FINA2303 Topic 06 Risk and Return

8/18/2019 FINA2303 Topic 06 Risk and Return

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Topic 6: Risk and Return


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Topic 6 Risk, Return and Cost of Capital M K Lai Page 2

Learning Outcomes

introduction to risk and return

historical risk and returns of stockshistorical tradeoff between risk and returncommon versus independent risk

diversification of stock portfoliosexpected return of a portfoliovolatility of a portfoliomeasuring systematic riskcapital asset pricing model (CAPM)

multi-factor models


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Introduction to Risk and Return

which of the following options do you choose?

receive $10,000 for suretake part in a game with 50% chance to win

$20,000 and 50% chance to win nothing(mean = $10,000)

risk preference : risk averse, risk neutral and riskloving (or risk-seeking)


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Introduction to Risk and Return: Risk

Averse Behavior

Have your familymembers ever

bought aninsurance policy?



Introduction to Risk and Return: Risk

Loving Behavior

Have your familymembers ever

bought a lotteryticket?



Risk Loving Behavior

jackpot *probability

of winning <$10

Why buy it?

jackpot

staff costs and other expenses

charitable activities

betting dutyto

government

$10 tobuy it




basic assumptions in finance

people are rationalpeople prefer more wealth to lesshigher expected return is better

people are risk averselower risk is better given the same expectedreturninvestors require compensation (known asrisk premium ) for bearing risk

relationship between risk and return(why?)




higher return reflects higher risk

risk-adjusted return= (nominal) risk-free rate + risk premium= real risk-free rate + inflation premium +risk premium

risk-free rate is estimated from .

real risk-free rate reflects .inflation premium reflects effect of .

government bond

postponed consumption

inflation




the more risky an investment, the are therisk premium and the the risk-adjustedreturn

from next graphsmall stocks accumulated the most wealth(return)small stocks experienced the largestfluctuations (risk)


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three issues to address

how to measure return ?

how to measure risk ?

how to consider a tradeoff between risk andreturn?


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Types of Returns

what is the difference between each pair of thefollowing?

nominal return vs. real returnhistorical return vs. expected return

unrealized return (paper return) vs. realizedreturn (both historical returns)

arithmetic average return vs. geometricaverage returnwhich in each pair is more important in finance?


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Return Measurement

return is benefits in excess of initial investment

total return with two componentsinterim income (Divt), e.g. dividendscapital gain/loss (P t – P t-1 ), e.g. change instock price

(rate of) return = (P t – P t-1 + Div t)/P t-1

where P t = current market value at t; P t-1 =original purchase price at t-1; Div t = interimincome received at t and (P t – P t-1 ) = capital

gain/loss


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Example: Historical Return

An investor bought a share of a company at $100a year ago. During the year, he had received anannual dividend of $4. The current stock price is$105. What is his historical total return on the

stock?

%9

100$

$4$100) -($105 returnofrate =

+=

capital gain yield

= 5%

dividend yield =

4%


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Example: Expected Return

An investor buys a stock at $25 now. She expectsto obtain an annual dividend of $1.25 and sell itat $30 in a year’s time. What is her expectedreturn on the stock?

%2525$

$1.25$25) -($30 returnofrate =

+=

expected capitalgain yield = 20%

expected dividendyield = 5%


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Annual Return

annual return can be calculated from periodicreturns

given that all quarterly dividends are immediately

reinvested and used to buy additional shares ofthe same stock

(1+R annual ) = (1+R 1)*(1+R 2)*(1+R 3)*(1+R 4)where R annual = annual return; R 1 , R2 , R3 and R 4are quarterly return in quarters 1, 2, 3 and 4

respectively


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Example: Annual Return

an analyst has collected the following quarterlydata:

Rannual = (1-4.26%)*(1+4.21%)*(1+6.42%)*

(1+7.11%) -1 = 4.52%

quarter price dividend quarterly return$15.25

1 $14.25 $0.35 -4.26%2 $13.25 $0.40 -4.21%3 $13.65 $0.45 6.42%4 $14.12 $0.50 7.11%


