Lecture 6 Foundations of Finance. Lecture 6: The Intertemporal CAPM (ICAPM): A Multifactor Model and Empirical Evidence

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1 Lecture 6 Foundatons of Fnance Lecture 6: The Intertemporal CAPM (ICAPM): A Multfactor Model and Emprcal Evdence I. Readng. II. ICAPM Assumptons. III. When do ndvduals care about more than expected return and standard devaton? IV. Examples V. Tastes and Preferences wth a Long-term Investment Horzon. VI. Portfolo Choce. VII. Indvdual Assets. VIII. CAPM vs ICAPM IX. Numercal Example. X. ICAPM Emprcally: The Fama and French [1993] 3-Factor Model Lecture 6: Valuaton Models (wth an Introducton to Captal Budgetng) XI. Readng. XII. Introducton. XIII. Dscounted Cash Flow Models. XIV. Expected Return Determnaton. XV. Constant Growth DDM. XVI. Investment Opportuntes. XVII. Relatve Valuaton Approaches. 0

2 Lecture 6 Corrected Foundatons of Fnance Lecture 6: The Intertemporal CAPM (ICAPM): A Multfactor Model and Emprcal Evdence I. Readng. A. BKM, Chapter 11, Sectons 11.1, 11.6 then B. BKM, Chapter 13, Sectons 13.2 and II. III. ICAPM Assumptons. 1. Same as CAPM except can not represent ndvdual tastes and preferences n {E[R], σ[r]} space. When do ndvduals care about more than expected return and standard devaton? A. sngle perod settng: 1. returns are not normally dstrbuted and ndvdual utlty depends on more than expected portfolo return and standard devaton. B. multperod settng: 1. returns are not normally dstrbuted and ndvdual utlty depends on more than expected portfolo return and standard devaton. 2. expected return and covarances of returns n future perods depends on the state of the world at the end of ths perod; e.g., predctable returns. 3. ndvdual preferences n the future depend on the state of the world at the end of ths perod. 4. ndvdual receves labor ncome. 1

3 Lecture 6 Corrected Foundatons of Fnance IV. Examples A. Predctable Returns. 1. It has been emprcally documented that expected stock returns over a perod depend on varables known at the start of the perod: e.g. dvdend yeld on the S&P 500 at the start of perod t, DP(start t): see Lecture A hgh S&P500 dvdend yeld at the start of ths month mples hgh expected returns on stocks ths month. 3. So a hgh S&P500 dvdend yeld at the end of ths month mples hgh expected returns on stocks next month. 4. Thus, S&P500 dvdend yeld at the end of ths month s a state varable that ndvduals care about when makng portfolo decsons today. B. Human Captal Value. 1. An unexpectedly poorer economy at the end of the month mples a negatve shock to human captal value over the month a. the negatve shock to human wealth s due to an ncreased probablty of a low bonus or, worse, job loss. 2. Thus, the state of the economy at the end of perod t s postvely related to the shock to human captal value over perod t. 3. Suppose a macroeconomc ndcator MI(end t) summarzes the state of the economy at the end of perod t: a. the economy at the end of perod t s better for hgher MI(end t). b. examples of such ndcators nclude # of help wanted postons and # of buldng permts ssued. 4. A suffcently rsk averse ndvdual lkes a portfolo whose return over perod t, R p (t), has a low or negatve covarance wth a. the shock to the ndvdual s human captal over perod t; b. the state of the economy at the end of perod t; c. MI(end t). 5. The macroeconomc ndcator, MI(end t), s a state varable the ndvdual cares about when makng portfolo decsons at the start of perod t. V. Tastes and Preferences wth a Long-term Investment Horzon. A. In general, f an ndvdual cares about a macroeconomc ndcator MI(end t) then can only fully represent an ndvdual s tastes and preferences for her perod t portfolo return usng {E[R(t)], σ[r(t)], cov[r(t), MI(end t)]}. B. Even more generally, f ndvduals care about a set of K state varables s 1 (end t),..., s K (end t), then can only fully represent an ndvdual s tastes and preferences for her perod t portfolo return usng E[R(t)], σ[r(t)], cov[r(t), s 1 (end t)],..., cov[r(t), s K (end t)]}. 2

