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D Arcy and Michael A. Dyer &UZS6Scope of Wacek DiscussionrD 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) .: WH7!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 >? g8"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;fSimplifying Assumptions in D Arcy Dyer Illustration44%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, H&=<Summary of Wacek DiscussionLPoint 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=6Shortcomings of D Arcy-DyerOption 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)6DD>Wacek Discussions,'Wacek Discussion Note 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~!k?Finding the OptionsY0 = initial invested assets = S0 + P0 Y1 = invested assets after 1 year Y1 is amount insurer has to pay claims, L > 0 Policyholders will recover L if Y1 > L, or Y1 if 0 < Y1 < L !Q @ Finding the OptionsFPolicyholders claim succinctly given as H1 = 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)Z$+(A Finding the OptionsThat 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)ZhD 8B 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 !C-JFinding the Options Put  Call Parity&zThe 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 l>8C Finding the OptionsHPut  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 %A  [GPricing ImplicationsbOPT 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) ~ H@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 J^kIjCalculation of Pre-Tax Shareholders Interest Example6."Let S0 = $100M, P0 = $160M, L = $150M, r = 4%, t = 1 year,  = 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 ^Z )  (6 ,;UJjCalculation of Pre-Tax Shareholders Interest Example6.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 c-},G}KPricing at EquilibriumD&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) ~Jk4L Pricing at Equilibrium - Exampleh 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 = $136.44MFUnique Feature of OPT Based PricingKClaim 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 leverageNZParadoxical Implications of OPT Based PricingInsurers 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 appropriate8Z(w =OTNote on Assumptions in D Arcy-Dyer ExampleInvestment volatility parameter, , of 50% extremely unrealistic Standard deviation of U.S. stock market returns (1900  2000) was 20.2% Hard to construct an investment portfolio with  = 50% No insurer would invest 100% of its investable assets in such a portfolio D Arcy & Dyer undoubtedly chose  = 50% to illustrate material default riskZ!9 j!, =*PMore Realistic Investment Volatility Assumption Shareholders InterestGGSuppose  = 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  = 50%) Default put option worth only $0.02 (vs. $5.53 with  = 50%)RQ (7 (e J=Q7More Realistic Investment Volatility Assumption Premium88dSuppose  = 20% All other assumptions the same Solve for fair value of P0 Now P0 = $144.07 (vs. $136.44 with  = 50%) Value of default put is $0.05 (vs. $7.68 with  = 50%) RB ( (` Jg8 RTaxesD&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 rateZ` ! $ `  h c `  h ) `  h  `  h  `  h ( ` S\Taxes in D Arcy-Dyer Example Wacek 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@`h`h`h`h&`TbTaxes in D Arcy-Dyer Example Authors Calculation22dAssume 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 incorrectZ_F!3UzEffect of Taxes in D Arcy-Dyer Example Shareholders Interest>>BCorrect value of shareholders interest, net of tax, is C0  T0 = $121.41 - $20.96 = $100.45 D Arcy-Dyer calculation was C0  T0 = $121.41 - $16.05 - $105.36Z:>V\Effect of Taxes in D Arcy-Dyer Example Premium//To find the fair (equilibrium) premium, we solve for P0 that yields C0  T0 = S0 P0 = 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@ 6 ,`h`h`h``ZW*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 removedX Symmetrical TaxationVGovernment s net tax position is given by T0* = 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) is V0*(160 | 150) = C0  T0* = $121.41 - $8.86 = $112.552,,   Y Z!Symmetrical TaxationFormula 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  r9["'Fair Premium Under Symmetrical Taxation((P0 = Le-rt  put0(y | Y0, L) + tax (1  e-rt) Authors example assumptions imply P0 = $138.80 b`h`$  `  ! ,:\#Stochastic ClaimsTreat 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: b!  ,^$Stochastic Claims8E[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 jP    g%#Stochastic Claims Formula Example 1$$xSuppose f(l) is lognormal with l = 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  "  " N"*2"* "f k&#Stochastic Claims Formula Example 2Suppose f(l) is lognormal with l = 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 T "  " N"*"*K"P $l'Stochastic ClaimsD 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 2m)YStochastic Claims with Symmetrical Taxation Fair Premium without Underwriting Risk Charge&Z9y P0 = $138.22 for example we have been following (vs. $138.80 for fixed L = $150) Still no underwriting risk charge <z r.Summary of Indicated Premiums Under Various Scenarios Based on Author s Assumptions2Up*VStochastic Claims with Symmetrical Taxation Fair Premium with Underwriting Risk Charge&W9Formula (16) below is implied by equilibrium condition E(V0*(P0)] = S0 +   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)!PZ; ( ( ( 7      4Jq+ 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 approachZ6/~     !"#$%&'()*+,  ` 3ff` @@̙f|` 4-̙w^33f` fx3` N3` 33f̙` ̙̙f3` Ab3 Ab3f3ffH` f4fZX>?" dd@?oUd@nP pF@n`o n?" dd@   @@``PR   @ ` `p>>  ,(  , , 68 " `0  T Click to edit Master title style! 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WACEK H ` 0޽h ? 33___PPT10i.#B+D=' = @B +   8B(  8x 8 c $& ,0    8 c $w,    BH 8 0޽h ? r3y`f___PPT10i..P+D=' = @B +   <0(  <x < c $H,0   x < c $,p  H < 0޽h ? r3y`f___PPT10i..+D=' = @B +   @0(  @x @ c $,0   x @ c $螐,`   H @ 0޽h ? r3y`f___PPT10i..0Z&+D=' = @B +  @0(  x  c $,0P0   x  c $@,`P0   H  0޽h ? 3f___PPT10i..S+D=' = @B +  P 0(   x   c $H, `0   x   c $,`P0   H   0޽h ? 3f___PPT10i..~+D=' = @B +  `$*(  $x $ c $ؐ,0   r $ S ِ, `0  H $ 0޽h ? 3f___PPT10i..+D=' = @B +  p((  (x ( c $$ߐ,0    ( <\ ?"6@ NNN?N & Despite these . . . shortcomings, the D Arcy Dyer paper is still a useful springboard for discussing the OPT approach. z zH ( 0޽h ? 3f___PPT10i..:+D=' = @B +  <(  x  c $(,`p     S , ,`   H  0޽h ? 3f___PPT10i.O46+D=' = @B +   00(  0x 0 c $ ,Pp0   x 0 c $ ,`P0    H 0 0޽h ? 3f___PPT10i.. o+D=' = @B +   40(  4x 4 c $4 ,P0   x 4 c $ ,0P0P    H 4 0޽h ? 3f___PPT10i..R#+D=' = @B +   80(  8x 8 c $ ,`0   x 8 c $@ ,P0    H 8 0޽h ? 3f___PPT10i.1 M:+D=' = @B +   <0(  <x < c $ ,`0   x < c $ ,0P0P    H < 0޽h ? 3f___PPT10i.1pf+D=' = @B +$  $(  r  S  ,`   r  S  ,P0   H  0޽h ? fx380___PPT10.18  @0(  @x @ c $ ,`0   x @ c $ ,P0    H @ 0޽h ? 3f___PPT10i.107j^+D=' = @B +  @\0(  \x \ c $ ,0   x \ c $ ,0P0P    H \ 0޽h ? 3f___PPT10i.2+D=' = @B +  ph0(  hx h c $ ,p0   x h c $l ,0P0P    H h 0޽h ? 3f___PPT10i.2+D=' = @B +  l0(  lx l c $H ,0 0`   x l c $ ,`    H l 0޽h ? 3f___PPT10i.2׭+D=' = @B +  p*(  px p c $< ,P0   r p S  ,`0   H p 0޽h ? 3f___PPT10i.20 G+D=' = @B +  t*(  tx t c $ ,@0   r t S | ,` `p   H t 0޽h ? 3f___PPT10i.2pqo+D=' = @B +  x<(  x~ x s * , `0   ~ x s * ,P0    H x 0޽h ? 3f___PPT10i.2pqo+D=' = @B +  |0(  |x | c $* ,`0   x | c $* ,`    H | 0޽h ? 3f___PPT10i.2P`^e+D=' = @B +  *(  x  c $P0 , P0   r  S $1 , `   H  0޽h ? 3f___PPT10i.2+D=' = @B +9  PH(  x  c $p> ,P`0   x  c $B ,0P0b      B ?"6@ NNN?NH  0޽h ? 3f___PPT10i.2€!+D=' = @B +  0(  x  c $H ,`   x  c $I ,P0   H  0޽h ? 3f___PPT10i.2 K+D=' = @B +  0(  x  c $@U ,   x  c $V ,P0   H  0޽h ? 3f___PPT10i.2„9+D=' = @B +|  #(  x  c $\^ , 0   r  S 0_ , `     N`  ?"6@ NNN?N @  5.  H  0޽h ? 3f___PPT10i.i30$f+D=' = @B +   0(  x  c $m ,`   x  c $n ,P0   H  0޽h ? 3f___PPT10i.k3 m+D=' = @B +  0*(  x  c $x ,`   r  S 4t , `   H  0޽h ? 3f___PPT10i.l3+D=' = @B +  @0(  x  c $ ,`p   x  c $Ċ ,P0   H  0޽h ? 3f___PPT10i.l3P+D=' = @B +  P6(  ~  s *d ,   x  c $8 ,   H  0޽h ? 3f___PPT10i.l3P+D=' = @B +   `0(  x  c $$ ,`P   x  c $ ,P0   H  0޽h ? 3f___PPT10i.m3 +D=' = @B + ! p*(  x  c $m,0   r  S m,`   H  0޽h ? 3f___PPT10i.n3f+D=' = @B + " 0(  x  c $@ ,0p0   x  c $ ,P0   H  0޽h ? 3f___PPT10i.p30_+D=' = @B + #  (  x  c $ ,0   x  c $p ,P0     <  ?"6@ NNN?NPP  3. $  H4  ?"6@ NNN?N  PS02     <  ?"6@ NNN?NnF h `1  tax6 b\  H  ?"6@ NNN?NVa  r= P.V. of Losses less Default Option plus taxes on P.V. of Investment Income on initial surplus (grossed up) s sH  0޽h ? 3f___PPT10i.p3Ү+D=' = @B +'  $ > 6  (  x  c $ ,0   x  c $h ,l       B ?"6@ NNN?N77 0  :0e0eA D    ? A@  A5% 8c8c     ?,A)BCD|E||# "0e@  `    @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abP `@  D  dH  0޽h ? 3f___PPT10i.u38+D=' = @B +# % :2 (  x  c $ ,0   x  c $ ,P0     B ?"6@ NNN?N  B ?"6@ NNN?N   B ?"6@ NNN?N  s ,A  a??"?@x ad# 0  @0e0eA l    ? @  5% 8c8c     ?,A)BCD|E||# "0e@  `    @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab @t l  dF @G A    ` R  TA m? ?"6@ NNN?N@p W@  m  d  ND  ?"6@ NNN?NG A  :(8.9*)   H  ?"6@ NNN?NF@ 8(14) H  0޽h ? 3f___PPT10i.x30_+D=' = @B +t &  t(  x  c $ ,0   x  c $l ,P    0 NA /? ?,  /   0 NA 0? ?, )u 0  H  0޽h ? 3f80___PPT10.3qk ' P(  ~  s *" , `0   ~  s *" ,`P     0 TA 1? ?,  1   0 TA 2? ?, )u 2  H  0޽h ? 3f80___PPT10.3qk0 ( `0(  x  c $* ,p0   x  c $+ ,P0   H  0޽h ? 