The Capital Asset Pricing Model. Chapter 9

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1 The Capital Asset Picing odel Chapte 9

2 Capital Asset Picing odel CAP centepiece of moden finance gives the elationship that should be obseved between isk and etun of an asset it allows fo the evaluation of: futue pices of stocks fai pice of stocks not tading yet IPOs deived using pinciples of divesification with simplified assumptions. 9-2

3 Assumptions individual investos ae pice takes = they act as if thei actions do not affect pices pefect competition single-peiod investment hoizon = all investos plan to hold assets fo the same peiod myopic behavio investments ae limited to taded financial assets ules out investment in human capital and boowing estictions no taxes and tansaction costs = no fees o commissions, o income taxes 9-3

4 Assumptions cont. investos ae ational mean-vaiance optimizes = they use the akowitz potfolio selection model infomation is costless and available to all investos thee ae homogeneous expectations = all investos shae the same view of the wold i.e., they deive the same efficient potfolio fontie 9-4

5 Optimal Risky Potfolio since all investos have the same potfolio fontie, all investos will hold the same potfolio fo isky assets the only diffeence is amount invested in it vs. the isk-fee asset sum up the holdings of all investos in the maket: boowing and lending cancel out net wealth of the economy all individuals hold the same isky potfolio popotion in potfolio = popotion in the maket 9-5

6 Resulting quilibium Condition All investos will hold the same potfolio fo isky assets, the maket potfolio maket potfolio = contains all secuities and the popotion of each secuity is its maket value pice times numbe of shaes as a pecentage of total maket value maket potfolio is not only efficient, it is also the tangency point to the optimal CAL the Capital aket Line becomes the best attainable CAL 9-6

7 Capital aket Line CL f σ σ 9-7

8 aket Potfolio the fact that all assets ae included and that the popotions in the individual isky potfolio and in the maket potfolio ae equal ae ensued by the picing mechanism mutual fund theoem: passive stategy of investing in the maket index is efficient anothe fom of the sepaation popety: boke finds the maket potfolio the optimal isky potfolio investos decide how much to invest in the maket potfolio vesus the isk-fee asset 9-8

9 Risk Pemium on the aket Potfolio depends on the aveage degee of isk avesion i.e., the degee of isk avesion of a typical investo ecall that p y = 0.01 Aσ since boowing and lending offset in the aggegate, y = 1 eaanging: f 2 p 2 f = 0.01 A σ 9-9

10 Risk and Retun of Individual Secuities what mattes is not individual secuity isk, but potfolio isk when assessing a secuity, what mattes is its contibution to potfolio isk potfolio isk with many assets: σ 2 p = n n i= 1 j= 1 w w Cov, i j the contibution of stock k to the potfolio: n w kwjcov k, j = j= 1 similaly, stock k s contibution to the isk pemium of the potfolio: w k [ p f ] i j w Cov k k, p 9-10

11 aket Risk and Retun so, when compaed to maket potfolio: Contibution to etun = w k [ f ] Contibution to isk = w k Cov k, hence, ewad-to-isk atio is: k Rewad to isk atio = Cov, the ewad-to-isk atio of the maket potfolio is called the maket pice of isk: aket pice of isk = f 2 σ k f 9-11

12 quilibium Rewad-to-Risk Ratio if an investment has lowe ewad-to-isk atio than anothe o the aveage, then investos would move away fom it pice falls etun inceases ewad-to-isk atio ises convesely, if an investment has a highe ewad-to-isk atio than anothe o the aveage, then investos would tilt towad it pice ises etun falls ewad-to-isk atio deceases in equilibium, all investments should offe the same ewad-to-isk atio 9-12

13 9-13 quilibium Risk-Retun Relationship hence, any stock should have the same ewad-to-isk atio as the maket potfolio: eaanging, this gives the equilibium isketun elationship: 2, f m k f k Cov σ = [ ] f m k f k Cov =, 2 σ

14 Beta the atio of the contibution of stock k to the maket potfolio isk to total isk is called beta: Cov β k = σ k 2, the usual expession of the capital asset picing model CAP is the elationship between expected etun and beta: k = f + β k [ f ] hence, the ight measue of isk in this famewok is beta 9-14

15 9-15 Potfolio Beta if the expected etun-beta elationship holds fo any individual asset, it has to hold fo any combination of assets as well: whee the potfolio beta is β p = w 1 β 1 + w 2 β w n β n [ ] [ ] [ ] [ ] f p f p f n n f n n n f f f f w w w w w w w w w + = + = + + = + + = β β β β

16 Intuition beta is popotional to the contibution of an asset to the isk of the optimal isky potfolio hence, beta is the ight measue of isk isk-avese investos evaluate assets based on thei isk isk pemium should be a function of the ight measue of isk this is the CAP: the isk pemium of an asset is popotional to its beta 9-16

17 Cautions impotant distinction between fim etun as measued by dividends, etc. and stock etuns as measued by the ate of etun on holding stocks if eveybody expects a company to do well and pay lage dividends as infomation is public, then pice inceases and expected etun stays the same only the isk of the company beta influences expected etuns 9-17