4 Lecture 6 Corrected Foundatons of Fnance VI. Portfolo Choce. A. Snce ndvdual s care about more than expected return and standard devaton of return, ndvduals no longer hold combnatons of the rskfree asset and the tangency portfolo: 1..e., ndvduals no longer hold portfolos on effcent part of the MVF for the N rsky assets and the rskless. 2..e., ndvduals no longer hold portfolos on the Captal Allocaton Lne for the tangency portfolo. B. Thus, n the ICAPM, snce ndvduals no longer necessarly hold combnatons of the rskfree asset and the tangency portfolo, the market portfolo s no longer necessarly the tangency portfolo. C. Example: Human Captal Value. 1. The tangency portfolo on the MVF for the N rsky assets may have return over perod t whose covarance wth MI at the end of perod t s hgh. 2. Thus, an ndvdual may prefer to hold a portfolo n perod t below the captal allocaton lne for the tangency portfolo but whch has a very low covarance wth MI over perod t. 3. It s possble to show all ndvduals hold combnatons of a. the rskfree asset. b. the market portfolo. c. a portfolo whose return R MI (t) hedges shocks to human captal value over perod t. D. More generally, t s possble to show that n equlbrum all ndvduals rrespectve of tastes and preferences hold a combnaton of: 1. the rskfree asset. 2. the market portfolo. 3. K hedgng portfolos, R h1, R h2,...,r hk, one for each state varable. E. Thus, the ICAPM s a generalzaton of the CAPM. 3

5 Lecture 6 Corrected Foundatons of Fnance VII. Indvdual Assets. A. Recall that the market portfolo s no longer necessarly the tangency portfolo: so the market need not le on the postve sloped part of the MVF for the N rsky assets. B. Mnmum varance mathematcs then tells us that there need not be a lnear relaton between expected return and Beta wth respect to the market portfolo;.e., assets need not all le on the SML: E[R ] - R f β,m {E[R M ] - R f }. C. Example (cont): Human Captal Value. 1. If ndvduals care about covarance of portfolo return over t wth MI(end t) and asset returns over t and MI(end t) are multvarate normally dstrbuted, the followng holds for all assets: E[R (t)] = R f + β *,M λ * M + β *,MI λ * MI where: λ * M = E[R M -R f ] = E[r M ] and λ* MI are constants that are the same for all assets and portfolos; and β *,MI and β *,M are regresson coeffcents from a multvarate regresson of r (t) on r M (t) and MI(end t): r (t) = a,0 + β *,M r M (t) + β *,MI MI(end t) + e (t). 2. The hedgng portfolo for MI(end t) can be used nstead of MI(end t): a. In the multple regresson to determne rsk loadngs, replace MI(end t) wth r MI (t) = [R MI (t)-r f ], the excess return on the portfolo that hedges shocks to human captal value over t: r (t) = a,0 + β *,M r M (t) + β *,MI r MI (t) + e (t). b. Then the followng expresson holds for all assets and portfolos of assets: E[R ] = R f + β *,M λ* M + β *,MI λ* hmi where: λ * M = E[R M -R f ] = E[r M ] and λ * hmi = E[R MI -R f ] = E[r MI ] are constants that are the same for all assets and portfolos. 4

