te Finance (4th Edition), July 2017.

Transcription

1 Tim-Varying Rats of Rturn and th Yild Curv Whn Rats of Rturn ar Diffrnt In this chaptr, w will mak th world a littl mor complx and a lot mor ralistic, although w ar still assuming prfct forsight and prfct markts. Th first assumption that w will abandon is that rats of rturn ar th sam no mattr what th invstmnt tim horizon is. In th prvious chaptrs, th intrst rat was th sam vry priod if a 30-yar bond offrd an intrst rat of 5% pr annum, so did a 1-yar bond. But this is not th cas in th ral world. Rats of rturn usually vary with th lngth of tim an invstmnt rquirs. Th U.S. Trasury Dpartmnt Rsourc Cntr informs you of this fact vry day. For xampl, on Dc 31, 2015, a U.S. Trasury paid 0.65% pr annum for a paymnt to b dlivrd in on yar (Dc 31, 2016), but 3.01% pr annum for a paymnt to b dlivrd in 30 yars (Dc 31, 2045): In Prcnt Pr Annum 1m 3m 6m 1y 2y 3y 5y 7y 10y 20y 30y nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition 12/31/ Is this stuff that just bond tradrs nd to know? Not at all. In fact, this stuff mattrs to you, too. Hav you vr wondrd why your bank s on-month CD offrd only 0.21% pr annum, whil thir fiv-yar CD offrd 0.8% pr annum? (Ths wr th avrag national rats on Dc 31, 2015.) Which should you choos? Why? And dos inflation mattr? CEOs must also know how to compar what thy should hav to pay for invstors willing to giv thm mony for a rturn promis in a 1-yar projct vs. a 30-yar projct. If th U.S. Trasury has to offr highr rats for lngthir invstmnt rturn priods, surly so will firms! In this chaptr, you will larn how to work with tim-dpndnt rats of rturn and inflation. In addition, this chaptr contains an (optional) sction that xplains th U.S. Trasury yild curv. 75 (4th Ed 5 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

2 76 Tim-Varying Rats of Rturn and th Yild Curv Intrst rats can diffr basd on th lngth of th commitmnt. A compounding xampl with tim-dpndnt rats of rturn. 5.1 Working with Tim-Varying Rats of Rturn In th ral world, rats of rturn usually diffr dpnding on whn th paymnts ar mad. For xampl, th intrst rat nxt yar could b highr or lowr than it is this yar. Morovr, it is oftn th cas that long-trm bonds offr diffrnt intrst rats than short-trm bonds. You must b abl to work in such an nvironmnt, so lt m giv you th tools. Compounding Diffrnt Rats of Rturn Fortunatly, whn working with tim-varying intrst rats, all th tools you hav larnd in prvious chaptrs rmain applicabl (as promisd). In particular, compounding still works xactly th sam way. For xampl, what is th two-yar holding rat of rturn if th rat of rturn is 20% in th first yar and 30% in th scond yar? (Th lattr is somtims calld th rinvstmnt rat.) You can dtrmin th two-yar holding rat of rturn from th two 1-yar rats of rturn using th sam compounding formula as bfor: (1 + r 0,2 ) = (1 + 20%) (1 + 30%) = (1 + 56%) (1 + r 0,1 ) (1 + r 1,2 ) = (1 + r 0,2 ) Subtract 1, and th answr is a total two-yar holding rat of rturn of 56%. If you prfr it shortr, r 0,2 = = = 56% Th calculation is not concptually mor difficult, but th notation is. You hav to subscript not just th intrst rats that bgin now, but also th intrst rats that bgin in th futur. Thrfor, most of th xampls in this chaptr must us two subscripts: on for th tim whn th mony is dpositd, and on for th tim whn th mony is rturnd. Thus, r 1,2 dscribs an intrst rat from tim 1 to tim 2. Asid from this xtra notation, th compounding formula is still th vry sam multiplicativ on-plus formula for ach intrst rat (subtracting 1 at th nd). nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Th gnral formula for compounding ovr many priods. You can also compound to dtrmin holding rats of rturn in th futur. For xampl, if th 1-yar rat of rturn is 30% from yar 1 to yar 2, 40% from yar 2 to yar 3, and 50% from yar 3 to yar 4, thn what is your holding rat of rturn for invsting bginning nxt yar for thr yars? It is Givn: r 1,2 = 30% r 2,3 = 40% r 3,4 = 50% (1 + r 1,4 ) = (1 + 30%) (1 + 40%) (1 + 50%) = ( %) (1 + r 1,2 ) (1 + r 2,3 ) (1 + r 3,4 ) = (1 + r 1,4 ) Subtracting 1, you s that th thr-yar holding rat of rturn for an invstmnt that taks mony nxt yar (not today!) and rturns mony in 4 yars (appropriatly calld r 1,4 ) is 173%. Lt s b clar about th timing. For xampl, say it was midnight of Dcmbr 31, 2016, right now. This would b tim 0. Tim 1 would b midnight Dcmbr 31, 2017, and this is whn you would invst your $1. Thr yars latr, on midnight Dcmbr 31, 2020 (tim 4), you would rciv your original dollar plus an additional $1.73, for a total rturn of $2.73. Intrst rats that bgin right now whr th first subscript would b 0 ar usually calld spot rats. Intrst rats that bgin in th futur ar usually calld forward rats. (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

