Capital Asset Pricing Model (CAPM) at Work


 Egbert Lambert Sullivan
 2 years ago
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1 Capital Asset Picing Model (CAPM) at Wok Some o the Intuition behind CAPM Applications o CAPM Estimation and Testing o CAPM
2 Intuition behind Equilibium Suppose you obseved the ollowing chaacteistics o two o you stocks within you potolio (completely ignoe isk o now). What do you do? Stock A Stock B Cuent Maket Pice $100 $50 Expected Selling Pice $108 $59 Expected Dividend pe Shae $2 $1 Expected (simple) Rate o Retun 10% 20% S S $100 $50 D D Stock A Stock B
3 S S $100 $50 D D Stock A Stock B I the pice o stock A dopped to $95.65 and stock B ose to $52.17, then what would be the esult? Stock A Stock B Cuent Maket Pice $95.65 $52.17 Expected Selling Pice $108 $59 Expected Dividend pe Shae $2 $1 Expected (simple) Rate o Retun 15% 15% How would things be dieent i stock A was moe isky than stock B? What i stock B was moe isky than stock A?
4 Suppose we ae holding welldivesiied potolios, then the elevant isk is what each stock contibutes to the potolio. M B B A A E E E ) ( ) ( ) ( L β β Thus, bidding up the pice o B dove down the expected etun. Deceasing the demand o A, loweing its pice, pushed up the expected etun. I the stocks have dieent betas, then the expected etuns will not be equal but athe popotional. B A B A E E β β ) ( ) ( Thus, the equilibium condition in the CAPM meely summaizes (and states pecisely) the vey old equilibium condition o competitive makets. Competitive makets ae in equilibium when the isk adjusted expected ates o etun ae equal.
5 Applications Assume the expected etun on the maket to be 15%, iskee ate o 8%, expected ate o etun on XYZ secuity o 17%, and beta o XYZ secuity o Within the context o CAPM, is XYZ ovepiced, undepiced, o aily piced?
6 Assume the expected etun on the maket to be 15%, iskee ate o 8%, expected ate o etun on XYZ secuity o 17%, and beta o XYZ secuity o Within the context o CAPM, is XYZ ovepiced, undepiced, o aily piced? The expected ate o etun based on CAPM would be CAPM XYZ 8 % (15% 8%) 16.75% Since the etun (17%) exceeds the CAPM, then we say that it is undepiced. People sometimes have touble thinking about the ove o unde piced asset in the CAPM context. I you do, ty changing the stoy a bit. Suppose you looked at the esults o XYZ o the past yea and ound that it did in act have a 17% ate o etun. You also go back and calculate what the CAPM ate o etun should have been o the past yea you calculate the 16.75%. Now, looking back, you might think I wish that I would have bought some o XYZ! in that case you ae eally saying that XYZ was undepiced duing the past yea. Remembe, othe things equal, i you begin puchasing the stock, you ll dive the pice upwad and as you do the actual ate o etun will all  ollowing CAPM you would do this until the etun ell to the CAPM level.
7 I CAPM is valid, then discuss which o the ollowing situations ae possible. a. Potolio (o, asset) Expected Retun Beta A 20% 1.4 B 25% 1.2 b. Potolio (o, asset) Expected Retun Beta A 30% 1.5 B 15%.75 c. Potolio (o, asset) Expected Retun Beta iskee 10% maket 18% 1 A 22% 1.5
8 I CAPM is valid, then discuss which o the ollowing situations ae possible. a. Potolio (o, asset) Expected Retun Beta A 20% 1.4 B 25% 1.2 This suely could not be coect accoding to CAPM. Accoding to CAPM, the geate the beta hence, the isk the highe the expected ate o etun. Since A has a highe systematic isk beta it should have a highe expected etun. Note, the only acto that dieentiates one asset s expected ate o etun om anothe is Beta. Conside the implications o this notion  nothing else mattes (e.g., chats o pice and volume movements, the undamentals o the issue, etc.)
