Why we use compounding and discounting approaches

Transcription

1 Comoudig, Discouig, ad ubiased Growh Raes Near Deb s school i Souher Colorado. A examle of slow growh. Coyrigh , Gary R. Evas. May be used for o-rofi isrucioal uroses oly wihou ermissio of he auhor. This maerial may o be used for ay oher urose wihou wrie ermissio of he auhor. Why we use comoudig ad discouig aroaches Whe we work wih valuaio mehods or he ricig of oios, fuures, or forward coracs, we are i a evirome where ayme is searaed by ime from delivery. Whe we work wih yield-bearig fiacial asses like bods (o direcly a subjec i his class) we are ryig o fid a value for a cashflow sream ha rojecs io he fuure. Comoudig ad (esecially) discouig formulas rovide us wih he ools ecessary o lace a curre value o hese coracs or asses.

2 Discouig summaio formulas These are used whe here are mulile aymes, yically of he same value, over may ime eriods. Examle: The Califoria loery ays he wier 0 equal aual isallmes (/0 of he aouced rize), he firs immediaely ad he las i 9 years. Wha would be he rese value of a $40 millio rize (which imlies a $ millio cash ayme aually) whe he risk free ieres rae is 6%? 9 i PV0 06 (. ) i0 This may also be wrie. 3 9 PV Saus check Wha we leared i E04 (or elsewhere).. I siuaios where we exec a discree (couable) umber of aymes i he fuure, we have see he formulas for esimaig heir rese value. Where we go ex.. We ow mus cosider he coiuous comoudig We ow mus cosider he coiuous comoudig case. Noe: if you have moey i a accou ha is comouded daily his is effecively a examle of coiuous comoudig.

3 Coiuous comoudig If we are hikig abou variables ha chage over ime, like he marke value of sock or he value of a home, we ca say ha he fuure value of he variable (yically a asse) is equal o he origial value of he variable imes a isaaeous growh rae exressed by he formula: e e = where e = r = coiuous growh rae = ime (i ay siulaed ierval, like days or years) = value of a ime 0 = iiial value of r Examle Wha is he value of a house if i was urchased a $0,000 ad has bee growig i value a he rae of 8% aually comouded daily for a eriod of 4 years ad 3 mohs? e Noe: Alhough comoudig daily is discree (here are Noe: oug co ou d g da y s d sc ee ( e e a e 365 aymes er year) he coiuous comoudig formula is accurae u o wo decimal ois. 3

4 Coiuous discouig Wha is he rese value of a asse or a corac ha will be delivered i he fuure if we use he coiuous comoudig ieres rae of r? where e,r, ad are he same 0 = rese value ow = delivery value i he fuure 0 e r Noe: This formula ad he coiuous comoudig formula are used exesively i he valuaio of fuures ad forward coracs, as well as geerally i fiacial valuaios. Examle Wha is he rese value of $00,000 o be delivered i four years ad six mohs if he rese shor-erm moey rae comouded daily is 3.45%? aswer 00e 0 $85,

5 Lookig ahead.. usig aural logs Rememberig he relaioshi bewee aural logs ad he magic umber e = (.78888): ad if if This meas ha if ad e he l Ae he l l A 0 e r he (l ) r l r Daa coversio: coiuous growh raes.. l f e r l r f 3. (l f l )( / ) r l f r also calculaed as l l may be a so rice, f may be a fuure so rice r is yically he aualized growh rae uless oherwise siulaed. If hese are weekly daa, his value is /(/5) = 5. If hese daa are for 3 years, his value is (/3) =

6 ... for examle Cosider aural gas, $ er MMBTU: march =.756 ad july = he l(3.058/.756) /4 = 3.9% aualized, [l(3.058) - l(.756)] /4 = 3.9% Coversio raes From coiuous o discree: r d me rc m ( ) From discree o coiuous: rd rc m l m From discree o discree, differe ime eriods: r m m m r m m m Referece slide, o required o memorize, is i he book. m = umber of eriods i a year; i.e. for quarerly m = 4. 6

7 Examle.... Cover a 6% ieres rae comouded aually o a coiuous rae.. Cover a 6% ieres rae comouded quarerly o a coiuous rae r c l(.06) 5.83% r 4l 5.955% 4 c Referece slide. Three aleraive growh raes (oi by oi esimaio from ime-series daa) Examle from s observaios o lef Discree: 0.5 Workig wih raw imeseries daa, we have o rasform i o growh raes, bu which? Geomeric Discree:.5 (log): l l l Coiuous 7