3 5.1. Working with Tim-Varying Rats of Rturn 77 Q 5.1. If th first-yar intrst rat is 2% and th scond yar intrst is 3%, what is th two-yar total intrst rat? Q 5.2. Although a two-yar projct had rturnd 22% in its first yar, ovrall it lost half of its valu. What was th projct s rat of rturn aftr th first yar? Q 5.3. From th closing of Dcmbr 31, 2009 to Dcmbr 31, 2015, Vanguard s S&P 500 fund (which rcivd and paid dividnds on th undrlying constitunt stocks to its fund invstors, but chargd administration fs) rturnd th following annual rats of rturn: % 2.1% 16.0% 32.3% 13.7% 1.3% What was th rat of rturn ovr th first 3 yars, and what was it ovr th scond 3 yars? What was th rat of rturn ovr th whol 6 yars? Was th ralizd rat of rturn tim-varying? Q 5.4. A projct lost on-third of its valu th first yar, thn gaind fifty prcnt of its valu, thn lost two-thirds of its valu, and finally doubld in valu. What was th avrag rat of rturn? What was th invstmnt s ovrall four-yar rat of rturn? If on is positiv, is th othr, too? Annualizd Rats of Rturn Tim-varying rats of rturn crat a nw complication that is bst xplaind by an analogy. Is a car that travls 163,680 yards in 93 minuts fast or slow? It is not asy to say, bcaus you ar usd to thinking in mils pr sixty minuts, not in yards pr ninty-thr minuts. It maks sns to translat spds into mils pr hour for th purpos of comparing thm. You can vn do this for sprintrs, who run for only 10 sconds. Spds ar just a standard masur of th rat of accumulation of distanc pr unit of tim. Pr-unit standard masurs ar statistics that ar concptual aids. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Th sam issu applis to rats of rturn: A rat of rturn of 58.6% ovr 8.32 yars is not as asy to compar to othr rats of rturn as a rat of rturn pr yar. Thrfor, most rats of rturn ar quotd as annualizd rats. Th avrag annualizd rat of rturn is just a convnint unit of masurmnt for th rat at which mony accumulats a sort-of-avrag masur of prformanc. Of cours, whn you comput such an annualizd rat of rturn, you do not man that th invstmnt arnd th sam annualizd rat of rturn of, say, 5.7% ach yar just as th car nd not hav travld at 60 mph (163,680 yards in 93 minuts) ach instant. If you wr arning a total thr-yar holding rat of rturn of 173% ovr th thr-yar priod, what would your annualizd rat of rturn b? Th answr is not th avrag rat of rturn of 173%/3 57.7%, bcaus if you arnd 57.7% pr yar, you would hav ndd up with %, not 173%. This incorrct answr of 57.7% ignors th compoundd intrst on th intrst that you would arn aftr th first and scond yars. Instad, to comput th annualizd rat of rturn, you nd to find a singl hypothtical annual rat of rturn that, if you rcivd it ach and vry yar, would giv you a thr-yar holding rat of rturn of 173%. How can you comput this? Call this hypothtical annual rat that you would hav to arn ach yar for thr yars r 3 (not th bar abov th 3 to dnot annualizd) in ordr to nd up with a holding rat of rturn of 173%. To find r 3, solv th quation or, for short, (1 + r 3 ) (1 + r 3 ) (1 + r 3 ) = ( %) (1 + r 3 ) (1 + r 3 ) (1 + r 3 ) = (1 + r 0,3 ) A pr-unit standard for rats of rturn: annualizd rats. Rturn to our xampl: You want to annualiz our thr-yar total holding rat of rturn. To find th t-yar annualizd intrst rat, tak th t-th root of th total rturn (t is numbr of yars). (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

4 78 Tim-Varying Rats of Rturn and th Yild Curv Math Background, Chaptr A, Pg.621. IMPORTANT (1 + r 3 ) 3 = ( %) (1 + r t ) t = (1 + r 0,t ) (5.1) In our xampl, th holding rat of rturn r 0,3 is known (173%) and th annualizd rat of rturn r 3 is unknown. Earning th sam rat (r 3 ) thr yars in a row should rsult in a holding rat of rturn of 173%. It is a smoothd-out rat of rturn of th thr yars rats of rturn. Think of it as a hypothtical, singl, constant-spd rat at which your mony would hav ndd up as quickly at 173% as it did with th 30%, 40%, and 50% individual annual rats of rturn. Th corrct solution for r 3 is obtaind by computing th third root of 1 plus th total holding rat of rturn: (1 + r 3 ) = ( %) (1 /3) Confirm with your calculator that r %, = % % (1 + r 0,t ) (1/t) = t 1 + r0,t = (1 + r t ) ( %) (1 + r 3 ) (1 + r 3 ) (1 + r 3 ) = (1 + r 0,3 ) In sum, if you invstd mony at a rat of 39.76% pr annum for 3 yars, you would nd up with a total thr-yar holding rat of rturn of 173%. As is th cas hr, for vry long priods, th ordr of magnitud of th annualizd rat will oftn b so diffrnt from th holding rat that you will intuitivly immdiatly rgistr whthr th quantity r 0,3 or r 3 is mant. In th ral world, vry fw rats of rturn, spcially ovr long horizons, ar quotd as holding rats of rturn. Almost all rats ar quotd in annualizd trms instad. Th total holding rat of rturn ovr t yars, calld r 0,t, is translatd into an annualizd rat of rturn, calld r t, by taking th t-th root: nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Translating long-trm dollar rturns into annualizd rats of rturn. (1 + r t ) = t 1 + r0,t = (1 + r 0,t ) 1/t Compounding th annualizd rat of rturn ovr t yars yilds th total holding rat of rturn. You also will oftn nd to comput annualizd rats of rturn from payoffs yourslf. For xampl, what annualizd rat of rturn would you xpct from a $100 invstmnt today that promiss a rturn of $240 in 30 yars? Th first stp is computing th total holding rat of rturn. Tak th nding valu ($240) minus your bginning valu ($100), and divid by th bginning valu. Thus, th total 30-yar holding rat of rturn is r 0,30 = $240 $100 $100 = 140% C 30 C 0 r 0,30 = C 0 Th annualizd rat of rturn is th rat r 30, which, if compoundd for 30 yars, offrs a 140% rat of rturn, Solv this by taking th 30th root, (1 + r 30 ) 30 = ( %) (1 + r t ) t = (1 + r 0,t ) (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