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Average Annual Returns

average annual return : the arithmetic average(AM) of an investment’s realized returns (R

1,

R2 , …, RT) for each year in T yearstry to estimate expected return over a future

horizon based on past performance(statistically, it is the true mean without bias)

TR...RR returnannualaverage T21 +++=


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Average Annual Returns

compound annual return : the geometric average(GM) of an investment’s realized returns (R

1,

R2 , …, RT) for each year in T yearstry to measure historical return as a

performance in the past (considercompounding effect)

1)R1(*...*)R1(*)R(1returnaverage

compound TT21 −+++=


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Example: Arithmetic Average and

Geometric AverageAn investor has gathered the annual returns onan investment fund for three years. Calculate thearithmetic average return and geometric averagereturn.

year annual return1 -5%

2 12%3 8%


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Example: Arithmetic Average and

Geometric Average

%74.41%)81(*%)121(*)5%(1GM

%53

8%12%5% -AM

3 =−++−=

=++

=


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Quotation about Risk from Mark Twain

“October. This is one of thepeculiarly dangerous

months to speculate instocks in. The others areJuly, January, September,

April, November, May,March, June, December,August and February.”


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Risk Measurement

risk vs. uncertainty (what is the difference?)

probability distribution of returns on an assetmutually exclusive and all exhaustivescenarios, e.g. state of the economy

probability for each scenariooutcome for each scenario

state of economy probability outcomebooming 20% 20%normal 50% 5%

recessionary 30% -10%


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Variance and Volatility of Returns

variance (of returns) : a statistical method tomeasure the variability of returns as averagesquared deviation of returns from the mean

standard deviation (of returns): positive squareroot of variance of returns (called volatility infinancial markets)

variance and standard deviation are both riskmeasures , including upside potential and

downside risk


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mean

return 1 higher than the

mean (upside potential)

return 2 lower than themean (downside risk)


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estimate variance and standard deviation ofreturns through realized returns

( )

)R(Var)R(SD1T

RR)R(arV

T

1tt

=−

−

=

∑=

where var(R) = variance of returns; R t = return forscenario t; R = arithmetic average return; T =number of realized returns; SD(R) = standarddeviation (volatility) of returns

2

E l V i d V l ili f


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Example: Variance and Volatility of

ReturnsAn investment has 4 years of annual returns of10%, 12%, -9% and 3% respectively. Calculatethe average return, the variance of returns andthe standard deviation of returns.

%49.9009.0)R(SD

0.00914

%)4%3(%)4%9(

%)4%12(%)4%10(

)R(Var

%44

%3%9%12%10R

22

22

==

=−

−+−−+

−+−

=

=+−+=


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Normal Distribution of Returns

prediction interval : a range of values that is likelyto include a future observation

68% prediction interval = average ±

1*standard deviation95% prediction interval = average ±2*standard deviations99.7% prediction interval = average ±3*standard deviations


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Example: Prediction Interval

An investment has an average return of 10% anda standard deviation of 12%. What is the 95%prediction interval for the future return?

95% prediction interval = from 10%-2*12% = -14% to 10%+2*12% = 34%

there is 95% of chance that the future return liesbetween -14% and 34%

Historical Tradeoff Between Risk and


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Returnconclusion

negative relationship between size and risk , i.e.large stocks have lower risk than small stocks

even large stocks are more volatile than aportfolio of large stocks, i.e. portfolio risk isless than individual stock riskall individual stocks have lower returns and/orhigher risk than portfolio



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Return


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Types of Risk

common risk : risk that is linked across outcomes;cannot be diversified away, e.g. risk ofearthquake

independent risk : risk that bear no relation toeach other; can be diversified away, e.g. risk oftheft

diversification : averaging of independent risks ina large portfolio, which renders portfolio risk less

than weighted average risk of items in portfolio


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Types of Risk

type of risk definition examplerisk diversified

in portfolio?

