A Universal Framework For Pricing Financial And Insurance Risks

Transcription

1 A Unversal Framework For Prcng Fnancal And Insurance Rsks Shaun S. Wang, Ph.D., FCAS, ASA SCOR Rensurance Company One Perce Place Itasca, Illnos PH: (630) E-mal: Abstract Ths paper presents a unversal framework for prcng fnancal and nsurance rsks. Examples are gven for prcng contngent payoffs, where the underlyng asset or lablty can be ether traded or not traded. The paper also outlnes an applcaton of the framework to prescrbe captal allocatons wthn nsurance companes, and to determne far value for nsurance labltes.

2 INTRODUCTION Currently there s a pressng need for a unversal framework for the determnaton of the far value of fnancal and nsurance rsks. In the nsurance ndustry, ths need s evdent n the Socety of Actuares Symposum on Far Value of Labltes, and n the Casualty Actuaral Socety s Rsk Premum Project and Task Force on Far Valung P/C Insurance Labltes. In the fnancal servces ndustry, ths pressng need s evdenced by the recent Basel Accords on regulatory rsk management that requre far value, analogous to market prces, to be appled to all assets or labltes, whether traded or not, on or off the balance sheet. In lght of all these current events, ths paper addresses a very tmely subject. The paper s comprsed of three parts, summarzed as follows: Part One: The Framework ntroduces a new transform and correlaton measure that extends CAPM to prce all knds of assets and labltes, havng any type of probablty dstrbuton, whether traded or underwrtten, n fnance or nsurance. Ths transform s just as easly appled to contngent payoffs that are co-monotone wth ther underlyng assets or labltes. In ts smplest form, the new transform reles on a parameter called the market prce of rsk, extendng a famlar concept n CAPM to rsks wth non-normal dstrbutons. The market prce of rsk can ether be appled to, or mpled from, a dstrbuton, n order to arrve at a rsk-adjusted prce for the underlyng rsk n queston. A market prce of rsk ncreases contnuously wth duraton, and s consstent at each horzon date between an underlyng and ts co-monotone contngent payoff. When returns for an underlyng asset have a normal dstrbuton, the new transform replcates the CAPM prce for that underlyng asset, and recovers the Black-Scholes prce for optons on that underlyng asset. Part Two: Examples of Prcng Contngent Payoffs llustrates applyng the new framework to prce call optons on traded stocks, and to prce weather dervatves. Part Three: Captal Allocaton and the Far Value of Labltes llustrates applyng the new framework to nsurance company captal allocatons, and to the determnaton of far value of nsurance labltes. In partcular, t addresses a challengng ssue concernng the long-term duraton of labltes. Also, the framework s equally applcable to prmary nsurance busness and excess-of-loss rensurance when calculatng far value of labltes. 2

3 Captal Asset Prcng Model PART ONE. THE FRAMEWORK CAPM s a set of predctons concernng equlbrum expected returns on assets. Classc CAPM assumes that all nvestors have the same one-perod horzon, and asset returns have multvarate normal dstrbutons. For a fxed tme horzon, let R and R M be the rateof-return for asset and the market portfolo M, respectvely. Classc CAPM asserts that E[ R ] = r + β { E[ RM ] r}, where r s the rsk-free rate-of-return and Cov[ R, RM ] β = 2 σ M s the beta of asset. Assumng that asset returns are normally dstrbuted and the tme horzon s one perod (e.g., one year), a key concept n fnancal economcs s the market prce of rsk: E[ R ] r λ =. σ In asset portfolo management, ths s also called the Sharpe Rato, after Wllam Sharpe. In terms of market prce of rsk, CAPM can be restated as follows: E[ R ] r Cov[ R, RM ] E[ RM ] r λ = = = ρ, M λm, σ σ σ M σ M where ρ s the lnear correlaton coeffcent between R and R M. In other words, the, M market prce of rsk for asset s drectly proportonal to the correlaton coeffcent between asset and the market portfolo M. CAPM provdes powerful nsght regardng the rsk-return relatonshp, where only systematc rsk deserves an extra rsk premum n an effcent market. However, CAPM and the concept of market prce of rsk were developed under the assumpton of multvarate normal dstrbutons for asset returns. CAPM has serous lmtatons when appled to nsurance prcng when loss dstrbutons are not normally dstrbuted. In the absence of an actve market for nsurance labltes, the underwrtng beta by lne of busness has been dffcult to estmate. Opton Prcng Theory Besdes CAPM, another major fnancal prcng paradgm s modern opton prcng theory, frst developed by Fscher Black and Myron Scholes (973). Some actuaral researchers have noted that the payoff functons of a European call opton and a stop-loss rensurance contract are smlar, and have proposed an optons prcng approach to prcng nsurance rsks. Unfortunately, the Black-Scholes formula only 3