5 5.1. Working with Tim-Varying Rats of Rturn 79 (1 + r 30 ) = ( %) 1/30 = % % (1 + r t ) = (1 + r 0,t ) 1/t = t 1 + r0,t Subtracting 1, you s that a rturn of $240 in 30 yars for an initial $100 invstmnt is quivalnt to a 2.96% annualizd rat of rturn. In th contxt of rats of rturn, compounding is similar to adding, whil annualizing is similar to avraging. If you arn 1% twic, your compoundd rat is 2.01%, similar to th rats thmslvs addd (2%). Your annualizd rat of rturn is 1%, similar to th avrag rat of rturn of 2.01%/2 = 1.005%. Th diffrnc is th intrst on th intrst. Now assum that you hav an invstmnt that doubls in valu in th first yar and thn falls back to its original valu. What would its avrag rat of rturn b? Doubling from, say, $100 to $200 is a rat of rturn of +100%. Falling back to $100 is a rat of rturn of ($100 $200)/$200 = 50%. Thrfor, th avrag rat of rturn would b [+100% + ( 50%)]/2 = +25%. But you hav not mad any mony! You startd with $100 and ndd up with $100. If you compound th rturns, you gt th answr of 0% that you wr intuitivly xpcting: ( %) (1 50%) = 1 + 0% r 0,2 = 0% (1 + r 0,1 ) (1 + r 1,2 ) = (1 + r 0,2 ) It follows that th annualizd rat of rturn r 2 is also 0%. Convrsly, an invstmnt that producs +20% followd by 20% has an avrag rat of rturn of 0% but lavs you with a loss: (1 + 20%) (1 20%) = (1 4%) r 0,2 = 4% (1 + r 0,1 ) (1 + r 1,2 ) = (1 + r 0,2 ) For vry $100 of your original invstmnt, you now hav only $96. Th avrag rat of rturn of 0% dos not rflct this loss. Both th compoundd and thrfor th annualizd rats of rturn do tll you that you had a loss: Compounding adding. Annualizing avraging. Avraging can lad to surprising rsults rturns that ar much highr than what you arnd pr yar. Look how dcptiv! nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition 1 + r 2 = (1 + r 0,2 ) = 1 4% = % r % If you wr an invstmnt advisor and quoting your historical prformanc, would you rathr quot your avrag historical rat of rturn or your annualizd rat of rturn? (Hint: Th industry standard is to quot th avrag rat of rturn, not th annualizd rat of rturn!) Mak sur to solv th following qustions to gain mor xprinc with compounding and annualizing ovr diffrnt tim horizons. Q 5.5. If you arn a rat of rturn of 5% ovr 4 months, what is th annualizd rat of rturn? Q 5.6. Assum that th two-yar holding rat of rturn is 40%. Th avrag (arithmtic) rat of rturn is thrfor 20% pr yar. What is th annualizd (gomtric) rat of rturn? Is th annualizd rat th sam as th avrag rat? Q 5.7. Is th compoundd rat of rturn highr or lowr than th sum of th individual rats of rturn? Is th annualizd rat of rturn highr or lowr than th avrag of th individual rats of rturn? Why? Q 5.8. Rturn to Qustion 5.3. What was th annualizd rat of rturn on th S&P 500 ovr th six yars in th tabl? Q 5.9. If th total holding intrst rat is 50% for a fiv-yar invstmnt, what is th annualizd rat of rturn? Q If th pr-yar intrst rat is 10% for ach of th nxt 5 yars, what is th annualizd fiv-yar rat of rturn? Mor about how arithmtic rturns ar too high in normally-distributd stock rturns, Pg.141. (4th Ed Do it! 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

6 80 Tim-Varying Rats of Rturn and th Yild Curv Duration and Maturity W somtims nd summary statistics of how long a bond or a projct will last. Maturity is th dat of th vry last paymnt of a bond. Howvr, you would probably not considr a bond that pays $1 in on yar and $1 in 30 yars a 30-yar bond it s mor lik a 15-yar bond. Th duration can b calculatd as 1 $1 + 2 $ $ $1 Duration( C 1 = $1, C 30 = $1 ) = = 15.5 yars $1 + $ $1 Duration( {C t } ) = t/w C t / t whr W is th sum of all paymnts, W = C t. Intuitivly, this bond has about a 15-yar t duration. Also intuitivly, a bond that pays $100 in on yar and $1 in 30 yars has a shortr duration, Duration( C 1 = $100, C 30 = $1 ) = 1 $ $1 $100 + $ A zro bond has a duration qual to its maturity. On important variation is th Macauly Duration, which ssntially rplacs th raw cash flow with its prsnt valu. Th intnt is to discount far-away payouts mor, which thus furthr shortns duration. For xampl, using our Dcmbr 2015 yild curv, Duration( C 1 = $1, C 100 = $1 ) = 1 $100/ $1/ $100/ $1/ Thr ar also othr masurs of duration, but thy ar all basically summary statistics of whn your avrag cash flow is going to occur. Prsnt Valus with Tim-Varying Intrst Rats nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Th PV formula still looks vry similar. Prsnt valus ar still alik and thus can b addd, subtractd, compard, and so on. Lt s procd now to nt prsnt valu with tim-varying intrst rats. What do you nd to larn about th rol of tim-varying intrst rats whn computing NPV? Th answr is ssntially nothing nw. You alrady know vrything you nd to know hr. Th nt prsnt valu formula is still NPV = PV C 0 + PV C1 + PV C2 = C 0 + = C 0 + = C 0 + C r 0,1 C r 1 C r 0,1 C r 0,2 C 2 (1 + r 2 ) PV C 3 (4th Ed + C r 0,3 C 3 (1 + r 3 ) 3 + C 2 (1 + r 0,1 ) (1 + r 1,2 ) + C 3 (1 + r 0,1 ) (1 + r 1,2 ) (1 + r 2,3 ) + Th only novlty is that you nd to b mor carful with your subscripts. You cannot simply assum that th multiyar holding rturns (.g., 1 + r 0,2 ) ar th squard 1-yar rats of rturn ((1 + r 0,1 ) 2 ). Instad, you must work with tim-dpndnt costs of capital (intrst rats). That s it. For xampl, say you hav a projct with an initial invstmnt of $12 that pays $10 in on yar and $8 in fiv yars. Assum that th 1-yar intrst rat is 5% and th fiv-yar annualizd intrst rat is 6% pr annum. In this cas, 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

7 5.1. Working with Tim-Varying Rats of Rturn 81 PV $10 in 1 yar = $ $9.52 PV $8 in 5 yars = $ $5.98 It follows that th projct s total valu today (tim 0) is $ If th projct costs $12, its nt prsnt valu is NPV = $12 + $ NPV = C C r 0,1 $ $3.50 C 5 = NPV 1 + r 0,5 You can also rwork a mor involvd projct, similar to that in Exhibit 2.3,. But to mak it mor intrsting, lt s now us a hypothtical currnt trm structur of intrst rats that is upward-sloping. Assum this projct rquirs an appropriat annualizd discount rat of 5% ovr on yar, and 0.5% mor for vry subsqunt yar, so that th cost of capital rachs 7% annualizd in th fifth yar. Th valuation mthod works th sam way as it did in Exhibit 2.3 you only hav to b a littl mor carful with th intrst rat subscripts. Th projct s valu is thus Projct Annualizd Com- Discount Tim Cash Flow Rat poundd Factor Valu 1 t C t r t r 0,t 1 + r 0,t PV C t Today $900 any $ Y1 +$ % 5.0% $ Y2 +$ % 11.3% $ Y3 +$ % 19.1% $ Y4 +$ % 28.6% $ Y5 $ % 40.3% $71.30 Hr is a typical NPV xampl. Hypothtical Projct Cash Flows, Exhibit 2.3, Pg.28. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Nt Prsnt Valu (Sum): $45.65 Q A projct costs $200 and will provid cash flows of +$100, +$300, and +$500 in conscutiv yars. Th annualizd intrst rat is 3% pr annum ovr on yar, 4% pr annum ovr two yars, and 4.5% pr annum ovr thr yars. What is this projct s NPV? (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

