Hedging the volatility of Claim Expenses using Weather Future Contracts

Transcription

1 Mrshll School of Business, USC Business Field Project t Helth Net, Inc. Investment Deprtment Hedging the voltility of Clim Epenses using Wether Future Contrcts by Arm Gbrielyn MSBA Cndidte co written by Emin Gbrielyn, PhD Pge of

2 Tble of Contents Overview...3 Influenz Like Illnesses ILI nd Temperture Correltion...4 HDD...6 ILI s Function of Temperture...7 Scling Epenses to HDD...8 Cost of Debt...9 Price probbility density trnsformtion...0 Neutrl right price of Futures... Probbility density function of insured costs nd futures with right prices...6 Finding the optiml strike price nd volume of futures...9 Files nd References...0 Pge of

3 Overview The purpose of the pper is to identify the correltion between wether temperture ptterns nd Helth Net, Inc. opertion epenses nd propose cost hedging strtegy using wether future contrcts. The strtegy is ment to decrese the voltility of Compny s clim epenses throughout the yer. Observtion of historicl dt indictes tht flu relted illnesses consistently strt from November nd lst until My of ech yer. The following chrt shows 008 to 009 flu trend in Cliforni. Chrt : Cliforni Flu Activity As you cn see the flu seson picks during the coldest months of the yer, December to Mrch. The following chrt shows the dily verge wether temperture trend in Cliforni for the sme period of the time. Chrt : Cliforni Averge Temperture Trend Pge 3 of

4 The tble bellow shows the number of Helth Net s provider clim receipts for Cliforni commercil members nd the number of Cliforni commercil members. Jul-08 Aug-08 Sep-08 Oct-08 Nov-08 Dec-08 Jn-08 Feb-08 Mr-08 Apr-08 My-08 Jun-08 Receipts 49,60 438,54 403,605 44,639 39, ,075 48,35 444,67 466, , ,68 44,40 Membership,96,95,9,0,90,667,86,349,8,48,8,007,33,490,38,908,33,63,304,34,306,500,303,30 Receipts per Member 33.09% 33.94% 3.7% 34.4% 30.49% 3.86% 3.5% 33.7% 35.5% 34.74% 33.4% 3.54% Tble : CA Commercil Clim Receipts nd Membership informtion Chrt 3: Clims per Member. Helth Net, Pln: CA, LOB: Commercil This nlysis indictes tht the number of receipts per member picks not in Jnury-Februry, the coldest months of the yer with the most flu relted illnesses but in Mrch. This perfectly correltes to the ssumption of incresed operting costs for mnged helth cre compnies during flu seson. There is lg time between doctor visit nd clim submitted to the helth insurnce compny by provider. This lg time verges t little bove one month. Influenz Like Illnesses ILI nd Temperture Correltion The regression nlysis of ILI s function of temperture indictes R0.69 nd R²0.48. Low R² is eplined by two fctors: the ILI informtion presented by Center for Disese Control nd Prevention CDC is presented in weekly brek downs while the temperture informtion is in dily formt. There is mrgin of error in converting dily temperture to weekly formt to be compred with ILI trend. Another fctor is tht individul observtion points re not necessrily Pge 4 of

5 moving in opposite directions. However, the correltion between Temperture nd ILI trend lines would hold significntly higher R². Chrt 4: ILI nd Temperture Correltion: ILI is represented s incrementl percentge from verge nd scled on the right scle. SUMMARY OUTPUT Regression Sttistics Multiple R R Squre Adjusted R Squre Stndrd Error Observtions 37 ANOVA df SS MS F Significnce F Regression E- Residul Totl Coefficients Stndrd Error t Stt P-vlue Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept E E Tble : ILI s function of Temperture: Regression Anlysis Pge 5 of

