Chapter 4 Return and Risk

Transcription

1 Chaper 4 Reur ad Risk The objecives of his chaper are o eable you o:! Udersad ad calculae reurs as a measure of ecoomic efficiecy! Udersad he relaioships bewee prese value ad IRR ad YTM! Udersad how obai a expeced securiy reur from poeial reurs ad associaed probabiliies! Defie ad measure risk! Udersad ad measure co-moveme 4.A. INTRODUCTION The purpose of measurig ivesme reurs is simply o deermie he ecoomic efficiecy of a ivesme. Thus, a ivesme's reur will express he profis geeraed by a iiial cash oulay relaive o he amou of ha oulay. There exis a umber of mehods for deermiig he reur of a ivesme. The measures preseed i his chaper are reur o ivesme ad ieral rae of reur. Arihmeic ad geomeric mea raes of reur o ivesme will be discussed alog wih ieral rae of reur ad bod reur measures. These mehods differ i heir ease of compuaio ad how hey accou for he imeliess ad compoudig of cash flows. 4.B. RETURN ON INVESTMENT: ARITHMETIC MEAN Perhaps he easies mehod o deermie he ecoomic efficiecy of a ivesme is o add all of is profis (π ) accruig a each ime period () ad dividig his sum by he amou of he iiial cash oulay (P 0 ). This measure is called a holdig period reur. To ease comparisos bewee ivesmes wih differe life expecacies, oe ca compue a arihmeic mea reur o ivesme (ROI) by dividig he holdig period reur by he life expecacy of he ivesme () as follows: (4.1) 1 ROI A P 0 The subscrip (A) afer (ROI) desigaes ha he reur value expressed is a arihmeic mea reur ad he variable (π ) is he profi geeraed by he ivesme i year (). Sice i is o always clear exacly wha he profi o a ivesme is i a give year, oe ca compue a reur 1

2 Chaper 4 based o periodic cash flows. Therefore, his arihmeic mea rae of reur formula ca be wrie: (4.2) ROI A 0 CF P 0 1 CF P 0 1 Noice ha he summaio i he firs expressio begis a ime zero, esurig ha he iiial cash oulay is deduced from he umeraor. (The cash flow [CF 0 ] associaed wih ay iiial cash oulay or ivesme will be egaive.) The primary advaage of Equaio (4.2) over (4.1) is ha a profi level eed o be deermied each year for he ivesme; ha is, he aual cash flows geeraed by a ivesme do o have o be classified as o wheher hey are profis or merely reur of capial. Muliplyig (P 0 ) by () i he deomiaor of (4.2) o aualize he reur has he same effec as dividig he eire fracio by () as i (4.1). I he secod expressio, he summaio begis a ime oe. The iiial oulay is recogized by subracig oe a he ed of he compuaio. For example, cosider a sock whose purchase price hree years ago was $100. This sock paid a divided of $10 i each of he hree years ad was sold for $130. If ime zero is he sock's dae of purchase, is arihmeic mea aual reur is 20%: ROI A Ideically, he sock's aual reur is deermied by (4.3): (4.3) ROI A 1 DIV P 0 P P0 P 0 where (DIV ) is he divided payme for he sock i ime (), (P 0 ) is he purchase price of he sock ad (P ) is he sellig price of he sock. The differece (P - P 0 ) is he capial gai realized from he sale of he sock. Cosider a secod sock held over he same period whose purchase price was also $100. If his sock paid o divideds ad was sold for $160, is aual reur would also be 20%: ROI A Therefore, boh he firs ad secod socks have realized arihmeic mea reurs of 20%. The oal cash flows geeraed by each, e of heir origial $100 ivesmes, is $60. Ye, he firs sock mus be preferred o he secod sice is cash flows are realized sooer. The arihmeic mea reur (ROI a ) does o accou for he imig of hese cash flows. Therefore, i evaluaes he wo socks 2

3 Reur ad Risk Applicaio 4.1: Fud Performace This applicaio is cocered wih how oe migh collec price daa for a fud ad compue reurs from ha daa. Suppose ha oe is ieresed i a give publically raded fud for which prices ad divideds are repored i sadard ews sources. Firs, rerieve fud price ad divided daa for each period uder cosideraio. This is easily doe wih he Wall Sree Joural or Yahoo Fiace as well as a variey of olie daa sources. Calculae periodic holdig reurs based o he followig: P P 1 DIV r P Dae P P -1 DIV r NOTES Jue Firs Moh July (55/50) - 1 =.10 Aug (50/55) - 1 = Sep (54/50) - 1 =.08 Oc ex-$2 divided; [(47+2)/54)]-1= Nov (51/47) - 1 =.081 Now, 5 mohly holdig period reurs have bee compued for he fud. The 5-moh holdig period reur migh be approximaed as he sum of idividual mohly reurs (hough we will discuss some problems wih his i Secio 4.B ad i Applicaio 4.2). The holdig period ad arihmeic mea reurs for he fud ca be compued wih a aleraive o Equaios 4.1 o 4.3 as follows: ROI A 1 r ROI 1 r H ideically eve hough he firs should be preferred o he secod. Because his measure of ecoomic efficiecy does o accou for he imeliess of cash flows, aoher measure mus be developed. 4.C. RETURN MEASUREMENT: GEOMETRIC MEAN The arihmeic mea reur o ivesme does o accou for ay differece bewee divideds (iermediae cash flows) ad capial gais (profis realized a he ed of he ivesme holdig period). Tha is, ROI A does o accou for he ime value of moey or he abiliy o re-ives cash flows received prior o he ed of he ivesme's life. I realiy, if a ivesor receives profis i he form of divideds, he has he opio o re-ives hem as hey are received. If profis are received i he form of capial gais, he ivesor mus wai uil he ed of his

