Synopsis of Various Rates of Return

Transcription

1 Syopss of Varous Raes of Reur (Noe: Much of hs s ake from Cuhberso) I he world of face here are may dffere ypes of asses. Whe aalysg hese, a ecoomc sese, we aemp o characerse hem by reducg hem o some of her more sale feaures,.e. we aemp o model hem. Thus, asses may vary he amou of moey requred o oba (ves ) hem, he amou of perods hese vesmes mus be made over, he rae a whch a vesme grows from purchase of he asse ul he po of redempo (maury), he lfe-me of he vesme, s lqudy (How quckly ca oe cover o cash?), or he rsk assocaed wh he reur o, ad value of, he asse. Some asses apprecae value (capal apprecao), some pay dvdeds; he omal value of some dvdeds s kow wh (ear) ceray (e.g. goverme bods), oher asses dvdeds are ucera eve omal erms (e.g. shares frms). I real erms, all reurs are ucera, for we mus adjus for he rae of flao (.e. oe may be able o buy less wh he proceeds of he vesme), or he exchage rae. We are eresed fdg he value of varous ypes of facal asses, gve hese characerscs of rsk, reur, lfespa, lqudy Usually reurs o vesmes are quoed smple aual erms,.e. o aeo s pad o compoud eres (eres o eres). I wha follows we wll cocer ourselves wh he fuure value,, of a vesme, ha may vary he frequecy of eres/dvded paymes, remag lfe-spa, rsk characerscs, ec.

2 If, a some me, we ves some amou over a gve me-perod, ad reap a amou + a me +, he our absolue omal ga (over ha me perod) s he dfferece bewee + ad. (Absolue ga), We ca also ascrbe a aural rae of growh, r, o hs vesme, by seg he absolue growh relave o he al value of he vesme,.e. by dvdg by he al value. Or, equvalely r + r Thus, f s vesed a a rae + r, we should reap + (+ r ). Naurally, f we revesed hs a me + a he prevalg rae of eres (or a projec ha would offer a growh rae of) r +, he we should have +2 (+ r + ) + (+ r + ) [(+ r ) ]. If r r + r (I hs case, we say he erm srucure of eres s fla), he perods from me we should have + (+ r). Thus, we are compoudg our eres paymes. By he reverse oke, f we kow he fuure value of some vesme a me +, +, ad we kow he growh (eres) rae ha exsed he perod pror o hs realsao, he we ca work ou he dscoued prese value, DV, a me of +. DV + ( + r) ( + Whch s evde by dvdg he foregog equao by (+r). + + r)

3 Assume we ves $x for years a a cera rae of aual eres, r. If eres s pad ou oly oce per aum (ad s mmedaely re-vesed a he same rae of eres), we compoud aually. Thus, he fuure value afer years s: $x (+r) If eres s pad ou m mes per year ad s mmedaely revesed a he same rae of reur as was he prcpal, he fuure value afer years wh m paymes per year s: m $x (+r/m) m, where r/m may be hough of as he perodc eres rae. Asde: I ca be show ha couous compoudg (le m progress oward fy) wll leave us wh he followg expresso: c $x e r [By akg logs] We ca see ha here exss a smple relaoshp bewee he quoed smple aual rae, r, wh m paymes per year ad he effecve aual rae, r f. By defo, he followg relaoshp mus hold: [+r f ] [+r/m] m *l[+r f ] *m*l[+r/m] +r f [+r/m] m Ad, r f > r sce r f akes compoudg o accou.

4 Dscoued rese Value, Aga Le he aual eres o a perfecly safe vesme over perods (years) be rs () [N.B.: a hs sage rs s o rased o he power of!!] Thus, rs () represes he aual eres ha prevals f we borrow for perods (years). We ves a amou $x ad receve a payme he fuure: $x(+rs () ) Thus, by verso, f oe were o receve wh ceray years me oe should be wllg o pay: DV ( + rs) I he forgog equao, ha amou should be precsely $x. The DV of a sream of receps, ( o ), whch are cera, s smply he sum of he dscoued prese values of each of he paymes (ha are cosue o hs sream). DV ( + rs) We have assumed he erm srucure of eres raes s fla,.e. hey are all he same. Assume a he me of he vesme, 0, we ca aval of a seres of eres raes rs (), whch deoes he eres rae from 0 o for, ad he rae from 0 o 2 for 2. Thus, Equvalely, for δ DV ( + rs () ( ) ( + rs ) ) + ( + rs 2 (2) ) ( + rs ( ) ) DV ( ) ( + rs ) δ