8 82 Tim-Varying Rats of Rturn and th Yild Curv Inflation is th incras in th pric of th sam good. Inflation mattrs whn contracts ar not writtn to adjust for it. Th CPI is th most common inflation masur. 5.2 Inflation Lt s mak our world vn mor ralistic and complx by working out th ffcts of inflation. Inflation is th procss by which th sam good costs mor in th futur than it dos today. With inflation, th pric lvl is rising and thus mony is losing its valu. For xampl, if inflation is 100%, an appl that costs $0.50 today will cost $1 nxt yar, a banana that costs $2 today will cost $4, and corporat financ txtbooks that cost $200 today will cost $400. Inflation may or may not mattr in a corporat contxt, dpnding on how contracts ar writtn. If you ignor inflation and writ a contract that promiss to dlivr brad for th pric of $1 nxt yar, it is said to b in nominal trms and you may hav mad a big mistak. Th mony you will b paid will b worth only half as much. You will only b abl to buy on appl for ach loaf of brad that you had agrd to sll for $1, not th two appls that anyon ls will njoy. On th othr hand, you could writ your contract in ral trms (or inflation-indxd trms) today, in which cas th inflationary pric chang would not mattr. That is, you could build into your promisd banana dlivry pric th inflation rat from today to nxt yar. An xampl would b a contract that promiss to dlivr bananas at th rat of four appls pr banana. If a contract is indxd to inflation, thn inflation dos not mattr. Howvr, in th Unitd Stats inflation oftn dos mattr, bcaus most contracts ar in nominal trms and not inflation-indxd. Thrfor, you hav to larn how to work with inflation. What ffct, thn, dos inflation hav on rturns? On (nt) prsnt valus? This is our nxt subjct. Masuring th Inflation Rat Th first important qustion is how you should dfin th inflation rat. Is th rat of chang of th pric of appls th bst masur of inflation? What if appls (th fruit) bcom mor xpnsiv, but Appls (th computrs) bcom lss xpnsiv? Dfining inflation is actually rathr tricky. To solv this problm, conomists hav invntd baskts or bundls of goods and srvics that ar dmd to b rprsntativ. Economists thn masur an avrag pric chang for ths itms. Th official sourc of most inflation masurs is th Burau of Labor Statistics (BLS), which dtrmins th compositions of a numbr of common bundls (indxs) and publishs th avrag total pric of ths bundls on a monthly basis. Th most prominnt such inflation masur is a hypothtical bundl of avrag houshold consumption, calld th Consumr Pric Indx (CPI). (Th CPI componnts ar roughly: housing 40%, food 20%, transportation 15%, mdical car 10%, clothing 5%, ntrtainmnt 5%, othrs 5%.) Th BLS offrs inflation data at and th Wall Strt Journal prints th prcnt chang in th CPI at th nd of its rgular column Mony Rats. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Th CPI mattrs vn if it is calculatd incorrctly. From Dc 2014 to Dc 2015, th inflation rat was a rmarkably low 0.7% pr annum. (And thr wr som months with ngativ inflation rats, too calld dflation!) Yar CPI A numbr of othr indxs ar also commonly usd as inflation masurs, such as th Producr Pric Indx (PPI) or th broadr GDP Dflator. Thy typically mov vry similarly to th CPI. Ovr short priods, on xpcts ths rats to mov fairly clos togthr (on avrag, not vry month); but ovr longr priods, thy can divrg. Thr ar also mor spcializd bundls, such as computr or flash mmory chip inflation indxs (thir prics usually dclin), or pric indxs for particular rgions. Th official inflation rat is not just a numbr that mirrors rality it is important in itslf, bcaus many contracts ar spcifically indxd to a particular inflation dfinition. For xampl, vn if actual tru inflation is zro, if th officially rportd CPI rat is positiv, th govrnmnt (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

9 5.2. Inflation 83 Th Grman Hyprinflation of 1922 Many conomists now bliv that a modst inflation rat btwn 1% and 3% pr yar is a halthy numbr. It s not so asy to maintain. Th Roman Empror Gallinus is still infamous two-thousand yars latr for having changd th valu of th Roman Dnarius by rducing its silvr contnt by a factor of 100. Th most famous pisod of xtrm inflation (hyprinflation) occurrd in Grmany from August 1922 to Novmbr Prics mor than quadrupld vry month. Th pric for goods was highr in th vning than in th morning! Stamps had to b ovrprintd by th day, and shopprs wnt out in th morning with bags of mony that wr worthlss by th nd of th day. By th tim Grmany printd 1,000 billion Mark Bank Nots, no on trustd Marks anymor. This hyprinflation was stoppd only by a drastic currncy and financial systm rform. But high inflation is not just a historic artifact. In 2015, Vnzula had an inflation rat of 180%. Don t trust Vnzulan Bolivars! Yt rcnt xprinc has humbld us conomists (furthr) by proving that it can also b difficult to push up inflation. In th Grat Rcssion (th financial crisis of ), th Fd trid to ful inflation by pushing mony into th hands of consumrs. Th ida was to gt thm to los just a littl trust in th currncy and spnd it, so as to rais th valu of much undrwatr ral stat. But consumrs turnd around and dpositd th mony back into thir banks, which in turn rdpositd th mony with you gussd it th Fd. must pay out mor to Social Scurity rcipints. Th lowr th official inflation rat, th lss th govrnmnt has to pay. You would thrfor think that th govrnmnt has th incntiv to undrstat inflation. But, strangly, this has not bn th cas. On th contrary, thr ar strong political intrst groups that hindr th BLS from vn fixing mistaks that vryon knows ovrstat th CPI that is, corrctions that would rsult in lowr official inflation numbrs. In 1996, th Boskin Commission, consisting of a numbr of minnt conomists, found that th CPI ovrstats inflation by about 74 basis points pr annum a hug diffrnc. Th main rasons wr and continu to b that th BLS has bn tardy in rcognizing th growing importanc of such factors as ffctiv pric dclins in th computr and tlcommunications industris and th rol of suprstors such as Wal-Mart and Targt. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Bfor w gt moving, a final warning: Th common statmnt in today s dollars can b ambiguous. Most popl man inflationadjustd. Som popl man prsnt valus (i.., compard to an invstmnt in risk-fr bonds ). Whn in doubt, ask! Q Rad th Burau of Labor Statistics wbsit dscriptions of th CPI and th PPI. How dos th CPI diffr concptually from th PPI? Ar th two official rats diffrnt right now? Ral and Nominal Intrst Rats To work with inflation and to larn how you would proprly indx a contract for inflation, you first nd to larn th diffrnc btwn a nominal rturn and a ral rturn. Th nominal rat is what is usually quotd a rturn that has not bn adjustd for inflation. In contrast, th ral rat of rturn somhow taks out inflation from th nominal rat in ordr to calculat a rturn as if thr had bn no pric inflation to bgin with. Th ral rturn rflcts th fact that, in th prsnc of inflation, dollars in th futur will hav lss purchasing powr than dollars today. IMPORTANT Nominal rturns ar what is normally quotd. Ral rturns ar adjustd for inflation. Thy ar what you want to know if you want to consum. (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