4 Chaper 4 ivesme holdig period o re-ives hem. The differece bewee hese wo forms of profis ca be accoued for by expressig compouded reurs. Tha is, he geomeric mea reur o ivesme will accou for he fac ha ay earigs ha are reaied by he firm will be auomaically re-ivesed, hus compouded. If reurs are realized oly i he form of capial gais, he geomeric mea rae of reur is compued as follows: (4.4) ROI P / P 1 g o For example, he geomeric mea reur o he secod sock whose price icreased from $100 o $160 discussed i Secio 3.B is 16.96%, deermied as follows: ROI g / If divideds or iermediae cash flows from he securiy are realized before he ed of he holdig period, reurs r should be compued for each period ad he averaged as follows: r P P 1 P DIV 1 (4.5) ROI (1 r ) 1 g 1 Suppose ha he sock i our previous example paid $20 i divideds i each of he hree years of he holdig period raher ha geeraig a $60 capial gai over he hree year period. The reur r for each period would be 20% ad he geomeric mea reur for he sock would be 20% compued as follows: ROI (1 r ) 1= 3 (1 r ) 1 3 (1.2)(1.2)(1.2) g Noe ha he geomeric mea reur is higher if profis ca be wihdraw from he ivesme durig he holdig period. 4

5 Reur ad Risk Applicaio 4.1: Fud Performace (Coiued) This applicaio coiues our discussio of how oe migh collec price daa for fuds, compue reurs from ha daa ad compare fuds. Here, we will compare he prices ad reurs of his fud (we will call i Fud A) o he mohly reurs ad prices o a secod Fud B: Fud A Fud B Dae P P -1 DIV r P P -1 DIV r Jue July Aug Sep Oc Nov Recall ha he arihmeic mea reur o ivesme for Fud A is The arihmeic mea reur o ivesme o Fud B is.12, compued as follows: r ROI B.12 5 This compuaioal procedure is obviously quie misleadig. Fud B paid o divideds ad eded he five-year period worh $5 less ha a he begiig of he five-year period. Despie he fac ha Fud B obviously los moey, is arihmeic mea reur was compued o be.12 over he five years. Fud A clearly ouperformed Fud B, ye is arihmeic mea reur appears o be lower. Aleraively, oe could compue he arihmeic mea rae of reur for Fud B usig he followig: P P ROI B.02 P0 550 While his seems more iuiively accepable, he reur for Fud A cao be compued exacly he same way. However, he geomeric mea reur o ivesme ca be compued for boh fuds: ROI g (1 r ) ROI g for A 5 (1 r ) 1 5 (1.1)(1.091)(1.08)(1.092)(1.081)

6 Chaper 4 5 ROI g for B 5 (1 r ) 1 5 (1.6)(1.5)(1.5)(1.5)(1.5) Wih he geomeric mea reur compuaio, chages i he fuds ivesme bases are accoued for. Tha is, as he values of he fuds chage, he amous o which reurs are compued vary. Hece, he geomeric mea rae of reur ca accou for compoudig ad he compariso of fud reurs is more meaigful. 4.D. INTERNAL RATE OF RETURN The primary sregh of he ieral rae of reur (IRR) as a measure of he ecoomic efficiecy of a ivesme is ha i accous for he imeliess of all cash flows geeraed by ha ivesme. The IRR of a ivesme is calculaed by usig a model similar o he prese value series model discussed i Secio 3.C: PV P 0 1 CF (1 r) or, CF (4.6) NPV 0 (1 r 0 ) where e prese value (NPV) is he prese value of he series e of he iiial cash oulay, ad (r) is he reur (or discou rae) ha ses he ivesme's NPV equal o zero. The ivesme's ieral rae of reur is ha value for (r) ha equaes NPV wih zero. There exiss o geeral forma allowig us o solve for he ieral rae of reur (r) i erms of he oher variables i Equaio (4.6); herefore, we mus subsiue values for (r) uil we fid oe ha works (uless a compuer or calculaor wih a buil-i algorihm for solvig such problems ca be accessed). Ofe, his subsiuio process is very ime-cosumig, bu wih experiece calculaig ieral raes of reur, oe ca fid shorcus o soluios i various ypes of problems. Perhaps, he mos impora shorcu will be o fid a easy mehod for derivig a iiial value o subsiue for (r) resulig i a NPV fairly close o zero. Oe easy mehod for geeraig a iiial value o subsiue for (r) is by firs calculaig he ivesme's reur o ivesme. If a ivesor waed o calculae he ieral rae of reur for he firs sock preseed i Secio 4.B, he may wish o firs subsiue for (r) he sock's 20% reur o ivesme: 6

7 Reur ad Risk NPV = r r r = r = 3.7 Sice his NPV is less ha zero, a smaller (r) value should be subsiued. A smaller (r) value will decrease he righ had side deomiaors, icreasig he size of he fracios ad NPV. Perhaps a feasible value o subsiue for (r) is 10%. The same calculaios will be repeaed wih he ew (r) value of 10%: NPV = (1.1) (1.1) Sice he ew NPV exceeds zero, he (r) value of 10% is oo small. However, because is closer o zero ha 22.54, he ex value o subsiue for (r) migh be closer o 20% ha o 10%. Perhaps a beer esimae for he IRR will be 18%. Subsiuig his value for (r) resuls i a NPV of.86: NPV (1.18) (1.18) = 0.86 This NPV is quie close o zero; i fac furher subsiuios will idicae ha he rue sock ieral rae of reur is approximaely %. These ieraios have a paer: whe NPV is less ha zero, decrease (r) for he ex subsiuio; whe NPV exceeds zero, icrease (r) for he ex subsiuio. This process of ieraios eed oly be repeaed uil he desired accuracy of calculaios is reached. Figure 4.1 describes he relaioship bewee he value seleced for r ad NPV.

