Technical details for computing de ation probabilities with TIPS prices. Pat Higgins

Transcription

1 echnical deails for compuing de aion probabiliies wih IPS prices. Pa Higgins By Summary of mehod:. Following Sack (2000, assume ha IPS marke paricipans discoun IPS coupon paymens and principal repaymens wih ed yields from a smoohed yield curve ed o observed reasury SRIPS yields. Also assume ha IPS marke paricipans are risk neural and value a IPS securiy as he discouned presen value of is expeced coupon and principal paymens. 2. Assume ha IPS marke paricipans expec ha monhly rend seasonally adjused annualized CPI in aion is an unknown consan where he consan has a Normal( ; 2 disribuion. Even if 2 = 0 and were known, here would be idiosyncraic uncerainy in fuure monhly in aion around he rend since monhly in aion has noise even when he rend is sable. We assume his idiosyncraic noise in annualized in aion has a Normal(0; 2 disribuion wih = 3:5% (approximaely he sandard deviaion of annualized monhly in aion since Assume ha rend in aion for an on-he-run 5-year IPS and an o -he-run 0-year IPS wih mauriy daes ha are only separaed by hree monhs e.g. he 5-year IPS issued in April 200 and due April 205 and he 0-year IPS issued in July 2005 and due July 205 have idenical Normal( ; 2 disribuions. Solve for he values of and 2 ha imply he expeced discouned coupon and principal paymens of boh he 5-year IPS and he 0-year IPS are equal o he observed prices. Wih and 2 in hand you can solve for he disribuion of in aion or probabiliy of de aion. Deailed Explanaion Using SRIPS yields and IPS prices o solve for a consan breakeven in- aion rae Sack (2000 oulines he following procedure o esimae breakeven in aion in he case where in aion is consan. Assume ha he discouned presen value of a zero-coupon bond paying $ in n years is worh d (n dollars. d (n can be compued from an n-year zero-coupon yield y zero (n wih he formula d (n = exp( y zero (nn Following Sack (2000, we esimae y zero (n by ing a smoohed yield curve o observed reasury SRIPS yields. For he funcional form of he smoohed yield curve, we use he Svensson (994 exension of he Nelson and Siegel (987 funcional form. For coupon paymens occuring in less han 3 monhs we use a cubic spline o inerpolae our yield curve wih he 4-week reasury yield.

2 Assuming > 0 is he consan annual in aion rae going forward [e.g. = :02 is 2% in aion] he discouned presen value of an N-year IPS issued oday ha promises a real paymen of $00 wih a coupon rae c [e.g. c = :02 denoes a 2% coupon rae] and semi-annual coupon paymens is P (N = 2NX n= c 2 00( + n=2 d ( n ( + N=2 d (N his formula is used by Sack (2000 o compue breakeven in aion. If < 0, hen he ( + N=2 erm vanishes since he principal repaymen canno be less han $00 and he discouned presen value of he IPS issue becomes. P (N = 2NX n= c 2 00( + n=2 d ( n d (N Formula for reference CPI and index raio for a IPS securiy Le IR +h; denoe he gross in aion rae (or plus he in aion rae from ime o ime + h in he reference CPI. he reference CPI is simply a daily inerpolaion of he monhly non-seasonally adjused CPI. Explicly if ime is in monh M and monh M has M Days in i and i is he day h of he monh (e.g. if = Sepember 28, 200 hen M = Sepember 200, M = Augus 200, M Days = 30, day = 28 hen REF cpi = day M Days CP INSA(M 2 + M Days day + M Days CP INSA(M 3 and IR +h; = REF cpi +h. IR REF cpi +h; is referred o as he index raio for a IPS ha was issued a ime. For IPS issues, he coupon and principal repaymen daes always ake place on he 5h day of a monh. Incorporaing uncerainy in in aion ino IPS prices Assuming risk neuraliy and a sochasic in aion rae implies a IPS expiring N years from now ha was issued M years ago has a ne presen value of 2(N+M X P IP S c (N = 2 00E [(IR M+ n 2 ; M ]d ( n 2 n= M + 00d (NE [max(; IR +N; M ] 2