10 84 Tim-Varying Rats of Rturn and th Yild Curv An xtrm 100% inflation rat xampl: Prics doubl vry yar. Hr is th corrct convrsion formula from nominal to ral rats. It masurs your trad-off btwn prsnt and futur consumption, taking into account th chang in prics. Start with a simpl xaggratd scnario: Assum that th inflation rat is 100% pr yar and you can buy a bond that promiss a nominal intrst rat of 700%. What is your ral rat of rturn? To find out, assum that $1 buys on appl today. With an inflation rat of 100%, you nd $2 nxt yar to buy th sam appl. Your gross rturn will b $1 ( %) = $8 for today s $1 of invstmnt. But this $8 will apply to appls costing $2 ach. Your $8 will buy 4 appls, not 8 appls. Your ral rat of rturn (1 appl yilds 4 appls) is thrfor r ral = (4 Appls for $8) (1 Appl for $2) (1 Appl for $2) = 300% For ach dollar invstd today, you will b abl to buy only 300% mor appls nxt yar (not 700% mor) than you could buy today. This is bcaus inflation will rduc th purchasing powr of your dollar by mor than on half. Th corrct formula to adjust for th inflation rat (π) is again a on-plus typ formula. In our xampl, it is ( %) = ( %) ( %) (1 + r nominal ) = (1 + r ral ) (1 + π) Turning this formula around givs you th ral rat of rturn, (1 + r ral ) = % % (1 + r ral ) = (1 + r nominal) (1 + π) = % In plain English, a nominal intrst rat of 700% is th sam as a ral intrst rat of 300%, givn an inflation rat of 100%. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition IMPORTANT For small rats, adding/ subtracting is an okay approximation. Bills, nots, and bonds, Th rlation btwn th nominal rat of rturn (r nominal ), th ral rat of rturn (r ral ), and th inflation rat (π) is (1 + r nominal ) = (1 + r ral ) (1 + π) (5.2) As with compounding, if th rats ar small, th mistak of just subtracting th inflation rat from th nominal intrst rat to obtain th ral intrst rat is not too grav. For xampl, with our (30-yar) 3% Trasury, if inflation wr to rmain 0.7%, th corrct ral intrst rat would b 2.28% and not 2.30%: Sct. 5.3, Pg.86. (1 + 3%) ( %) ( %) % + 0.7% Adding or compounding intrst rats, and th cross-trm, Pg.20. (1 + r nominal ) = (1 + r ral ) (1 + π) = 1 + r ral + π + r ral π }{{} cross-trm Th diffrnc btwn th corrct and th approximation, i.., th cross-trm, is trivial, and asily swampd by masurmnt nois in th currnt inflation rat and uncrtainty about futur inflation rats. Howvr, whn inflation and intrst rats ar high as thy wr, for xampl, in th Unitd Stats in th lat 1970s thn th cross-trm can b important. (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

11 5.2. Inflation 85 A positiv tim valu of mony that th sam amount of mony today is worth mor than tomorrow is only tru for nominal quantitis, not for ral quantitis. Only nominal intrst rats ar (usually) not ngativ. In th prsnc of inflation, ral intrst rats not only can b ngativ, but oftn ar ngativ. For xampl, in Dcmbr 2015, th on-month Trasury paid 0.14% whil th inflation rat was 0.7%, implying a ral intrst rat of about 0.56%. Evry dollar you invstd in such U.S. Trasuris would b worth lss in ral purchasing powr on month latr. You would hav ndd up with mor cash but also with lss purchasing powr. Gold Ral intrst rats can b ngativ. Somtims, th pric of gold has bn usd as a masur of inflation. Although gold is not a grat masur of purchasing powr in gnral, it dos mak it asy to conduct long-run comparisons. A Roman lgionary was paid th quivalnt of 2.31 oz of gold a yar. A U.S. Army privat nowadays is paid th quivalnt of oz about fiv tims as much. A Roman cnturion was paid oz. A U.S. army captain is paid 27.8 oz, about a quartr lss. Thus, as an asst class, an invstmnt in gold would hav arnd a rat of rturn roughly in lin with th incom growth ovr two millnia. Erb and Harvy (2013) Q From mmory, writ down th rlationship btwn nominal rats of rturn (r nominal ), ral rats of rturn (r ral ), and th inflation rat (π). Q Th nominal intrst rat is 20%. Inflation is 5%. What is th ral intrst rat? Inflation in Nt Prsnt Valus Whn it coms to inflation and nt prsnt valu, thr is a simpl rul: Nvr mix appls and orangs. Th bauty of NPV is that vry projct s cash flows ar translatd into th sam units: today s dollars. Kp vrything in th sam units in th prsnc of inflation, so that this NPV advantag is not lost. Whn you us th NPV formula, always discount nominal cash flows with nominal discount rats, and ral (inflation-adjustd) cash flows with ral (inflation-adjustd) discount rats. Th most fundamntal rul: nvr mix appls and orangs. Nominal cash flows must b discountd with nominal intrst rats. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Lt s rturn to our appl xampl. With 700% nominal intrst rats and 100% inflation, th ral intrst rat is ( %)/( %) 1 = 300%. What is th valu of a projct that givs 12 appls nxt yar, givn that appls cost $1 ach today and $2 ach nxt yar? Thr ar two mthods you can us: 1. Discount th nominal cash flow of 12 appls nxt yar ($2 12 = $24) with th nominal intrst rat. Thus, th 12 futur appls ar worth Nominal Cash Flow 1 + Nominal Rat = $ % = $3 2. Discount th ral cash flows of 12 appls nxt yar with th ral intrst rat. Thus, th 12 futur appls ar worth Our xampl discountd both in ral and nominal trms. Discount nominal cash flows with nominal rats. Discount ral cash flows with ral rats. (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

