Chapter 13: The Capital Asset Pricing Model

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

Objective

1 •The Theory of the CAPM •Use of CAPM in benchmarking • Using CAPM to determine correct rate for discounting

Chapter 13 Contents

13.1 The Capital Asset Pricing Model in Brief 13.2 Determining the Risk Premium on the Market Portfolio 13.3 Beta and Risk Premiums on Individual Securities 13.4 Using the CAPM in Portfolio Selection 13.5 Valuation & Regulating Rates of Return 2

Introduction

• CAPM is a theory about equilibrium prices in the markets for risky assets • It is important because it provides – a justification for the widespread practice of passive investing called indexing – a way to estimate expected rates of return for use in evaluating stocks and projects 3

13.1 The Capital Asset Pricing Model in Brief

• Developed in the 1960’s by Sharp, and independently by Lintner, and Mossin • It answers the question – What would equilibrium risk premiums be if • people had the same set of forecasts of expected returns, risk, and correlations • all chose their portfolios according the principles of efficient diversification 4

So what’s wrong with

ms

analysis

• The assumptions of the last chapter appeared fully acceptable – In fact it may appear to be pedantic to mention them at all • Why develop a new model for risk-return if the present model ain’t broke?

5

ms

-analysis: Estimation

• We did not spell it out, but if you recall the mnemonic for obtaining the portfolio volatility in the ms -model, (given n shares in the portfolio,) we needed – n-means (no problem) – n-standard deviations (no problem) – n*(n-1)/2 correlations (? problem) 6

ms

-analysis: Estimation

• All parameters need estimation, and there are n*(n+1)/2 + n parameters • Assume a portfolio of, say, 2,000 shares represent the market, then we need to estimate more than 2,000,000 parameters, most of which are correlations 7

ms

-analysis: Estimation

• Recall that when you estimate parameters, it is done with only a given level of confidence • Confidence improves with the number of observations • In practice the parameters have time dependence, so old data introduces error • For 2,000 shares, and a 99% confidence, about 20,000 parameters will be in error 8

ms

-analysis: Estimation

– The errors may, or may not, be significant to your investment decision, but their existence calls for further analysis – In any case, the data collection, verification, and processing, is a significant use of analytical resources 9

ms

-analysis: Wishes

• After we have the estimated parameters, finding the optimal portfolio requires quadratic programming, and this again requires heavy use of computational resources – The problem is similar to knowing the position and velocity of every star in the Milky Way, and attempting to predict their futures by computing individual interactions

ms

-analysis: Guidance Principles for Simplification

• An important principle of financial modeling is to create equations that capture the key factors parsimoniously • Another important principle is to attempt to develop simple models – Linear models are then preferred to quadratic models 11

The Astrophysics of Finance

– In the Milky Way problem, an astronomer should specify exactly what needs to be predicted, and give attention to the variables that most affect it – So, if he wants to know when the next star will come close enough to Sol to disturb the Oort cloud then – close stars need individual analysis – distant stars may be treated homogeneously 12

Specifying the Model

• In the last chapter we examined diversifying a homogenous portfolio, and we observed that there were two kinds of risk – diversifiable or individual risk – Nondiversifiable or market risk 13

Specifying the Model

• We also observed that in the limit as the number of securities becomes large, we obtained the formula s

portfioio

 s

exemplar



exemplar i ,exemplar j

– This formula tells us that the correlations are of crucial importance in the relationship between a portfolio risk and the stock risk 14

Specifying the Model

• In the homogenous model, we saw that there was individual- and market-risk • Assume that each equity’s return is the composition of two random variables: – one associated with the market’s return – one associated with the company-specific return 15

Specifying the Model: Assumptions

• Company-specific return on any stock x – is not correlated to the company-specific return on any other stock y – is correlated with the market return • The risk-free rate is constant during the investment the period 16

Assumptions

– Investors forecasts agree with respect to expectations, standard deviations, and correlations of the returns of risky securities – Therefore all investors hold risky assets in the same relative proportions – Investors behave optimally • In equilibrium, prices adjust so that aggregate demand for each security is equal to its supply 17