4 apples to lognormal dstrbutons of market returns, whereas actuares work wth a large array of dstrbutonal forms. Furthermore, there are subtle dfferences between opton prcng and actuaral prcng (see Mldenhall, 999). One way to better apprecate the dfference between fnancal asset prcng and nsurance prcng, s to recognze the dfference n types of data avalable for prcng. Optons prcng s performed n a world of Q-measure, where the avalable data conssts of observed market prces for related fnancal assets. On the other hand, actuaral prcng s conducted n a world of P-measure, where the avalable data conssts of projected losses, whose amounts and lkelhood need to be converted to a far value prce (see Panjer et al, 998). Because of ths dfference, modern opton prcng s mostly concerned wth the mnmal cost of settng up a hedgng portfolo, whereas actuaral prcng s based on actuaral present value of costs, wth addtonal adjustments for correlaton rsk, parameter uncertanty and cost of captal. A Unversal Prcng Method Consder a fnancal asset or lablty over a tme horzon [0,T]. Let X=X T denote ts future value at tme t=t, wth a cumulatve dstrbuton functon F(x)=Pr{X x}. In Wang (2000), the author proposed a unversal prcng method based on the followng transform: F *( x) = Φ[ Φ ( F( x)) + λ], () where Φ s the standard normal cumulatve dstrbuton. The key parameter λ s called the market prce of rsk, reflectng the level of systematc rsk. The transform () s now better known as the Wang Transform among fnancal engneers and rsk managers. The Wang Transform was partly nspred by the work of several promnent actuaral researchers, ncludng Gary Venter (99, 998) and Robert Butsc (999). For a gven asset X wth F(x), the Wang Transform wll produce a rsk-adjusted cumulatve probablty dstrbuton F*(x). The mean value under F*(x), denoted by E*[X], wll defne a rsk-adjusted far value of X at tme T, whch can be further dscounted to tme zero, usng the rsk-free nterest rate. The Wang Transform s farly easy to numercally compute. Many software packages have both Φ and Φ as bult-n functons. In Mcrosoft Excel, Φ(y) can be evaluated by NORMDIST(y,0,,) and Φ (y) can be evaluated by NORMINV(y,0,). One fortunate property of the Wang Transform s that normal and lognormal dstrbutons are preserved: If F has a Normal(µ,σ 2 ) dstrbuton, F* s also a normal dstrbuton wth µ* = µ λσ and σ* = σ. If F has a lognormal(µ,σ 2 ) dstrbuton such that ln(x) ~ Normal(µ,σ 2 ), F* s 4

5 another lognormal dstrbuton wth µ* = µ λσ and σ* = σ. Stock prces are often modeled by lognormal dstrbutons, whch mples that stock returns are modeled by normal dstrbutons. Equvalent results can be obtaned by applyng the Wang Transform ether to the stock prce dstrbuton, or, to the stock return dstrbuton. Consder an asset on a one-perod tme horzon. Assume that the return R for asset has a normal dstrbuton wth a standard devaton of σ. Applyng the Wang Transform to the dstrbuton of R we get a rsk-adjusted rate-of-return: E *[ R ] = E[ R ] λσ. In a compettve market, the rsk-adjusted return for all assets should be equal to the rskfree rate, r. Therefore we can nfer that λ=(e[r ] r)/ σ, whch s exactly the same as the market prce of rsk n classc CAPM. Wth λ beng the market prce of rsk for an asset, the Wang Transform replcates the classc CAPM. Unfed Treatment of Assets & Labltes A lablty X can be vewed as a negatve asset Y= X, and vce versa. Mathematcally, f a lablty has a market prce of rsk λ, when treated as a negatve asset, the market prce of rsk wll be λ. That s, the market prce of rsk wll have the same value but opposte sgns, dependng upon whether a rsk vehcle s treated as an asset or lablty. For a lablty X, the Wang Transform has an equvalent representaton. S * ( x) = Φ[ Φ ( S( x)) + λ], (2) where S(x)= F(x). If a lablty has a Normal(µ,σ 2 ) dstrbuton, the Wang Transform wll produce another normal dstrbuton wth µ* = µ+λσ and σ* = σ. Thus, for a lablty wth a normal dstrbuton, the Wang Transform recovers the tradtonal standard-devaton loadng prncple, wth the parameter λ beng the constant multpler. A New Measures of Correlaton Accordng to CAPM, the market prce of rsk λ should reflect the correlaton of an asset wth the overall market portfolo. When we generalze the concept of market prce of rsk to assets and labltes wth non-normal dstrbutons, the Pearson lnear correlaton coeffcent becomes an nadequate measure of correlaton. Examples can be constructed such that a determnstc relatonshp has a Pearson correlaton coeffcent close to zero. Such an example was provded n Wang (998): Consder the case where X lognormal(0,) and Y=X σ. Despte ths determnstc relatonshp, the lnear correlaton coeffcent between X and Y approaches zero as σ ncreases to nfnty. That s, ρ X,Y! 0 as σ ncreases. 5

6 Ths also mples that correlaton should not be estmated by runnng lnear regresson, unless all of the varables have normal dstrbutons. Now we show a new way to extend the Pearson correlaton coeffcent to varables wth non-normal dstrbutons. For any par of varables {X, Y} wth dstrbutons F X and F Y, we transform them nto standard normal varables : U=Φ [F X (X)], and V=Φ [F Y (Y)]. We next defne a new measure of correlaton between {X, Y} as the Pearson lnear correlaton coeffcent between these transformed standard normal varables {U, V}: ρ * Cov( U, V ) X, Y = = Cov( U, V ) σ ( U ) σ ( V ). Now, let us reconsder the case where X lognormal(0,) and Y=X σ. Consstent wth ths determnstc relatonshp, ths new measure of correlaton between X and Y s always. * That s, ρ for all σ values. X, Y = Usng our new measure of correlaton we may extend classc CAPM as follows: λ = ρ *, M λm, where λ and λ M are the respectve market prces of rsk n the Wang Transform, wthout assumng normalty. Prcng of Contngent Payoffs For an underlyng rsk X and a functon h, we say that Y = h(x) s a dervatve (or contngent payoff) of X, snce the payoff of Y s a functon of the outcome of X. If the functon h s monotone, we say that Y s a co-monotone dervatve of X. For example, a European call opton s a co-monotone dervatve of the underlyng asset; n (re)nsurance, an excess layer s a co-monotone dervatve of the ground-up rsk. Theoretcally, the underlyng rsk X and ts co-monotone dervatve Y should have the same level of systematc rsk, λ, smply because that they have the same correlaton (as shown by usng our new measure of correlaton) wth the market portfolo. In prcng a contngent payoff Y = h(x), there are two ways of applyng the Wang Transform. Method I: Apply the Wang Transform to the dstrbuton F X of the underlyng rsk X. Then derve a rsk-adjusted dstrbuton F from * Y F usng Y* = h(x*). Method II: Frst derve ts own dstrbuton F Y for Y = h(x). Then apply the Wang Transform to F Y drectly, usng the same λ as n Method I. Mathematcally t can be shown that these two methods are equvalent. Ths mportant result valdates usng the Wang Transform for rsk-neutral valuatons of contngent payoffs. * X 6