9 b. Potolio (o, asset) Expected Retun Beta A 30% 1.5 B 15%.75 Suppose the iskee ate is 12% and the maket etun is 24%, then o A, E( ) + β [ E( ) ] 12% + 1.5(24 12) A A M 30% What about B? E( ) + β [ E( ) ] 12% +.75(24 12) B B M 21% So, in act, the etuns ae not coect by CAPM standads. On the othe hand, i the 30% and 15% had been Expected Excess Retuns (i.e., expected etun minus the iskee ate), then
10 things would be ine accoding to CAPM in act, oten times it is easie to think in tems o excess etuns. It s easie because you see that you can diectly tanslate elative betas into elative excess etuns. Fo example, solve the CAPM o beta. M A A E E ) ( ) ( β M B B E E ) ( ) ( β Now, we can simply state the elative betas 2 15% 30% ) ( ) ( ) ( ) ( ) ( ) ( B A M B M A B A E E E E E E β β Thus, the elative excess etuns must be equal to the elative betas. It is not tue that the elative ates o etun will be equal to elative betas.
11 c. Potolio (o, asset) Expected Retun Beta iskee 10% maket 18% 1 A 22% 1.5 Using the CAPM equation, E( ) + β [ E( ) ] 10% + 1.5(18% 10%) M 22% So, yes, this is coect accoding to CAPM.
12 Suppose the equity holdes (i.e., ownes o the stock) investment in the publicly egulate utility company is $100 million and beta is I the iskee ate is 3% and the expected ate o etun on the maket is 5%, then calculate the ai accoding to CAPM  ate o etun o investos and thei poits.
13 Suppose equity holdes investment in the publicly egulate utility company is $100 million and beta is.60. I the iskee ate is 3% and the expected ate o etun on the maket is 5%, then calculate the ai accoding to CAPM  ate o etun o investos and thei poits. Fist, we calculate the CAPM expected ate o etun. E( ) + β [ E( ) ] 3% +.6(5% 3%) M 4.2% Notice, the CAPM ate o etun o the utility company is less than the maket etun. Why? Because the beta o 0.60 is less than 1 tells us that the utility s etun luctuates less than the maket it is less isky than the maket. On the othe hand, it is geate than the iskee ate because it is still isky. Second, the investment is $100 million, so we can easily calculate the ai poits. $100 million x.042 $4,200,000 The egulatoy agency will now allow the company to set a pe unit pice to taget the poit igue.
14 A judge uses CAPM to detemine the damages in a copyight iningement case. The judge has been advised that a beta o 2.0 is typical o the publishing industy. What would be the ai accoding to CAPM  ate o etun to be used to detemine damages? [assume a maket etun o 14% and iskee ate o 4%]
15 A judge uses CAPM to detemine the damages in a copyight iningement case. The judge has been advised that the a beta o 2.0 is typical o the publishing industy. What would be the ai accoding to CAPM  ate o etun to be used to detemine damages? [assume a maket etun o 14% and iskee ate o 4%] E ( ) 4% + 2(14% 4%) 24%
16 The owne o an investment im is deciding on whethe o not to pay a bonus to his two potolio manages. One manage (M. Mako) was able to obtain a 22% ate o etun on his potolio the potolio had a beta o 1.5. The othe manage (M. Witz) was able to ean a 12% ate o etun on his potolio the potolio had a beta o 1.0. Assuming the owne uses a CAPM citeia to detemine bonuses, which manage  i any deseves the bonus? [assume a maket etun o 14% and iskee ate o 4%]
17 The owne o an investment im is deciding on whethe o not to pay a bonus to his two potolio manages. One manage (M. Mako) was able to obtain a 22% ate o etun on his potolio the potolio had a beta o 1.5. The othe manage (M. Witz) was able to ean a 12% ate o etun on his potolio the potolio had a beta o 1.0. Assuming the owne uses a CAPM citeia to detemine bonuses, which manage  i any deseves the bonus? Calculate the expected ates o etun based on CAPM. E ( ) 4% + 1.5(14% 4%) 19% Mako E ( ) 4% + 1.0(14% 4%) 14% Witz Now, note that based on the isk o Mako s potolio he should have eaned only 19%. In act, Mako has been able to beat the maket o beat CAPM by eaning a 22% ate o etun. Gaphically, you can pictue Mako s etuns as lying above the secuity maket line (SML). Witz on the othe hand should have been able to ean a 14% ate o etun on his potolio accoding to CAPM. He actually only eaned a 12% ate o etun his potolio lies below the SML (in saying this, please do not conuse the eeence to the SML hee with ou use o it when detemining when a paticula asset is unde o ove piced). M. Mako deseves the bonus not because he achieved a highe etun than Witz, but athe because he obtained a highe etun than expected by CAPM! Given Mako s iskie potolio we would expect him (on aveage, ove time) to ean a highe etun than Witz howeve, we would not expect him to consistently beat CAPM. By doing so, especially i this was on a longtem basis, we would say that Mako is supeio at picking stocks something we ll begin to check out late.