12 86 Tim-Varying Rats of Rturn and th Yild Curv IMPORTANT Usually, it is bst to work only with nominal quantitis. Ral Cash Flow 12 Appls = 1 + Ral Rat % = 3 Appls in today s appls. Bcaus an appl costs $1 today, th 12 appls nxt yar ar worth $3 today. Both th ral and th nominal mthods arriv at th sam NPV rsult. Th opportunity cost of capital is that if you invst on appl today, you can quadrupl your appl holdings by nxt yar. Thus, a 12-appl harvst nxt yar is worth 3 appls to you today. Th highr nominal intrst rats alrady rflct th fact that nominal cash flows nxt yar ar worth lss than thy ar this yar. As simpl as this may sound, I hav sn corporations first work out th ral valu of thir goods in th futur, and thn discount this with standard nominal intrst rats. Just don t! Discount nominal cash flows with nominal intrst rats. Discount ral cash flows with ral intrst rats. Eithr works. Nvr discount nominal cash flows with ral intrst rats, or vic-vrsa. If you want to s this in algbra, th rason that th two mthods com to th sam rsult is that th inflation rat cancls out, PV = = $ % = 12A % = 12A ( %) ( %) ( %) N = 1 + r nominal R = 1 + r ral R (1 + π) (1 + r ral ) (1 + π) whr N is th nominal cash flow, r is th ral cash flow, and π is th inflation rat. Most of th tim, it is asir to work in nominal quantitis. Nominal intrst rats ar far mor common than ral intrst rats, and you can simply us publishd inflation rats to adjust th futur pric of goods to obtain futur xpctd nominal cash flows. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Th simplst and most important bnchmark bonds nowadays ar Trasuris. Thy hav known and crtain payouts. U.S. Trasury bills, nots, and bonds hav diffrnt maturitis. Q If th ral intrst is 3% pr annum and th inflation rat is 8% pr annum, thn what is th prsnt valu of a $500,000 nominal paymnt nxt yar? 5.3 U.S. Trasuris and th Yild Curv It is now tim to talk in mor dtail about th most important financial markt in th world today: th markt for bonds issud by th U.S. govrnmnt. Ths bonds ar calld Trasuris and ar prhaps th simplst projcts around. This is bcaus, in thory, Trasuris cannot fail to pay. Thy promis to pay U.S. dollars, and th U.S.-controlld Fdral Rsrv has th right to print mor U.S. dollars if it wr vr to run out. Thus, thr is no uncrtainty about rpaymnt for Trasuris. (In contrast, som Europan countris or U.S. stats that borrow in currncis that thy cannot crat may wll not hav th mony to pay and thrfor dfault.) Th shorthand Trasury coms from th fact that th dbt itslf is issud by th U.S. Trasury Dpartmnt. Thr ar thr main typs: 1. Trasury bills (oftn abbrviatd as T-bills) hav maturitis of up to on yar. 2. Trasury nots hav maturitis btwn on and tn yars. 3. Trasury bonds hav maturitis gratr than tn yars. (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

13 5.3. U.S. Trasuris and th Yild Curv 87 Th 30-yar bond is oftn calld th long bond. Togthr, th thr ar usually just calld Trasuris. Concptually, thr is rally no diffrnc among thm. All ar rally just obligations issud by th U.S. Trasury. Indd, thr can b Trasury nots today that ar du in 3 months such as a 9-yar Trasury not that was issud 8 yars and 9 months ago. This is rally th sam obligation as a 3-month Trasury bill that was just issud. Thus, w shall b (vry) casual with nam distinctions. In lat 2015, th Unitd Stats Fdral Govrnmnt owd ovr $18 trillion in Trasury obligations (on a GDP of about $17.5 trillion). With a population of 322 million, this dbt translatd to ovr $55,000 pr prson. With 125 million housholds and 121 million full-tim workrs, it rprsntd about $150,000 pr houshold or workr. (A wors problm, howvr, is that th Unitd Stats has alrady promisd bnfits to futur rtirs that far xcd this numbr.) But th Unitd Stats also has assts. It owns mor than $100 trillion in land, infrastructur, and minral rights undr th land. Aftr Trasuris ar sold by th govrnmnt, thy ar thn activly tradd in what is on of th most important financial markts in th world today. It would not b uncommon for a ddicatd bond tradr to buy $100 million of a Trasury not originally issud 10 yars ago that has 5 yars rmaining, and 10 sconds latr sll $120 million of a thr-yar Trasury not issud 6 yars ago. Larg buyrs and sllrs of Trasuris ar asily found, and transaction costs ar vry low. Trading volum is hug: Around 2015, it was about $500 billion pr trading day. Thrfor, th annual trading volum in U.S. Trasuris about 252 $500 billion $130 trillion, whr 252 is th approximat numbr of trading days pr yar was about an ordr of magnitud largr than th annual U.S. gross domstic product (GDP). Who owns thm? About $6 trillion is owd to forignrs, with th Chins (our largst crditor) holding about $1.2 trillion of our bonds. about $12 trillion is owd to ourslvs. Th U.S. Fdral Rsrv stimatd that domstically about 20% wr hld by individuals, 25% by banks and mutual funds, 15% by public and privat pnsion funds, 15% by stat and local govrnmnts, and 25% by othr invstors. Intrst rats on Trasuris chang vry momnt, dpnding on thir maturity trms. Fortunatly, you alrady know how to handl tim-varying rats of rturn, so w can now put your knowldg to th tst. Th principal tool for working with Trasury bonds is th yild curv (or trm structur of intrst rats or just th trm structur). It is a graphical rprsntation, whr th tim to maturity is on th x-axis and th annualizd intrst rats ar on th y-axis. Thr ar also yild curvs on non-trasury bonds, but th Trasury yild curv is so prominnt that unlss clarifid furthr, th yild curv should b assumd to man invstmnts in U.S. Trasuris. (A mor prcis nam would b th U.S. Trasuris yild curv. ) This yild curv is so important that most othr dbts in th financial markts, lik mortgag rats or bank lnding rats, ar bnchmarkd rlativ to th Trasury yild curv. For xampl, if your firm wants to issu a fiv-yar bond, your crditors will want to compar your intrst rat to that offrd by quivalnt Trasuris, and oftn will vn dscrib your bond as offring x basis points abov th quivalnt Trasury. Magnituds Th Trasuris markt is on of th most important financial markts in th world. Who owns thm? Th yild curv shows th annualizd intrst rat as a function of bond maturity. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Q What ar th thr typs of Trasuris? How do thy diffr? (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