7 Impled λ and the Effect of Duraton For a traded asset, the market prce of rsk λ can be estmated from observed market data. We shall now take a closer look at the mpled market prce of rsk and how t vares wth the tme horzon under consderaton. Consder a contnuous tme model where asset prces are assumed to follow a geometrc Brownan moton (GBM). Consder an ndvdual stock, or a stock ndex,. The asset prce X (t) satsfes the followng stochastc dfferental equaton: dx ( t) X ( t) = µ dt + σ dw, (4) where dw s a random varable drawn from a normal dstrbuton wth mean equal to zero and varance equal to dt. In equaton (4), µ s the expected rate of return for the asset, and σ s the volatlty of the asset return. Let X (0) be the current asset prce at tme zero. For any future tme T, the prospectve stock prce X (T) as defned n equaton (4) has a lognormal dstrbuton (see Hull, 997, p. 229): 2 ( T 0.5σ T, T ) X ( T ) / X (0) ~ lognormal µ σ. (5) Next we apply the Wang Transform to the dstrbuton of X (T) n (5) and we get X * 2 ( µ T λσ T 0.5σ T, T ) ( T ) / X (0) ~ lognormal σ. For any fxed future tme T, a no arbtrage condton (or smply, the market value concept) mples that the rsk-adjusted future asset prce, when dscounted by the rsk-free rate, must equal the current market prce. In ths contnuous-tme model, the rsk-free rate r needs to be compounded contnuously. As a result, we have an mpled parameter value: ( µ r ) λ = λ ( T ) = T = T λ (). (6) σ The mpled λ n (6) concdes wth the market prce of rsk of asset as defned n Hull (997, p. 290). Ths mpled λ s also consstent wth Robert Merton s nter-temporal, contnuous-tme CAPM (see Merton, 973). It s nterestng to note that the market prce of rsk λ ncreases as the tme horzon lengthens. Ths makes ntutve sense snce the longer the tme horzon, the greater the exposure to unforeseen changes n the overall market envronment. Ths nterestng result has applcatons n prcng long-taled nsurance where losses are not reported or settled untl many years after the polcy perod expres. 7

8 If the evoluton of ncurred loss resembles geometrc Brownan moton, the parameter λ should be proportonal to the square root of the tme perod from polcy ncepton to the date of loss settlement. The relatonshp (6) between λ and duraton T s very useful n calculatng far value of nsurance labltes (ncludng loss reserve dscountng) and optmzng captal allocatons wthn an nsurance company. Applyng the Wang Transform wth the λ n equaton (6), asset has a rsk-adjusted dstrbuton * 2 X ( T ) / X (0) ~ logormal( rt 0.5σ T, σ T ), where both the market prce of rsk λ, and the expected stock return µ have dropped out from the transformed dstrbuton F*(x). Recovery of the Black-Scholes Formula A European call opton on an underlyng stock (or stock ndex) wth a strke prce K and exercse date T s defned by the followng payoff functon Y= Call( K) = X 0, ( T ) K, when X ( T ) K, when X ( T ) > K. Beng a non-decreasng functon of the underlyng stock prce, the opton payoff, Call(K), s co-monotone wth the termnal stock prce, X (T); thus t has the same level of systematc rsk as the underlyng stock. Therefore, the same λ as n equaton (6) should be used to prce the opton Call(K). In other words, the prce of a European call opton s the expected payoff under the transformed (rsk-neutral) stock prce dstrbuton F*(x), where the expected stock return µ s replaced by the rsk-free rate r. The resultng opton prce s exactly the same as the Black-Scholes formula. There s an analogy between an unlmted stop-loss cover wth retenton K, and a European call-opton wth strke prce K. Both are co-monotone dervatves of the underlyng (loss or asset) varable. By applyng the Wang Transform to the stop-loss varable, we get a stop-loss premum as the expected stop-loss value under the transformed ground-up loss dstrbuton. Lkewse, the prce for a European call opton can be evaluated as the expected opton payoff under the transformed (rsk-neutral) dstrbuton for the underlyng stock prce, where the expected stock-return µ does not appear n the optons prcng model. Thus the Wang Transform adds a new perspectve to the well-known rsk-neutral valuaton methodology of optons (see Cox and Ross, 976). Equlbrum and Replcaton Perspectves Recall that CAPM provdes an equlbrum perspectve of asset prces n lght of ts correlaton wth the market portfolo. Wth the equlbrum perspectve, an opton s co- 8

9 monotone wth the underlyng asset, thus have the same market prce of rsk. Usng the same market prce of rsk, the Wang Transform produces an opton prce as the expected opton payoff under a transformed rsk-neutral asset dstrbuton where the expected rate-of-return s equal to the rsk-free rate-of-return. On the other hand, modern fnance presents the Black-Scholes formula va a replcaton perspectve. The replcaton approach reles on the ablty to create a contnuous rskless hedge. If asset prces change n small amounts, t s possble to smultaneously buy an opton and sell a quantty of the underlyng asset, so that the combned portfolo has no rsk. Note that the nstantaneous hedge s possble only because the opton s a comonotone dervatve of the underlyng asset. Emanuel Derman (996), who had worked closely wth Fscher Black, commented that Deep nsde, Fscher seemed to rely on the equlbrum approach of the captal asset prcng model as the source for hs ntuton about optons prcng. I beleve ths s the way the Black-Scholes equaton was orgnally derved, although the frst dervaton of the optons prcng formula n the Black-Scholes artcle s based on valuaton by replcaton. The Wang Transform takes the equlbrum perspectve of CAPM, and yet s able to reproduce the Black-Scholes prce for optons on underlyng assets wth lognormal dstrbutons. The Wang Transform thus formalzes an ntrnsc relatonshp between CAPM and the Black-Scholes formula, along the lnes of Fscher Black s reported nsghts. Adjust for Parameter Uncertanty Before applyng the Wang Transform, some adjustments for parameter uncertanty must be ncorporated. Ths can be done explctly on a case-by-case bass usng Bayesan-type pror mxng parameters. One non-bayesan method of adjustng for parameter uncertanty uses a postve parameter b, and modfes the objectve probabltes as follows: F * ( x) = Φ b Φ ( F( x)). (7) [ ] For adjustment (7), the mean value of the dstrbuton may not be preserved for nonsymmetrc dstrbutons. The composte of transforms (7) and () ncorporates both systematc rsk and parameter uncertanty, and produces a more generalzed verson of the Wang Transform: [ b Φ ( F( x)) + λ] F * ( x) = Φ, (8) When a marketplace s ratonal and ambguty-averse, the value of b should be greater than for assets, and less than for labltes. The adjustment factor b may be tweaked for the value of F(x), wth further adjustments possble for extremes, lke for way out-ofthe-money contngent clams, or way-beyond-a-horzon-date clam settlements, where 9