6 HDD Wether Futures re trded in Chicgo Mercntile Echnge CME. The contrcts re on the dily cumultive Heting Degree Dys HDD nd Cooling Degree Dys CDD for month observed t wether sttion. HDDm 0,65-A nd CDDm 0,A-65 Where: A is the verge of dily minimum nd mimum tempertures. One Future contrct is on $00 times the cumultive HDD or CDD for one full month. The historicl temperture verge provided in the Tble is mesured in Los Angeles Civic Center. Historicl trend is relible forecst for the future yers since Wether Future Contrcts hve no systemtic risk. This cn be further djusted for globl wrming effect verges in Los Angeles Jul Aug Sep Oct Nov Dec Jn Feb Mr Apr My Jun Averge M Averge Min Averge Temperture HDD CDD Tble 3: Historicl Temperture nd CDD/HDD. For Emple, if we purchsed one HDD future contrct for Jnury t strike price of 00 nd if the verge temperture held our py off would hve been *$00$5,960. Historicl verge of dily HDD rnges from 0 to 9 nd for month 0 to 70. However, HDD cn rech s high s 750 in coldest prts of the world. Bellow is the temperture chrt nd chrt of devition of my HDD clcultion nd WRCC HDD clcultion bsed on historicl dily verge tempertures from906 to 008. Pge 6 of

7 Chrt 5: verge dily minimum nd mimum verge temperture Chrt 6: verge HDD ILI s Function of Temperture Incrementl percentge of ILI s function of temperture is t its pick of 3.4% when the verge of dily min nd m temperture is 5 degrees of Fhrenheit. The drop in bellow 5 F is dictted by lck of dt. Pge 7 of

8 Chrt 7: incrementl ILI s Function of Temperture: ILI is represented s incrementl percentge from verge Similrly ILI incrementl s function of HDD is mirrored imge of ILI s function of temperture. As shown in Chrts 5 nd 6, HDD increses s the temperture decreses [HDD m 0,65-A]. Chrt 8: incrementl ILI s Function of HDD: ILI is represented s incrementl percentge from verge Scling Epenses to HDD The purpose of the project is to hedge the flu seson ILI relted clim epenses using HDD Wether Futures. Since our gol is to clculte the most optiml mount of etr epense to be hedged considering the cost of the future contrcts, we need to convert HDD metrics into Dollr scle to be ble to design the optiml cost-benefit probbility model. Due to confidentility of Pge 8 of

9 clim epense informtion I will use nominl vlues s Helth Net s ILI relted clim costs. Plese note tht clim epense informtion is vilble for ech month with doctor visit reson, visit dte nd clim dte detils. Let's ssume tht the etr clim epense during flu seson hs $ mil historicl men µ nd $3. mil stndrd devition. The historicl HDD µ during Jnuries is 60 with 60. Therefore, $ mil corresponds to 60 HDD nd one degree of HDD corresponds to $3. mil/60 $53,333. Now to cover ny epense over $ mil etr epense, the 3th million nd bove for emple, we will need to purchse one HDD t strike price 60 in our terms, which corresponds to $53,333/$00533 HDD's, gin, t 60 strike price. Going forwrd we will discuss the project in dollr terms, which cn be converted into rel HDD terms using the scle mtching strtegy in this section. Cost of Debt Our objective is to find futures' optiml strike price nd volume to hedge the enterprise epenses ginst voltility. For this purpose we introduce the voltility cost of enterprise. On the other hnd we shll introduce the profit mrgin of the issuer of futures. In our model the verge mount of ILI relted epenses µ is equl to $ mil nd the vrince is equl to $3. mil. The cost of the voltility is introduced vi two prmeters. A debt cost debtc in the cse when the enterprise epenses end up t their men µ, nd multipliction fctor debtk by which the debt cost of enterprise would increse if the compny ends up with n epense higher from the verge by i.e. µ. The higher is debtk the higher is the interest to sty closer to µ the verge being equl to µ. We ssume tht the defult debt cost debtc is equl to $ the defult price pplicble for ll scenrios where epenses end up t the verge of $ mil. The multipliction fctor debtk is prmeter chnging from.5 to 5. For instnce if this fctor is, it mens tht if the enterprise ends up Pge 9 of

10 with n epense of $5. mil i.e. one wy from the µ, it will cost etr $ to the compny e.g. due to short-term debt interests. Price probbility density trnsformtion In this section we nlyze trnsformtions of probbility density of n opertion cost, cused by purchses of futures issued on the sme level of the cost. We ssume two free vribles in the purchsing of futures, the strike price nd the quntity of futures. We ssume tht ll futures re purchsed with the sme strike price. When the opertion cost eceeds the strike price, the futures cover the given percentge of the opertion cost ecess. Tht percentge depends on the mount of futures purchsed. It cn vry from 0% to 00% underinsurnce or cn be more thn 00% over-insurnce. The choice of the quntity of purchsed futures determines the insurnce rtio. The probbility density of the hedged price is computed s follows: c k c c < k p, if ep ep π 0, if c π Where: k, 0, if if c k k c k < k c is the cost of purchsed futures is the strike price k is the insurnce rtio corresponds to 00% Obviously: p d π d ep The following nimtion shows how the probbility curve of price chnges when the rtio of insurnce chnges from 0% to 50%. The strike price is the sme in ll smples. The cost of futures is computed with very simplified empiric formul it is sufficient for this demonstrtion. As epected, this Pge 0 of