Market Portfolio

• Since every investor’s relative holdings of the risky security is the same, the only way the asset market can clear is if those optimal relative proportions are the proportions in which they are valued in the market place • Market Portfolio 18

CML and the CAPM

• CAPM says that in equilibrium, any investor’s relative holding of risky assets will be the same as in the market portfolio • Depending on their risk aversions, different investors hold portfolios with different mixes of riskless asset and the market portfolio 19

CAPM Formula

m

r

 m

m

s 

m

r

f

s

r



r

f

slope

 m

m

s 

m

r

f

20

Active v. Passive Management

• CAPM implies that, on average, the performances of active portfolio managers is equal to that of passive managers employing just the market portfolio and the risk-free security • Diligent managers do outperform passive managers, but only to the degree that their diligence is rewarded 21

Reward Only for Market Risk

• The risk premium on any individual security is proportional only to its contribution to the risk of the market portfolio, and does not depend on its stand-alone risk • Investors are rewarded only for bearing market risk 22

13.2 Determining the Risk Premium on the Market Portfolio

• CAPM states that – the equilibrium risk premium on the market portfolio is the product of • variance of the market, s 2 M • weighted average of the degree of risk aversion of holders of risk, A m

r m



r f



A

s 2

M

23

Comment

– CAPM explains the difference between the riskless interest rate and the expected rate of return on the market portfolio, but not their absolute levels – The absolute level of the equilibrium expected rate of return on the market portfolio is determined by such factors as – expected productivity – household inter-temporal preferences for consumption 24

Example: To Determine ‘A’

m m

r M r M A

  0 .

14 , 

r f

 s

r M A

s 2

r M

 0 .

20 ,

r f



A

  0 .

06 , m

r M

s 2 

r M r f

0 .

14  0 .

06  2 .

0 0 .

20 2 25

13.3 Beta and Risk Premiums on Individual Securities

– If risk is defined as that measure such that as it increases, a risk-averse investor would have to be compensated by a larger expected return in order that she would continue to hold it in her optimal portfolio, then the measure of a security’s risk is its beta, b b tells you how much the security’s rate of return changes when the return on the market portfolio changes 26

Comment:

b

= 1

• A security with a b = 1 on average rises and falls with the market – a 10% (say) unexpected rise market return premium will, on average, result in a 10% rise (fall) (fall) in the in the security’s return premium 27

Comment:

b 

1

• A security with a b  1 on average rises and falls more than the market – With a (fall) b = 1.3, a 10% (say) unexpected rise in the market return premium will, on average, result in a 13% rise (fall) in the security’s return premium • Such a security is said to be aggressive 28

Comment:

b 

1

• A security with a b  1 on average rises and falls less than the market – With a (fall) b = 0.7, a 10% (say) unexpected rise in the market return premium will, on average, result in a 7% rise (fall) in the security’s return premium • Such a security is said to be defensive 29

CAPM Risk Premium on any Asset

• According the the CAPM, in equilibrium, the risk premium on any asset is equal the product of – b (or ‘Beta’) – the risk premium on the market portfolio m

r i



r f

  m

m



r f

 b

i

 m

r i



r f

  m

m



r f

 b

i

30

Security Market Line

– The plot of a security’s risk premium (or sometimes security returns) against security beta • Note that the slope of the security market line is the market premium • By CAPM theory, all securities must fall precisely on the SML (hence its name) 31

Practical Example

– Some simulated data was generated under the assumptions that: • the market portfolio return has an expected value of 0.15, a volatility of 0.20, and index 0 = 50 the share z has a return of 0.12, a volatility of 0.25, and price 0 = 30 (no dividends) • the correlation between the returns is 0.90; and the risk-free rate is 0.05

32

Security Prices

70 60 50 40 30 20 Market_Price Stock_Z_Price 10 0.000 0.083 0.167 0.250 0.333 0.417 0.500 0.583 0.667 0.750 0.833 0.917 1.000