6 Lecture 6 Corrected Foundatons of Fnance D. Generally, f ndvduals care about the covarance of portfolo return wth a set of state varables s 1, s 2,...,s K, returns and the state varables are multvarate normally dstrbuted then can show that the followng holds for all assets and portfolos of assets: E[R ] = R f + β *,M λ * M + β *,s1 λ * s1 + β *,s2 λ * s β *,sk λ * sk where: λ * M, λ * s1, λ * s2,..., λ * sk are constants that are the same for all assets and portfolos; and β *,sk for k=1,2,...,k, and β *,M are regresson coeffcents from a multvarate regresson of r on r M, s 1, s 2,... and s K : r = a,0 + β *,M r M + β *,s1 s 1 + β *,s2 s β *,sk s K + e E. Note: 1. r = R - R f and r M = R M - R f. 2. β *,sk for k=1,2,...,k, and β *,M are referred to as rsk loadngs and vary across assets; they measure the senstvty of asset to each of the rsks that ndvduals care about. 3. λ * M, λ * 1, λ * 2,..., λ * K are referred to as rsk prema and measure the expected return compensaton an ndvdual must receve to bear one unt of the relevant rsk. 4. λ* M = E[R M ]-R f = E[r M ] snce when r M s regressed on r M, s 1, s 2,... and s K get β* M,M = 1 and β* M,s1 = β* M,s2 =... =β* M,sK = 0. F. The K hedgng portfolos can be used nstead of the K state varables: 1. Replace s 1, s 2,...,s K wth [R h1 -R f ], [R h2 -R f ],...,[R hk -R f ] n the multple regresson. 2. Wth ths substtuton, we get rsk prema that satsfy: a. λ * h1 = E[R h1 ]-R f, λ * h2 = E[R h2 ]-R f,..., λ * hk = E[R hk ]-R f. VIII. CAPM vs ICAPM A. It can easly be seen that the CAPM s a specal case of ths ICAPM model. B. In partcular, the expresson for expected return on any asset n VII. D. above reduces to the CAPM when K=0;.e., when ndvduals only care about E[R] and σ[r]. 5

7 Lecture 6 Corrected Foundatons of Fnance IX. Numercal Example. Let GIP(Jan) be the January growth rate of ndustral producton. Suppose each ndvdual cares about {E[R p (Jan)], σ[r p (Jan)], σ[r p (Jan), GIP(Jan)]} when formng hs/her portfolo p for January. The followng addtonal nformaton s avalable: E[R (Jan)] β*,m β*,gip Pnk 1.73% Grey 1.34% Black? where β *,M and β *,GIP are regresson coeffcents from a multple regresson (tme-seres) of R (t) on R M (t) and GIP(t): r (t) = φ,0 + β *,M r M (t) + β *,GIP GIP(t) + e (t). Also know that rskless rate for January, R f (Jan), 0.7%. 1. What s the rsk premum for bearng β *,M rsk? Know ICAPM holds. So all assets le on E[R (Jan)] = R f (Jan) + β*,m λ * M + β*,gip λ * GIP where λ * M = E[R M (Jan)] - R f (Jan). Usng ths formula for Pnk and Grey: Pnk: 1.73 = λ* M λ* GIP Grey: 1.34 = λ* M λ* GIP Now Pnk Y λ* GIP = 4 (1.03% λ* M ) whch can be substtuted nto Grey to obtan 1.34 = λ* M x 4 (1.03% λ* M ). It follows that λ * M = 0.6% and λ * GIP = 1%. So the rsk premum for bearng β *,M rsk λ * M s 0.6%. 2. What s the expected January return on the market portfolo E[R M (Jan)]? E[R M (Jan)] = λ * M + R f (Jan) = 0.6% + 0.7% = 1.3%. 6

8 Lecture 6 Corrected Foundatons of Fnance 3. What s the rsk premum for bearng β *,GIP rsk? From above, the rsk premum for bearng β *,GIP rsk λ * GIP s 1%. 4. Is the market portfolo on the mnmum varance fronter of the rsky assets n the economy? Why or why not? Not necessarly. The reason s that ndvduals care about more than just E[R] and σ[r]. Know that Black satsfes: 5. What s the expected return on Black? E[R Black (Jan)] = R f (Jan) + β * Black,M λ * M + β * Black,GIP λ * GIP = β* Black,M β* Black,GIP 1 = x x 1 = 1.29% 7