11 nimtion shows tht the more you increse the insurnce rtio towrd 00%, the nrrower the voltility of your prices becomes. We lso see tht the price, when over insured, never eceeds the strike price plus the cost of insurnce. Animtion link: Neutrl right price of Futures Let us hve n opertion price probbility, subject to norml Gussin distribution see norml distribution: p ep π The following chrt shows such distribution, where the men vlue is equl to $4 000 nd the vrince is equl to $ 000: Pge of

12 We seek to compute the right price of futures issued for our opertion cost s function of the strike price. Such futures py-off ech dditionl $ of opertion cost eceeding the selected strike price. The sttisticl mrgin-less cost of such futures must be equl to: F p d The cost computed in such wy contins neither security nor beneficiry mrgin of the issuer of futures. Using the bove formul the joined Ecel file computes the cost of futures stright-forwrdly by summing refund clims weighted by their probbilities. For ech price eceeding the strike price, the formul tkes the difference between the price nd the strike-price, multiplies it by the probbility of the price, nd dds ll of them together. The finl sum is multiplied by the distnce between the smple points on the is. Our objective is to find the nlyticl formul of the price of futures nd if not nlyticl then one using well known functions, such s the error function. Pge of

13 Pge 3 of d p F Due to the symmetric bell shpe of the norml distribution: d p d p Therefore: d p cdf d p d p d p d p F CDF is the cumultive distribution function [see norml distribution]. It represents the probbility tht the price will fll below its rgument. dt t p cdf

14 [ls] Considering the product rule see derivtive: f g f g f g We cn write tht: p d cdf cdf d Therefore: F cdf cdf d p d CDF cn be epressed vi error function s follows see norml distribution: Pge 4 of

15 Pge 5 of erf cdf ERF Ecel function is implemented so no pproimtions or simultions is needed. It is known tht see list of integrls: erf e d erf π Tking into ccount the chin rule see derivtive: g g h f g h f The CDF integrl is computed s follows: ep π erf d cdf The mrgin-less price of futures cn be therefore epressed s follows: ep π erf F The following nimtion shows the price of futures s function of the strike price where the vrince, i.e. the voltility, chnges from $00 to $6500 over the time. The men vlue is lwys the sme nd is equl to $4000:

16 Pge 6 of Animtion link: Probbility density function of insured costs nd futures with right prices In Price probbility density trnsformtion [09087 ii] we developed the probbility density function of opertion costs insured by futures. This density is lso function of selected strike price nd of insurnce rtio k representing the py-off rtio of cost units eceeding the strike price, or otherwise the mount of purchsed futures. In the document we ssumed the cost of futures c s n input prmeter. No formul ws provided for computing the cost of futures. In Neutrl right price of Futures [09089 ii] we developed formul for computing the right cost of futures for given strike price. Under term right or neutrl we understnd the price, such futures would cost to n insurnce compny tking into ccount the voltility of the cost. This shll be n integrl of ecess of opertion prices eceeding the strike price weighted by the norml distribution of the opertion price see norml distribution. d k F k c ep π The formul of futures price tht we developed in [09089 ii], is fully nlyticl, with the eception of error function ERF implemented in Ecel see error function. ep π erf F Becuse the price of futures is right, if computed s follows:

17 c k F The following must hold for ll nd k p d, k In order to vlidte both of our formuls, we use n Ecel model. The men of our opertion cost t $ mil nd the vrince is $3. mil. For two-dimensionl set of prmeters, strike prices vrying from $4 mil to $0 mil nd b the insurnce rtios vrying from 0% to 00%, we compute right prices of futures using our formul k F. The blow chrt shows this function for the two dimensionl set of input prmeters: Sub-sequentilly, bsed on the right price, we compute probbility density vlues with our formul p k, by loding results into n rry with prices rnging from $0 to $ Pge 7 of