3f80___PPT10.3wد ) (  x  c $43 ,`   x  c $4 ,`    sF P   P    >A  n? ?"?P n  d   N6  ?"6@ NNN?NF E (15)$ H  0޽h ? 3f80___PPT10.3`Fi , !p`(  `r ` S tG ,    ` HH  ?"6@ NNN?N0i  O Fixed Stohastic Claims1 Claims1, 2 No Tax $136.44 Asymmetrical Tax $159.33 $158.89 Symmetrical Tax $138.80 $138.22v Z, ^ ` HT  ?"6@ NNN?N & s  <1 No underwriting risk charge included 2 Claims lognormal with  = 11%^H  ' "*,@H ` 0޽h ? fx380___PPT10.8(* * (  x  c $ ,`   x  c $ ,P0     B ?"6@ NNN?N48 @h 8@  s ,A 9??"?@h~x 9d  N  ?"6@ NNN?NF 6(16) H  0޽h ? 3f80___PPT10.3 1* + *(  x  c $lb ,p0   r  S c , `p   H  0޽h ? 3f80___PPT10.3v 0 (  X  C *     S Tu  0   TThe second paper I am presenting today is a discussion of Stephen D Arcy s and Michael Dyer s 1997 Proceedings paper entitled  Ratemaking: A Financial Economics Approach H  0޽h/. ? 3380___PPT10.6`- 0 x(  X  C *   x  S   0   The D Arcy-Dyer paper is viewed by the CAS Exam Committee as an important paper, and accordingly, it appears on the syllabus for Exam 9. One of the sections that students are advised will be fair game for direct testing is Section 8, which describes a ratemaking method based on Option Pricing Theory (OPT for short). My discussion focuses exclusively on Section 8.ppH  0޽h/. ? 3380___PPT10.&8 0 nf(  X  C *   f  S   0   D Arcy and Dyer based their summary of OPT ratemaking on the work of Neil Doherty and James Garven published in the Journal of Finance in 1986 and updated by Garven in the CAS Forum in 1988. The OPT approach stems from D&G s insight that since insurers invest their assets mainly in tradable securities on which it would be possible to price an option, option theory might be useful in valuing an insurance company and setting fair insurance prices.>\<9H  0޽h/. ? 3380___PPT10.&80R 0 b(  X  C *     S  , P,   d0In particular, D&G showed that the pretax value of an insurer is equivalent to a call option on its assets, where the strike price (or exercise price) is equal to the total amount of insurance claims. Since insurance claims are variable, this makes for an unusual call option, since normally an option strike price is fixed, but conceptually there is no reason why a call option can t have a variable strike price. D&G also showed that the government s income tax claim is a call option, provided no credits are allowed for losses. The appropriate rate level is the one that yields equality between beginning surplus and the value of the call option, net of tax.H  0޽h/. ? 3380___PPT10.'8Iov 0 (  X  C *     S  \ P   For the sake of illustration of the principles of the OPT approach, D Arcy and Dyer introduced some simplifying assumptions. I recap the important ones here. Policies are written for a one-year term at a common date Claims totaling a fixed (and known) amount, 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 entirely in taxable assets initially valued at Y0 x T  D lH  0޽h/. ? 3380___PPT10.(8ep= 0 jb(  X  C *   b  S   ,   The D Arcy-Dyer introduction to OPT ratemaking is a good springboard for discussion of that approach, but it contains a number of shortcomings. The purpose of my discussion is to point out those shortcomings, and to correct, clarify and extend the authors treatment of this ratemaking approach. I rework and extend the D Arcy-Dyer examples and expand the exposition to allow for (a) symmetrical taxation and (b) stochastic claims. Ultimately, I show that the OPT method is less exotic than it seems at first blush and that under realistic conditions it reduces to a fairly conventional ratemaking approach. eeH  0޽h/. ? 3380___PPT10.*8P;E  0   (  X  C *     S   ,   @What are the shortcomings in D Arcy-Dyer? Well, they make some basic mistakes with options. In particular, they wrongly describe the policyholders claim on the insurer as a call option. Then, they wrongly parameterize the call option representing the government s income tax claim. Second, their notation (which is largely borrowed from D&G, so perhaps D Arcy & Dyer should not be blamed for it) is cumbersome and inconsistent. I have sought to tidy up the notation in my discussion. Third, surprisingly for a paper on ratemaking, they give no example or formula for the calculation of the fair premium. Finally, while treating the claims amount as a fixed and known quantity, essentially as a loan, is helpful in illustrating the option ideas, D Arcy and Dyer never successfully get beyond that. Their brief attempt to describe D&G s treatment of stochastic claims is seriously flawed (and not even faithful to D&G!).}!HTH  0޽h/. ? 3380___PPT10.