9 Lecture 6 Corrected Foundatons of Fnance X. ICAPM Emprcally: The Fama and French [1993] 3-Factor Model: A. ICAPM nterpretaton of CAPM s emprcal falure: 1. Interpret sze and book-to-market for asset as proxyng for rsk loadngs (β*,sk s) on state varables that ndvduals care about. B. Two hedgng portfolos n Fama-French model: excess returns 1. SMB zero-nvestment portfolo: long small and short bg stocks, whle beng book-to-market neutral. 2. HML zero-nvestment portfolo: long hgh and short low book-to-market stocks, whle beng sze neutral. C. Implcatons of Fama-French model for expected returns: 1. All assets satsfy: E[R ] = R f + β *,M E[r M ] + β *,SMB E[r SMB ] + β *,HML E[r HML ] where: r (t) s the excess return on portfolo n month t. r M (t) s the excess return on market portfolo n month t. r SMB (t) s the return on the SML portfolo n month t. r HML (t) s the return on the HML portfolo n month t. β h,m, β h,smb, and β h,hml are the regresson coeffcents from the followng regresson: r (t) = α,3 + β *,M r M (t) + β *,SMB r SMB (t) + β *,HML r HML (t) + u (t) 2. Intuton: a. Value stocks are poor hedges aganst rsks that typcal nvestors care about (β *,HML for value stocks s hgh): so value stocks requre a hgher expected return than growth stocks (all else equal) to nduce nvestors to hold them. b. Small stocks are poor hedges aganst rsks that typcal nvestors care about (β *,SMB for small stocks s hgh): so small stocks requre a hgher expected return than large stocks (all else equal) to nduce nvestors to hold them. 8

10 Lecture 6 Corrected Foundatons of Fnance D. 25 portfolos: 1. quntle break-ponts calculated on the bass of sze and book-to-market. 2. form 25 value-weghted portfolos based on these breakponts. E. Results 1. Devatons from the Fama-French ICAPM expected return equaton much smaller than devatons from the SML a. Value portfolos: largest devaton only 0.13% per annum (compared to a largest devaton of 0.57% per annum from SML) 9

11 Lecture 6 Foundatons of Fnance Lecture 6: Valuaton Models (wth an Introducton to Captal Budgetng). I. Readng. A. BKM, Chapter 18, except Secton B. RWJ, Chapter 8, Secton 8.1 and skm Sectons II. Introducton. A. Defnton of Valuaton. 1. Valuaton s the art/scence of determnng what a securty or asset s worth. a. sometmes we can observe a market value for a securty and we are nterested n assessng whether t s over or under valued (e.g., stock analysts). b. sometmes there s no market value and we are tryng to construct one for barganng or transacton purposes (e.e., a corporaton s nterested n sellng a dvson.). c. sometmes we have a project that we are decdng whether to accept or reject. 2. The value of a securty or asset s gong to depend crucally on the asset prcng model we choose. (The effect s through the approprate dscount rate.) 3. The most common knds of valuaton problem are a. equty valuaton. (1) seasoned equty. (2) IPOs. b. frm valuaton. c. captal budgetng: project valuaton. B. Three Valuaton Approaches. 1. Dscounted Cash Flow (DCF) Models: values an asset by calculatng the present value of all future cash flows 2. Relatve Valuaton: values an asset by lookng at the prces of comparable assets and usng multples such as prce/earnngs (P/E). 3. Contngent Clam Valuaton: uses opton prcng tools to value assets wth opton features. 10

12 Lecture 6 Foundatons of Fnance III. Dscounted Cash Flow Models. A. General Approach. 1. The ntrnsc value of an asset P t s the present value of expected cash flows E[D t ] on the asset dscounted by the requred rate of return on the asset E[R ]: 0 ' E[CF 1 ] 1 % E[R ] % E[CF2 ] (1 % E[R ]) %... % E[CFτ ] %... 2 (1 % E[R ]) τ P ' j 4 τ'1 E[CF τ ] (1 % E[R ]) τ B. Two tems affect the ntrnsc value of an asset. 1. Expected Return on the asset. 2. Stream of Expected Cash Flows on the asset. C. Dscusson. 1. Ths formula hghlghts the relaton between expected return and prce and why we call a model that tells us somethng about expected return an asset prcng model. 2. We can see that holdng expected cash flows fxed, asset prce today s decreasng n expected asset return; the hgher the expected return needed to compensate for the asset s rsk the lower the asset s prce.. 11