5 [Asde: For rsky vesmes we may adjus δ ( ) ( ) ( + rs + rp ) ake rsk o accou,.e. we arbue a rsk premum, rp () o perod.] order o Ivesg rojecs If a projec has a al vesme cos KC, he a erepreeur should oly ves f he dscoued value of he come sream he secures wh hs vesme s a leas as large as he capal cos requred order o ves hs scheme. DV KC Or, equvalely f he e prese value, NV, mus be o-egave NV DV KC 0. Gve KC s cosa ad DV falls as r rses (WHY??: dscou facor!), we ca deerme a level of r such ha NV 0. A hs rae of eres, he reur o he vesme jus allows us o pay eres ad repaymes. Le us deoe ha level of eres as y, he Ieral Rae of Reur (IRR). KC ( + y) Yeld o Maury Zero Coupo Bods A zero-coupo, or pure dscou, bod s a asse ha has a fxed lfespa (me o maury), a whch po provdes a oce-off payme, s par value, ypcally deoed by M. Thus, f M s pad ou afer oe perod, he zero has a oe-perod rae of reur (equvale o above), whch s defed by M ad he prce pad for,. rs () (M )/

6 Hece, s he prce pad a 0 ad M s he recep a maury, whe. Vewed erms of DV, M/(+y) or y (M-)/ Usually, zeroes (zero coupo bods) have a shor lfespa (3 mohs, e.g. T- Blls, o a year), ad are hghly lqud moey marke srumes. The reur o a zero coupo bod ca also be called spo rae, whch s decal o s eral rae of reur. Coupo ayg Bods A level (o-callable) -perod, coupo payg bod pays fxed coupos, C, perodcally (ofe b-aually) ul redempo,, ad pays a redempo prce a exprao dae, M. M s kow wh (assumed) ceray, ad C s defed as a perceage of hs par value, M. Hece, C s also kow wh (assumed) ceray. [N.B.: Eve govermes ca defaul o deb, or ca drecly do so by allowg her exchage rae o deprecae)] Gve he marke prce perods from maury, (), we ca solve for y by usg he followg equao. ( ) C ( + y) C ( + y) C ( + y) C + ( + y) M + ( + y) The yeld o maury s ha cosa rae of dscou, whch, a a po me, equaes he DV of fuure paymes wh he curre marke prce; herefore, s also a form of eral rae of reur. Implcly, we assume he rae of revesme s equal o hs dscou rae. If creases; Why? () falls, y

7 The rug yeld of a bod s defed as a sgle coupo payme relave o he marke prce [*00], whch may o be oo useful. However, for a perpeuy, a bod ha s o redeemable ad pays s coupos for ever, he rae of reur s gve by exacly ha equao. Noe, ha he YTM mplcly assumes he revesme rae s decal o he YTM self. To see hs, cosder a wo-perod, coupopayg bod. Is eral rae of reur, y, s gve by: C C M y ( + y) ( + y) 2 ( + y) C( + y) + ( C + M ) Ths clearly shows ha, he fuure value of he al oulay a a growh rae of y s equal o a seres of paymes, he frs of whch mus be revesed a a rae equal o y. 2 Asde: I he BKM book, you wll oe ha he geeral expresso s broke o wo pars: T C M BodValue + ( + r) ( + r) The lef par of he sum s he summao of he prese value of a T-perod auy. The rgh par of he summao s he prese value of a lump sum payme T perods me (be careful o dsgush he, whch represes a seres of perods, from he T, whch represes a specfc me-perod,.e. he fal perod. T How ca hs be used o smplfy calculaos geeral?

8 We ca calculae he value of a -perod auy. Cosder he value of a perpeuy, whch we have show s approxmaely equal o C r. Furhermore, he prese dscoued value of a perpeuy ha begs payg ou T-perods from ow should be C r ( + r) T. Thus, he dfferece prese value bewee hese vesmes should be equal o he prese value of a T-perod auy: T C M BodValue + ( + r) ( + r) T C r ( r) T +. Curre Yeld The curre yeld s smply measured by he rao of omal aual coupo come o he prce pad for he bod: C Aual Thus, eglecs o ake explc accou of he horzo o maury or growh raes. Realsed Compoud Yeld The realsed compoud yeld provdes us wh a ex-pos performace measure,.e. provdes us wh a rae of growh for our al vesme, akg explc accou of he acual revesme raes ha prevaled over he lfe-spa of he bod. I measures by wha rae of compouded eres he al vesme would have o have grow order o provde us wh he same amou we obaed from holdg he bod ad revesg he.