+8p(Z 0 j(  X  C *     S   0   l8I don t want to be too hard on D Arcy and Dyer. Despite these shortcomings, their paper is still a useful springboard for discussing the merits of the OPT approach. I m glad they included this approach in their discussion of ratemaking methods based on ideas in financial economics.H  0޽h/. ? 3380___PPT10.,8 wT` , 0   p (  X  C *     S  \ ,   r I mentioned that the D Arcy-Dyer notation is problematical. While I have retained it as much as possible I have refined and augmented it to some extent in order to enhance clarity. For example, in my discussion, my numerical subscripts have a consistent temporal meaning  0 means inception, 1 means expiry in 1 year. I have introduced lower case variable names, l and y, to designate random variables for the claim amount and invested assets at expiry, respectively. Capital L and Capital Y, represent specific values of these random variables. I have also introduced notation for puts and calls that I believe helps to keep the option parameters straight. For example, the call expression shown here denotes the expiry value (note the subscript 1) of a one-year European call option on y (which represents invested assets at expiry), given a  price of y=Y, and  exercise price of l = L. I am going to leave it at that. For more on my notation and the problems I see with the D Arcy-Dyer notation, see my discussion.Lo H  0޽h/. ? 3380___PPT10.,8Дw 0 4,@(  X  C *    ,  S ,   0    Let s move on to the OPT approach itself and start looking for the options. Let Y0 represent the beginning invested assets, comprising the beginning surplus, S0 and the premium receipts (net of expenses), P0 . Those invested assets will grow to some value, Y1, by the end of one year. Note that Y1 is the maximum amount the insurer has available to pay claims (which, remember, all are payable after 1 year). That means that if claims are less than Y1, policyholders receive L, and if claims are greater than Y1, they receive Y1. Effectively, claims are capped at Y1. This notation is the same as D Arcy-Dyer.hRM.4&;&-H  0޽h/. ? 3380___PPT10.-8c4X  0 Ph(  X  C *      S 3  0    jWe can describe the value of policyholders interest at expiry using the expression for H1 shown here. This is the D Arcy-Dyer formula (8.3), and it is correct. However, they describe H1 as equivalent to the expiration payoff to the owner of a European call option with an exercise price of L. I don t know what caused them to write that. I am 100% certain that D Arcy and Dyer know that statement is incorrect. The call option they describe has a payoff equal to the greater of Y1  L or zero.fY a ) H  0޽h/. ? 3380___PPT10.08P$C  0 ,$`(  X  C *    $  S dE  ,    NThat call option belongs to the shareholders not the policyholders. That is one of the key insights of Doherty & Garven and clearly stated in their work. To recap, the sale of insurance policies is (collectively) equivalent to a sale of the insurer s invested assets to policyholders in exchange for a call option given to the insurer s shareholders to reacquire the assets at a price of L, the total claims. If invested assets at expiry exceed the total claims, the insurer (on behalf of its shareholders) will exercise the option to reacquire the assets, resulting in a gain of Y1  L. If invested assets will not cover the claims in full, the option will not be exercised.2H _r0H  0޽h/. ? 3380___PPT10.08  0 F>p(  X  C *    >  S   0    HThe pre-tax shareholders interest at expiry, symbolized here by C1, can be summarized as shown in the call option notation I described earlier. The policyholders interest at expiry is equivalent to a long position in y and a short position in the call. Note that I deviate from the D Arcy-Dyer notation here. They use C1 to refer to the pre-tax shareholders interest at inception, whereas I use C1 to refer to its value at expiry. I do not know why the letters C and H were chosen to refer to shareholders interest and policyholders interest.t%B  M  H  0޽h/. ? 3380___PPT10.180(o;  0 (  X  C *      S f  0    HNeither D&G nor D Arcy & Dyer mention this in their papers but we can use put-call parity to derive an alternative and very useful characterization of policyholders interest. Put-call parity implies the policyholders interest at expiry can be expressed as the total claims L, less the payoff value of a European put option on invested assets, given such assets at expiry of Y1 and a strike price of L. The value of the put option at expiry is the expected value of the claims the policyholder won t recover due to the insolvency of the insurer.2%z H  0޽h/. ? 