14 88 Tim-Varying Rats of Rturn and th Yild Curv Annualizd rat, r, in % Yild Curv Shaps Normal (upward): May 2011 Flat: January Maturity (in months) nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Annualizd rat, r, in % (4th Ed Annualizd rat, r, in % Maturity (in months) Invrtd (downward): Dcmbr 1980 Humpd: Jun Maturity (in months) Maturity (in months) Exhibit 5.1: Various Historical Yild-Curv Shaps. Th upward slop is so common that it is considrd th normal shap. In 2016, th yild curv was normal, but not as stp as that in May A downward slop is somtims calld invrtd. Annualizd rat, r, in % 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

15 5.3. U.S. Trasuris and th Yild Curv 89 Exhibit 5.1 shows som historical yild curvs. Thy ar commonly classifid into four basic shaps: 1. Flat: Thr is littl or no diffrnc btwn annualizd short-trm and long-trm rats. A flat yild curv is basically th scnario that was th subjct of th prvious chaptr. It mans you can simplify (1 + r 0,t ) (1 + r) t. 2. Upward-sloping ( normal ): Short-trm rats ar lowr than long-trm rats. This is th most common shap. It mans that longr-trm intrst rats ar highr than shortr-trm intrst rats. Sinc 1934, th stpst yild curv (th biggst diffrnc btwn th long-trm and th short-trm Trasury rats) occurrd in Octobr 1992, whn th long-trm intrst rat was 7.3% and th short-trm intrst rat was 2.9% just as th conomy pulld out of th rcssion of As of 2016, th yild curv has bn upward-sloping sinc th Grat Rcssion. 3. Downward-sloping ( invrtd ): Short-trm rats ar highr than long-trm rats. 4. Humpd: Short-trm rats and long-trm rats ar lowr than mdium-trm rats. Invrtd and humpd yild curvs ar rlativly rar. Macroconomic Implications of Diffrnt Yild Curv Shaps Yild curvs ar oftn, but not always, upward-sloping. Economists and pundits hav long wondrd what thy can larn from th shap of th yild curv about th futur of th conomy. It appars that th yild curv shap is a usful though unrliabl and noisy signal of whr th conomy is hading. Stp yild curvs oftn signal mrgnc from a rcssion, as you will s in Exhibit 5.4 on Pag 96. Invrtd yild curvs oftn signal an impnding rcssion. But can t th Fdral Rsrv Bank control th yild curv and thrby control th conomy? It is tru that th Fd can influnc th yild curv and sinc 2008, it has workd on influncing it lik nvr bfor. But ultimatly th Fd dos not control it instad, it is th broadr dmand and supply for savings and crdit in th conomy that dtrmins it. Economic rsarch has shown that th Fd typically has a good dal of influnc on th short nd of th Trasury curv by xpanding and contracting th supply of mony and short-trm loans in th conomy but not much influnc on th long nd of th Trasury curv, spcially in th long-run. And vn in th financial crisis of 2008, th Fd s influnc on th short nd was ultimatly limitd, too th nominal rat alrady stood at 0% and thr was littl th Fd could do to drop it furthr. (Larg ngativ nominal rats ar not possibl.) In fact, by flooding th conomy with chap mony, th Fd was trying to push banks to lnd and popl to spnd mony but popl instad just dpositd th cash right back into th banks! nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition If you want to undrtak your own rsarch, you can find historical intrst rats at th St. Louis Fdral Rsrv Bank at Thr ar also th Trasury Managmnt Pags at Or you can look at SmartMony.com for historical yild curvs. PiprJaffray.com has th currnt yild curv as do many othr financial sits and nwspaprs. Financ.yahoo.com/bonds provids not only th Trasury yild curv but also yild curvs for many othr typs of bonds. An Exampl: Th Yild Curv on Dcmbr 31, 2015 Lt s focus on working with on particular yild curv. Exhibit 5.2 shows th Trasury yilds on Dcmbr 31, This yild curv had th most common upward slop. Th curv tlls you that if you had bought a 3-month Trasury at th nd of th day on Dcmbr 31, 2015, your annualizd intrst rat would hav bn 0.16% pr annum. (A $100 invstmnt would turn into $100 ( %) 1 /4 $ $ on March 31, 2016.) If you had Common data sourcs for intrst rats. W will analyz th Trasury yild curv at th nd of Dcmbr (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

16 90 Tim-Varying Rats of Rturn and th Yild Curv You can intrpolat annualizd intrst rats on th yild curv. Annualizd rat, r, in % bought a thiry-yar bond, your annualizd intrst rat would hav bn 3.01% pr annum, or $ on Dcmbr 2045 for a $100 invstmnt. Somtims it is ncssary to dtrmin an intrst rat for a bond that is not listd. This is usually don by intrpolation. For xampl, if you had wantd to find th yild for a 4- yar bond, a rasonabl guss would hav bn an intrst rat halfway btwn th 3-yar bond and th 5-yar bond. In Dcmbr 2015, this would hav bn an annualizd yild of (1.31% %)/2 1.54%. (This is not xact, as you can guss by noticing that th yild curv looks mor concav.) nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Maturity (in months) (4th Ed Months Yars /31/ Exhibit 5.2: Th Trasury Yild Curv on Dcmbr 31, Ths rats ar annualizd yilds to maturity (intrnal rats of rturn) calculatd from Trasury prics. If thy wr truly Trasury zro-bonds, thy would just b th standard discount rats computd from th final paymnt and today s pric, but w ignor th dtails hr. Such yild curvs can b found on many wbsits. Th yild curv changs vry day although day-to-day changs ar usually small. Our xampl works primarily with this particular yild curv. Sourc: Fdral Rsrv, Th Dcmbr 2015 yild curv was upward-sloping: Annualizd intrst rats wr highr for longr maturitis. As notation for th annualizd horizon-dpndnt intrst rats, w continu using our arlir mthod. W call th two-yar annualizd intrst rat r 2 (hr, 1.06%), th thr-yar Dpr: Thr ar som small inaccuracis in my dscription of yild curv computations. My main simplification is that U.S. yild curvs ar basd on smi-annually-compoundd coupon bonds in ral lif, whras our txtbook prtnds that th yild is quotd on a zro bond. In corporat financ, th yild diffrnc btwn annual compounding and smi-annual compounding is almost always inconsquntial. Howvr, if you want to bcom a fixd-incom tradr, you cannot tak this approximation litrally. Consult a ddicatd fixd-incom txt instad. 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