8 Chaper 4 Figure 4.1: The relaioship bewee NPV ad r for sock oe. The primary advaage of he ieral rae of reur over reur o ivesme is ha i accous for he imeliess of all cash flows geeraed by ha ivesme. However, IRR does have hree major weakesses: 1. As we have see, IRR akes cosiderably loger o calculae ha does ROI. Therefore, if ease of calculaio is of primary imporace i a siuaio, he ivesor may prefer o use ROI as his measure of efficiecy. As discussed i he appedix o his chaper, here are calculaors ad compuer programs ha will compue IRR very quickly. 2. Someimes a ivesme will geerae muliple raes of reur; ha is, more ha oe (r) value will equae NPV wih zero. This will occur whe ha ivesme has associaed wih i more ha oe egaive cash flow. Whe muliple raes are geeraed, here is ofe o mehod o deermie which is he rue IRR. I fac, oe of he raes geeraed may make ay sese. Whe he IRR is ifeasible as a mehod for comparig wo ivesmes, ad he ivesor sill wishes o cosider he ime 8

9 Reur ad Risk value of moey i his calculaios, he may simply compare he prese values of he ivesmes. 3. The ieral rae of reur is based o he assumpio ha cash flows received prior o he expiraio of he ivesme will be re-ivesed a he ieral rae of reur. Tha is, i is assumed ha fuure ivesme raes are cosa ad equal o he IRR. Obviously, his assumpio may o hold i realiy. 4.E. BOND YIELDS By coveio, raes of reur o bods are ofe expressed i erms somewha differe from hose of oher ivesmes. For example, he coupo rae of a bod is he aual ieres payme associaed wih he bod divided by he bod's face value. Thus, a 4-year $1000 corporae bod makig $60 aual ieres paymes has a coupo rae of 6%. However, he coupo rae does o accou for he acual purchase price of he bod. Corporae bods are usually raded a prices ha differ from heir face values. The bod's curre yield accous for he acual purchase price of he bod: (4.7) INT cy P 0 If his bod were purchased for $800, is curre yield would be 7.5%. The formula for curre yield, while easy o work wih, does o accou for ay capial gais (or losses) ha may be realized whe he bod maures. Furhermore, curre yields do o accou for he imeliess of cash flows associaed wih bods. The bod's yield o mauriy, which is esseially is ieral rae of reur does accou for ay capial gais (or losses) ha may be realized a mauriy i addiio o he imeliess of all associaed cash flows: CF INT (4.8) NPV 0 P 0 (1 y) (1 y) 0 1 (1 ) The yield o mauriy (y) of he 4-year $1000 corporae bod above is 6% if i were purchased for $1,000; he yield o mauriy would be 12.7% if he purchase price were $800. Regardless, (y) is ideical o he bod's ieral rae of reur. If he bod makes semiaual ieres paymes, is yield o mauriy ca be more accuraely expressed: 2 CF INT / 2 (4.9) NPV 0 P 0 (1 y) (1 y / 2) F y 0 1 (1 ) Here, we are cocered wih semiaual ieres paymes ad (2 ) 6-moh ime periods where () is he umber of years o he bod's mauriy. While yield o mauriy is perhaps he mos widely-used of he bod reur measures, i sill assumes a fla yield curve. This meas ha coupo F y

10 Chaper 4 paymes received prior o bod mauriy will be ivesed a he same rae as he bod s yield, a urealisic assumpio whe ieres raes are expeced o chage sigificaly over ime. 4.F. INTRODUCTION TO RISK Whe a firm ivess, i subjecs iself o a leas some degree of uceraiy regardig fuure cash flows. Maagers cao kow wih ceraiy wha ivesme payoffs will be. This chaper is cocered wih forecasig ivesme payoffs ad reurs ad he uceraiy associaed wih hese forecass. We will defie expeced reur i his chaper, focusig o i as a reur forecas. This expeced reur will be expressed as a fucio of he ivesme's poeial reur oucomes ad associaed probabiliies. The riskiess of a ivesme is simply he poeial for deviaio from he ivesme's expeced reur. The risk of a ivesme is defied here as he uceraiy associaed wih reurs o ha ivesme. Alhough oher defiiios for risk such as he probabiliy of losig moey or goig bakrup ca be very useful, hey are ofe less complee or more difficul o measure. Our defiiio of risk does have some drawbacks as well. For example, a ivesme which is cerai o be a complee loss is o regarded here o be risky sice is reur is kow o be -100% (hough we oe ha i probably would o be regarded o be a paricularly good ivesme). 4.G. EXPECTED RETURN Cosider a ecoomy wih hree poeial saes of aure i he ex year ad Sock A whose reur is depede o hese saes. If he ecoomy performs well, sae oe is realized ad he sock ears a reur of 25%. If he ecoomy performs oly saisfacorily, sae wo is realized ad he sock ears a reur of 10%. If he ecoomy performs poorly, sae hree is realized ad he sock achieves a reur of -10%. Assume ha here is a 20% chace ha sae oe will occur, a 50% perce chace ha sae wo will occur ad a 30% chace ha sae hree will occur. The expeced reur o he sock will be 7%, deermied by Equaio (4.10): (4.10) ERA i1 E[R A ]= (.25.20) + (.10.50) + ( ) =.07, where (R i ) is reur oucome (i) ad (P i ) is he probabiliy associaed wih ha oucome. Therefore, our forecased reur is 7%. The expeced reur cosiders all poeial reurs ad weighs more heavily hose reurs ha are more likely o acually occur. Alhough our forecased reur level is seve perce, i is obvious ha here is poeial for he acual reur oucome o deviae from his figure. This poeial for deviaio (variaio) will be measured i he followig secio. R Ai P i 10