3380___PPT10.18Bʕ- 0 f^(  X  C 4    ^  S t  0    There is an important relationship between puts and calls known as put-call parity. The combination of a share of stock and a put option on the stock has the same value at expiry as a call option and a T-bill in the amount equal to the strike price of both options. In our example and notation where Y1 plays the role of the stock and L plays the role of the strike price, this can be represented symbolically as shown here.2/ {H  0޽h/. ? 3380___PPT10.18 0 (  X  C *      S   0    (OPT-based ratemaking indicates the premium rate should be reduced by the value of the put. Ignoring taxes for now, this implies P0 = H0, where H0 is the present value at inception of the policyholders interest, which in turn is the present value of claims less the value of the put at inception.f)    H  0޽h/. ? 3380___PPT10.28 0 <4(  X  C *    4  S    `,    pWe previously discussed the value of the pre-tax shareholders interest at expiry. Here we show the value of that interest at inception, expressed in three different, but equivalent ways. Note that D Arcy & Dyer use C1 to denote the shareholders interest at inception, which I think is confusing. We prefer to use C0 in order to make it clear this is a value measured at inception. Since y represents assets whose behavior meets B-S conditions (an assumption we described in one of the early slides), we can use the B-S formula to calculate the value of the call.B9cH  0޽h/. ? 3380___PPT10.38TF  0 V(  X  C *      S <  `P,    XD Arcy & Dyer provide an example of the calculation of the pre-tax shareholders interest. Suppose beginning surplus is $100M, net premium receipts are $160M, claims total a fixed $150M, the risk fee interest rate is 4%, insurance policies are for a one-year term and all claims are paid one year from inception, and the annualized volatility of the return on invested assets is 50%. Then the value of the pre-tax shareholders interest at inception, C0, can be calculated from the B-S formula as $121.41M. I have no quarrel with any of that. D Arcy & Dyer go on to observe that the $121.41M is surprisingly high, since adding the initial equity to the underwriting profit totals only $110. 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 and thus incorrectly value the default option. ,H  0޽h/. ? 3380___PPT10.38s   0 &(  X  C *      S  \ xh6    (It should be clear that it doesn t make sense to compare the $110 to the $121.41. The $110 is valued at the end of the period but without interest, while the second number is valued at inception and incorporates interest. Let s use the formula for the pre-tax shareholders interest that involves the put option. Remember, a few slides back, we showed three different but equivalent equations for C0. If the put option has a value of zero, then the shareholders interest is the difference between initial invested assets (which, remember, encompass the premium and the beginning surplus) and the present value of claims. That amount is $115.88 and it is that amount that should be compared to $121.41. The value of the default option is the value of the put, which is $5.53. In this case the value of the put option accrues as a windfall benefit to shareholders.@gH  0޽h/. ? 3380___PPT10.480ᛝ 0 (  X  C *      S  \ `\    FD&G observe that in equilibrium the P.V. of the shareholders interest at inception must equal the initial surplus. The value of the insurer must equal the value of the insurer. The previous example was not at equilibrium. In the pre-tax case equilibrium implies that C0 = S0. From that equilibrium condition it is easy to derive a formula for the indicated premium, which in our notation is the present value of claims less the inception value of the default put. D Arcy and Dyer allude to such a formula, but neither give one nor give an example. Since this is a ratemaking discussion, we think going through that calculation is important, so we will give an example.L  H  0޽h/. ? 3380___PPT10.48 0 ((  X  C *      S   0    *We calculate the indicated premium, P0, using the authors assumptions as $136.44. That amount is made up of $144.12 which is the present value of the $150 in claims, less the inception value of the put option, which is worth $7.68. Notice that at equilibrium the value of the put is a credit to policyholders rather than an extra profit margin for shareholders. 2o% IH  0޽h/. ? 3380___PPT10.68p  0 "(  X  C *     S   ` ,   $A unique feature of the OPT approach to ratemaking is that the claim default risk (i.e., the risk of insurer insolvency) is automatically factored into the insurance rate. Other ratemaking methods ignore the possibility that the insurer will go insolvent. As you saw in the example we just finished, the OPT method reduces the insurance rate by the value of the default option rather than allowing shareholders to reap a windfall by pursuing a risky investment or highly leveraged underwriting strategy.H  0޽h/. ? 