13 Lecture 6 Foundatons of Fnance R ' D 1 %P 1 D. Equty Valuaton: Dvdend Dscount Model (DDM) 1. DDM s an example of a dscounted cash flow model. 2. DDM assumes that the stock s bought, held for some tme (dvdends are collected), and then sold. 3. The share s valued as the present value of the expected dvdends and the expected proceeds from the sale. 4. Assume that dvdends are pad annually and that the tme 0 dvdend has just been pad. 5. If the stock s held one year, the return on the stock s P 0 & 1 P 0 ' E[D 1 % P 1 ]. 1 % E[R ] where D t s frm s dvdend per share at tme t and P t s the stock prce of the frm at t. Takng expectatons and rearrangng gves 6. Notce that E[R ] here apples to the frm s equty not the frm s assets. 7. If the stock s held for two years, the present value s gven by 0 ' E[D 1 ]. 1 % E[R ] % E[D2 2 ] (1 % E[R ]) 2 P P 8. If the stock s held untl the company s lqudated, the present value s gven by 0 ' E[D 1 ] 1 % E[R ] % E[D2 ] (1 % E[R ]) %... % E[Dτ ] %... 2 (1 % E[R ]) τ ' j 4 τ'1 E[D τ ] (1 % E[R ]) τ whch s known as a dvdend dscount model (DDM).. 12

14 Lecture 6 Foundatons of Fnance E. Captal Budgetng Decsons. 1. A frm s constantly decdng whether to undertake varous projects avalable to t: these are captal budgetng decsons. 2. The frm s clamholders want the frm to undertake a project f ts value to the frm s postve. 3. The dscounted cash flow approach says to dscount the expected cash flows from the project to the present usng the approprate expected return for the cash flows. 4. The sum of present values (ncludng any ntal outlay) s known as the net present value NPV of the project: NPV 0 ' CF 0 % E[CF ' j 4 τ'0 1 ] 1 % E[R ] % E[CF E[CF τ ] (1 % E[R ]) τ 2 ] τ ] (1 % E[R ]) %... % E[CF %... 2 (1 % E[R ]) τ 5. The frm should only undertake projects wth postve NPVs. 6. The project cash flows nclude all ncremental cash flows as a result of undertakng the project. 7. The approprate expected return for dscountng back the expected project cash flows depends on ther rskness. 8. Example. ZDF Co. s consderng an expanson nto a new lne of busness producng wdgets. The new lne wll requre an outlay of $20 mllon today and wll generate an expected net cash flow of $5 mllon per year for the next 10 years. Each year s cash flow wll be receved at the end of the year. The requred return on the frm s equty s currently 20% p.a. whle the requred return on the frm s assets s 18% p.a.. The requred return on the assets for frms currently producng wdgets s 15% p.a. Headquarters expects to use all ts current dle admnstratve capacty to oversee the new busness lne. Headquarter overhead s $6 mllon per year and dle capacty s currently 25% of total capacty. Should ZDF Co. expand nto the new lne of busness? a. Use the requred return on assets n the new busness lne of 15% p.a.. b. Ignore headquarter overhead snce t s a sunk cost. c. Calculate the NPV of the expanson: NPV 0 = PVAF 15%,10 = x = d. Snce the NPV 0 >0, ZDF Co. should undertake the expanson.. 13