18 An entepeneu is consideing puchasing a business. The entepeneu expects the business to have yealy poits oeve o $1 million. The iskee ate is cuently 8% and the expected ate o etun on the maket is 18%. The entepeneu believes that the beta o this business is How much should the entepeneu pay o this business?
19 An entepeneu is consideing puchasing a business. The entepeneu expects the business to have yealy poits oeve o $1 million. The iskee ate is cuently 8% and the expected ate o etun on the maket is 18%. The entepeneu believes that the beta o this business is How much should the entepeneu pay o this business? Fist, we calculate the CAPM expected ate o etun. E( ) + β [ E( ) ] 8% (18% 8%) 19.5% M Next, we calculate the pesent value o a steam o income o $1 million annually oeve. I attempted to simpliy the poblem by assuming the income would be the same oeve. This makes the investment appea much like a consol. In capital budgeting poblems, such as this one, one can use the CAPM expected ate o etun as the appopiate discount acto. $1,000,000 P $5,128,205 This is the appoximate pice, ounding to the neaest dolla..195 Notice, when one ist leans to use pesent value omulas, some sot o inteest ate is typically used as the discount acto. This inteest ate could be the inteest ate on the loan o the iskee ate (as the pue oppotunity cost o the money). In any case, this sot o pocedue ignoes the elevant isk o the investment. I we had used the iskee ate as the discount acto, then the pice we would be willing to pay o a $1 million annual income steam would be the ollowing. P $1,000, $12,500,000
20 Estimation & Testing o CAPM E( ) + β [ E( M ) ] What ae some o the diiculties in estimating CAPM? Histoy vs. Expectations? Time Fame? What is the maket? What is the iskee ate? {o any students that will eve be in a situation o using the CAPM, please note that thee exists moe than one deinition o beta (e.g., leveed vs. unleveed betas) and many ways to estimate a beta}
21 Secuity Maket Line (SML) E() SML E( M ) Slope E( M )  β1 β Now, ewite the CAPM in tems o Excess Retuns and gaph E( ) β[ E( M ) ]
22 E()  Slope β E( M )  In this om, we can begin to think about how to estimate beta. Take a look a plot o GIS and S&P 500.
23 15.00% 10.00% Excess Retuns o GIS 5.00% 0.00% % % % 5.00% 0.00% 5.00% 10.00% 15.00% 5.00% % % Excess Retuns o S&P The above data comes om YahooFinance monthly etuns and uses the Tbill ate om August 2002 to August Can you pictue dawing a staight line to it the data? Thus, we use the egession technique to it the best staight line to the data. What ae the two items o most inteest to us? 1. The slope. It will be the point estimate o beta 2. Whethe the yintecept is statistically signiicant. I so, then CAPM is wong! I not, then CAPM might just be ight. Why?
24 SUMMARY OUTPUT Regession Statistics Multiple R R Squae Adjusted R Squae Standad Eo Obsevations Coeicients Standad Eo t Stat Pvalue Lowe 95% Uppe 95% Intecept Excess Retuns o S&P yintecept Beta E( ) β [ E( M ) ].2179[ E( M ) ] What else can we say?