10 markets are llqud, benchmark data sparse, negotatons dffcult, and the cost of keepng captal reserves s hgh. For a lognormal dstrbuton, the transform (8) amplfes the volatlty parameter after a locaton parameter shft, along lnes suggested by Butsc (999). Gary Venter, n a prvate communcaton, has also reported to the author that John Major had ftted transform (8) to emprcally observed property CAT treaty prces. Extrapolaton of Tal Probabltes Usng transform (8) to adjust for parameter uncertanty does not always work n all stuatons. For nstance, an nsurance contract mght offer a $00M lmt, wth no data ndcatng hstorcal losses greater than $50M, even after trendng. In such a case, tal probabltes for losses greater than 50M need to be extrapolated from the estmated probabltes for losses below 50M. The Extreme Value Theory may be a useful technque for the extrapolaton (see Embrechts, et al, 997). The Wang Transform can be appled to the extrapolated tal probabltes. Prcng Versus Portfolo Management Prcng s a result of collectve market behavors. The market prce of rsk reflects not only the correlaton wth the overall market portfolo, but also an average cost of captal commtment for a gven ndustry. Portfolo management nvolves an actve selecton (deleton) of the most (least) proftable busness n relaton to the ncremental rsk to the exstng portfolo. Fortunately, the Wang Transform s as useful n portfolo management as t s n prcng. For example, the Wang Transform s more precse than a standard devaton as an underlyng axs for plottng an effcent fronter, or for calculatng an optmal portfolo. The Wang Transform can also be used by a portfolo manager to dentfy good/bad rsks by comparng ther respectvely mpled lambdas wth ther own benchmarks for rsk and return. 0

11 PART TWO. EXAMPLES OF PRICING CONTINGENT PAYOFFS A contngent payoff s a contractual agreement between counter-partes, whose payment trgger and amount are determned by observed outcomes of the underlyng varable. A contngent payoff s a more general type of fnancal nstrument than an opton, snce the underlyng varable can nclude non-traded assets or labltes, statstcal ndces, or even physcal events. Most underlyng varables do not follow a lognormal dstrbuton, makng the Black-Scholes formula napproprate for benchmark prcng. In contrast, the Wang Transform s applcable to any dstrbutonal form, and can be used as a unversal method for prcng all knds of contngent payoffs. An Example of Prcng Optons Asset prcng s based on antcpated future prce movements. Hstorcal returns may or may not be a good ndcator of future prce movement. For llustraton purposes, we assume the avalablty of a robust stock prce projecton model utlzng hstorcal prce data and other avalable nformaton. Such a stock prce projecton may be based on a GARCH model wth due consderatons to mean-reverson and other economc factors. For our llustraton, such a model has produced the below sample of outcomes wth equal probablty weghts. The underlyng s a stock ndex wth a current prce of $ Our model has produced 20 outcomes (partally based on 5-year hstory of quarterly returns): 28.7, 309.5, , , 58.84, , 45.00, , , 89.37, , , 358.4, 49.09, 550.2, , , , , The stock ndex return has a mean of 4.08% and a standard devaton of 8.07%. Assumng that the 3-month rsk-free rate s.5%. The emprcal Sharpe Rato for the 3- month tme-horzon s 0.32=(4.08%.5%)/8.07%. We want to prce a 3-month European call opton on ths stock wth a strke prce of $375. We wll apply the Wang Transform to the sample stock ndex dstrbuton, wthout assumng a lognormal dstrbuton. Detaled steps are shown n Table wth further explanatons below: Column. Sort the sample of projected outcomes n ascendng order, from worst to best. In ths example, the underlyng s consdered to be an asset, so the worst outcome here s the lowest number, and best s the hghest number. Column 2. Assgn objectve probabltes f(x) to each projected outcome x. In ths example, the objectve probablty s based on 20 equally weghted observatons, so each assgned probablty s /20. Column 3. Add up the ndvdual objectve probabltes f(x) to yeld a seres of cumulatve probabltes F(x). Column 4. Usng the emprcal Sharpe Rato (0.32) as a starter lambda value, apply

12 the Wang Transform to the cumulatve probabltes F(x), to yeld F*(x). Recall that the Sharpe Rato assumes a normal dstrbuton, so we may need to tweak our lambda value later, to account for a possble non-normal dstrbuton. Column 5. De-cumulate the transformed probabltes F*(x) to recover f*(x). Evaluate the mean value of ths projected sample usng probablty weghts f*(x). If the dscounted mean value s greater (or less) than the current market value, tweak upward (or downward) the lambda value. Repeat the process of columns 4-5 untl the dscounted mean value matches the current market prce. In ths example, the starter lambda value of has been tweaked to 0.342, n order to match the current prce of $ The values of F*(x) and f*(x) shown n columns 4 and 5 are thus the fnal transformed probabltes usng λ= Now we proceed to columns 6-8. Column 6. For a gven strke prce ($375 n ths example), calculate the opton payoff for each projected future prce for the stock. That s, y(x)=max(x 375, 0). Column 7. Calculate the expected payoff by multplyng the values of the opton payoff functon n Column 6 by the objectve probabltes n Column 2. In ths example, the resultng expected payoff s $4.53 before dscountng, and $40.93 after dscountng. Column 8. Calculate the rsk-adjusted payoff usng the transformed dstrbuton. We do that by multplyng Column (5) by Column (6). The resultng opton prce s $25.35 before dscountng, and $24.98 after dscountng. 2