3380___PPT10.68S 0 ,$  (   X   C *   $   S   `,   bWhile that feature has obvious appeal, it has some paradoxical implications. It means that insurers most at risk of insolvency are required to charge premiums that are less than the expected present value of their claim obligations! That only makes their demise even more likely, and cannot represent good public policy. It would be far better for regulators to correct an insurer s financial weakness or overly risky investment strategy (so that the insolvency risk is negligible) rather than to require a weak or poorly managed company to reduce its rates. Moreover, if regulation is effective, then the conventional ratemaking assumption that default risk is zero seems appropriate.@0H   0޽h/. ? 3380___PPT10.680C 0 ^V0(  X  C *   V  S ( \ `\   hYou might have been surprised that the value of the default option in our two examples was as high as it was. Those high values stems from D Arcy & Dyer s choice of an investment volatility assumption of 50%, which is extremely unrealistic. The standard deviation of the U.S. stock market returns from 1900  2000 was about 20%. A recent check of implied volatility in the S&P 500 option market yielded a figure just under 19%. It is difficult to construct an investment portfolio with volatility of 50%. Moreover, no insurer would invest 100% of its investable assets in such a portfolio. D Arcy and Dyer undoubtedly chose  = 50% in order to be able to illustrate material default risk&u@">, ?>H  0޽h/. ? 3380___PPT10.>8A 0 *"@(  X  C *   "  S X;  0   Let s rework the examples using a more realistic (but still aggressive) volatility assumption of 20%. Leave all other assumptions alone. Now the pre-tax shareholders interest is $115.90 (compared to $121.41 with volatility at 50%). The default put option is worth only 2 cents (compared to $5.53 with volatility at 50%).EEH  0޽h/. ? 3380___PPT10.>8^ 0 @8P(  X  C *   8  S PE \ P   Let s also recalculate the indicated fair premium using the same assumptions. The indicated premium is $144.07 (vs. $136.44 with volatility at 50%). The value of the default put is 5 cents, compared to $7.88 with volatility at 50%. For volatility less than 20% the value of the default option in these examples quickly goes to zero.PPH  0޽h/. ? 3380___PPT10.?8B  0   `( (  X  C *     S P \ `   *Now let s talk about income taxes. For reasons I do not know, D&G and hence D Arcy & Dyer, made the assumption that taxes apply only to positive income. If an insurer loses money, it simply pays no tax. It does not accrue credits against future taxes. That is an unrealistic assumption, but it does allow the government s tax claim to be characterized as a call option. To see this, first observe that income, I1, at the end of the period is equal to the growth in the value of invested assets plus underwriting income. By means of a little arithmetic manipulation, we see that I1 can be characterized as ending invested assets less initial surplus and claims. Now if we focus only on the set of positive income outcomes, i.e., it has the same payoff profile at expiry as a call option on invested assets with a strike price equal to S0 + L. So the value of the government tax claim at inception is equal to the tax rate times the value of the call option at inception. h0          H  0޽h/. ? 3380___PPT10.?8Sj 0  p (   X   C *      S w  0   LAssume the tax rate is 35% and applicable to all income. In the original example with a premium of $160, the government tax claim at inception, denoted as T0, is worth 35% of the call on invested assets where the initial invested assets total $260 and the strike price is $250. The B-S value of the government tax claim is $20.96.4M0 H   0޽h/. ? 3380___PPT10.@8p`h, 0 $<(  $X $ C *    $ S  \ `\   >lLet s look at the D Arcy-Dyer tax calculation. They use the same 35% tax rate, but they come up with a value for T0 of $16.05. The problem is the parameters they use in the B-S formula do not make sense. Their parameters and the $16.05 correspond to 35% of the call shown here. However, that implies the value of invested assets at inception is Y1  Y0 + P0, when by definition it must be Y0. They seem to have lost track of which asset the call option refers to. The upshot is that their use of the B-S formula and hence their calculation of T0 is incorrect.70s    !  H $ 0޽h/. ? 3380___PPT10.