fina2303 topic 06 risk and return
TRANSCRIPT
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Topic 6: Risk and Return
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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?
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Introduction to Risk and Return: Risk
Loving Behavior
Have your familymembers ever
bought a lotteryticket?
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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
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Introduction to Risk and Return
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?)
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Introduction to Risk and Return
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
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Introduction to Risk and Return
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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Introduction to Risk and Return
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Introduction to Risk and Return
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Introduction to Risk and Return
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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Variance and Volatility of Returns
mean
return 1 higher than the
mean (upside potential)
return 2 lower than themean (downside risk)
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Variance and Volatility of Returns
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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Historical Tradeoff Between Risk and
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
Historical Tradeoff Between Risk and
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Historical Tradeoff Between Risk and
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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Total Risk/Volatility of Portfolio
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)
Total Risk/Volatility of Portfolio
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Total Risk/Volatility of Portfolio
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 )
Total Risk/Volatility of Portfolio
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Total Risk/Volatility of Portfolio
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
Total Risk/Volatility of Portfolio
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Total Risk/Volatility of Portfolio
)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
=
+
+=
= ∑∑= =
Total Risk/Volatility of Portfolio
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Total Risk/Volatility of Portfolio
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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Example: Risk of Portfolio
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%
Example: Risk of Portfolio
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Example: Risk of Portfolio
%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?)
Market Risk and Beta
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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)
Market Risk and Beta
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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
Market Risk and 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)
Trade-Off Between Risk and Return
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expected return
efficientfrontier
A B C D E1
23
4
mvp
Trade-Off Between Risk and Return
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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.
Challenging Questions
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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.
Challenging Questions
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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%
Challenging Questions
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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?
Challenging Questions
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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?
Challenging Questions
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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?
Challenging Questions
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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?