11 Reur ad Risk 4.H: VARIANCE AND STANDARD DEVIATION The saisical cocep of variace is a useful measure of risk. Variace accous for he likelihood ha he acual reur oucome will vary from is expeced value; furhermore, i accous for he magiude of he differece bewee poeial reur oucomes ad he expeced reur. Variace ca be compued wih Equaio (4.11): 1 2 (4.11) ( R ER i1 ) 2 i P i The variace of sock A reurs preseed i Secio 4.G is.0156: σ 2 = ( ) ( ) ( ) 2.3 =.0156 The saisical cocep of sadard deviaio is also a useful measure of risk. The sadard deviaio of a sock's reurs is simply he square roo of is variace: σ= Thus, he sadard deviaio of reurs o he sock described i Secio 4.F is 12.49%. Σ i=1 (R i -E[R]) 2 P i Table 4.2: Expeced Reur, Variace ad Sadard Deviaio of Reurs for Sock A i R i P i R i P i R i - E[R a ] (R i - E[R a ]) 2 (R i - E[R a ]) 2 P i E[R a ]=.07 σ 2 a = σ a = Readers wihou a backgroud i saisics may wish o cosul he saisics review i he o-lie Elemeary Mahemaics Review.

12 Chaper 4 Cosider a secod securiy, Sock B whose reur oucomes are also depede o ecoomy oucomes oe, wo ad hree. If oucome oe is realized, Sock B aais a reur of 45%; i oucomes wo ad hree, he sock aais reurs of 5% ad -15%, respecively. From Table 4.3, we see ha he expeced reur o Sock B is seve perce, he same as for Sock A. However, he acual reur oucome of Sock B is subjec o more uceraiy. Sock B has he poeial of receivig eiher a much higher or much lower acual reur ha does Sock A. For example, a ivesme i Sock B could lose as much as fifee perce, whereas a equal ivesme i Sock A cao lose more ha e perce. A ivesme i Sock B also has he poeial of aaiig a much higher reur ha a ideical ivesme i Sock A. Therefore, reurs o Sock B are subjec o greaer variabiliy (or risk) ha reurs o Sock A. The cocep of variace (or sadard deviaio) accous for his icreased variabiliy. The variace of Sock B (.0436) exceeds ha of Sock A (.0156), idicaig ha Sock B is riskier ha Sock A. Table 4.3: Expeced Reur, Variace, ad Sadard Deviaio of Reurs for Sock B i R i P i R i P i R i - E[R b ] (R i - E[R b ]) 2 (R i - E[R b ]) 2 P i E[R b ]=.070 σ 2 = σ b =.2088 Wih he expeced reur ad sadard deviaio of reurs of a ivesme, we ca esablish rages of poeial reurs ad probabiliies ha acual reurs will fall wihi hese rages if i appears ha poeial reurs for ha ivesme are ormally disribued. For example, cosider a hird sock wih ormally disribued reurs wih a expeced level of 7% ad a sadard deviaio of 10%. From Table V i he ex appedix, we see ha here is a 68% probabiliy ha he acual reur oucome o his sock will fall bewee -.03 ad.17: (See he Saisics Review i he ex appedix.) E[R] - 1σ < R i < E[R] + 1σ 12

13 Reur ad Risk ( ) < R i < ( ) A similar aalysis idicaes a 95% probabiliy ha he acual reur oucome will fall bewee -.13 ad.27: E[R] - 2σ < R i < E[R] + 2σ ( ) < R i < ( ) Obviously, a smaller sadard deviaio of reurs will lead o a arrower rage of poeial oucomes give ay level of probabiliy. If a securiy has a sadard deviaio of reurs equal o zero, i has o risk. Such a securiy is referred o as he risk-free securiy wih a reur of (r f ). Therefore, he oly poeial reur level of he risk-free securiy is (r f ). No such securiy exiss i realiy; however, shor erm Uied Saes reasury bills are quie close.the U.S. goverme has prove o be a exremely reliable debor. Whe ivesors purchase reasury bills ad hold hem o mauriy, hey do receive heir expeced reurs. Therefore, shor-erm reasury bills are probably he safes of all securiies. For his reaso, fiacial aalyss ofe use he reasury bill rae (of reur) as heir esimae for (r f ) i may impora calculaios. 4.I: HISTORICAL VARIANCE AND STANDARD DEVIATION Empirical evidece suggess ha hisorical sock reur variaces (sadard deviaios) ca be reasoable idicaors of fuure variaces (sadard deviaios). Tha is, a sock whose previous reurs have bee subjec o subsaial variabiliy probably will coiue o realize reurs of a highly volaile aure. Therefore, pas riskiess is ofe a good idicaor of fuure riskiess. A sock's hisorical reur variabiliy ca be measured wih a hisorical variace: (4.12) h ( R R 1 ) where (R ) is he sock reur i ime () ad (_R) is he hisorical average reur over he () ime period sample. The sock's hisorical sadard deviaio of reurs is simply he square roo of is variace. If a ivesor deermies ha he hisorical variace is a good idicaor of is fuure variace, he may eed o o calculae poeial fuure reurs ad heir associaed probabiliies for risk esimaes; he may prefer o simply measure he sock's riskiess wih is hisorical variace or sadard deviaio give as follows:

14 Chaper 4 2 ( Ri R) 1 H Table 4.4 demosraes hisorical variace ad sadard deviaio compuaios for Sock D. Table 4.4: Hisorical Variace ad Sadard Deviaio of Reurs of Sock D _ R R - R D (R - R D ) 2 (R - R D ) 2 1/ _R d =.16 σ 2 = σ d = J. COVARIANCE Sadard deviaio ad variace provide us wih measures of he absolue risk levels of securiies. However, i may isaces, i is useful o measure he risk of oe securiy relaive o he risk of aoher or relaive o he marke as a whole. The cocep of covariace is iegral o he developme of relaive risk measures. Covariace provides us wih a measure of he relaioship bewee he reurs of wo securiies. Tha is, give ha wo securiies reurs are likely o vary, covariace idicaes wheher hey will vary i he same direcio or i opposie direcios. The likelihood ha wo securiies will comove i he same direcio (or, more accuraely, he sregh of he relaioship bewee reurs o wo securiies) is measured by Equaio (4.13): R ER (4.13) k, j Cov[ k, j] Rk, i ERk 1 14 j, i j P i where (R ki ) ad (R ji ) are he reur of socks (k) ad (j) if oucome (i) is realized ad (P i ) is he probabiliy of oucome (i). E[R k ] ad E[R j ] are simply he expeced reurs of securiies (k) ad (j). For example, he covariace bewee reurs of socks A ad B is: cov(a,b) = {( ) ( ).20} + {( ) ( ).50} + {( ) ( ).30}

15 Reur ad Risk = {.01368} + {-.0003} + {.01122} = Sice his covariace is posiive, he relaioship bewee reurs o hese wo securiies is posiive. Tha is, he larger he posiive value of covariace, he more likely oe securiy will perform well give ha he secod will perform well. A egaive covariace idicaes ha srog performace by oe securiy implies likely poor performace by he secod securiy. A covariace of zero implies ha here is o relaioship bewee reurs o he wo securiies. Table 4.5 deails he soluio mehod for his example. Table 4.5: Covariace bewee Reurs o Socks A ad B i R ai R bi P i R ai -E[R a ] R bi -E[R b ] (R ai -E[R a ])(R bi -E[R b ])P i COV(A,B) =.0246 Empirical evidece suggess ha hisorical covariaces are srog idicaors of fuure covariace levels. Thus, if oe is uable o associae probabiliies wih poeial oucome levels, i may cases he may use hisorical covariace as his esimae for fuure covariace. Table 4.6 demosraes how o deermie hisorical covariace for wo hypoheical socks D ad E. Table 4.6: Hisorical Covariace bewee Reurs o Socks D ad E R d R e (R d -R d ) (R e -R e ) (R d -R d )(R e -R e )1/ R D =.16.20=R E COV(D,E) =.00140

16 Chaper 4 4.K: COEFFICIENT OF CORRELATION The coefficie of correlaio provides us wih a meas of sadardizig he covariace bewee reurs o wo securiies. For example, how large mus covariace be o idicae a srog relaioship bewee reurs? Covariace will be smaller give low reurs o he wo securiies ha give high securiy reurs. The coefficie of correlaio (ρ kj ) bewee reurs o wo securiies will always fall bewee -1 ad If securiy reurs are direcly relaed, he correlaio coefficie will be posiive. If he wo securiy reurs always covary i he same direcio by he same proporios, he coefficie of correlaio will equal oe. If he wo securiy reurs always covary i opposie direcios by he same proporios, (ρ k,j ) will equal egaive oe. The sroger he iverse relaioship bewee reurs o he wo securiies, he closer (ρ k,j ) will be o egaive oe. If (ρ k,j ) equals zero, here is o relaioship bewee reurs o he wo securiies. The coefficie of correlaio (ρ k,j ) bewee reurs is simply he covariace bewee reurs o he wo securiies divided by he produc of heir sadard deviaios: (4.14) COV ( k, j) kj Equaio (4.14) implies ha he covariace formula ca be rewrie: k j (4.15) COV(k,j) = σ k σ j ρ k,j If a ivesor ca access oly raw daa peraiig o securiy reurs, he should firs fid securiy covariaces he divide by he producs of heir sadard deviaios o fid correlaio coefficies. However, if for some reaso he ivesor kows he correlaio coefficies bewee reurs o securiies, he ca use his value alog wih sadard deviaios o fid covariaces. The coefficie of correlaio bewee reurs o socks A ad B is.96: kj This value ca be squared o deermie he coefficie of deermiaio bewee reurs of he wo securiies. The coefficie of deermiaio (ρ k,j 2 ) measures he proporio of variabiliy i oe securiy's reurs ha ca be explaied by or be associaed wih variabiliy of reurs o he secod securiy. Thus, approximaely 92% of he variabiliy of sock A reurs ca be explaied by or associaed wih variabiliy of sock B reurs. The coceps of covariace ad correlaio are 1 May saisics exbooks use he oaio (r i,j ) o desigae he correlaio coefficie bewee variables (i) ad (j). Because he leer (r) is used i his ex o desigae reur, i will use he lower case rho (ρ ij ) o desigae correlaio coefficie. 16