3 For noaional conveinance we de ne d (x = 0 for x 0 already been paid are no assigned a posiive value. so ha coupons ha have In he case of a zero-coupon IPS, P IP S (N = 00d (NE [max(; IR +N; M ]. In his case he zero-coupon IPS pays $00 real dollars if here is no gross de aion since he issue dae and a larger real paymen if here is gross de aion. Assume ha monhly in aion for he seasonally-adjused CPI measured in logarihims, is an iid random normal variable wih he disribuion 2 log(cp ISA(M + h=cp ISA(M + h ~Normal( ; 2 where Normal( ; 2 denoes a normal disribuion wih mean and variance 2 For example using he mean and sandard deviaion of monhly (annualized in aion for he seasonally adjused CPI since 994 corresponds o = 2:4%, = 3:5%. In wha follows below, we assume ha = 3:5%. We inerpre as he idiosyncraic uncerainy in monhly in aion ha would sill be presen even if we knew wha rend in aion was going forward. We also assume ha IPS marke paricipans expec wih cerainy ha he curren seasonal adjusmen facors CPI will coninue o evolve as hey have over he pas 2 monhs [i.e. we do no incorporae uncerainy abou he evoluion of seasonaliy]. hen IR ; M is a known consan, so we can wrie P IP S (N given as 2(N+M X (P IP S c (Nj = 2 00(IR ; ME [(IR M+ n 2 ; j ]d ( n 2 M+00d (NE [max(; IR +N; M j ] n= Gross de aion over he enire life of he IPS occurs if IR +N; M <. We assume ha ~Normal( ; 2, i.e. IPS marke paricipans do no know wha rend in aion is going forward, bu hey have a subjecive probabiliy disribuion for i. We will use prices of he mos recenly issued 5-year IPS and he o -he-run 0-year IPS ha maures 3 monhs afer he on-he-run 5 year IPS maures o derive wha and 2 are. is Given our assumpion abou he evoluion of in aion, he expeced value of P IP S (Nj Z (2E [P IP S (Nj ] = P IP S (Nj ( d where denoes he sandard normal pdf. he probabiliy of gross de aion for he on-he-run 5-year IPS ha we assume was issued M years ago is 3

4 Z (3E [Pr(IR+5 M; M < j ] = Pr(IR+5 M; M < j ( d We will discuss how o compue (, (2 and (3 in he appendixes. Under he assumpion of risk-neuraliy, we have P IP S (N = E [P IP S (Nj ]. For he 5-year IPS mauring in April 205 and he 0-year IPS mauring in July 205, we will assume ha rend in aion has idenical means and variances for boh disribuions, i.e. boh July205 and April205 have idenical disribuions Normal( ; 2. Le he observed values of he April 205 and July 205 IPS bond prices be P April5 IP S and P July5 IP S respecively. We solve for he values of and 2 April5 IP S ha saisfy P (N = P April5 IP S July5 IP S and P (N = P July5 IP S. For each realizaion of, rend in aion is a consan so ha he uncerainy of -monh in aion in Sepember 200 is he same as he uncerainy in -monh in aion in Sepember 204. his is somewha unappealing as he uncerainy in monhly in aion should probably increase wih he forecas horizon. We experimened wih allowing rend in aion o evolve as a random walk process. I.e. +h+ = +h + +h+, where he sandard deviaion of +h+ is assumed o be consan across forecas horizons and calibraed wih hisorical CPI daa. he de aion probabiliies calculaed in his manner were similar o hose compued wih he model we used, where +h+ 0. We show a graph of his probabiliy in gure along wih a lower bound on he probabiliy of de aion proposed by Wrigh (2009 and a modi ed lower bound ha adjuss for seasonaliy in he CPI. Discussion of resuls here are more han a few pifalls in calculaing a de aion probabiliy as we have. For example, he 5-year IPS mauring in April 205 and he 0-year IPS mauring in July 205 do no have he same mauriy daes, coupon raes, or coupon paymen daes. his makes i challenging o disinguish he price di erence in he securiies due o hese feaures and he price di erence due o he di eren de aion safeguards. Furhermore IPS marke paricipans may believe ha he probabiliy disribuion for fuure in aion has faer ails han he normal disribuion. Figure 2 plos he cumulaive disribuion funcion for average CPI in aion over he nex 5-years implied by our IPS pricing model along wih a second cumulaive disribuion funcion based on he hisorical forecas errors of a very simple forecasing model. he model is a varian of he Akeson- Ohanian model ha uses he curren 2-monh in aion rae for he core CPI as a forecas of he average in aion rae over he nex 5 years using all forecass since 958 (he core CPI sars in 957. he hisorical forecas errors are added o he curren 2-monh core CPI in aion rae (0.89% o ge a disribuion for he 5-year in aion rae. he 4

