Show that under realistic conditions OPT reduces to more conventional ratemaking approach

  7. Shortcomings of D’Arcy-Dyer • Option mistakes • Policyholders’ claim wrongly described as a call option • Tax claim (correctly) described as call option, but wrongly parameterized • Notation is cumbersome and inconsistent • No formula or example given for calculation of fair premium (surprising for a paper on ratemaking!) • Most of discussion treats claims amount as fixed, known quantity, essentially as a loan • Attempt to address stochastic claims scenarios is seriously flawed (and not faithful to D&G)

  8. Wacek Discussion “Despite these . . . shortcomings, the D’Arcy–Dyer paper is still a useful springboard for discussing the OPT approach.”

  9. Wacek DiscussionNote on Wacek Notation • Discussion uses D’Arcy-Dyer notation with some refinements • We use numerical subscripts only to refer to time: 0 = inception, 1 = expiry in 1 year • l represents the random variable for claims. L refers to a specific claim amount • y represents the random variable for invested assets at expiry. Y1 refers to a specific value • call1(y | Y1, L) refers to the expiry value of a one-year European call option on y, given a “price” at expiry of y = Y1 and “exercise price” of l = L

  10. Finding the Options • Y0 = initial invested assets = S0 + P0 • Y1 = invested assets after 1 year • Y1 is amount insurer has to pay claims, L > 0 • Policyholders will recoverL if Y1> L, orY1 if 0 < Y1 < L

  11. Finding the Options • Policyholders’ claim succinctly given asH1 = max [min (Y1, L), 0] (8.3) • D’Arcy & Dyer describe H1 as “equivalent to the expiration payoff to the owner of a European call option with an exercise price of L.” • That is incorrect . . . Option they describe has payoff equal to max (Y1 – L, 0)

  12. Finding the Options • That call option belongs to shareholders, not policyholders • Sale of insurance policies is equivalent to sale of insurer’s assets to policyholders in exchange for a call option to reacquire the assets at price of L • If Y1> L, insurer will exercise option to reacquire assets for a gain of Y1 – L • If Y1 < L, insurer will not exercise option (no gain or loss)

  13. Finding the Options • Shareholders’ interest (pre-tax) at expiry can be summarized as C1 = max(Y1 – L, 0) = call1(y | Y1, L) • Policyholders’ interest is equivalent to a long position in y (random variable for invested assets) and short position in the call:H1 = Y1 – call1(y | Y1, L) = Y1 – C1

  14. Finding the OptionsPut – Call Parity • The combination of a stock and a put (on same stock) has same expiry value as the combination of a T-bill (in an amount equal to the option strike price) and a call option (on the same stock) • In our example and notation:Y1 + put1(y | Y1, L) = L + call1(y | Y1, L) • This relationship is known as “put-call parity”

  15. Finding the Options • Put – call parity implies:Y1 – call1(y | Y1, L) = L – put1(y | Y1, L) • Policyholders’ interest can alternatively be expressed as:H1 = L – put1(y | Y1, L) • put1(y | Y1, L) denotes payoff value of European put option on y, given invested assets at expiry of Y1 and option strike price of L

  16. Pricing Implications • OPT–based ratemaking indicates the premium rate should be reduced by the value of the put • Ignoring taxes for now, this implies:P0 = H0 = Le-rt – put0(y | Y0,L)

  17. Shareholders’ Interest (Pre-Tax) • The value of the pre-tax shareholders’ interest at policy inception is:C0 = Y0 – H0 = Y0 – Le-rt + put0(y | Y0,L) = call0(y | Y0,L) • Since y represents assets whose behavior meets “Black-Scholes conditions” the value of C0 can easily be calculated

  18. Calculation of Pre-Tax Shareholders’ InterestExample • Let S0 = $100M, P0 = $160M, L = $150M, r = 4%, t = 1 year, σ = 50% • D’Arcy–Dyer calculate C0 = $121.41M from B-S formula • They note $121.41M is surprisingly high, since “adding the initial equity to the underwriting profit totals $110M” • They attribute the difference to the “default option” considered by the OPT approach • Qualitatively, they are on to something, but they compare the wrong numbers

  19. Calculation of Pre-Tax Shareholders’ InterestExample • Wrong to compare $110 to $121.41 since first number is valued at end of period but ignores interest and second number is valued at inception • C0 = Y0 – Le-rt + put0(y | Y0,L) = $121.41 • If default (put) option = 0, C0 = Y0 – Le-rt = $115.88 • $121.41 should be compared to $115.88 (not $110) • Value of default option (windfall to shareholders) is $5.53

