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FINANCIAL MANAGEMENT

Lecture 6: Risk and return– Capital Asset Pricing Model

Assoc.Prof.Dr. NGUYEN THU THUYFaculty of Business Administration FOREIGN TRADE UNIVERSITY

Contents

1. Relation between risk and return2. Estimation of risk/return of a security3. Evaluation of risk/return of a

portfolio4. Capital Asset Pricing Model (CAPM)

and its applications5. WACC – cost of capital

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Holding Period Returns - HPR

l One share of stock is purchased today for $100. One year later the stock price is $116.91 and it has paid a dividend of $4.50

l Dollar profit = dividend + capital gainl Dollar profit = dividend + [ending price – beginning

price] = dividend + [P1 – P0]= 4.50 + [116.91 – 100] = 4.50 + 16.91 = $21.41

l Total one-year HPR = [dividend + P1 – P0] / P0

= 4.50/100 + (116.91 – 100)/100 = 0.045 + 0.1691= 0.2141 or 21.41%

l HPR consists of 4.5% dividend yield and 16.91% capital gains yield

Introduction to risk and return

l Stock and bond returns: usually stated in terms of mean return & standard deviation (= measure of risk or dispersion of individual returns around the mean)

Year U.S. T-bills S&P 50019811982198319841985

0.14710.10540.08800.09850.0772

-0.04910.21410.22510.06270.3216

Mean return 10.324% 15.488%

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Introduction to risk and return

l Variance

l Standard deviation

1

)(1

2

2

-

-=å=

n

rrn

ii

s

1

)(1

2

-

-=å=

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Introduction to risk and return

l Calculate the std.deviation of S&P500 1981-1985 à σ = 14.69% (mean = 15.49%)

l T-bills: defined as riskless investment (mean = 10.32%)

l Realized risk premium for S&P500: 15.49% - 10.32% = 5.17% per year

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Historical record of investment classes and returns in the U.S. during 1926-2002

Investment Mean returns Std. deviation

Treasury billsLong-term T-bondsLong-term Corp. bondsLarge firm stocksSmall firm stocks

3.8%5.8%6.2%

12.2%16.9%

3.2%9.4%8.7%

20.5%33.2%

Inflation 3.1% 4.4%

Compared to T-bills, large firm stocks earned an average realized risk premium = 12.2 – 3.8 = 8.4%

The normal distribution & stock returns

Histograms of annual returns for both large and small firms closely resemble a normal distribution or bell curve.

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The normal distribution & stock returns

The normal distribution & stock returns

- With the normal distribution, we can expect that:- 68.26% of the annual returns are in the range of +/-

one std.dev. of the mean return- 95.44% of the annual returns are in the range of +/-

two std.dev. of the mean return- 99.74% of the annual returns are in the range of +/-

three std.dev. of the mean return- During 1926-2002: large firm returns are

described by a mean return of 12.2% and a std.dev. of 20.5%- We expect about 68% of the annual returns fall within

the range from -8.3% to +32.7%

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A single security’s return

l Expected (ex ante) return:¡ Future possible states “s” of the economy¡ ps: probability of state “s” occurring¡ Rs: return on the security if state “s” occurs

å=

==K

sss RpRRE

1)(

l Variance å=

-=K

sss RRp

1

22 )(s

iancevar=sl Std. dev.

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Two securities

l Covariance between any two securities:

l Correlation coefficient between any two securities:

å=

--=K

sBBsAAssAB RRRRp

1,, ))((s

)/( BAABAB sssr =

Risk and return of a two stock portfolio

Outcomes Probability ps RA RB

BoomNormalBust

0.250.500.25

20%10%0%

5%10%15%

Let’s calculate the expected return, covariance and correlation of the portfolio

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Risk and return of a two stock portfolio

Expected return: 10.0== BA RR

Std. deviation: %071.7=As%536.3=Bs

Covariance:

Correlation coefficient:

%25.0-=ABs

0.1)03536.0)(07071.0/(0025.0)/( -=-== BAABAB sssr

Security B: very unusual à highest return in downturns, lowest returns in boom times à concept of diversification

Diversification

l Suppose we invest $100 in A and $200 in B. Dollar returns under each possible outcome:

Outcome Prob. CF on $100 in A

CF on $200 in B

Total CF % return on $300 in A&B

BoomNormalBust

0.250.500.25

$120$110$100

$210$220$230

… …

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Diversification

l Suppose we invest $100 in A and $200 in B. Dollar returns under each possible outcome:

Outcome Prob. CF on $100 in A

CF on $200 in B

Total CF % return on $300 in A&B

BoomNormalBust

0.250.500.25

$120$110$100

$210$220$230

$330$330$330

10%10%10%

- Expected return = 10%- Variance = 0.00- Std.dev = 0.00 (no risk)

Diversification

l Works in many casesl Correlation between two securities:

¡ Positively correlated: 0 < ρAB < 1¡ Perfectly positively correlated: ρAB = 1¡ Negatively correlated: -1 < ρAB < 0¡ Perfectly negatively correlated: ρAB = -1¡ Uncorrelated: ρAB = 0

à In which case, will diversification not work?Only when ρAB = 1

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Return and risk for portfolios

l Expected return of a portfolio:

å=

==N

iiiPP RXRRE

1)(

å åå= ¹==

+=N

i

N

jijijji

N

iiiP XXX

1 ,11

222 2 sss

l Variance of a portfolio:

Xi = % (or weight) of the portfolio in security “i” N = number of securities in the portfolio.Sum of weights = 1

The case of portfolio of 2 securities

)()()( BBAAP REXREXRE +=

ABBABBAAP XXXX ssss 222222 ++=

l The above example:

10.0)( =PRE02 =Ps

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Efficient sets and diversification

Efficient sets and diversification

l As long as ρ < 1, std.dev. of a portfolio is lessthan the weighted average of std.dev. of the individual securities à that’s why diversification works

l Efficient sets = portfolios within the investment opport. set that represent the best return-risk combinations

à the portfolios must have the highest expected return at a given std.dev. relative to all other portfolios in the invest.opport.set

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Efficient set for many securities

CALs = Capital Allocation Lines = infinite linear combinations of the riskless asset and a portfolio of risky assets.

Capital market equilibrium & CAPM

CML = identical for all rational investors

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Systematic vs. unsystematic risk

l A rational investor will only hold the well-diversified Market portfolio

l Diversification eliminates part of the risk of an individual security

l The only relevant risk = a security’s contribution toward the std.dev. of the well-diversified portfo.

Total risk of indivi.securi. = systematic risk + unsystematic (diversifiable) risk

Systematic vs. unsystematic risk

l A typical stock: 25% (75%) of its total return variance is driven by systematic (unsystematic) risk.

l The 75% is specific/unique to the firm à firm-specific risk à unsystematic, diversifiable.

l A market portfolio of stocks: 100% of return variance is due to risk that is macroeconomic or market in nature ànondiversifiable risk.

l Because firm-specific risk is diversifiable, it should be irrelevant.

l Only the market or systematic risk is now relevant as it affects all stocks in some degree.

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Relation between portfolio risk and number of securities in a portfolio

CAPM modell All rational investors identify exactly the same ORP

portfolio of risky assets à the only to choose is the Market portfolio.

l Beta (β): relevant risk of any security as contribution to the risk of Market portfolio.

l Market portfolio has Beta = 1, by definition. Ideally, it is a global wealth portfolio of all assets à certainly unobservable à a stock market index, e.g., S&P500, is usually used

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CAPM – definition of Beta

l Linear relationship between required return and Beta for an individual security i under CAPM:

2, / MMii ssb =

[ ]FMiFi RRRR -+= b

RM – RF = market risk premium à extra compensation for risk that risk adverse investors require for holding the market portfolio