18 The following nimtion shows tht the choice of nd k cn chnge the shpe of price distribution significntly: Animtion link: By hving, for ech shpe dictted by pir of strike price nd insurnce rtio k of the probbility density rry we compute the men of the insured opertion cost: N i p i, k i m N min Irrespectively how wildly the probbility density shpe is chnged by the choice of two prmeters of futures, the Ecel simultion shows tht the computed men of insured opertion costs is lwys equl to the men of the uninsured costs, i.e. to µ with eceptions of precision errors. This suggests no hidden costs in prices of futures. Pge 8 of

19 The results of this simultion vlidte our two formuls for clcultion of the right price of futures k F nd subsequent clcultion of the insured costs' probbility density p, k. Finding the optiml strike price nd volume of futures In our model we del with future contrcts tht cn py-off portion of enterprise epenses eceeding pre-selected strike price. The choice of the py-off portion of the ecess with respect to the strike price is mde by the choice of the mount of purchsed futures. This frction cn be less thn, more thn, or equl to. This model is pplicble to more generl cse, where insted of futures issued directly on the enterprise epenses, the futures re issued on nother underlying instrument highly correlted to the enterprise epenses. HDD future contrcts re good emple of such instruments. Our objective is to find the optiml strike price nd volume of futures, purchsed for hedging the enterprise epenses ginst voltility see Cost of Debt section. In [09087 ii] we developed price probbility density trnsformtion chieved by the purchse of futures. In [09089 ii] we developed formul for computing the neutrlly or right price of futures. In [09089b ii] we showed tht the men of insured epenses does not chnge, if the futures re bought t neutrlly right prices. This mens tht if the mrket price of futures is lwys right, we re definitely interested in purchsing futures ll the time, even if the voltility cost is low. Futures re nrrowing the probbility density of epenses nd re therefore minimizing the voltility relted debt costs. However the plesure of hving future bsed insurnce shll often hve mrket cost eceeding its neutrlly right price [09089 ii]. The difference between the mrket price of futures nd the right price computed by our formuls [09089 ii] is precisely the cost of this plesure. This cost must be Pge 9 of

20 counterblnced with svings chieved on the voltility side. The mrgin dded on the right price of futures is represented in percentges. We nlyze rnge from 0.5% to 90%. In the following chrt we ssume tht the defult debt cost debtc is equl to $ the defult price pplicble for ll scenrios where epenses end up t the verge of $ mil. The multipliction fctor debtk is prmeter chnging from.5 to 5. For instnce if this fctor is, it mens tht if the enterprise ends up with n epense of $5. mil i.e. one vrince wy from the men, it will cost etr $ to the compny e.g. due to short-term debt interests see Cost of Debt section. The surfce shown in the nimtion below represents the overll cost the voltility debt cost together with the profit mrgins pid to issuers of futures s function of the strike price in the rnge from $4 mil to $0 mil nd the insurnce rtio in the rnge from 0% to 00%. The nimtion shows the chnges in the shpe when chnging the issuer's profit mrgin firness nd the debt cost fctor dependence. For ech frme i.e. for ech pir of firness of futures nd dependence of enterprise from the voltility, the tet dt pnel shows the best recommended strike price nd insurnce rtio. This nimtion demonstrtes the concve shped form of the surfce for ll pirs of firness nd dependence showing tht there is lwys n optiml choice to mke depending on the mrket prices of futures nd the etr cost of the voltility of the enterprise epenses. Animtion link: Files nd References Hedging the voltility of Clim Epenses using Wether Future Contrcts this document: gbrielyn/public/09083 hedging with wether futures/ Pge 0 of

21 gbrielyn/public/09083 hedging with wether futures/ This document in different formts [pdf], [htm], [doc], [doc] Finding the optiml strike price nd volume of futures for hedging ginst the voltility of enterprise epenses: gbrielyn/public/09080 best strike/ gbrielyn/public/09080 best strike/ Vlidtion of the formul of the right price of futures nd of the function of the insured price s probbility density: gbrielyn/public/09089 verge of hedged cost/ gbrielyn/public/09089 verge of hedged cost/ Right price of futures s function of the strike price: gbrielyn/public/09089 neutrl price vs Blck Scholes/ gbrielyn/public/09089 neutrl price vs Blck Scholes/ Probbility density of hedged price: gbrielyn/public/09087 hedging cost with futures/ gbrielyn/public/09087 hedging cost with futures/ * * * Pge of