  20. Pricing at Equilibrium • D&G observe that in equilibrium the P.V. of shareholders’ interest at inception must equal the initial surplus • Previous example not at equilibrium • In pre-tax case equilibrium implies:C0 = S0 • D’Arcy-Dyer allude to this, but do not derive the indicated pricing formula, which in our notation is:P0 = Le-rt – put0(y | Y0,L)

  21. Pricing at Equilibrium - Example • The indicated premium at equilibrium in the authors’ example (which they do not calculate) is $136.44 • P.V. of losses = $144.12 • Value of put at inception = $7.68 • At equilibrium, value of the put is credit to policyholders rather than extra margin for shareholders • Fair premium = $144.12 - $7.68 = $136.44

  22. Unique Feature of OPT–Based Pricing • Claim default risk is automatically incorporated into insurance rate • Other ratemaking methods ignore possibility of claim default by insurer • OPT method reduces insurance rate by value of the default option rather than allowing shareholders to reap windfall arising from risky investment strategy and/or high underwriting leverage

  23. Paradoxical Implications of OPT–Based Pricing • Insurers most at risk of insolvency are required to charge premiums that are less than the expected present value of their claim obligations! • That makes their demise even more likely! • Not good public policy • Far better for regulators to correct an insurer’s financial weakness or investment strategy (so that default risk is negligible) rather than to require it to reduce its rates • If regulation is effective, conventional ratemaking assumption that default risk is zero seems appropriate

  24. Note on Assumptions in D’Arcy-Dyer Example • Investment volatility parameter, σ, of 50% extremely unrealistic • Standard deviation of U.S. stock market returns (1900 – 2000) was 20.2% • Hard to construct an investment portfolio with σ = 50% • No insurer would invest 100% of its investable assets in such a portfolio • D’Arcy & Dyer undoubtedly chose σ = 50% to illustrate material default risk

  25. More Realistic Investment Volatility Assumption Shareholders’ Interest • Suppose σ = 20% (still very aggressive for an insurer) • All other assumptions the same (P0 = $160, L = $150, etc.) • Now shareholders’ interest, C0 = $115.90 (vs. $121.41 with σ = 50%) • Default put option worth only $0.02(vs. $5.53 with σ = 50%)

  26. More Realistic Investment Volatility Assumption Premium • Suppose σ = 20% • All other assumptions the same • Solve for fair value of P0 • Now P0 = $144.07 (vs. $136.44 with σ = 50%) • Value of default put is $0.05 (vs. $7.68 with σ = 50%)

  27. Taxes • D&G / D’Arcy-Dyer made assumptions that taxes apply only to income and no tax credits arise from losses • Then government tax claim can be characterized as a call option • Income = I1 = (Y1 – Y0) + (P0 – L) • Since Y0 = S0 + P0, I1 = Y1 –(S0 + L) • Focusing on positive outcomes, max(I1, 0) is the payoff profile at expiry of a call option on invested assets, y, with strike price of S0 + L • Government tax claim is: tax call0(y | Y0, S0 + L), where “tax” denotes the tax rate .

  28. Taxes in D’Arcy-Dyer ExampleWacek Calculation • Assume tax rate is 35% and applicable to all income • Tax call, T0 = 0.35 call0(y | Y0, S0 + L) • B-S value of T0 = 0.35 call (y | $260, $250) = $20.96

  29. Taxes in D’Arcy-Dyer ExampleAuthors’ Calculation • Assume tax rate is 35% and applicable to all income • D’Arcy-Dyer use B-S formula to calculate T0 = $16.05 • However, the parameters they use in the B-S formula do not make sense • Their parameters and the $16.05 correspond to 0.35 call0(y | Y1 – Y0 + P0, L) • But that implies the value of invested assets at inception is Y1 – Y0 + P0, when by definition it must be Y0 • Their use of B-S formula and hence calculation of T0 is incorrect

  30. Effect of Taxes in D’Arcy-Dyer ExampleShareholders’ Interest • Correct value of shareholders’ interest, net of tax, isC0 – T0 = $121.41 - $20.96 = $100.45 • D’Arcy-Dyer calculation was C0 – T0 = $121.41 - $16.05 - $105.36

  31. Effect of Taxes in D’Arcy-Dyer ExamplePremium • To find the fair (equilibrium) premium, we solve for P0 that yields C0 – T0 = S0P0 = Le-rt – put0(y | Y0, L) + tax .call0(y | Y0, S0 + L) • This implies a premium of P0 = $159.33 using authors’ example assumptions • Note: D’Arcy-Dyer did not solve for fair premium