13 Table. Prcng of Call-Opton Usng the Wang Transform (λ=0.342) () (2) (3) (4) (5) (6) (7) (8) Sorted Objectve Transformed Contngent Weghted Rsk Sample Probablty Probablty Payoff Value Adjusted x f(x) F(x) F*(x) f*(x) y(x) f(x) y(x) f*(x) y(x), , , , , , , , , , , , , , , , , , , , Values Expected,380.5, Dscounted,359.60, Here are some related comments: The market prce of rsk, as calculated by (E[R]-r)/σ, s precse only when the underlyng asset has a normal dstrbuton. The Wang Transform, on the other hand, can terate a precse market prce of rsk for underlyng assets or labltes wth any type of dstrbuton. The Wang Transform can be used to allevate some of the volatlty smles n observed opton prces. Ths s because the Wang Transform can automatcally ncorporate any devatons (lke skewness and kurtoss) from normalty n the asset return dstrbuton. Wth the Wang Transform, we can take advantage of a good prce projecton model ncorporatng stochastc volatltes for underlyng asset. 3

14 An Example of Prcng Weather Dervatves For most weather dervatves, a payoff s contngent upon the number of observed Heatng-Degree-Days (HDD) for the wnter months, or Coolng-Degree-Days (CDD) for the summer months, multpled by some notonal amount. The underlyng varables of weather dervatves, namely HDDs and CDDs, are not traded assets by themselves. Ths s n contrast to equty dervatves, where the underlyng stock s usually a traded asset. Therefore, to prce weather dervatves, an equlbrum approach makes more sense than a replcaton approach. In wnter months, extreme cold weather drves up the cost for heatng. The recent energy crss has boosted the demand for call optons on HDDs, to hedge rsng heatng costs. The wrters of such optons need to set asde captal to fund potental payouts. We gve an example of usng the Wang Transform to prce weather dervatves. Here we use Chcago Mercantle Exchange Weather Data --- Monthly Aggregate from //979 to //200. Table 2 s the aggregate HDDs for months of December observed at the Chcago O Hare Staton. Note that there are a total of 22 observatons wth a mean of 54.7 and a standard devaton of Table 2. Monthly Aggregate Data for Chcago O Hare Staton, Date Dec-79 Dec-80 Dec-8 Dec-82 Dec-83 Dec-84 Dec-85 Dec-86 HDD Date Dec-87 Dec-88 Dec-89 Dec-90 Dec-9 Dec-92 Dec-93 Dec-94 HDD Date Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00 HDD

15 Table 3. Prcng Weather Dervatves Usng the Wang Transform (λ= 0.25) () (2) (3) (4) (5) (6) (7) (8) Sorted Objectve Transformed Contngent Weghted Rsk Sample Probablty Probablty Payoff Value Adjusted x f(x) F(x) F*(x) f*(x) y(x) y(x) f(x) y(x) f*(x) Total Exp.Value Prce For llustraton, the call opton on Dec-200-HDDs s assumed to have a strke prce of 350. Thus, the payoff functon can be expressed mathematcally as max(hdd 350, 0). In order to apply the Wang Transform, we frst sort the annual December HDDs n an ascendng order and assgn objectve probabltes. Here we use hstorcal data wthout adjustng for on-gong trends or cycles of weather condtons. In real lfe applcatons, of course, such trends and cycles can be consdered. The key to the applcaton of the Wang Transform bols down to the selecton of the lambda value. Ths lambda value should be decded from observed market prces wherever possble. In ths case, the lambda value s not known, and the table of outcomes only reflects HDDs as a denomnatonal unt of value. No face value or notonal amount has been attached to HDDs as yet. 5

16 Table 3 shows the applcaton of the Wang Transform. For the strke level of 350 Dec- 200-HDDs, the call opton has an expected payoff of Dec-200-HDDs, usng objectve probabltes. However, the far value of the opton s Dec-200-HDDs, usng the transformed dstrbuton. We only need to apply a common notonal amount, denomnated n a common currency, and a rsk-free rate for that currency, to arrve at a dscounted cash prce for ths opton. Applyng a face value of $ for each HDD, we can apply the same lambda of 0.25, so that call optons wth dfferent strke prces can be evaluated and compared (see Table 4). Table 4. Prce of Optons at Varous Strke Prces (λ= 0.25) Strke Exp. Payoff $ $ $ $ $.50 $ 4. Prce $ 68.2 $ $ $ $ 7.8 $ 6.59 Loadng 43% 43% 44% 45% 49% 60% Hstorcal data may not guarantee an accurate ndcaton of future weather condtons, gven the rapd envronmental changes. There are many ways of adjustng for parameter uncertanty. Here we consder one adjustment usng transform (8) wth a constant multpler b=0.95. To somewhat offset the extra loadng, we adjusted the lambda value downward to As shown n Table 5, wth adjustment for parameter uncertanty, the relatve loadng ncreases faster wth strke prces. Table 5. Prce of Optons wth adjustment for parameter uncertanty (λ= 0.20, b=0.95) Strke Exp. Payoff $ $ $ $ $.50 $ 4. Prce $ $ $ $ $ 7.55 $ 6.93 Loadng 43% 43% 45% 47% 53% 69% Comments: The lognormal assumpton s absolutely not needed here. The methodology s analogous to the Black-Scholes valuaton for optons. 6