17 Reur ad Risk crucial o he developme of porfolio risk ad relaive risk models preseed i laer chapers. Hisorical evidece suggess ha covariaces ad correlaios bewee sock reurs remai relaively cosa over ime. Thus, a ivesor ca use hisorical covariaces ad correlaios as his forecased values. However, i is impora o realize ha hese hisorical relaioships apply o sadard deviaios, variaces, covariaces ad correlaios, bu o o he acual reurs hemselves. Tha is, we ca ofe forecas fuure risk levels ad relaioships o he basis of hisorical daa, bu we cao forecas reurs o he basis of hisorical reurs. Thus, las year's reur for a give sock implies almos ohig abou ex year's reur for ha sock. The hisorical covariace bewee reurs o securiies (i) ad (j) ca be foud by solvig Equaio (4.16): (4.16) ij ( Ri Ri )( RJ, 1, 1 R j ) where he sample daa is ake from () years. See Table (4.6). 4.L: THE MARKET PORTFOLIO As we shall see i he ex chaper, a porfolio is simply a collecio of ivesmes. The marke porfolio is he collecio of all ivesmes ha are available o ivesors. Tha is, he marke porfolio represes he combiaio or aggregaio of all securiies (or oher asses) ha are available for purchase. Ivesors may wish o cosider he performace of his marke porfolio o deermie he performace of securiies i geeral. Thus, he reur o he marke porfolio is represeaive of he reur o he "ypical" asse. A ivesor may wish o kow he marke porfolio reur o deermie he performace of a paricular securiy or his eire ivesme porfolio relaive o he performace of he marke or a "ypical" securiy. Deermiaio of he reur o he marke porfolio requires he calculaio of reurs o all of he asses available o ivesors. Because here are hudreds of housads of asses available o ivesors (icludig socks, bods, opios, bak accous, real esae, ec.), deermiig he exac reur of he marke porfolio may be impossible. Thus, ivesors geerally make use of idices such as he Dow Joes Idusrial Average or he Sadard ad Poor's 500 (S&P 500) o gauge he performace of he marke porfolio. These idices merely ac as surrogaes for he marke porfolio; we assume ha if he idices are icreasig, he he marke porfolio is performig well. For example, performace of he Dow Joes Idusrials Average depeds o he performace of he hiry socks ha comprise his idex. Thus, if he Dow Joes marke idex is performig well, he hiry securiies, o average are probably performig well. This srog performace may imply ha he marke porfolio is performig well. I ay case, i is easier o measure he performace of a porfolio of hiry or five hudred socks (for he Sadard ad Poor's 500) ha i is o measure he performace of all of he securiies ha comprise he marke porfolio. 4.M. CONCLUSION Risk meas differe higs o differe people. To some, i meas poeial for o earig

18 Chaper 4 profis, for ohers i meas poeial for bakrupcy. I his ex, we defie risk o be uceraiy. There also exis umerous measures of risk. Amog he more popular measures of risk is sadard deviaio. Sadard deviaio is a paricularly useful idicaor of risk for several reasos: 1. Sadard deviaio ca accommodae all poeial oucomes associaed wih a ivesme. Each oucome which varies sigificaly from he mea or expeced value icreases sadard deviaio. 2. Poeial oucomes which are more likely affec sadard deviaios more ha oucomes which are less likely. For example, a very likely oucome which deviaes subsaially from expeced value will icrease sadard deviaio. 3. The ormal disribuio, which describes he frequecy of such a large umber of fiacial pheomea, is defied i erms of sadard deviaio. Hisorical sadard deviaio is useful whe he aalys believes ha he hisorical volailiy of a ivesme is a good idicaor of is fuure uceraiy. Co-moveme saisics such as covariace are quie useful i deermiig he relaioship bewee various fiacial daa. The sig of covariace idicaes he direcio of co-moveme ad is useful i measureme of porfolio risk ad i he compuaio of relaive risk saisics o be discussed laer. Coefficie of correlaio is esseially a sadardized covariace; is absolue value idicaes he iesiy of co-moveme. The coefficie of deermiaio idicaes he proporio of variabiliy i oe daa se which ca be explaied by variabiliy i a secod daa se. 18

19 Reur ad Risk QUESTIONS AND PROBLEMS 4.1. A ivesor purchased oe share of Drysdale sock for oe hudred dollars i 2003 ad sold i exacly oe year laer for $200. Calculae he ivesor's arihmeic mea reur o ivesme A ivesor purchased oe share of Wilso Compay sock i 2006 for $20 ad sold i i 2013 for $40. Calculae he followig for his ivesor: a. arihmeic mea reur o ivesme. b. geomeric mea reur o ivesme. c. ieral rae of reur A ivesor purchased oe hudred shares of Mahewso Compay sock for $75 apiece i 1995 ad sold each share for $80 exacly six years laer. The Mahewso Compay paid aual divideds of $8 per share i each of he six years he ivesor held he sock. Calculae he followig for he ivesor: a. arihmeic mea reur o ivesme. b. ieral rae of reur The Paige Bakig Compay is cosiderig he purchase of a ove for $100,000 whose oupu will yield he compay $20,000 i aual afer-ax cash flows for each of he ex five years. A he ed of he fifh year, he ove will be sold for is $40,000 salvage value. Calculae he followig for he machie Paige is cosiderig purchasig: a. arihmeic mea reur o ivesme. b. ieral rae of reur Wha is he e prese value of a ivesme whose ieral rae of reur equals is discou rae? 4.6. A ivesor purchased oe hudred shares each of Grove Compay sock ad Dea Compay sock for $10 per share. The Grove Compay paid a aual divided of oe dollar per share i each of he eigh years he ivesor held he sock. The Dea Compay paid a aual divided of $.25 per share i each of he eigh years he ivesor held he sock. A he ed of he eigh year period, he ivesor sold each of his shares of Grove Compay sock for $11 ad sold each of his shares of Dea Compay sock for $18. a. Calculae he sum of divideds received by he ivesor from each of he compaies. b. Calculae he capial gais realized o he sale of sock of each of he compaies. c. Calculae he reur o ivesme for each of he wo compaies' sock usig a arihmeic mea reur. d. Calculae he ieral rae of reur for each of he wo socks. e. Which of he wo socks performed beer durig heir holdig periods? 4.7. The Lemo Compay is cosiderig he purchase of a ivesme for $100,000 ha is