15 Lecture 6 Foundatons of Fnance IV. Expected Return Determnaton. A. Approaches: 1. In a CAPM framework, use the SML; ths approach allows you to explctly make adjustments to your Beta estmate to reflect your assessment of the future Beta of the stock. 2. If valung exstng equty, can also use a hstorcal average return as an estmate of expected return. 3. Can also adjust the estmate to take nto account the predctablty of returns and to allow for the senstvty of the stock to other sources of rsk (n an I-CAPM) settng; we wll not focus on these adjustments here. 4. For smplcty, we wll gnore tax consderatons. B. Equty Beta versus Frm Beta. 1. Can thnk of the frm as a portfolo of assets/projects or a portfolo of clams on those assets: V = A 1 + A A J and V=S+B where V s the value of the frm; A j s the value of the jth asset of the frm; S s the market value of the frm s equty; B s the market value of the frm s debt. 2. Recall that Beta wth respect to the market for a portfolo s a weghted average of the Betas of the assets that comprse the portfolo where the weghts are the portfolo weghts. In an I-CAPM context, the same s true for Beta wth respect to other varables ndvduals care about. 3. It follows that for Beta wth respect to any varable (whch of course ncludes Beta wth respect to the market): β V ' A 1 V β A1 % A 2 V and β V ' S V β S % B V β B β A2 %... % A J V β AJ where β V s the Beta of the frm; β Aj s the Beta of the jth asset of the frm; β S s the Beta of the frm s equty; β B s the Beta of the frm s debt.. 14

16 Lecture 6 Foundatons of Fnance 4. Note a. If the frm s assets are unchanged, then frm Beta wth respect to any varable s unchanged. b. Each asset or project of the frm can have a dfferent Beta from the frm Beta. c. Equty Beta can be calculated by rewrtng the above formula: β S ' V S β V & B S β B ' β V % B S [β V & β B ] d. Can see that equty Beta depends on: (1) the Betas of the frm s assets; (2) the level of debt of the frm; and (3) the Beta of the frm s debt. e. If the frm s debt s rskless, debt Beta wth respect to any varable s 0 and so equty Beta can be calculated: β S ' V S β V 5. Decdng whch Beta to use for a partcular valuaton problem. a. When usng DCF methods to value the frm drectly, use frm Beta to calculate the expected return on the frm s assets. b. When usng DCF methods to value the frm s equty drectly, use equty Beta to calculate the expected return on the frm s equty. c. When valung or evaluatng a specfc project, always use the Beta of the project (whch could be dfferent from the frm s Beta). C. Examples. 1. Suppose ZX company has a two assets. The frst has a Beta wth respect to the market of 1.5 whle the second has a Beta wth respect to the market of 0.9. The frst asset s worth $12M and the second s worth $8M. The frm has $4M of rskless debt. The CAPM holds for the economy, the rskless rate s 5% p.a. and the expected return on the market portfolo s 13% p.a. What s the expected return on ZX s equty? a. Frst, get the Beta of the frm: β V,M ' A 1 V β A1,M % A 2 V β A2,M ' 12 12%8 1.5 % ' %8 b. Second, get the Beta of the equty: β S,M ' V S β V,M ' 20 20& '

17 Lecture 6 Foundatons of Fnance c. Thrd, use the SML to get the expected return on the equty: E[R S ] = R f + {E[R M ]- R f }β S,M = 5% + {13%-5%} = 17.6%. 2. Project Beta s lkely to dffer from frm Beta when: a. The frm s a conglomerate. b. The project represents an entry nto a new ndustry by the frm. 3. IBM Example: Value IBM equty as of the end of 12/04. a. To use DCF technques, need an estmate of the expected return on IBM stock: E[R S-IBM ] b. CAPM (1) Inputs: (a) market expected return (E[R M ]) ) average monthly return on the S&P 500 for the perod 1/65 to 12/04 s 0.94% ) annualzed gves 11.88% = ( ) (b) R f : based on yeld curve at end of 12/04: use 2.74%. (2) Usng the SML: E[R S-IBM ] = 2.74% x ( )% = 11.88% (c) β S-IBM,M can be obtaned from the Bloomberg screen: 1.01 s the Beta obtaned usng weekly data from 1/03 to 12/04. c. Could also use an estmate of expected equty return based on hstorcal average return. 16