17 5.3. U.S. Trasuris and th Yild Curv 91 annualizd intrst rat r 3 (hr, 1.31%), and so on. It is always ths ovrlind-subscript yilds that ar graphd in yild curvs. Lt s work with this particular yild curv, assuming it is basd xclusivly on zro-bonds, so you don t hav to worry about intrim paymnts. Holding rats of rturn First, lt s figur out how much mony you will hav at maturity. That is, how much dos an invstmnt of $500,000 in U.S. two-yar nots (i.., a loan to th U.S. govrnmnt of $500,000) on Dcmbr 31, 2015, rturn on Dcmbr 31, 2017? Us th data in Exhibit 5.2. Bcaus th yild curv prints annualizd rats of rturn, th total two-yar holding rat of rturn (as in Formula 5.1) is th twic compoundd annualizd rat of rturn, so your $500,000 will turn into r 0,2 = % r 0,2 = (1 + r 2 ) (1 + r 2 ) 1 C 2 ( %) $500,000 $510,656 C 2 = (1 + r 0,2 ) C 0 on Dcmbr 31, (In th ral world, you might hav to pay a commission to arrang this transaction, so you would nd up with a littl lss.) What if you invst $500,000 into 30-yar Trasuris? Your 30-yar holding rat of rturn is r 0,30 = % r 0,30 = (1 + r 30 ) 30 1 Thus, an invstmnt of C 0 = $500,000 in Dcmbr 2015 turns into a rturn of C 30 $1.2 million by Dcmbr Computing th holding rat of rturn for 2-yar and 30-yar Trasuris. Annualizing, Formula 5.1, Pg.78. nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Forward rats of rturn Scond, lt s figur out what th yild curv in Dcmbr 2015 implid about th 1-yar intrst rat from Dcmbr 2016 to Dcmbr This would b bst namd r 1,2. It is an intrst rat that bgins in on yar and nds in two yars. As alrady mntiond, this is calld th forward rat. Th 1-yar annualizd intrst rat is r 1 = 0.65%. Th two-yar annualizd rat of rturn is r 2 = 1.06%. You alrady know that you can work out th two holding rats of rturn, r 0,1 = 0.65% and r 0,2 = (1 + r 2 ) %. You only nd to us th compounding formula to dtrmin r 1,2 : ( %) = ( %) (1 + r 1,2 ) r 1,2 1.47% (1 + r 0,2 ) = (1 + r 0,1 ) (1 + r 1,2 ) Not that this forward rat r 1,2 is highr than both r 1 and r 2 from which you computd it. Exhibit 5.3 summarizs our two-yar calculations, and xtnds thm by anothr yar. (This hlps you to chck your rsults in an xrcis blow.) On qustion you should ask yourslf is whthr I us so many subscripts in th notation just bcaus I njoy torturing you. Th answr is an mphatic no: Th subscripts ar thr for good rason. Whn you look at Exhibit 5.3, for xampl, you hav to distinguish btwn th following: th thr holding rats of rturn, r 0,t (0.65%, 2.13%, and 3.98%) th thr annualizd rats of rturn, r t (0.65%, 1.06%, and 1.31%) Lt s work out on forward rat implid by th Dcmbr 2015 yild curv. Is th prolifration of subscripts tortur or ncssity? (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi

18 92 Tim-Varying Rats of Rturn and th Yild Curv Rats of Rturn Maturity Total Holding Annualizd Compoundd Rats 1 Yar ( %) = ( %) 1 = ( %) (1 + r 0,1 ) = (1 + r 1 ) 1 = (1 + r 0,1 ) 2 Yars ( %) ( %) 2 ( %) ( %) (1 + r 0,2 ) = (1 + r 2 ) 2 = (1 + r 0,1 ) (1 + r 1,2 ) 3 Yars ( %) ( %) 3 ( %) ( %) ( %) (1 + r 0,3 ) = (1 + r 3 ) 3 = (1 + r 0,1 ) (1 + r 1,2 ) (1 + r 2,3 ) Exhibit 5.3: Rlation btwn Holding Rturns, Annualizd Rturns, and Yar-by-Yar Rturns on Dcmbr 31, 2015, by Formula. Th individually compoundd rats ar th futur intrst rats. Thy ar implid by th annualizd rats quotd in th middl column. Th txt workd out th two-yar cas. You will work out th thr-yar cas in Qustion th thr individual annual rats of rturn r t 1,t (0.65%, 1.47%, and 1.81%), whr th scond and third forward rats bgin at diffrnt momnts in th futur. In ral lif, you hav not just thr yarly Trasuris, but many Trasuris btwn 1 day and 30 yars. Anyon daling with Trasuris (or CDs or any othr fixd-incom invstmnt) that can hav diffrnt maturitis or start in th futur must b prpard to suffr doubl subscripts. If Trasuris offr diffrnt annualizd rats of rturn ovr diffrnt horizons, do corporat projcts hav to do so, too? Almost surly ys. If nothing ls, thy compt with Trasury bonds for invstors mony. And just lik Trasury bonds, many corporat projcts do not bgin immdiatly, but may tak a yar or mor to prpar. Such projct rats of rturn ar ssntially forward rats of rturn. Doubl subscripts yiks, but somtims thr is no way out of painful notation in th ral world! nc tio Ivo W. Ivo Wl17. I inanc (nc ( lch, Codition Ys, corporat projcts hav doubl subscripts, too! Q Comput th thr-yar holding rat of rturn on Dcmbr 31, Thn, using th two-yar holding rat of rturn on Dcmbr 31, 2015, and your calculatd thr-yar holding rat of rturn, comput th forward intrst rat for a 1-yar invstmnt bginning on Dcmbr 31, 2017, and nding on Dcmbr 31, Ar ths th numbrs in Exhibit 5.3? Q Rpat th calculation with th fiv-yar annualizd rat of rturn of 1.76%. That is, what is th fiv-yar holding rat of rturn, and how can you comput th annualizd forward intrst rat for a two-yar invstmnt bginning on Dcmbr 31, 2018, and nding on Dcmbr 31, 2020? (4th Ed 7. Ivo Wlch, Co (4t017. Ivo Wl17. orat Finan h, Corpora or ion), h Edition rpora Ivo Wlch 017. orat Fi