5 probabiliy of in aion being below 3% is abou 6% according o he emprical disribuion and jus slighly more han % according o he IPS model. he de aion proecion of he 5-year IPS will urn ou o be more valuable ex-pos if in aion urns ou o be -3% as opposed o, say, -0.5%. Hence i is possible o ge a lower probabiliy of de aion and have he IPS prices be consisen wih he preferences of a risk-neural invesor by, say, aking some of he probabiliy mass assigned o an in aion oucome beween -% and 0% and assigning par of i o an in aion oucome above 0% and he oher par o an in aion oucome below -%. As a robusness check, we used a varian of an in aion model proposed by Sock and Wason, speci cally, heir unobserved componens wih sochasic volailiy model o generae a probabiliy disribuion for in aion over he nex 5-years. he sochasic volailiy feaure of he Sock and Wason (2007 model generaes faer ails for in aion han he normal disribuion. his model says ha here is abou a 0% chance ha average in aion over he nex 5 years will be below 0. Bu he model also says ha expeced in aion is abou.8%; if we change his expecaion o he IPS model expecaion of.% for Ocober 4 200, hen he probabiliy of de aion increases o 20%. In any case, hese empirical checks sugges ha he IPS model may be oversaing he probabiliy of de aion a leas somewha. As shown in gure, he probabiliy of de aion has been above 25% since June. Our model could also be oversaing he probabiliy of de aion if invesors are willing o pay a higher premium for he enhanced de aion safeguard of he on-he-run ve-year IPS han a risk neural invesor would. Neverheless, concluding ha IPS marke paricipans hink here is virually no probabiliy of de aion would require some explanaion as well. Figure 3 plos breakeven in aion raes for various IPS issues using he same adjusmen for he seasonaliy in he CPI as in Appendix B. he breakeven in aion raes for he veyear IPS clearly lie above he breakeven raes for he 0-year IPS. Wihou a posiive probabiliy of de aion, his occurrence would be di cul o explain unless here is a liquidiy premium of he ve-year IPS over he 0-year IPS. Before he recen nancial crisis, when he probabiliy of de aion was presumably low, here was lile evidence of such a premium. Appendix A: More on accouning for seasonaliy. In March, April and May nonseasonally CPI in aion is higher han seasonally adjused in aion. Since IPS paymens are indexed o he NSA CPI, he seasonal paern in hese 3 monhs will push up he value of he o -he-run 0-year IPS mauring in July 205 relaive o he on-he-run IPS mauring in April 205. If IPS marke paricipans expeced he monhly seasonally adjused in aion rae o be a single consan for all forecas horizons, seasonal facors would cause he breakeven in aion rae for he July 205 IPS o be abou 0.5 percenage poins higher han he breakeven rae for he April 205 IPS. Hence, no adjusing for seasonaliy in he CPI will resul in undersaing he probabiliy of de aion since he value of he enhanced de aion safeguard of he on-he-run 5-year IPS mauring 5