  32. Symmetrical Taxation of Profits and Losses • D&G (and hence D’Arcy-Dyer) ignored tax loss carry-forward and carry-back provisions • Easy to deal with in D’Arcy-Dyer framework • Tax credits equivalent to a long put owned by shareholders (short put on part of government) • Note that if an insurer becomes insolvent, it won’t be in position to use the tax credit – that portion must be removed

  33. Symmetrical Taxation • Government’s net tax position is given byT0*= tax[call0(y | Y0, S0 + L) – (put0(y | Y0, S0 + L) – put0(y | Y0,L))] • In authors’ example, T0* would be $20.96 - $14.03 + $1.93 = $8.86 • After-tax value of shareholders’ interest, V0*(P0 | L) isV0*(160 | 150) = C0 – T0* = $121.41 - $8.86 = $112.55

  34. Symmetrical Taxation • Formula for T0* can be simplified • T0* = tax (Y0 - (S0 + L)e-rt + put0(y | Y0, L)) • First term combines tax debits and credits • Second term corrects for unusable tax credit due to insolvency

  35. Fair Premium Under Symmetrical Taxation . S0 • P0 = Le-rt – put0(y | Y0, L) + tax (1 – e-rt) • Authors’ example assumptions imply P0 = $138.80 1 –tax = P.V. of Losses less Default Option plus taxes on P.V. of Investment Income on initial surplus (grossed up)

  36. Stochastic Claims • Treat claims as random variable, l, rather than fixed amount, L • Begin with authors’ asymmetrical taxation • Ignore underwriting risk load issues for now • Expected value E[V0(P0)] of after-tax shareholders’ interest is:

  37. Stochastic Claims (8.9*) (14) • E[V0(P0)] expands to • Compare to D’Arcy-Dyer stochastic claims formula (8.9): • Correcting the second term (as discussed) and restating in our notation with i = 1, (8.9) becomes • 14 and 8.9* are different • (8.9*) does not actually reflect a stochastic claims assumption

  38. Stochastic Claims Formula Example 1 • Suppose f(l) is lognormal with σl= 11% and E(l) = $150 • All other assumptions same as authors’ • Our formula (14) indicates P0 = $158.89 • D’Arcy-Dyer formula (8.9*) indicates P0 = $159.33

  39. Stochastic Claims Formula Example 2 • Suppose f(l) is lognormal with σl = 15% and E(l) = $150 • All other assumptions same as authors’ • Our formula (14) indicates P0 = $158.80 (indicating a slightly higher default risk given greater underwriting volatility) • D’Arcy-Dyer formula (8.9*) still indicates P0 = $159.33, even though this scenario has greater underwriting volatility

  40. Stochastic Claims • D’Arcy-Dyer claim their formula (8.9) is based on D&G’s work • In fact, D&G formula, while superficially similar, is quite different • D&G define a special call option with a random variable strike price and embedded underwriting risk charge

  41. Stochastic Claims with Symmetrical TaxationFair Premium without Underwriting Risk Charge (15) • P0 = $138.22 for example we have been following (vs. $138.80 for fixed L = $150) • Still no underwriting risk charge

  42. Summary of Indicated Premiums Under Various ScenariosBased on Author’s Assumptions Fixed Stohastic Claims1Claims1,2 No Tax $136.44 Asymmetrical Tax $159.33 $158.89 Symmetrical Tax $138.80 $138.22 1 No underwriting risk charge included 2 Claims lognormal with σ = 11%

  43. Stochastic Claims with Symmetrical TaxationFair Premium with Underwriting Risk Charge (16) • Formula (16) below is implied by equilibrium condition E(V0*(P0)] = S0 + λ • λ is the after-tax underwriting risk charge at inception • The only option in (16) is the put representing the credit for insurer insolvency • If that put is zero (as it should be under effective regulation), (16) reduces to D’Arcy-Dyer’s DCF formula (6.1)!

  44. Conclusions • D’Arcy-Dyer concluded OPT approach more complex than CAPM or DCF, but avoids CAPM issue of estimating betas • We hope our presentation makes clear that if taxation is symmetrical and default risk is zero (both realistic) and underwriting risk is treated in a conventional way, OPT = DCF • Difference from DCF in D&G’s OPT framework is that they based their underwriting risk charge on the correlation between insurance claims and the stock market (making it similar to the CAPM approach and subject to same problems) • OPT approach is interesting (if impractical) application of option theory, but much less exotic than it appears from either D&G or D’Arcy-Dyer • In fact little or no need to resort to the option approach

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