20 Chaper 4 expeced o pay off $50,000 i wo years, $75,000 i four years ad $75,000 i six years. I he hird year, Lemo mus make a addiioal payme of $50,000 o susai he ivesme. Calculae he followig for he Lemo ivesme: a. Reur o ivesme usig a arihmeic mea reur. b. The ivesme ieral rae of reur. c. Describe ay complicaios you ecouered i par b A $1,000 face value bod is currely sellig a a premium for $1,200. The coupo rae of his bod is 12% ad i maures i hree years. Calculae he followig for his bod assumig is ieres paymes are made aually: a. Is aual ieres paymes. b. Is curre yield. c. Is yield o mauriy Work hrough each of he calculaios i Problem 4.8 assumig ieres paymes are made semi-aually The Nichols Compay ivesed $100,000 io a small busiess wey years ago. Is ivesme geeraed a cash flow equal o $3,000 i is firs year of operaio. Each subseque year, he busiess geeraed a cash flow which was 10% larger ha i he prior year; ha is, he busiess geeraed a cash flow equal o $3,300 i he secod year, $3,630 i he hird year, ad so o for ieee years afer he firs. The Nichols Compay sold he busiess for $500,000 afer is weieh year of operaio. Wha was he ieral rae of reur for his ivesme? Suppose ha a muual fud ivesig o behalf of shareholders yielded he followig price ad price performace resuls: Dae P P -1 DIV Jue July Aug Sep Oc Nov Calculae for his fud mohly reurs for each of he five mohs July o November ad compue a geomeric mea reur over his 5-moh period Mack Producs maageme is cosiderig he ivesme i oe of wo projecs available o he compay. The reurs o he wo projecs (A) ad (B) are depede o he sales oucome of he compay. Mack maageme has deermied hree poeial sales oucomes (1), (2) ad (3) for he compay. The highes poeial sales oucome for Mack is oucome (1) or $800,000. If his 20

21 Reur ad Risk sales oucome were realized, Projec (A) would realize a reur oucome of 30%; Projec (B) would realize a reur of 20%. If oucome (2) were realized, he compay's sales level would be $500,000. I his case, projec (A) would yield 15%, ad Projec (B) would yield 13%. The wors oucome (3) will resul i a sales level of $400,000, ad reur levels for Projecs (A) ad (B) of 1% ad 9% respecively. If each sales oucome has a equal probabiliy of occurrig, deermie he followig for he Mack Compay: a. he probabiliies of oucomes (1), (2) ad (3). b. is expeced sales level. c. he variace associaed wih poeial sales levels. d. he expeced reur of Projec (A). e. he variace of poeial reurs for Projec (A). f. he expeced reur ad variace for Projec (B). g. sadard deviaios associaed wih compay sales, reurs o Projec (A) ad reurs o Projec (B). h. he covariace bewee compay sales ad reurs o Projec (A). i. he coefficie of correlaio bewee compay sales ad reurs o Projec (A). j. he coefficie of correlaio bewee compay sales ad reurs o Projec (B). k. he coefficie of deermiaio bewee compay sales ad reurs o Projec (B) Which of he projecs i Problem 4.12 represes he beer ivesme for Mack Producs? Hisorical perceage reurs for he McCarhy ad Also Compaies are lised i he followig char alog wih perceage reurs o he marke porfolio: Year McCarhy Also Marke Calculae he followig based o he precedig diagram: a. mea hisorical reurs for he wo compaies ad he marke porfolio. b. variaces associaed wih McCarhy Compay reurs ad Also Compay reurs as well as reurs o he marke porfolio. c. he hisorical covariace ad coefficie of correlaio bewee reurs of he wo securiies. d. he hisorical covariace ad coefficie of correlaio bewee reurs of he McCarhy Compay ad reurs o he marke porfolio. e. he hisorical covariace ad coefficie of correlaio bewee reurs of he Also Compay ad reurs o he marke porfolio Forecas he followig for boh he McCarhy ad Also Compaies based o your calculaios i Problem 4.14:

22 Chaper 4 a. variace ad sadard deviaio of reurs. b. coefficie of correlaio bewee each of he compaies' reurs ad reurs o he marke porfolio The followig able represes oucome umbers, probabiliies ad associaed reurs for sock A: oucome (i) reur (R i ) Probabiliy (P i ) ? Thus, here are e possible reur oucomes for Sock A. a. Wha is he probabiliy associaed wih Oucome 9? b. Wha is he sadard deviaio of reurs associaed wih Sock A? The Durocher Compay maageme projecs a reur level of 15% for he upcomig year. Maageme is ucerai as o wha he acual sales level will be; herefore, i associaes a sadard deviaio of 10% wih his sales level. Maagers assume ha sales will be ormally disribued. Wha is he probabiliy ha he acual reur level will: a. fall bewee 5% ad 25%? b. fall bewee 15% ad 25%? c. exceed 25%? d. exceed 30%? Wha would be each of he probabiliies i Problem 4.17 if Durocher Compay maageme were cerai eough of is forecas o associae a 5% sadard deviaio wih is sales projecio? Uder wha circumsaces ca he coefficie of deermiaio bewee reurs o wo securiies be egaive? How would you ierpre a egaive coefficie of deermiaio? If here are o circumsaces where he coefficie of deermiaio ca be egaive, describe why Sock A will geerae a reur of 10% if ad oly if Sock B yields a reur of 15%; Sock B will geerae a reur of 10% if ad oly if Sock A yields a reur of 20%. There is a 50% probabiliy ha Sock A will geerae a reur of 10% ad a 50% probabiliy ha i will yield 20%. a. Wha is he sadard deviaio of reurs for Sock A? 22