17 PART THREE. CAPITAL ALLOCATION & FAIR VALUE OF LIABILITIES Next we dscuss the applcaton of the Wang Transform to nsurance captal allocaton and to the calculaton of the far value of labltes. Consder an nsurance company wrtng multple lnes of busness. Assume that we already know the overall economc captal for the company, or alternatvely, we have derved a total requred economc captal for the company based on ndustry benchmarks. Our goal s to allocate the cost of captal to dfferent lnes-of-busness and ndvdual contracts. Gven the long-taled nature of nsurance payment patterns, nsurers are requred to contnuously hold captal to support the reserve labltes. The underlyng ssue s how to approprately reflect the duraton of nsurance labltes. Some actuares suggest that captal needs to be commtted each year n proporton to all remanng unpad losses, wthout consderaton of the dversfcaton effect among development years. Others argue that the dversfcaton beneft between development years should be consdered. Some even go to the extreme that only a one-tme allocaton s needed n the frst year to account for the uncertantes assocated wth the present value of reserves. Dramatcally dfferent mplcatons of these vewponts present a challengng ssue assocated wth nsurance captal allocaton. The rest of the paper s devoted to tacklng ths ssue usng the prcng framework n Part One. Avalable Data We frst consder ground-up or prmary busness only. We shall use the followng data: Based on hstorcal accdent-year ultmate loss ratos, we have estmates of the loss rato volatlty for each lnes of busness, denoted by σ AY. We have estmates of the loss payment pattern for each lne of busness, wth an average duraton, denoted by D GU. Let R(t) be the porton of losses remanng unpad by tme t. We have D GU = R( t) dt. 0 Assumptons for the Evoluton of Losses. The best-estmate of remanng unpad losses evolves wth the passage of tme as more nformaton becomes avalable. Durng each tme perod (e.g. one-year), the revsed estmates of loss reserve may go up or go down, wth a random nature. 2. There are two opposng arguments regardng the relatve uncertanty of the remanng reserve: (a) t should ncrease wth tme as more rsky clams are settled later; (b) t should decrease wth tme as more nformaton becomes avalable. Here we assume that the relatve uncertanty (coeffcent of varaton) of remanng reserves remans constant over tme. See Phlbrck (994) for further dscusson of ths ssue. 3. Based on the above consderatons, we assume a geometrc Brownan moton process for the loss reserve evoluton over tme. Assumng that the nstantaneous per annum volatlty s a constant σ, we have 7

18 2 AY = 2 2 σ σ R( t) dt = σ D GU. 0 Thus we can estmate the per-year volatlty as σ AY σ =. DGU In practce, the geometrc Brownan moton assumpton can be further refned to reflect more closely the true underlyng process. The nstantaneous volatlty σ(t) may change wth tme. For property lnes of busness, σ(t) may be hgher for the prcng rsk (0<t<) than the reservng rsk (t>), as new nformaton wll emerge durng the contract perod regardng catastrophe actvtes. For casualty lnes, σ(t) may be hgher for IBNR reserves than that for case reserves. In the case of changng σ(t), an average per annum volatlty can be calculated by σ ( t) R( t) dt 0 σ =. R( t) dt 0 Although the mentoned refnements can be ncorporated n the calbraton of the nsurance company captal allocaton, here we wll restrct ourselves to the geometrc Brownan moton assumpton wth σ(t)= σ. Measure of Rsk We defne a rsk-measure to approxmate the cost of captal commtment, based on the followng assumptons: (a) The cost of captal per-year s proportonal to the underlyng per-year volatlty σ. (b) Accordng to the mult-perod CAPM, the magntude of systematc rsk ncreases wth the tme horzon. Let parameter λ be the per-year systematc rsk. The multperod CAPM says that the systematc rsk for tme horzon T s: λ T = λ T. Intutvely ths makes sense. For lablty nsurance, the longer the duraton, the hgher the uncertanty, especally wth respect to judcal changes and court rulngs. (c) The cost of captal s proportonal to the systematc rsk for the underlyng busness. Fnancal economsts may thnk of systematc rsk as the correlaton rsk relatve to the market portfolo. Under fnancal theory, nsurance nsolvency can be automatcally reflected n the market nsurance prces, the requred captal would be much less than that by ratng agences and regulators. However, n the presence of transacton costs, the systematc rsk should reflect not only the correlaton rsk, but also the true cost of dong busness ncludng the cost of captal. As a result, we assume that most lnes of busness have a smlar level of systematc rsk, namely, λ, except for property catastrophe and mass-torts n whch hgher λ values should be used. For each $ of expected loss for a lne of busness wth per-annum volatlty σ and average duraton of D GU, the total rsk measure for ground-up nsurance coverage s: 8

19 where λ GU λ = λ σ AY σ DGU = λ DGU = λ σ AY DGU = λgu σ AY, DGU D GU. Ths relaton can also be verfed usng a steady-state model. Prcng Ground-up Insurance Contracts For a gven lne of busness, to calculate the nsurance premum for each $ ground-up expected loss, we wll do the followng: (). Calculate the dscount factor PV GU (): Use market rsk-free nterest rate and the ground-up loss payment pattern. (2). Apply rsk loadng to derve a pure premum: PVGU ( ) { + λσ D GU }. The factor λ should be calbrated from total portfolo re-balancng based on a target return-on-equty (TROE). In other words, for the aggregate nsurance portfolo, the rato of the total rsk load plus nvestment return to the total economc captal should produce a target return-on-equty. The total allocated captal over the lfetme of ths $ lablty s λ σ D GU ( + r) /( TROE r), For year j, the allocated captal s λ σ P j ( + r) /( TROE r), where P j s the expected percentage of losses to be pad wthn year j. (3). Load for expenses: suppose the total expense factor s θ, we can load the pure premum by a factor of /( θ). (4) Knowng the amount of allocated captal, we can calculate the actual ROE for any gven quoted premum rate. A remark on the relaton to the Wang Transform: Assume that ground-up accdent-year loss rato follows a Brownan moton process wth a total volatlty σ AY. Formula (4) s an approxmaton to the resultng premum usng the Wang Transform wth λ GU = λ DGU. Thus, for ground-up busness, our rsk load (and captal allocaton) methodology s shown to be an approxmate result of the Wang Transform. Prcng Aggregate Stop-Loss Insurance Contracts The Wang Transform can be appled to prcng aggregate stop-loss layers, as a drect analogy to the Black-Scholes formula for prcng optons. Aggregate covers are usually appled on a mult-lne bass. One needs to blend the volatlty and payment patterns by lne of busness, and take nto consderaton that aggregate stop-loss contracts may be assocated wth hgher parameter uncertanty and moral hazards. Ths may mply that a hgher λ value and a large parameter adjustment b need to be used. 9