18 Lecture 6 Foundatons of Fnance V. Constant Growth DDM. A. Assumpton: 1. g, the growth rate of the expected dvdend, s assumed constant. 2. So E[ D 1 ] = D 0 {1 + g }, E[ D 2 ] = E[D 1 ] {1 + g },..., E[ D τ+1 ] = E[D τ ] {1 + g }. B. DDM can be wrtten: 0 ' D 0 (1 %g ) E[R ] & g P ' E[D 1 ] E[R ] & g whch s vald so long as E[R ] > g. C. IBM Example: 1. Am s to value IBM stock as at 12/31/ Note: a. The total dvdends pad n 2004 were $0.70 per share. b. Usng the CAPM gves a dscount rate of 11.88% p.a. c. The Earnngs Estmates table ndcates a growth rate n annual earnngs over the next 5 years of 10.68%: ths wll be our estmate of g IBM. 3. Inputs: a. D IBM 0 = $0.70 b. E[R IBM ] = 11.88% c. g IBM = 10.68% 4. Usng the constant growth DDM: P 0 IBM = D 0 IBM {1 + g IBM } / {E[R IBM ] - g IBM } = $0.70x ( ) / ( ) = whch can be compared to the prce of IBM at the end of December 2004 of $ If we had full fath n our valuaton we would consder IBM to be overvalued and ssue a sell recommendaton. 17

19 Lecture 6 Foundatons of Fnance D. Other Implcatons of the Constant Growth DDM. 1. Can rewrte the basc model: E[R ] ' E[D 1 ] P 0 % g E[P 1 ] ' P 0 (1 %g ). whch breaks requred return nto the expected dvdend yeld plus expected captal gan. 2. The expected captal gan on the stock s g (assumng no stock splts or stock dvdends): 3. If we assume the stock s correctly valued, we can use the stock s dvdend yeld and earnngs growth rate to calculate an estmate of expected return. 18

20 Lecture 6 Foundatons of Fnance VI. K τ Investment Opportuntes. A. Introducton. 1. Let K τ be the book value of a share of equty at tme τ. 2. The book value per share evolves through tme n the followng way: = K τ-1 + (E τ -D τ ) where E τ are frm s earnngs (after nterest) per share n perod τ. Any earnngs not pad out as a dvdend get added to the book value. B. Assumptons. 1. The constant growth DDM correctly values stock: so g, the growth rate of the expected dvdend, s assumed constant. 2. Each year frm s assets generate an expected after nterest cash flow whch s a constant fracton ROE of the book value of the equty at the start of the year: E[E τ+1 ] = K τ ROE where ths ROE s known as the expected return on book equty. 3. Frm pays a constant fracton (1-b ) of ts earnngs as a dvdend. So D τ+1 = (1-b ) E τ+1 for any τ. E[E 1 ] E 0 a. (1-b ) s called the payout rato. b. b s called the plowback or retenton rato. C. Implcatons. 1. Constant payout rato means expected earnngs per share also grow at g : ' 1 % g E[K 1 ] K 0 ' 1 % g. 2. Fxed ROE means book value per share s also expected to grow at g : b ROE = g. 3. Earnngs growth depends on retenton rate and ROE: 19

21 Lecture 6 Foundatons of Fnance D. IBM Example. 1. Note: a. Takng the earnngs growth estmate as our estmate of g IBM s consstent wth ths constant payout model. b. Earnngs per share for IBM for 2004 was $ Inputs: a. E IBM 0 = $4.94 b. IBM D 0 = $0.70. c. g IBM = 10.68%. 3. Calculatons: b IBM : (1-b IBM ) = D 0 IBM / E 0 IBM = 0.70/4.94 = 14.17% and b IBM = 85.83%. E[E 1 IBM ]: E[E 1 IBM ] = E 0 IBM [1+g IBM ] = 4.94 [ ] = ROE IBM : ROE IBM = g IBM / b IBM = / = 12.45%. 4. K 0 IBM mpled by the model: K 0 IBM = E[E 1 IBM ] / ROE IBM = $5.48/ = $44.02 whch can be compared wth the actual book value at the end of 2004 of $ The dfference between the two s a measure of the extent to whch the assumptons about the evoluton of book value hold for IBM. 20