CAPM - example

l Apple Computer has a Beta of 0.8; l RM (S&P500) = 10%; RF = 5%

à RAPPL = 0.05 + 0.8*(0.10 – 0.05) = 0.09 or 9%

Note:(0.10 – 0.05) = 5% = market risk premium0.8*(0.10 – 0.05) = 4% = risk premium for Apple Computer stock

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Security Market Line - SML

SML refers to the relationship of the required return of an individual asset to its Beta = graphical depiction of the linear CAPM equation

CAPM, SML & mispricing of stocks

l Under CAPM: any asset’s required return should be a function of its Beta à its return should fall exactly on the SML.

l If not on the SML à the stock is mispriced.

l Example: You’re an analyst and know the true beta of IBM.True βIBM = 2.0; RM = 10%; RF = 6%

à RIBM = 0.06 + 2.0*(0.10 – 0.06) = 14% per yearDividend next year = $1.00 per share, with permanent growth rate g = 6% per year

à The correct price should be:P0 = Div/(r-g) = 1.00/(0.14-0.06) = $12.50 per share

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Example of mispricing (cont.)l If the market has priced the stock incorrectly (perhaps

because the market or investors estimate the risk incorrectly)¡ The stock is priced by the market to yield an expected return of r =

16% per year (too high rate of return) à mispriced and selling atP0 = 1.00/(0.16-0.06) = $10.00 per shareà The stock is undervalued at $10.00, and you would recommend that this stock should be purchased.

¡ The stock is priced by the market to yield an expected return of r = 12% per year (too low rate of return) à mispriced and selling atP0 = 1.00/(0.12-0.06) = $16.67 per shareà The stock is overvalued at $16.67, and you would recommend that this stock should be sold or perhaps even shorted.

Mispricing should be enventually corrected

l The stock lies above SML: undervaluedThe numerous purchasing activities force the stock prices go up à back to its correct value of $12.5 per share

l The stock lies below SML: overvaluedThe numerous selling activities force the stock price go down à back to its correct value of $12.5 per share

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Drawbacks of CAPM

l Problems while using CAPM¡ Size effect: stocks of companies with small

market cap outperform those with large market cap (ceteris paribus)

¡ The effect of P/E and M-to-B ratio: stocks of companies with low P/E and MB outperform those with high P/E and MB cao (ceteris paribus)

¡ The effect of month/day/season à not very consistent over time.

Multifactor models

l Fama-French three-factor model (FF model): Fama, Eugene F.; French, Kenneth R. (1993). "Common Risk Factors in the Returns on Stocks and Bonds". Journal of Financial Economics 33 (1): 3–56¡ Introducing 2 more factors (besides market factor) to

the asset pricing model: size (SMB factor) and MB (HML factor)

¡ Explaining more variance of the stock returns

l Other authors try to add other different things/factors

l Momentum (past/historical stock prices) is something important too

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Estimating equity Betas for publicly traded firmsl Betas are traditionally estimated from a

linear regression of the firm’s stock returns on the returns of a market index.

l OLS to estimate the Beta of General Electric’s common stock:

GE example (cont.)

l GE’s monthly common stock excess returns for Jan.1997 – Dec.1999 are regressed against the excess returns to the CRSP value weighted market index¡ Riskfree rate = return on one-month T-bills¡ CRSP = Center for Research in Security Prices at the

Uni. of Chicago

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GE example (cont.)

GE example (cont.)

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Some notes on WACC

l Cost of preferred stockl Another method to estimate the

cost of common stock

Cost of preferred stock

kp= Dp/P0

kp: The cost of preferred sstock

Dp: Dividend on the preferred stock

P0: Net sales from issuing the preferred stock (Price – Issuing costs if any)

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Cost of common stock – another look

Dividend Growth Model

• g : Constant rate of dividend growth• ke: Required rate of return• D1: Expected dividend on the common stock

in the next period• P0: Current share price

D1P0

ke = + g