8 DWBH... calculaig geomeric mea ad geomeric sadard deviaio (from discree-covered covered daa) Geomeric Mea value of growh rae: Geomeric sadard deviaio of he growh rae: gx i x i or gx e i l x i Remember he growh rae will be exressed as.5 raher ha 0.5. Tyically used o calculae he mea ad disersio of a log-ormal disribuio, here we are saig ha his mus be used o he same for ime-series daa rasformed o discree raher ha coiuous growh raes.... calculaig mea, sadard deviaio ad variace from he log-covered daa Mea value of growh rae: Variace of he growh rae: R i R i V R R i i Noe: hese esimaors assume ha you are usig a Gaussia (ormal) disribuio. Equally imora hese classical esimaors cao be used o daa rasformed io discree growh raes! Sadard deviaio of growh rae: SD V Noe: A good source for reviewig saisical alicaios is he Naioal Isiue of Sadards ad Techology Egieerig Saisics Hadbook, available olie a h:// 8

9 A roblem ha we eed o hik abou... Whe you rasform raw ime-series daa o discree or geomeric discree growh raes, boh he radiioal arihmeic mea ad sadard deviaio are biased dowward ad herefore uusable. Why? You have covered he daa io a geomeric series ad because of ha his ecessary codiio for he absece of bias is violaed: For observaios 0 (he firs) o f (he fial) where α i is he discree growh rae for ime i ad μ is he arihmeic mea of hese umbers (he alhas): However, if we ake he arihmeic mea ad sadard deviaio of he coiuous growh rae, he esimaors are o biased because he followig codiio is me (where α is ow he coiuous growh rae): Remember: Why we ca use he arihmeic mea: Suose you had a $00 ivesme ha eared, over five years, hese yields: % 0% 30% -0% 5% The arihmeic mea of hese reurs is 4%. So will your ivesme a he ed of five years be worh = $9.54? No. Usig he umbers direcly, y,your ivesme will be worh: 00 (.5) (.0) (.30) (0.80) (.5) = $79.40 You ca ake he mea of he hisorical value of discree growh raes o calculae he acual growh from begiig o ed (or esimae is coiuaio). I will be biased! 9

10 Oe more examle: Suose a sock idex we from 00 o 50, which we would describe as a erceage fall of 50%, he from 50 o 75, which we would describe as a erceage icrease of 50%. So because i we dow by 50% he u by 50%, are we back where we sared? Hardly! Noe ha: ad The coveie addiive ad mea- calculaio feaure of log growh raes. 0 e r r r r r r. e 0e e 0e ec. geerally... ri 3. 0 e 0 e Mea r which meas ha we ca ake mea coiuous growh raes ad comoud ad discou, icludig comoud forward for growh esimaes, wih bias. 0

11 Firs ses i esimaig measures of daily coiuous growh raes Give some samle from a oulaio se of rices, such as he daily closig rice of CSCO, for us he firs se is o cover he daa o coiuous growh raes: CGR x R P l P for each aired observaio. I Excel: Samle Price CGRP Origial samle (i ar) Covered daa Period Coversio (of growh raes) I his class we are usually usig daily daa ad are coe o esimae ad use daily coiuous growh raes, which requires o furher coversio. If you are usig weekly daa he you are calculaig weekly coiuous growh raes. Also, much of he lieraure ad Hull s book cosisely refers o aual coiuous growh raes (a aroach I do use of like). Use he followig coveio for eriod coversio. Imora oe: If he daa shows oly 5 observaios er week or 5 observaios er year, he use hose umbers raher ha 7 or 365. Period GR = [Mea] Daily GR Days i Period Aual GR = 5 [M]DGR

12 Head-sar o he large HW se As soo as you lear abou risk roxies, our ex segme, you will be give a very large HW se i Excel. I ha you will be asked o dowload oe year of daily daa i SPY, he cover i o daily coiuous growh raes, which you should ow be able o do. You eed o kow how o dowload hisorical daa from fiace.yahoo. Wach his video o see how o do ha (ad o cover i o he roer forma): h:// Toical... so far, sluggish marke i 04 Earigs we sellar i 03 ad margis were fa, ad he larges cororaios are cash-rich ($7)! Growh is ickig u very slighly bu he oulook is sill ucerai, ad he FRS is suosed o sar aerig QE3. Some of he emergig aios, i aricular Brazil, Argeia, Idia, Turkey, lus Ausralia ad maybe Caada, are sufferig from caial flow ad exchage rae roblems. El-Eria qui Pimco?? We reur o hese laer.