Years

33

Data Set Used

• In order to display the material clearly, only one year of data is generated, and is collected monthly, resulting in 13 sets of prices • In a real simulation, must be collected in order to provide an adequate confidence interval for parameter estimates much more data 34

Transformation of Prices into Returns

– The prices are transformed into monthly holding period returns (mhpr_Ind, and mhpr_Z) – The mhprs are transformed into annual rates, compounded annually – The annual rates compounded annually are transformed to annual rates compounded continuously 35

Table of Prices

month 0 Mkt_Price Z_Price 50.00

hpr_Mkt 30.00 hpr_Mkt 9 10 11 12 1 2 3 4 5 6 7 8 55.84

52.87

58.19

60.33

56.97

51.52

52.80

55.04

55.76

62.20

56.84

55.30

33.87

33.65

39.19

41.30

38.93

34.20

35.88

38.24

40.64

46.26

41.01

39.54

11.68% -5.32% 10.07% 3.66% -5.57% -9.56% 2.47% 4.24% 1.32% 11.55% -8.62% -2.71% hpr_Z hpr_Z 12.90% -0.64% 16.47% 5.38% 132.55% -65.59% 115.15% 43.19% 145.56% -7.75% 182.98% 62.90% 176.82% -64.54% 155.62% 67.97% -5.74% -68.71% -70.89% -68.35% -12.15% -120.56% -155.40% -131.50% 4.91% 6.56% 29.32% 49.83% 57.54% 76.22% 51.08% 76.06% 6.28% 13.83% 15.70% 131.12% 73.08% 155.46% 34.48% 175.09% -11.34% -108.23% -144.43% -116.49% -3.58% -32.93% -43.78% -24.76% an_an_fact 1.105934 1.318151

an_cont_rate 0.10069

0.27623

mu sig rho beta 36 10.07% 27.62% 0.259099 0.325796

0.968777

1.218157

-150%

Regression of Returns of Z on Market

-100% 200% 150% 100% -50% 50% 0% -50% 0% -100% -150% -200%

Market Return

37 50% 100% 150%

Financial Calculators

• Everything could have been done using a modern standard-issue financial or scientific calculator – Remember, the correct rate to use is the annual rate compounded continuously, and that month-to-year conversions of standard deviation involve a square root of 12 – Take care to enter the market rate as the independent variable, x 38

Accuracy Issue

– We assumed that the s ’s and  ’s are constants, but they are random variables too – In order to achieve adequate confidence, a large sample is needed – Small movements in price are masked by transaction prices • The result is a compromise between currency and confidence 39

Model and Measured Values of Statistical Parameters

m m s m m z s z  b modl 15% 20% 12% 25% 90% 1.13

Meas 10% 26% 28% 33% 97% 1.22

40

Comment

• The illustrated trajectory is typical for monthly data collected over a year • Caution: avoid using small data sets to estimate CAPM parameters 41

Regression Line

• The slope of the regression line of dependent stock against independent market returns is beta 42

-2.0

-1.5

-1.0

Security Market Line

20% 15% 10% 5% -0.5

0% -5% 0.0

-10% -15% -20%

Beta (Risk)

43 0.5

1.0

Market Portfolio 1.5

2.0

Observation

• All securities, (not just efficient portfolios) plot onto the SML, if they are correctly priced according to the CAPM 44

The Beta of a Portfolio

• When determining the risk of a portfolio – using standard deviation results in a formula that’s quite complex s

w

1

r

1 

w

2

r

2  ...



w n r n

  

i

  1 ,

n



w i

s

r i

 2  2  

i j



w i w j

s

r i

s

r j



i

,

j

   1 2 – using beta, the formula is linear b

w

1

r

1 

w

2

r

2  ...



w n r n



w

1 b

r

1 

w

2 b

r

2  ...



w n

b

r n

 

i w i

b

r i

45

Computing Beta

• Here are some useful formulae for computing beta b b

i i

  b

i

,

M

m

r i

m

M

 

r f



r f

s s

i

,

M

2

M

 s

i

s s

M

2

M



i

,

M

 s

i

s 

M i

,

M

46

13.4 Using the CAPM in Portfolio Selection

• Whether or not CAPM is a valid theory, indexing is attractive to investors because – historically it has performed better than most actively managed portfolios – it costs less to implement that active management 47