18 Secuity aket Line the expected etun-beta elationship can be intepeted as a line in the expected etunbeta plane, called Secuity aket Line SL slope of the SL = [ f ] beta of maket potfolio is β Cov = σ, beta of the isk-fee asset is β f Cov = σ σ 2 = = 2 2 σ, 0 2 σ f = =

19 Secuity aket Line SL Slope = [ f ] f β = 1 β 9-19

20 Secuity aket Line vs. Capital aket Line What is plotted CL plots efficient potfolios, i.e. combinations of the isky potfolio and the isk-fee asset it is not valid fo individual assets SL plots individual assets and potfolios easue of isk fo CL standad deviation because welldivesified potfolios fo SL beta because individual assets 9-20

21 xample of SL = 11% f = 3% aket isk pemium = f = 11 3 = 8% β x = 1.25 x = f + β x [ f ] = = 13% β y = 0.6 y = f + β y [ f ] = = 7.8% 9-21

22 Secuity aket Line x = 13% = 11% y = 7.8% f β y = 0.6 β = 1 β x = 1.25 β 9-22

23 Usefulness of CAP Stock Picing SL gives the fai etun and hence pice of a stock, given its isk beta in pactice, assets might not lie exactly on the SL because of picing eos an undepiced asset would give a highe expected etun than pedicted by SL, hence it would be plotted above the line convesely, an ovepiced asset gives a lowe expected etun than pedicted by the SL and would plot below the SL 9-23

24 Alpha the diffeence between the actual and the fai expected ates of etun on an asset is called alpha: α = a if α > 0, the stock is undepiced hence desiable to invest in if α < 0, the stock is ovepiced hence undesiable to invest in 9-24

25 xample of Calculating Alphas = 11% f = 3% aket isk pemium = f = 11 3 = 8% β x = 1.25 a x = 15% x = f + β x [ f ] = [11 3] = 13% α y = a x x = = 2% hence, stock X is undepiced 9-25

26 Secuity aket Line a x = 15% x = 13% α x = 11% f β = 1 β x = 1.25 β 9-26

27 Usefulness of CAP Budgeting Decisions CAP gives the equied ate of etun of an investment poject, given its isk, so that investos find it acceptable manages can use CAP to find the cutoff intenal ate of etun 9-27

28 xtensions of the CAP No Boowing CAP is based on the sepaation pinciple: all investos find the same isky potfolio to be optimal when boowing is esticted, the sepaation pinciple fails maket potfolio is not the common optimal isky potfolio anymoe hence, CAP fails Black 1972 povides a model that extends the CAP to cases whee boowing is patially o completely esticted i.e., no iskfee asset 9-28

29 Zeo-Beta odel Implications: any combination i.e., potfolio of efficient potfolios is also efficient fo evey efficient potfolio thee is an inefficient potfolio on the mean-vaiance fontie with which it is uncoelated the zeo-beta potfolio the expected etun of any asset can be found using any two fontie potfolios P and Q: Cov k, P Cov P, Q k = Q + 2 P Q σ Cov, P P Q [ ] 9-29

30 Finding Zeo-Beta Potfolios P Q ZQ ZP ZQ ZP σ ΖQ σ ΖP σ 9-30

31 xample No isk-fee asset suppose thee is no isk-fee asset then investos cannot boow o lend at the iskfee ate investos will want to invest in efficient potfolios since any potfolio can be witten as a combination of 2 fontie potfolios, any potfolio the investos choose can be witten as a combination of the maket potfolio and its zeobeta countepat 9-31

32 9-32 xample No isk-fee asset cont. in this case, the equation detemining the expected etun of an asset becomes: this is like the CAP equation, but with Z instead of f [ ] [ ] [ ],,,, 2 2 Z k Z Z k Z Z Z Z k Z k Cov Cov Cov Cov + = + = + = β σ σ

33 Zeo-Beta odel No Risk-fee Asset P ZP C ZP σ ΖP σ 9-33

34 CAP and Lifetime Consumption anothe extension concens the time hoizon investos conside investos may not wish to hold assets fo just one peiod, o may have diffeent holding peiods Fama 1970 showed that the single-peiod CAP is appopiate even in a multipeiod setting, unde cetain assumptions 9-34

35 CAP and Liquidity yet anothe extension elates to the liquidity pemium liquidity = the cost and ease with which an asset can be conveted into cash i.e., sold eseaches found that liquidity isk i.e., the isk of not being able to sell apidly and cheaply the asset is systematic, hence difficult to divesify less liquid asset should offe a liquidity pemium ove moe liquid assets 9-35

36 CAP and Liquidity cont. hence, need to modify the CAP famewok since illiquid assets offe a liquidity pemium, in the long un they offe highe ates of etuns than thei liquid countepats the isk pemium will take liquidity into account: k f = β k [ f ] + fc k whee fc k is the liquidity pemium as a function of the tansaction costs of asset k fc k is inceasing in c k, but at a deceasing ate a useful measue of liquidity is the bid-ask spead: moe liquid assets have lowe speads 9-36

37 Liquidity and Aveage Retuns Aveage onthly Retun % Bid-Ask Spead % 9-37

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