20 Prcng Excess-of-Loss Insurance Contracts For excess busness, we need more data than for ground-up (or prmary) busness. In addton to the requred data for ground-up busness, we need the followng: A severty curve based on ndustry data or theoretcal loss dstrbutons. Loss payment pattern for the excess cover wth an average duraton D XOL, whch s generally longer than the ground-up payment duraton. From the perspectve of a top-down approach, ths nvolves an allocaton of overall rsk load to varous layers. We apply the Wang Transform to the severty curves to derve rsk load relatvty by layer. If we fx our base layer as (0, M), we can calculate a relatvty factor for any layer (a,b] as follows: layer _ relatvty = relatvty _ loadng _ for _ layer( a, b] relatve _ loadng _ for _ base _ layer We prce the excess of loss layer as follows: (). Calculate the dscount factor PV XOL (): Use market rsk-free nterest rate and the excess-layer loss payment pattern. (2). Apply rsk loadng to get a pure premum: PVXOL( ) { + λ σ DXOL ( layer _ relatvty) }. The factor λ should be calbrated from total portfolo re-balancng based on a target return-on-equty (TROE). In other words, for the aggregate nsurance portfolo, the rato of the total rsk load plus nvestment ncome to the total economc captal should produce a target return-on-equty. The total allocated captal over the lfetme of ths $ lablty s λ σ D XOL ( layer _ relatvty)( + r) /( TROE r), For year j, the allocated captal s λ ( σ layer _ relatvty) Pj ( + r) /( TROE r), where P j s the expected percentage of excess-layer losses to be pad n year j. (3). Load for expenses: suppose the total expense factor s θ, we can load the pure premum by a factor of /( θ). (4). Knowng the amount of allocated captal, we can calculate the actual ROE for any gven quoted premum rate. Prcng Varable-Ratng Excess-of-Loss Insurance Contracts For varable ratng terms such as sldng scales and retrospectve ratng, we can make some adjustments for the magntude of rsk-reducton. Ths can be accomplshed by applyng the Wang Transform to the Lee dagram (see Lee, 988). We wll leave ths adjustment method for future dscussons. 20

21 Prcng of Property Catastrophe Covers Although the geometrc Brownan moton assumpton mght apply to the evoluton of lablty losses, t probably does not apply to property catastrophe losses. Here the loss emergence may follow some jump process contngent upon the occurrence of some events. For property catastrophe covers, modern CAT modelng technques often use a bottomup perspectve. Gven geographc spread and amount of nsurance data, commercal CAT models can provde us wth a fnal aggregate loss dstrbuton for any gven layer of coverage. Ths fnal loss dstrbuton already takes nto account the potental frequency and severty of CAT events, as well as the correlaton (concentraton) of the book of busness. Ideally, prcng of CAT covers should be based on such bottom-up nformaton. If a fnal aggregate CAT loss dstrbuton s avalable, we can apply the Wang Transform drectly to t. Theoretcally, the parameter λ would be much hgher than for the non-cat counterpart, to reflect a hgher correlaton rsk. Due to the nature of the CAT loss modelng, parameter uncertanty s present and should be reflected n a hgher value of b. Here are some addtonal remarks: In most cases we suggest the use of top-down approach, whch utlzes ndustry data by lne of busness (loss volatlty, severty curve, loss payment pattern). The topdown approach s based on the prncple of CAPM. In other words, wth the top-down approach, only the systematc rsk (ncludng the cost of captal) s prced nto the contracts. Bault (995) argued why ndustry data, rather than ndvdual company data, should be used for prcng purposes. In some stuatons, a bottom-up approach s warranted, especally for prcng property catastrophe covers, where a fnal loss dstrbuton to the CAT layer can be obtaned from avalable CAT-Models. The outlned approach s based on the prcng framework usng the Wang Transform. For prcng ground-up busness, the Wang Transform extends the classc CAPM n that the parameter λ can now be calbrated from overall ndustry captal requrements. For prcng excess-of-loss layers, the Wang Transform mples rsk-load relatvty by layer, n parallel to the Black-Scholes formula for prcng optons. For both prmary and excess layers, the Wang Transform prescrbes a method to account for the duraton of labltes. The above calculatons dd not account for federal ncome tax. But tax calculatons can be ncorporated. 2

22 Loss Reserve Dscountng Consder the loss reserve lablty for a gven lne of busness. The prcng approach can be equally appled to valuaton of reserve labltes. It should be kept n mnd that the reservng rsk, n terms of σ(t), may dffer from the prcng rsk. Here we provde an alternatve (and more drect) approach to the dscount of loss reserves. Agan we assume that the loss reserve evoluton follows a geometrc Brownan moton. For $ loss reserve lablty, the ncurred losses at tme T has a dstrbuton 2 X ( T ) ~ logormal( µ T 0.5σ T, σ T ). Assume that the rsk-free rate s a constant r. The present value of the ncurred losses has a dstrbuton: 2 X ( T ) exp( rt) ~ logormal ( µ r) T 0.5σ T, σ T. ( ) Let λ be the market prce of rsk for ths lne of busness wth one-year tme horzon. For tme horzon T, the market prce of rsk should be λ T. Applyng the Wang Transform to the dstrbuton for the dscounted reserves, we get another lognormal dstrbuton: 2 X * ( T ) exp( rt) ~ logormal ( µ r + λ σ ) T 0.5σ T, σ T. (9) ( ) From relaton (9) we nfer that applyng the Wang Transform s equvalent to usng the followng dscount rate: = r λ σ. (0) In relaton to equaton (0) we make the followng observatons: The dscount rate n equaton (0) s the mrror formula of CAPM for assets. It s also n lne wth a reserve-dscount formula proposed by Butsc (988) and D Arcy (988). The per annum volatlty σ for product lablty should be hgher than that for worker s compensaton. As a result, a lower dscount rate should be used for product labltes. For Worker s Compensaton lfetme-penson cases, the per annum volatlty σ should be neglgble and the dscount rate should be close to the rsk-free rate. Wth ths outlned approach, the key parameter λ can be (and should be) calbrated from aggregate ndustry captal allocatons for each sector of underwrtten busness. Ths s n contrast to the tradtonal CAPM method where underwrtng beta s derved from runnng lnear regressons of equty prces of nsurance frms. The cost-ofcaptal calbraton of λ should be more robust than the tradtonal estmaton of underwrtng beta. 22