22 Lecture 6 Foundatons of Fnance E. Uses of the Model. 1. Valuaton. a. Can easly show that the followng formula must hold for the stock prce of frm : 0 ' E (1 % b ROE )(1 & b ) E[E 1 0 ' ](1 & b ). E[R ] & b ROE E[R ] & b ROE P P 2. Optmal Plowback Rato. a. If frm pad out all ts earnngs as a dvdend (b =0), ts stock prce at tme zero would be E[E 1 ]/E[R ]. The dfference between ths value and the constant growth DDM value s due to growth. Thus, 0 ' E[E 1 ] E[R ] % PVGO 0 ' E 0 {1 % g } E[R ] % PVGO 0 where PVGO 1 s the value at tme 0 of frm s growth opportuntes. b. Note that: (1) If E[R ] > ROE : (a) PVGO 1 # 0; and, (b) b = 0 maxmzes P 1. (2) If E[R ] < ROE : (a) PVGO 1 $ 0; and, (b) P 1 s ncreasng n b. (3) If E[R ] = ROE : (a) PVGO 1 = 0; and, (b) P 1 s unaffected by choce of b. F. IBM Example. 1. Inputs. a. IBM P 0 = $ b. E[E IBM 1 ] = $5.48. c. E[R IBM ] = 11.88%. d. ROE IBM = 12.45%. 2. Can calculate PVGO IBM 0 : PVGO IBM 0 = P IBM 0 - {E[E IBM 1 ] / E[R IBM ]} = $ $5.48/ = $ $46.13 = $ Note that PVGO 0 IBM $ 0 as would be expected snce 11.88% = E[R IBM ] < ROE IBM = 12.45%.. 21

23 Lecture 6 Foundatons of Fnance E[E 1 ] E 0 G. Proofs of mplcatons and valuaton formula: 1. Snce dvdend per share s a fxed fracton of earnngs per share, t follows that expected earnngs per share also grow at g : ' E[D 1 ](1 & b ) (1 & b )D 0 ' E[D 1 ] D 0 ' (1 % g ) E[K 1 ] K 0 2. Can show that the book value per share s also expected to grow at g : ' E[E 2 ]ROE ROE E[E 1 ] ' E[E 2 ] ' 1 % g. E[E 1 ] 3. What s the expected book value per share at tme 1? E[K 1 ] = E[K 0 + (E 1 -D 1 )] = K 0 + K 0 ROE - K 0 ROE (1-b ) = K 0 (1 + ROE b ). Thus, have shown that b ROE = g. 4. Can easly show that the followng formula must hold for the stock prce of frm : 0 ' D 0 {1 %g } ' E {1 %g }{1 & b }. E[R ] & g 0 ' E[E 1 ]{1 & b } E[R ] & b ROE E[R ] & b ROE P 22

24 Lecture 6 Foundatons of Fnance VII. P 0 E 0 Relatve Valuaton Approaches. A. Defnton of P/E rato. 1. The Prce/Earnngs or P/E rato s defned as the prce per share dvded by the earnngs per share (after nterest). 2. IBM Example: As of the end of December 2004, the P/E rato for IBM s gven as by Bloomberg. 3. The P/E rato s sometmes used to descrbe the prce as IBM s sellng at tmes earnngs. B. Logc: 1. Investment opportuntes model above mples: ' (1 %g )(1 & b ) E[R ] & g. P 0 ' E 0 (P/E) C. 2. Consder stock and let (P/E) C be the P/E rato of a comparable frm where comparable means a. Smlar requred return. b. Smlar payout rate. c. Smlar growth rate for expected dvdends. 3. Then: C. Use of P/E rato. 1. The P/E rato s sometmes used to get a rough measure of the ntrnsc value of a company that s not publcly traded. 2. When valung the equty of a frm, the approach requres a set of comparable frms to be dentfed. 3. An average P/E rato s calculated for the set of comparable frms. 4. The current earnngs of the frm are multpled by ths average P/E to obtan an estmate of the frm s ntrnsc value. D. Advantages of relatve valuaton. 1. Smple and quck. E. Dsadvantages of relatve valuaton. 1. Defnton of a comparable frm s subjectve. 2. Accountng earnngs are subject to dstortons across frms due to unstable accountng practces. 23

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