25 GIST GIST Pedicted GIST S&PT The above is the actual plot o the data and estimated egession line.
26 Thee is anothe method to calculate Beta diectly (without unning a egession). Howeve, ist let s do the egession method once again using Taget (TGT) as ou company. I ll spell out the steps moe diectly hee. (1) Collect data on the adjusting closing pice (which includes any dividends i paid) o the stock. I am using monthly data hee. (2) Collect data on the maket (I use the S&P GSPC) (3) Collect data on the inteest ate o the isk ee asset (I m using 4 week Tbills hee). Hee s the ist ew lines o my aw data table. DATE TGT S&P TBill Aug Jul Jun May Ap
27 (4) Calculate the Retuns o the stock and the maket. This is the simple pecentage change between two consecutive months (nothing ancy hee). Hee ae the ist ew lines o the etuns table. Monthly Retuns o Stocks DATE TBill S&P TGT Aug % 4.20% 9.09% Jul % 7.41% 10.51% Jun % 0.02% 0.43% May % 5.31% 4.37% Ap % 9.39% 19.99% Ma % 8.54% 21.47% (5) We calculate the excess etuns by meely subtacting the iskee ate (in my case Tbills) om the maket etun (S&P) and the stock etun (TGT). Fist ew lines again. TGTT S&PT 8.97% 4.08% 10.36% 7.26% 0.33% 0.08% 4.51% 5.17% 19.90% 9.30%
28 (6) Once we have the data in excess etuns omat, we can un a egession by going to [Tools] menu at the top, then choosing the [Data Analysis], and inally choosing [egession]. Once you have the egession window open: the yaxis will be you column o stock (TGT) excess etuns and the xaxis will be you column o maket excess etuns. It s actually bette i you include the label at the top o the column as well (e.g., TGTT, S&PT), and make sue you click on the Label button within the egession window. Finally, i you want a nice pictue then click on the lineit plot button nea the bottom. S&PT Line Fit Plot TGTT % 30.00% 20.00% 10.00% 0.00% % 0.00% % % % % % TGTT Pedicted TGTT S&PT What is the estimated slope o that line?
29 SUMMARY OUTPUT Regession Statistics Multiple R R Squae Adjusted R Squae Standad Eo Obsevations 85 Coeicients Standad Eo t Stat Pvalue Lowe 95% Uppe 95% Intecept S&PT Now, let s suppose we wanted to value Taget stock using the Godon Model. Note, hee I m just going to make up some numbes to illustate the pocedue. In ode to use the Godon model we need next yea s dividend, the expected gowth ate o the dividend, and the appopiate discount ate (o, equied ate o etun). Suppose the ollowing Next Dividend pe Shae $4 Expected Gowth o the Dividend pe yea 1% Thus, we have the ollowing so a
30 Value $4.01 What should the discount ate be? We can use CAPM to come up with this because it takes into account the isk o the stock. Thus, suppose we look at the data and ind the ollowing. Riskee ate 2% Expected Maket Retun 8% Thus, we get a CAPM ate o etun o 2% (8%  2%) 2% % 9.38% Value $ $47.73 What i you wanted to see a ange o values? Look back at the egession table
31 SUMMARY OUTPUT Regession Statistics Multiple R R Squae Adjusted R Squae Standad Eo Obsevations 85 Coeicients Standad Eo t Stat Pvalue Lowe 95% Uppe 95% Intecept S&PT We could use the lowe bound o the conidence inteval and uppe bound o the conidence inteval o the estimate o beta. I so, we could aive at a sot o ange o estimates o the value o Taget (though keep in mind, we ve aleady aived at ou best guess using the point estimate o beta).
32 Lowe Bound CAPM and Value 2% (8%  2%) 2% % 7.72% Value $ $59.52 Uppe Bound CAPM and Value 2% (8%  2%) 2% % 11.05% Value $ $39.80 The highe the beta, the lowe the value. That is, you need to be able to buy the stock at the lowe pice in ode to obtain the highe equied ate o etun.
33 Diect Method o Estimating Beta σ β σ 1M 1 2 M Co vaiance Vaiance (1) Go back to you data o the etuns o the stock and the maket (you can also use the excess etuns i you pee) (2) Calculate the Covaiance between the stock and the maket. You can ind this unction in the statistical section o the unctions: COVAR(Data,Data) (3) Calculate the Vaiance o the Maket etuns: VAR(Data) (4) Divide the covaiance by the vaiance. That is, you beta.
34
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