@8{Cj 0 (z(  (X ( C *    ( S 4  0   |HThe correct value of the shareholders interest in D Arcy & Dyer s main example is $121.41 less $20.96, or $100.45 rather than the $105.36 they give in their paper.H ( 0޽h/. ? 3380___PPT10.A8U$#` 0 ,p(  ,X , C *    , S Ğ \ ,   rTo find the equilibrium premium when taxes apply, we solve for the value of P0 that yields C0  T0 = S0, which gives us the formula shown here. The indicated premium is equal to the present value of losses less the inception value of the default option plus the inception value of the government tax claim. This implies a premium of $159.33 using the D Arcy-Dyer assumptions, not including any risk load to reflect underwriting risk. Note D Arcy-Dyer did not solve for the fair premium. 0M    H , 0޽h/. ? 3380___PPT10.A8`H6 0 0F(  0X 0 C *    0 S |  0   HD&G and D Arcy-Dyer ignored tax loss carry forwards and carry backs. Let s consider their impact, which is easy to do within the D Arcy-Dyer framework. Tax credits are equivalent to a long put owned by the shareholders (or a short put on the part of the government). Note that if an insurer becomes insolvent, it won t be in a position to use the tax-credit - that portion must be removed.0H 0 0޽h/. ? 3380___PPT10.A8PZRn  0 4~(  4X 4 C *    4 S   `,   (If taxation is symmetrical, by which I mean positive income generates tax debits and negative income generates tax credits, then the government s tax position at inception is given by the formula for T-nought-star given here. In the authors example, the government s total tax claim amounts to $20.96 for the call related to positive income less $14.03 for the put related to negative income plus $1.93 for the put related to insolvency, for a grand total of $8.86. The after-tax value of the shareholders interest is $112.55. 0EH 4 0޽h/. ? 3380___PPT10.A8p%! 0 og8(  8X 8 C *   g 8 S   0   Under symmetrical taxation, we can actually simplify the formula for the tax claim. We simply apply the tax rate to income valued at inception and adjust for the unusable tax credit related to insolvency.H 8 0޽h/. ? 3380___PPT10.B8+T" 0 <d(  <X < C *    < S ؝ \ `   fThe formula for fair premium under symmetrical taxation is given here. In words, the indicated premium is equal to the present value of losses, less the inception value of the default put, plus taxes on the p.v. of investment income on beginning surplus, this last item grossed up for tax. Using the authors assumptions, we calculate a fair premium of $138.80 in the symmetrical tax case.0H < 0޽h/. ? 3380___PPT10.B8`# 0  @(  @X @ C *     @ S  \ P   Until now claims have been treated as a fixed, known amount. It is time to consider the effect of treating the claims amount as a random variable, which we denote with a lower case l. We will begin with the authors asymmetrical taxation assumption and we will ignore the need for an underwriting risk load for now. Under a stochastic claims scenario, the expected value of the after-tax shareholders interest is given by the integral shown here, where it should be understood that C0 and T0 are functions of l.00  H @ 0޽h/. ? 3380___PPT10.C8x >x$ 0 D(  DX D C *    D S   \ `P,   VThis expected value expands to formula (14) shown on this slide. Compare formula (14) to D Arcy & Dyer s formula (8.9) for the stochastic claim case. Correcting the second term (the tax term) and restating it in our notation with i = 1, (8.9) becomes formula (8.9*) as shown. It is clear that formulas (14) and (8.9*) are different. In particular, (8.9*) does not actually reflect a stochastic claims assumption at all. It makes use of the expected value of claims as the strike price, which is no different from assuming the claims amount is fixed. ,0,H D 0޽h/. ? 3380___PPT10.C81% 0 JBH(  HX H C *   B H S   0   To illustrate the difference, let s assume the claims amount is lognormally distributed with a sigma of 11% and a mean of $150. All other assumptions are the same as the authors . Formula (14) indicates a premium of $158.89. The D Arcy-Dyer formula indicates a premium of $159.33. Our premium is less than D Arcy-Dyer s because the variability of claims increases the value of the default option, resulting in a bigger credit to the policyholders.0@ yH H 0޽h/. ? 3380___PPT10.D8೶^& 0   L(  LX L C *    L S  \ `P,   |Suppose the claim volatility is changed from 11% to 15%. All other assumptions, including the expected value of claims, remain the same. Our formula (14) now indicates a premium of $158.80 (slightly lower than before, reflecting a slightly higher default risk due to greater underwriting volatility). The D Arcy-Dyer formula yields the same premium of $159.33, since it depends only on the expected value of claims, not the whole distribution.0H L 0޽h/. ? 