9 coupos. Assume we held a wo perod bod, smple aual coupo, wh a revesme rae of 0% over he secod year, radg a par value: C 0. M 000 r ( + y RCY ) 2 00(.) y RCY s he realsed compoud yeld. I hs case s equal o he revesme rae ad he coupo rae, whch s also equal o he yeld o maury. I should be oed ha f he coupo raes ad revesme raes dffer he he YTM wll o equal he realsed compoud yeld. If C r > M Why? I hs case he bod s a compeve vesme, sce C s far compesao for he me-value of moey. Each coupo s r of M, ad each coupo s revesed a r. Hece, o furher compesao should be ecessary. Wha f C < r? Wha s o M. Expla hs. Apprecao/Deprecao of vs-à-vs M wll compesae ( ecoomc erms) for dffereces r ad C. Thus, he prce wll chage over me order o do hs. No-Callable Zero Coupo Bod s wll apprecae over me ul M a he po of maury. Ths adjusme wll crease over me, as s expoeal.

10 To see hs, smply work ou he prese value of a Zero some perods from maury (e.g. 0 perods) whe eres raes are assumed o be kow. The Oe-erod Holdg erod Reur The oe-perod holdg perod reur, H +, s defed as H D , where he frs eleme he sum capures he capal ga (loss) o he asse, whle he laer capures he proporoae dvded yeld. Ths mples, D H Thus, ex pos, for a -perod horzo, wh al vesme A, he oal value accrued (gve all proceeds ca be revesed a a rae ha s decal o he performace of he asse self) should equal: Y A[ + H + ][ + H + 2 ]...[ + H + For a -perod, coupo payg bod: > Y A [ + ] H + ] where () H + C ( ) ( ) + +, ( ) ( ) s he marke prce a me for a bod wh a remag lfespa of -perods. Ths s referred o as he holdg perod yeld, or HY, whch s useful for he evaluao of socks ad bods. YTM vs HR If he YTM says uchaged ha he HR ad he YTM are decal.

11 If yelds flucuae, so wll he rae of reur va uexpeced chages he eres rae. If YTM > HR < YTM al YTM f(, M, C), whch are all kow a me (assumg o defaul). HR g(, +, C), Whch are o all observable a me. YTM ca be used ow, whereas HR ca a bes be esmaed, sce fuure prces are a fuco of uexpeced chages he eres rae. Wha abou defaul? Rsk prema compesae for cred rsk. [oe: ECB saeme o coury deb.] Socks Socks are facal asses ha have cera hgs commo wh bods, ad cera dsgushg feaures. Esseally, socks do o provde perodc paymes (dvdeds) of a fxed, omal sze. I coras, mos (goverme) bods do provde cera, fxed coupo paymes, a leas wh close o ceray well-developed ecoomes. Ther lfespa s o lmed, ulke bods, whch have a specfed perod of me ul redempo. Hece, socks provde he holder wh he rgh o a ucera come sream. Uceray mples varao reur, or rsk. (As we shall see,

12 sascal mehods are ofe employed o evaluae/descrbe ucera reurs by employg expecaos of reur o measure expeced proceeds ad he varace of reurs order o capure he rsk (varao or uceray) assocaed wh hese expecaos. Sce socks are herely rsky, her reurs should o be dscoued he same fasho as are he cera reurs o (ear) rsk-less asses, such as (goverme) bods. I s dffcul o fd a pere measure for he reur o hese asses. However, f here exss a me a expecao of he oe perod HY for +, q E H +, he he fudameal value of ha sock may be vewed as he DV of he expeced fuure dvdeds E D +j, deflaed by dscou facors (rsk prema). The, he fudameal value a me s D + D + 2 V E , ( + q) ( + q)( + q2 ) where q s he oe-perod reur bewee me perod +- ad +. Wh perfec formao ad raoal ages here should be o profable sysemac opporues for prof makg,.e. V. If o, profs are possble by arbrage.