6 in April 205 will be o se somewha by he more favorable seasonable facors of he o he-run 0-year IPS mauring in July 205. In a commen on a paper Campbell, Shiller and Viceira (2009, Wrigh (2009 derives a simple formula for nding a lower bound on he probabiliy of de aion. We will show o adjus his lower bound for seasonaliy in he CPI. For now, we x rend in aion a any paricular value. Suppose he laes reading for he monhly CPI we have is for monh (e.g. as of his wriing on Sepember 28h, 200 he mos recen CPI is for Augus 200. Leing CP I NSA denoe he level (index reading of he seasonally unadjused CPI and leing CP I SA denoe he level of he seasonally adjused CPI, for any horizon h monhs in he fuure, we can wrie or +h CP ISA +h = ( CP I SA [( CP ISA ( CP INSA +h ] CP I SA +h log( CP INSA +h CP ISA = log( +h CP I SA hx CP INSA +n log( CP I SA +n n= Our assumpion ha 2 log(p+h SA SA =P+h is normally disribued wih mean ( h 2 and variance h( is iid Normal( ; 2 CP ISA implies ha log( +h CP I SA 2 2. Furhermore we assume ha IPS marke paricipans forecas he following evoluion of seasonaliy log( CP INSA +n CP I SA +n CP INSA = log( log( 2+n CP I SA 2+n CP INSA +n CP I SA +n (( 2 X m=0 log( CP INSA = log( +n 2 CP I SA +n 2 CP INSA m CP I SA m (for n 2 (for n > 2 For each of he rs 2 monhs ou, IPS marke paripans expec he log change in he seasonal facor o equal whaever he change in he seasonal facor was 2 monhs before. here is also a small correcion so ha he forecased average 2-monh in aion rae for he SA and NSA CPI for 2, 24, 36,... monhs ou are he idenical. For example, he SA CPI increased 0.3% in July 200 while he NSA CPI was unchanged. In each of July 2008 and July 2009 he SA CPI -monh in aion rae was also 0.3 percenage poins higher han he -monh NSA CPI in aion rae. So i is assumed ha IPS marke paricipans also expec ha in July 20 he -monh SA CPI in aion rae will exceed he -monh NSA CPI in aion rae by 0.3 percenage poins. hus, we can ieraively forecas CumSA +h = hx CP INSA +n log( CP I SA +n n= 6

7 so ha a given value of rend in aion implies hus +h log( CP INSA +h ~Normal(( h 2 + CumSA +h ; h( 2 2 (A is log-normally disribued. Appendix B: More on calculaing he expeced value of IPS reurns. As in he above Appendix A, we x rend in aion a any paricular value. Suppose he laes reading for he monhly CPI we have is for monh (e.g. as of his wriing on Sepember 28h, 200 he mos recen CPI is for monh = Augus 200 and suppose we are ineresed in he expeced value of he reference CPI for a coupon or principal repaymen ha will ake place on he 5h day of monh + h. For example, for he 0-year July 205 IPS, he nex coupon paymen is on January 5, 20, which is for monh + 5 in his example. he reference CPI for he coupon or principle repaymen is AvgCP I 5h +h = ( 4 + ( 4 d NSA +h +h 3 where is he number of days in monh + h (e.g. = 3 in January 20 and CP In NSA is he index level of he NSA CPI in monh n. hen AvgCP I 5h +h is he sum of wo log-normal random variables (see Appendix A and from (A is expeced value is E [AvgCP I 5h +h] = [f( 4 exp(( h 2 2 (B +CumSA + ( h 2 ( g +f( 4 exp(( h 3 2 +CumSA +h 3 + ( h 3 ( g] his uses he fac ha if log(x~normal(; 2 2 +, hen E[X] = e 2. We have ignored he case where h = or h = 2. In hese special cases E [AvgCP I 5h +h] = ( 4 E [AvgCP I 5h +h] = ( 4 + ( days +h 4 days +h + ( (for h= (for h=2