A Paradox Resolved

• The last chapter posed a paradox with two securities co-existing, one having a lower standard deviation and higher return than the other • If we accept the CAPM as a valid theory, we have a resolution • Both securities lie on the SML, and both securities lie below the CML 48

s

risk and

b

risk

• A security has two kinds of risk: risk that may be diversified away, and risk that is associated with the market – The CAPM theory states that the lower return on the s riskier security implies that it has a lower level of market b risk, and this is the only relevant risk – The s riskier security contains relatively more (irrelevant) security-specific risk 49

A Brand Manager

• Most investors have the opportunity to eliminate most individual risk from their portfolio; but consider a product manager’s exposure to risk • If a brand manager’s products • perform well, promotion, higher salary, and greater autonomy follow • perform badly, humiliation, unemployment and poverty follow 50

A Brand Manager

• Now assume that a new product is available for inclusion in the brand, but given its b -risk and expected return, it falls below the sml, and hence is not in the investors’ interests – The manager discovers that the new product reduces his total risk, and acts in his own interests (rather than the investors’), and accepts the product (agency problem) 51

The Portfolio Manager

• Remember (last chapter) we had not resolved the issue how to evaluate the performance of a portfolio manager, but given the CAPM a resolution is at hand • If your portfolio is producing actual returns with a lower beta than the sml specifies (with statistical significance), then you should certainly not be fired 52

The Portfolio Manager

• The further a well diversified portfolio consistently lies above (below) the sml, the better (worse) the fund manager’s performance – There are several measures of this distance, but this topic is better left for another day 53

How to Win Investment Games

– You may have been asked to take part in an investment game where you ‘given’ $100,000 to manage for a semester; winner takes all – The overwhelming chances are that the winning student uses poor financial practices 54

How to Win Investment Games (Continued)

– The criteria of success for the game differs significantly from real-life investing, so your strategy for winning is likely to be different – If you diversify away unsystematic risk--even if you have some kind of informational advantage over your competition--you are very unlikely to win the game – To win, you crowd need individual risk to separate you from the – Unlike a real investor • you don’t have real downside-risk • your upside-potential materializes only by being first 55

13.5 Valuation and Regulating Rates of Return

• Beta may be used to obtain the discount factor for a project • Assume a project is similar to the projects undertaken by another firm, ‘Betaful’ • Betaful is financed by 20% short-term debt, and 80% equity, and its b is 1.3 (assume debt is risk-free) • Your optimal capital structure is 40% (risk free) debt, and 60% equity 56

Valuation and Regulating Rates of Return

• Assume the market rate is 15%, and the risk free rate is 5% • Compute the beta of Betaful’s operations b

company

b

company

b

company



w equity

b

equity

 1 .

04 

w

bond b

bond

 0 .

80 * 1 .

3  0 .

20 * 0 57

Valuation and Regulating Rates of Return

• Beta of Betaful’s operations is equal to the beta of our new operation • To find the required return on the new project, apply the CAPM m

r

  0

f

 .

05 b  

r m

1 .

 04 

r f

 0 .

15  0 .

05   15 .

4 % 58

Valuation and Regulating Rates of Return

• Assume that your company is just a vehicle for the new project, then the beta of your unquoted equity is b

company

1 .

04  

w equity

b

equity

0 .

60 * b

equity



w

bond b

bond

 0 .

40 * 0 b

equity

 1 .

73 59

Valuation and Regulating Rates of Return

• Assume that your company has an expected dividend of $6 next year, and that it will grow annually at a rate of 4% forever, the value of a share is

p

0 

r D

1 

g

 0 .

154 6  0 .

04  $ 52 .

63 60

Valuation and Regulating Rates of Return

• Regulators use the CAPM to establish a ‘fair’ rate of return on invested capital in public utilities, given the level of risk 61