common risklinked across

outcomes

risk of

earthquakeno

independentrisk

risks that bear norelation to each other

risk of theft yes


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Risk of Securities

total risk (volatility, standard deviation of returns)= systematic risk + unsystematic risk

security prices are affected by two types of news

company or industry-specific newsunsystematic risk : fluctuations of securityreturns due to company or industry-specificnews representing independent riskscan be diversified away

give some examples


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Risk of Securities

market-wide newssystematic risk : fluctuations of securityreturns due to market-wide newsrepresenting common risk

cannot be diversified awaygive some examples

when forming a portfolio of securities ,unsystematic risk will be diversified away

for a well-diversified portfolio , only systematic

risk remains, i.e. not risk-free


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Risk of Securities

risk premium of a security is not affected by itsunsystematic (diversifiable) risk , i.e. investorsare not compensated with higher return forbearing unsystematic riskrisk premium of a security is determined by itssystematic risk onlythere is no relationship between volatility andaverage returns for individual securitiespositive relationship between systematic riskand average returns for individual securitiesand portfolios


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Risk of Securities

if returns per year are independent, an investorcan diversify the risk he faces by investing formany years – do you agree?

it is true that the volatility of average annual

returns will decline with the number of yearshe investshowever, the volatility of cumulative return

grows with investment horizonthis is known as the fallacy of long-rundiversification (or time diversification )


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Return and Risk of Portfolio

if we hold several individual assets at the sametime, this combination of individual assets formsa portfolio

estimate return and risk of a portfolio through thereturn of the individual assets, the risk of theindividual assets and the correlations among the

individual assets in the portfolio

d f f l


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Expected Return of Portfolio

return of a portfolio is weighted average ofreturns of individual assets in portfolio and theweights are percentage of individual asset valueto portfolio value (historical return)

expected return of a portfolio is weighted averageof expected returns of individual assets in

portfolio and the weights are percentage ofindividual asset value to portfolio value (expectedreturn)

E d R P f li


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Expected Return on Portfolio

portfolio weight : the fraction of the totalinvestment in a portfolio held in each individualinvestment of the portfolio

portfolio weight of individual asset i, w i =

market value of individual asset i/total marketvalue of portfolioall weights add up to 1 (why?)

E d R P f li


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Expected Return on Portfolio

∑

∑

∑

=

=

=

=

=

=

n

1ii

n

1iiip

n

1iiip

1w

)R(E*w)R(E

R*wR

where R p = historical return on portfolio; n = numberof individual assets in portfolio; w i = weight ofindividual i in portfolio; R j = historical return ofindividual asset i; E(R p) = expected return onportfolio; E(R j) = expected return of individual asset i

E l R t f P tf li


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Example: Return of Portfolio

An investor bought a portfolio of two stocks Aand B one year ago with the followinginformation. Assume that they do not provide anydividends. Calculate the portfolio weight in eachstock and the return of the portfolio.

E l R t f P tf li


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Example: Return of Portfolio

A B portfolionumber of shares 1,000 500

purchase price per share $10 $20original market value $10,000 $10,000 $20,000original portfolio weight 0.50 0.50 1.00

current price per share $9 $24return -10% 20% 5%new market value $9,000 $12,000 $21,000

new portfolio weight 0.43 0.57 1.00

%5%20*5.0%)10(*5.0Rp =+−=

E ample: E pected Ret rn of Portfolio


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Example: Expected Return of Portfolio

An investor buys a portfolio of three stocks A, Band C with the following expected returns andportfolio weights. Calculate the expected returnof the portfolio.

%70.12%15*5.0%12*3.0%8*2.0)R(E p =++=

A B Cexpected return 8% 12% 15%portfolio weight 0.2 0.3 0.5

Total Risk/Volatility of Portfolio


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correlation : a statistical measure of the degree towhich returns share common risks

covariance is a statistical measure to show therelationship between two variables R i and R j

Covar(R i, R j) = Σ(Ri – R i)*(R j – R j)/(T-1)where T is the number of observationscorrelation [Corr(.)] calculated as covariance ofreturns [Covar(.)] divided by product ofstandard deviation of each return

Corr(R i, R j) = Covar(R i, R j)*SD(R i)*SD(R j)



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sign shows direction of co-movementfigure shows magnitude of co-movementmust lie between –1 ≤ Corr(R i, R j) ≤ 1