23 Fnal Comments of Part Three Most of the applcatons shown are equally applcable to banks and other fnancal nsttutons. Our approach s manly a top-down approach, whch s consstent wth CAPM. The topdown approach uses ndustry aggregate data, rather than relyng solely on ndvdual rsk dstrbutons. The top-down approach also elmnates any possble nconsstences related to the treatment of frequency/severty (see Venter, 998). In the CAS Whte Paper on Far Valung Property/Casualty Insurance Labltes, several methods of estmatng rsk adjustments are surveyed and compared. The Whte Paper dscussed the advantages and dsadvantages of the Dstrbuton Transform Method, ncludng the PH-transform method. The Wang Transform can overcome most of the dsadvantages lsted n the Whte Paper: As shown earler, the Wang Transform can be used for producng prces or rsk loads on prmary busness. In fact, under some common assumptons, the Wang Transform reproduces the CAPM method and the Rsk-Adjusted Dscountng Method, whch have both been used n prcng prmary busness. Unlke many other possble transforms ncludng the PH-transform, the smple and more generalzed famles of the Wang Transform are probably the only ones that bulds drectly upon CAPM and Black-Scholes Theory. The key parameter n the Wang Transform s the market prce of rsk. Ths has been a famlar concept to fnancal economsts. Better yet, the market prce of rsk n the Wang Transform s not subject to the same drawbacks of underwrtng beta. Ths s because that the market prce of rsk can be easly calbrated from ndustry captal requrements. Ths calbraton s more robust than hstorcal estmates of the underwrtng beta. Acknowledgments Durng the development of the transform n queston, the author has benefted from comments by Phelm Boyle, John Kulk, Stephen Mldenhall, and Gary Venter. The author also thanks Jula Wrch for edtoral suggestons and especally Adam Burczyk for some helpful edtng. Any errors or omssons reman wth the author alone. 23

24 REFERENCES Black, F., and M. Scholes, 973, The prcng of optons and corporate labltes, Journal of Poltcal Economy, May-June, 8: Bault, T. 995, Dscusson of Feldblum (990), PCAS, LXXXII, pp Butsc, R.P., 988, Determnng the proper nterest rate for loss reserve dscountng: an economc approach, CAS Dscusson Paper Program, pp It s avalable for download from webste: Butsc, R.P., 999, Captal Allocaton for property-lablty nsurers: a catastrophe rensurance applcaton, Casualty Actuaral Socety Forum, Sprng 999 Rensurance Call Papers, -70. CAS Task Force on Far Value Labltes, 2000, Whte paper on far valung property/casualty nsurance labltes, Casualty Actuaral Socety. Cox, J.C. and S.A. Ross, 976, The valuaton of optons for alternatve stochastc processes, Journal of Fnancal Economcs, 3, D Arcy, S.P., 988, Use of CAPM to dscount Property-Lablty loss reserves, Journal of Rsk and Insurance, Vol. 55:3, pp Derman, E. 996, Reflectons on Fscher, The Journal of Portfolo Management, Specal Issue, December 996. Avalable on the Goldman Sachs webste, Embrechts, P., Kluppelberg, C., and Mkosch, T., 997, Modellng Extreme Events for Insurance and Fnance, Sprnger-Verlag. Hull, J. 997, Optons, Futures, and Other Dervatves, 3 rd edton, Prentce Hall, Upper Saddle Rver, New Jersey. Lee, Y.S., 988, The mathematcs of excess of loss coverage and retrospectve ratng --- a graphcal approach, Proceedngs of the Casualty Actuaral Socety, Vol. LXXV, Ths paper s avalable for download on webste Merton, R.C., 973, An ntertemporal captal asset prcng model, Econometrca, 4, Mldenhall, S., 999, Dscusson of Mchael Wacek's 997 PCAS Paper --- Applcaton of the opton market paradgm to the soluton of nsurance problems, Proceedngs of the Casualty Actuaral Socety, to appear. It wll be avalable for download on webste 24

25 Panjer, H.H. (edtor). 998, Fnancal Economcs, The Actuaral Foundaton, Schaumburg, IL. Phlbrck, S., 994, Accountng for rsk margns, Casualty Actuaral Socety Forum, Sprng 994. It wll be avalable for download on webste Venter, G.G., 99, Premum mplcatons of rensurance wthout arbtrage, ASTIN Bulletn, 2, No. 2: Venter, G.G., 998, Dscusson of Implementaton of PH-transforms n ratemakng by S.S. Wang, Proceedngs of the Casualty Actuaral Socety, Vol. LXXXV, avalable for download on webste Wang, S.S., 998, Aggregaton of correlated rsks: models and algorthms, Proceedngs of the Casualty Actuaral Socety, Vol. LXXXV, It s avalable for download on webste Wang, S.S., 2000, A class of dstorton operators for prcng fnancal and nsurance rsks, Journal of Rsk and Insurance, Vol. 67, No.,