8 Looking back a equaion (, we can see how o derive he expeced value of he coupon paymens. he expeced value of a coupon paymen depended on E [E [(IR M+ n 2 ;j ]] where IR o ime M+ n 2 ; = REF cpi M+ n 2 is he gross in aion rae in he reference CPI from ime REF cpi M + n. Bu we can rewrie his erm as 2 E [E [(IR M+ n 2 ; j ]] = ( E REF cpi [E [(AvgCP I 5h +hj ]] E [(AvgCP I 5h +h j ] can be read o from (B, (B2 or (B3 depending on he value of h, and is expeced value can be solved by inegraing over he possible values of since ~Normal( ; 2. Because of he de aion proecion for he principal repaymen for IPS, we also need o know he enire disribuion of AvgCP I 5h +h in order o compue he expeced value of he principal repaymen aking place in monh + h [we can safely assume h > 2]. We can wrie AvgCP I 5h +h = ( 4 CP I NSA +h 3+( 4 CP I NSA +h 2 = CP I NSA +h d 3[+( 4 (exp( ] +h where = log( CP INSA CP I NSA +h 3. hus log(avgcp I 5h +h log(cp I NSA = log(cp I NSA +h 3[ + ( 4 (exp( ] log( CP INSA +h 3 = log( + log([ + ( 4 (exp( CP I NSA ] CP INSA +h 3 log( + ( 4 (exp( CP I NSA CP INSA +h 3 log( + ( 4 CP I NSA wih CP INSA Noe ha log( +h 3 4 and ( days +h are independen random normal variables CP INSA log( +h 3 ~Normal(( h CumSA +h 3 ; (h 3( 2 2 8

9 and ~Normal(( 2 + CumSA CumSA +h 3 ; ( 2 2 [see A a end of Appendix A]. So log(avgcp I 5h +h is approximaely a linear combinaion of wo independen random normal variables, so i is also approximaely a random normal variable wih a mean of log AvgCP Ij = log( + ( h ( 4 ( 2 +CumSA +h 3 ( 4 [CumSA CumSA +h 3 ] (B4 and a variance of 2 log AvgCP Ij = (h 3( [( 4 ( 2 ]2 (B5 his implies ha IR +N; M = ( REF cpi M AvgCP I 5h +h, is approximaely a lognormal random variable. hen he E [max(; IR +N; M j ] erm in equaion ( deermining he value of he expeced principal is E [max(; IR +N; M j ] = Z REF cpi M f(xj dx + ( REF cpi M Z REF cpi M xf(xj dx where f(xj is he probabiliy densiy funcion for AvgCP I 5h +h which we know since log AvgCP I 5h +h is (approximaely normal wih known mean and variance in (B4 and (B5. Explicily f(xj = ( exp( x q2 2log AvgCP Ij (log x log AvgCP Ij log AvgCP Ij (B6 he probabiliy of gross de aion given is hen Pr[x < REF cpi M ] = R REF cpi M f(xj dx Appendix C: Solving for he probabiliy of de aion. 9

10 For any given value of, we can solve for P IP S (Nj in equaion ( using he calculaions in he above 2 appendixes. For any proposed values of and 2 where ~Normal( ; 2 IP S, we approximae he expeced value of P (Nj using he formula X9999 E [P IP S (Nj ] = f P IP S (Nj = :50 + n=0 4 [( n= :50 + (n + :5=0 4 ( :50 + (n :5=0 4 ]g +P IP S (Nj = :50( :50 + :5=0 4 +P IP S (Nj = :50( ( :50 :5=04 where is he cumulaive disribuion funcion for a sandard normal variable. his is jus numerical inegraion wih grid poins for rend in aion of = -50%, %, %,..., 49.99%, 50%. April205 IP S Once we solve for he values of and 2 which equae E [P (Nj ] and July205 IP S E [P (Nj ] wih heir observed prices, we can solve for he probabily of de aion using he formula. E [Pr(IR+5 M; M < j ] = X9999 f n= Z REF cpi M f(xj = :50 + n=0 4 dx[( :50 + (n + :5=0 4 ( :50 + (n :5=0 4 ] +P IP S (Nj = :50( :50 + :5=0 4 +P IP S (Nj = :50( ( :50 :5=04 where f(xj dx is he probabiliy of de aion for a known value of shown in (B6. Appendix D: Adjusing Wrigh s model for seasonaliy in he CPI. 0