Corr(R i, R j) = 1 ( perfectly positively

correlated )Corr(R i, R j) = 0 ( uncorreled )

Corr(R i, R j) = -1 ( perfectly negativelycorrelated )



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risk diversification

(independent) risk can be reduced throughdiversification by combining stocks into a

portfolio

the amount of risk that is eliminated in aportfolio depends on the degree to which thestocks face common risks and move together



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)R(Var)SD(R

)R(SD*)R(SD*)R,R(Corr*w*w*2

)SD(R*w)SD(R*w)Var(R

assetsindividualtwoofportfolioafor

)R(SD*)R(SD*)R,R(Corr*w*w)R(Var

pp

212121

22

22

21

21p

n

1i

n

1 j ji ji jip

=

+

+=

= ∑∑= =



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where Var(R p) = variance of returns of portfolio; n= number of individual assets in portfolio; w i =weight of individual i in portfolio; SD(R i) =standard deviation of returns on individual asseti and Corr(R i,R j) = correlation between individualassets i and j; SD(R p) = standard deviation ofreturns of portfolio

Example: Risk of Portfolio


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Given the following information about theexpected returns and standard deviations ofreturns for two assets, calculate the expectedreturn and standard deviation of a portfolio thatis 50% invested in asset 1 and 50% in asset 2.The correlation between the returns on the twoassets is 0.4.

individual asset weight in portfolio expected return risk1 50% 10% 20%2 50% 15% 28%



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%20.200408.0)R(SD

0408.0

%28*%20*4.0*%50*%50*2

%28*%50%20*%50)R(Var

%5.12%15*%50%10*%50)R(E

p

2222

p

p

==

=+ +=

=+=

Diversification


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Diversification

portfolio risk is less than weighted average riskof the individual assets

risk reduction process is known as diversificationdiversification effect comes from imperfect co-movements among different assets, measuredthrough correlationswhen returns on two individual assets have acorrelation of 1 , they are the same and hencethere is no diversification effectas long as average correlation < 1 , diversificationeffect takes place

Diversification


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Diversification

when average correlation is closer to ,diversification effect will be greaterunsystematic risk is diversifiable as the factorsare independent across companies

systematic risk is non-diversifiable as the factorsare market-wide to affect all companiesin other words, a well-diversified portfolio is stillsubject to the systematic risk , i.e. not risk-free

Diversification


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Diversification

number of securities in portfolio

portfolio risk

unsystematic risk

systematic risk

30

Market Portfolio


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Market Portfolio

market portfolio : the portfolio of all riskyinvestments held in proportion to their valuesmeasured through market capitalization (numberof shares * stock price)

market proxy : a portfolio whose return shouldclosely track true market portfolio


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Market Risk and Beta


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relationship between individual security returnand market portfolio (or market proxy) return isused to measure systematic or market risk ofthat security (measured by beta, β)

beta : percentage change in return of a securityfor a 1% change in return of market portfolio

(βMkt = 1),β = 1.25 (what does it mean?)



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in practice, use regression of security returnagainst market return and the slope (regression

coefficient) is the beta of that securityRi = α + β*R Mkt + εI (simple linear regressionmodel)

where where R i = security return; R Mkt =market return; α = intercept, β = regressioncoefficient (beta) and ε = error term

in formula, β i = Covar(R i, RMkt )/Var(R Mkt ) =SD(R

i)*Corr(R

i, R

Mkt)/SD(R

Mkt)



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take variance on the regression model22

Mkt

2

ii

2

i)R( εσ+σβ=σ

total risk = systematic + unsystematicrisk risk

for simplicity, systematic risk is measured bybeta only and each investment has its own beta



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0

+

-

- +

security return

market return

best-fitting

regression lineslope = beta

Portfolio Beta


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portfolio beta is the weighted average of thebetas of individual assets in the portfolio

Example: A portfolio consists of equally-weighted

individual assets with betas of 0.5 and 1.2respectively. What is the portfolio beta?

portfolio beta = 0.5*50% + 1.2*50% = 0.85

Risk Measures


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source: quamnet

HSI beta = 1

HSI volatility = 38.23%

How to interpret? Does itmean the risk is high or low?