3380___PPT10.D8P"' 0 0P2(  PX P C *    P S ! , `P,   4D Arcy and Dyer claim their formula is based on D&G s work. If so, they misread Doherty & Garven. While their formula is superficially similar to D&G s, it is, in fact, quite different. D&G define a special call option with a random variable strike price and embedded underwriting risk charge that cannot be valued using Black-Scholes.T0TP0&4H P 0޽h/. ? 3380___PPT10.E8Y) 0 @T(  TX T C *    T S -  0   _Just touching on symmetrical taxation in the stochastic claims scenario, the indicated premium in that situation is given by formula (15). Given underwriting volatility of 11%, and all other assumptions unchanged, we find a premium of $138.22. This compares to $138.80 in the fixed claims case. There is no underwriting risk charge in this example.`0`H T 0޽h/. ? 3380___PPT10.E8#* 0 ldPX(  XX X C *   d X S :  0   zIf we add an underwriting risk charge,  we arrive at formula (16) shown here. Notice that the only option in this formula is the put representing the credit for insurer insolvency. If that put is zero (as it should be under an effective regulatory regime) formula (16) reduces to D Arcy-Dyer s DCF formula (6.1)! 4>0'",'H X 0޽h/. ? 3380___PPT10.E80 + 0   `\(  \X \ C *    \ S G , `,   XTime to wrap up this presentation. D Arcy & Dyer concluded that the OPT approach, while more complex than CAPM or DCF, had the advantage of avoiding the CAPM issue of estimating betas. I hope this presentation makes clear that if taxation is symmetrical and default risk is zero (both reasonably realistic assumptions) and underwriting risk is treated in a conventional way, then the OPT approach is the same as the DCF approach. The difference from DCF in D&G s original OPT framework is that they based their underwriting risk charge on the correlation between insurance claims and the stock market, which contrary to D Arcy & Dyer s assertion actually makes it subject to the same beta estimation problem as CAPM. The OPT approach is an interesting (if impractical) application of option theory, but it is less exotic than D&G and D Arcy-Dyer make it appear and bottom line is that there is little or no need to resort to this approach.0 H \ 0޽h/. ? 3380___PPT10.F8'. 0 OGh(  hX h C *   G h S l1  0   This is a summary of most of the indicated fair premiums we have calculated in the course of this discussion. 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DArcy and Michael A. Dyer           .. ?2 %Stephen P. DArcy and Michael A. Dyer           .-@Arial-. 42 EDISCUSSION BY MICHAEL G. WACEK       .. 42 DDISCUSSION BY MICHAEL G. WACEK       .-՜.+,0@      On-screen Show OdysseyReSho,,{ 3ArialTimes New Roman Wingdings Mistral AVSimSunCascadeMicrosoft Equation 3.0:CAS 2004 Spring Meeting Presentation of Proceedings PaperScope of Wacek Discussion"Origin of OPT Ratemaking Approach#Summary of OPT Ratemaking Approach4Simplifying Assumptions in DArcyDyer IllustrationSummary of Wacek DiscussionShortcomings of DArcy-DyerWacek Discussion(Wacek Discussion Note on Wacek NotationFinding the OptionsFinding the OptionsFinding the OptionsFinding the Options&Finding the Options Put Call ParityFinding the OptionsPricing Implications!Shareholders Interest (Pre-Tax)6Calculation of Pre-Tax Shareholders Interest Example6Calculation of Pre-Tax Shareholders Interest ExamplePricing at Equilibrium!Pricing at Equilibrium - Example$Unique Feature of OPTBased Pricing.Paradoxical Implications of OPTBased Pricing+Note on Assumptions in DArcy-Dyer ExampleGMore Realistic Investment Volatility Assumption Shareholders Interest8More Realistic Investment Volatility Assumption PremiumTaxes/Taxes in DArcy-Dyer Example Wacek Calculation2Taxes in DArcy-Dyer Example Authors Calculation>Effect of Taxes in DArcy-Dyer Example Shareholders Interest/Effect of Taxes in DArcy-Dyer Example Premium+Symmetrical Taxation of Profits and LossesSymmetrical TaxationSymmetrical Taxation(Fair Premium Under Symmetrical TaxationStochastic ClaimsStochastic Claims$Stochastic Claims Formula Example 1$Stochastic Claims Formula Example 2Stochastic ClaimsZStochastic Claims with Symmetrical Taxation Fair Premium without Underwriting Risk ChargeUSummary of Indicated Premiums Under Various Scenarios Based on Authors AssumptionsWStochastic Claims with Symmetrical Taxation Fair Premium with Underwriting Risk Charge Conclusions  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles,"_m BDuckworthBDuckworth  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSUVWXYZ[]^_`abcefghijkpRoot EntrydO)PicturesCurrent UserdSummaryInformation(TPowerPoint Document(DocumentSummaryInformation8\