11 Wrigh derives he following formula for calculaing a lower bound on he probabiliy of de aion. In order o keep he noaion consisen wih he above secions, we modify his noaion slighly. hen Wrigh s formula is Pr(IR +5 M; M < = ( r((5 M + 8 IP S0yr log(ir =IR +5 M+ : 4 ; M 5+ : 4 +5 M; M where r is he spread beween he real yield on he o -he-run 0-year IPS and onhe-run 5-year IPS, M is he number of years ago ha he on-he-run 5-year IPS was issued [currenly April 5, 200], (5 M is he number of years unil he on-he-run 5-year IPS issue expires, and (5 M + is he number of years unil June of he year ha 8 he 5-year IPS maures. he IR erm refers o he curren index-raio of he on-he-run 5-year IPS and IR +(5 M; M IP S0yr +5 M+ : 4 ; M 5+ : 4 is he curren index-raio of he o -he-run 0-year IPS ha maures 3 monhs afer he 5-year IPS maures (ha is why a : erm 4 appears. he raio of he index raios in he denomanaior is also he reference CPI on he day he 5-year IPS was issued divided by he reference CPI on he day he 0-year IPS was issued. Wrigh s formula can herefore be rewrien as Pr(IR+5 M; M < = ( rm log(re fcp I IssueDae5yr = Re fcp I IssueDae5yr where M = (5 M +. Our modi caion of Wrigh s formula will adjus r o 8 rsa by doing he following for boh he 5-year and 0-year IPS issues. (A Se = = 0, so ha here is no uncerainy in in aion, bu coninue o assume ha non-seasonally adjused CPI in aion varies from monh o monh according o he seasonal paern described in Appendix B. Given a single IPS price eiher he 5-year IPS or 0-year IPS we can hen solve for using he same formulas in appendix B. (B Using he value of solved for in sep (, compue wha he value of he IPS issue would be if here were no seasonaliy in he CPI. his can be seing CumSA +h = 0 for h > 0, and plugging ino equaion ( o solve for he nominal IPS price, i.e. wha you would pay in acual dollars for he IPS securiy. (C Conver he nominal IPS price o a real IPS price using he sandard conversion formula provided a p/news/ips&zcbrevised.pp (see slide 4. (D Compue he real yield from he real bond price.

12 r SA is hen calculaed as he spread beween he modi ed real yields of he o -he-run 0-year IPS and he on-he-run 5-year IPS calculaed from he above four seps. he modi ed lower bound on he probabiliy of de aion is hen +5 M; M < r SA M SA = ( log(re fcp I IssueDae5yr = Re fcp I IssueDae5yr Pr(IR he exac probabiliy calculaed wih he IPS pricing model described above along wih he Wrigh lower bound and seasonally adjused Wrigh lower bound are shown in he gure below. References Akeson, Andrew, Ohanian Lee. "Are Phillips Curves Useful for Forecasing In aion," FRB Minneapolis Quarerly Review, Winer 200, pp.2-. Campbell, John Y, Shiller, Rober J., and Viceira Luis M. "Undersanding In aion- Indexed Bond Markes," Brookings Papers on Economic Aciviiy, Spring 2009, pp Gurkayank Refe, and Sack, Brian, and Wrigh Jonahan. "he IPS Yield Curve and In aion Compensaion," American Economic Journal: Macroeconomics, 200, 2:, pp Nelson, Charles R., and Andrew F. Siegel. Parsimonious Modeling of Yield Curves. Journal of Business, 987, 60(4: Sack, Brian. "Deriving In aion Expecaions From Nominal and In aion-indexed reasury Yields.", Federal Reserve Board Finance and Economics Discussion Series, , pp.-24. Sock, James H., and Wason, Mark W. "Why Has U.S. In aion Become Harder o Forecas?", Journal of Money, Banking and Credi, Vol. 39, No., February 2007, pp Svensson, Lars E. O. Esimaing and Inerpreing Forward Ineres Raes: Sweden Naional Bureau of Economic Research Working Paper 487, 994. Wrigh Jonahan. "Commen on Undersanding In aion-indexed Bond Markes," Brookings Papers on Economic Aciviy, Spring 2009, pp

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