Trade-Off Between Risk and Return


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modern portfolio theory (portfolio optimizer)

portfolio with lowest risk given expected returnportfolio with highest expected return given

riskthe chosen portfolios are called efficientportfoliosthe curve joining all efficient portfolios is calledthe efficient frontier starting from the

minimum variance portfolio (mvp)



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Topic 6 Risk, Return and Cost of Capital M K Lai Page 67risk

expected return

efficientfrontier

A B C D E1

23

4

mvp



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dominance principle (portfolio optimizer)portfolio with lowest risk given expected return

portfolio with highest expected return given riskthe chosen portfolios are called efficient portfolios andthe curve joining all efficient portfolios is called the

efficient frontierfor financial instruments with different risks and returns,we have to use modern financial theories to consider

their trade-offone widely used financial theory is the capital assetpricing model (CAPM)

Capital Asset Pricing Model (CAPM)


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assume that everybody holds a well-diversifiedportfolio and hence are concerned about the

systematic risk only

E(Ri) = expected return on asset or portfolio i; r f =risk-free rate; β i = systematic risk of asset or

portfolio i and R Mkt = expected market return

[ ]fMktifi r)R(E*r)R(E −β+=

systematic risk

nominal risk-

free rate

risk premium

for i


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Capital Asset Pricing Model (CAPM)


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in practice, use a stock market index as a proxyfor the market portfolio

in other words, the return on the stock marketindex represents the market returnE(Ri) is also called required (rate of) return , i.e.expected return of an investment that isnecessary to compensate for the risk ofundertaking the investment

positive relationship between return andsystematic riskapplicable to both individual securities andportfolios (why?)

Example: CAPM


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An asset has a beta of 1.25. The risk-free rate is3% and the expected market return is 15%.

What is the expected return on the asset?

E(Ri) = 3% + 1.25*(15%-3%) = 18%

Example: Market/Equity Risk Premium


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A stock has a beta of 0.95. The risk-free rate is3.25% and the market/equity risk premium is 7%.

What is the expected return on the stock?

E(Ri) = 3.25% + 0.95*7% = 9.90%

Security Market Line (SML)


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if plotting expected return against beta, getstraight line known as security market line (SML)

which is the graphic representation of the capitalasset pricing model

Security Market Line (SML)


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risk-free rate

risk premium

β

E(R)

0 1

RMkt

rf

marketportfolio

securitymarket linei

βi

Ri

Summary of CAPM


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investors require a risk premium proportional tothe amount of systematic risk they are bearing

we can measure the systematic risk of aninvestment by its beta , which is the sensitivity ofthe investment return to the market returnthe most common way to estimate a stock’s betais to regress its historical returns on the market’s

historical returncompute expected or required return for anyinvestment by E(R i) = r f + β i*[E(R Mkt ) – r f)]

Problems of CAPM


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researchers have found that a simple marketproxy (e.g. the stock market index) has led to

consistent pricing errors from the CAPMin CAPM, there is only one systematic riskfactor captured by the market proxysmall stocks , stocks with high book-to-marketratios and stocks that have recently performed

extremely well have consistently earned higherreturns than the CAPM would predict

Problems of CAPM


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momentum strategy : good and badperformance continues, and buy the winner

and sell the loser (empirical studies showthat it works in short run)contrarian strategy : buy the loser and sellthe winner (empirical studies show that itwins out in the long run)

it gives rise to an idea that there may be othersystematic risk factors not captured by themarket proxy


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Multi-Factor Models


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different multi-factor modelsarbitrage pricing theory (APT): a multi-factormodel relies on the absence of arbitrage toprice securities (similar to valuing a couponbond with zero-coupon prices/yields)Fama-French-Carhart (FFC) factor specification :a multi-factor model of risk and return in which

the factor portfolios are the market, small-minus-big, high-minus-low, and prior 1-yearmomentum portfolios