23 Reur ad Risk b. Wha is he covariace of reurs bewee Socks A ad B? A ivesor has he opporuiy o purchase a risk-free reasury bill yieldig a reur of 10%. He also has he opporuiy o purchase a sock which will yield eiher 7% or 17%. Eiher oucome is equally likely o occur. Compue he followig: a. he variace of reurs o he sock. b. he coefficie of correlaio bewee reurs o he sock ad reurs o he reasury bill The followig daily prices were colleced for each of hree socks over a welve day period. Compay X Compay Y Compay Z DATE PRICE DATE PRICE DATE PRICE 1/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Based o he daa give above, calculae he followig: a. Reurs for each day o each of he hree socks. There should be a oal of e reurs for each sock - begiig wih he dae 1/10. b. Average daily reurs for each of he hree socks. c. Daily reur sadard deviaios for each of he hree socks.

24 Chaper 4 APPENDIX 4.A RETURN AND RISK SPREADSHEET APPLICATIONS Table 4.A.1 coais spreadshee eries for compuig sock variaces, sadard deviaios ad covariaces. The able liss daily closig prices for Socks X, Y ad Z from Jauary 9 o Jauary 20 i Cells B3:B14, E3:E14 ad H3:H14. From hese prices, we compue reurs i Cells B19:B29, E19:E29 ad H19:H29. Variace, sadard deviaio ad covariace saisics i Rows 30 o 38 are compued from formulas displayed i Table 4.A.2. Table 4.A.1: Sock Prices, Reurs, Risk ad Co-moveme A B C D E F G H 1 CORP. X CORP. Y Z CORP. 2 DATE DATE DATE PRICE PRICE PRICE 3 9-Ja Ja 20 9-Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja CORP. X CORP. Y CORP. Z 17 DATE DATE DATE RETURN RETURN RETURN 18 9-Ja N/A 9-Ja N/A 9-Ja N/A Ja 0 10-Ja 0 10-Ja Ja Ja Ja Ja 0 12-Ja Ja Ja Ja Ja Ja Ja 0 14-Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Ja Mea Mea Mea Variace Variace

25 Reur ad Risk Variace 32 Variace (P) Variace (P) Variace (P) S.D S.D S.D. 34 S.D S.D. (P) S.D. (P) (P) 35 COV(X,Y)= COV(Y,Z)= COV(X,Z)= CORR(X,Y)= CORR(Y,Z)= CORR(X,Z)= Formulas for compuig reurs are give i Rows 19 o 29 i Table 4.A.2. Meas, variaces, sadard deviaios, covariaces ad correlaio coefficies are compued i Rows 30 o 38. Row 30 compues he arihmeic mea reur for each of he hree socks. Table 4.A.2 liss formulas associaed wih he values i cells A30:H38. The =(Average) fucio may be yped i direcly as lised i Table 4.A.2 Row 30 or obaied from he Pase Fucio buo (f x ) meu uder he Saisical sub-meu. Ery isrucios are give i he dialogue box obaied whe he Average fucio is seleced. The variace formulas i Row 31 are based o he Sample formula; he Variace (P) formulas i Row 32 are based o he populaio formula. Sadard deviaio sample ad populaio resuls are give i Rows 33 ad 34. Covariaces ad correlaio coefficies are give i Rows 35 o 38. Table 4.A.2: Sock Reurs, Risk ad Co-moveme: Formula Eries 16 A B C D E F G CORP. X CORP. Y CORP. Z H 17 DATE RETURN DATE RETURN DATE RETURN 18 9-Ja N/A 9-Ja N/A 9-Ja N/A Ja =B4/B Ja =E4/E Ja =H4/H Ja =B5/B Ja =E5/E Ja =H5/H Ja =B6/B Ja =E6/E Ja =H6/H Ja =B7/B Ja =E7/E Ja =H7/H Ja =B8/B Ja =E8/E Ja =H8/H Ja =B9/B Ja =E9/E Ja =H9/H Ja =B10/B Ja =E10/E Ja =H10/H Ja =B11/B Ja =E11/E Ja =H11/H Ja =B12/B Ja =E12/E Ja =H12/H Ja =B13/B Ja =E13/E Ja =H13/H Ja =B14/B Ja =E14/E Ja =H14/H Mea =AVERAGE(B19:B29) Mea =AVERAGE(E19:E29) Mea =AVERAGE(H19:H29) 31 Variace =VAR(B19:B29) Variace =VAR(E19:E29) Variace =VAR(H19:H29) 32 Variace (P) =VARP(B19:B29) Variace (P) =VARP(E19:E29) Variace (P) =VARP(H19:H29) 33 S.D. =STDEV(B19:B29) S.D. =STDEV(E19:E29) S.D. =STDEV(H19:H29) 34 S.D. (P) =STDEVP(B19:B29) S.D. (P) =STDEVP(E19:E29) S.D. (P) =STDEVP(H19:H29) 35 =COVAR(B19:B29,E19:E29) COV(Y,Z)= =COVAR(E19:E29,H19:H29) COV(X,Y)= =COVAR(B19:B29,H19:H29)

26 Chaper 4 36 COV(X,Z)= 37 CORR(X,Y)= =CORREL(B19:B29,E19:E29) CORR(Y,Z)= =CORREL(E19:E29,H19:H29) 38 CORR(X,Z)= =CORREL(B19:B29,H19:H29) 26