Fama-French-Carhart FactorSpecification


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add three additional factor portfolios (risk factors)apart from the stock market index (Mkt)

buying small firms and sell large firms (assmall firms generate higher returns), known asthe small-minus-big (SMB) portfolio

buy high book-to-market firms and sell lowbook-to-market firms (high book-to-marketfirms generate higher return), known as high-minus-low (HML) portfolio

Fama-French-Carhart FactorSpecification


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buy stocks that have recently done extremelywell and sell those that have done extremely

poor, known as prior 1-year (PR1YR)momentum portfolio

[ ])R(E*)R(E*

)R(E*r)R(E*r)R(E YR1PR

YR1PRiHML

HMLi

SMB

SMB

ifMkt

Mkt

fi i

β+β+β+−β+=

where β iMkt , β iSMB, β iHML and β iPR1YR , are thefactor betas of stock i and measure thesensitivity of the stock return to each portfolio(risk factor)

Example: Fama-French-Carhart FactorSpecification


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An analyst wants to estimate the expected returnon a stock. He collects the following information

on a monthly basis:risk-free rate = 0.125%

equity risk premium = 0.61%expected return on SMB = 0.25%expected return on HML = 0.38%expected return on PR1YR = 0.70%stock market beta = 0.687

SMB beta = -0.299

Example: Fama-French-Carhart FactorSpecification


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HML beta = -0.156PR1YR beta = 0.123

Find the expected return on the stock based onthe FFC factor specification.

0.496%

%70.0*123.0%38.0*156.0

%25.0*299.0%61.0*687.0%125.0)R(E i

=+−

−+=


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Challenging Questions


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4. Why do investors demand a higher return wheninvesting in riskier securities ?

5. For a lay investor, he usually considers it as riskwhen the actual return falls short of hisexpectation or is negative. Explain the difference

between this risk concept and the use ofstandard deviation of returns as a risk measurein the financial market.

6. Explain how a commercial bank makes use theconcept of diversification in carrying out its loanbusiness. And how an insurance company makes

use of it in carrying out its insurance business.



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7. Given a positive investment in every asset in aportfolio, is it possible for the standard deviation

of returns on the portfolio to be less than that onevery asset in it?

8. Given a positive investment in every asset in aportfolio, is it possible for the beta on theportfolio to be less than that non every asset init?

9. Explain why an individual stock is never chosenas an efficient portfolio under the modern

portfolio theory.



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10.Under the modern portfolio theory, which of theportfolios is an efficient portfolio? Why?

A. a portfolio with expected return of 15%and standard deviation of returns of 25%B. a portfolio with expected return of 12%and standard deviation of returns of 25%A. a portfolio with expected return of 15%

and standard deviation of returns of 28%A. a portfolio with expected return of 12%and standard deviation of returns of 28%



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11.“I originally owned the shares of Company Xwith a beta of 1.5. I sold 50% of Company X’s

shares and used the proceeds to buy the sharesof Company Y with a beta of 0.8. The beta of myportfolio of the shares of Company X andCompany Y is 1.15 now. By forming a portfolio, Ican reduce the beta from 1.5 to 1.15. This riskreduction process is known as diversification.”Do you agree? Why or why not?



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12.“When the returns on two assets have acorrelation of zero, there is no relationship

between them at all. In other words, when theaverage correlation among the returns ofindividual assets in a portfolio is zero, thediversification effect is the greatest.” Do youagree? Why or why not?

13.If an investor is holding a well-diversified

portfolio, she wants to buy an additional stock .Which type of risks (total risk, systematic riskand unsystematic risk) should she be concernedabout with respect to the stock?



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14.What determines how much risk will beeliminated by combining stocks in a portfolio?

15.If an analyst estimates the expected return on astock lies above the security market line (SML) ,what should be his investment recommendationon the stock? Explain.

16.If an investment has a positive NPV , does its

expected return lie below or above the securitymarket line (SML)? Why?



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17.“Diversification reduces risk. Therefore,companies ought to favour capital investments

with low correlations with their existing lines ofbusiness.” Do you agree? Why or why not?

FINA2303 Topic 06 Risk and Return

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