lecture 1 - nov. 3, 2011 (1)
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Em irics of Financial Markets
Patrick J. Kelly, Ph.D.
MICEX
1500
2000
2500
ex
Level
MICEXLevels
2011 Patrick J. Kelly 2
0
500
1000
September97
February98
July98
December98
May99
October99
March00
August00
January01
June01
November01
April02
September02
February03
July03
December03
May04
October04
March05
August05
January06
June06
November06
April07
September07
February08
July08
December08
May09
October09
March10
August10
January11
June11
Ind
MICEX returns
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WHY?
MICEX + constituents: levels
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MICEX + constituents: returns
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Asset Pricing Models in General
Asset pricing models relate expected stock returns tofactors. Typically, these are written as linear models(because easier than non-linear):
These linear models are an approximation of marginalutility growth.
This says: Discounted aggregate marginal utility growthapproximately follows some function of factors (f)
2011 Patrick J. Kelly 7
What Are Factors?
These factors are proxies for marginal utility growth:
Factors signal current (or forecast future) marginalutility growth.
What grows or stunts marginal utility?
States of the economy: consider when the economy goes bad, put
extra value on assets/portfolios with high payoffs in these bad states
Such portfolios will have high prices and low returns
In some models factors can be those that forecast future
marginal utility growth, but they shouldnt predict too well(otherwise the models will predict larger than factual interest variation)
This is why we use changes, not levels: returns, not prices
2011 Patrick J. Kelly 8
Asset Pricing Models:
Different ways to model marginal utility growth
Capital Asset Pricing Model (CAPM)
One factor
Intertemporal CAPM (ICAPM)
CAPM + 1 factor to capture changes in investmentopportunities
Arbitrage Pricing Theory (APT)
Multi-factor
2011 Patrick J. Kelly 9
CAPM
rw is the return to current wealth NOT just themarket. Includes:
Labor income
Real estate
Any private property
All public property (parks, lakes, roads, bridges)
One period model (ignores time)
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The foundations of CAPM
Sharpe, Lintner and Treynor (separately) groundedasset pricing in a single market factor
Because they created models of asset prices that built on theintuition in Markowitzs portfolio theory, which
simply notes that
Investors do not (should not?) care about any one asset in theirportfolio of assets, but they should only care about the risk and
reward of the entire portfolio.
and demonstrates
The power of diversification across many assets
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This class
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Goal of Course
Briefly introduce key theories in finance
Mostly related to
Asset pricing
Market efficiency
And what the data tell us about the theories
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What we will cover
Portfolio theory
With liquidity and short sale constraints
Asset Pricing
Event studies
Return predictability
At long and short horizons
Behavioral Finance
2011 Patrick J. Kelly 14
Syllabus
40% of class 3 projects
Find one partner
E-mail me if you have problemspkelly@nes.ru
I expect you know Gauss and Excel
If you dont, rethink taking this class
60% of grade final exam
In English
You will have at least one sample exam
Few surprises
Please pay attention to my.nes.ru!
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Portfolio Selection
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Whats next?
We are going to run through about 1/4th to 1/3rd thematerial I cover in a typical MBA Investments andPortfolio Management class
in about 45 min to an hour.
The point:
To give you some of the basic intuition before you startapplying the reasoning to the data in your assignments
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Portfolio Selection
Marokowitz (1952)
Prior belief: investors should solely maximize the discountedvalue of cash flows
Two assets with the same discounted cash flows are equally good
regardless of risk
Proposes/Assumes the Mean-Variance Criterion
Investors care about both risk and return
Shows that through diversification investors can get amaximum return for a minimum of risk
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Mean-Variance Criterion
Investors prefer more to less and dislike risk
From this, we can build a theory of investment choice basedon the expected (mean) return of an investment (higher =better) and its risk as measured by the variance of returns(lower = better)
This is the mean-variance criterion.
Critical assum tion:the variance ofreturns isa oodcharacterization
2008 Patrick J. Kelly 19 2008 Patrick J. Kelly 19
of the investment risk that investors care about
rr
s.d. s.d.
Mean Variance Criterion provides
Intuitive assessment of the relative merits of assets andportfolios of assets
Mean variance space is spanned by only two funds(portfolios).
A risk free and an efficient risky portfolio
Translates directly to investor Utility as a function ofportfolio return and variance
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We can show our preferences (Utility)
E[r]
Q
S
Higher
Return
2
2
1][ ArEU
Indifference
Curve
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P
R
More
Risk
Critical Assumption
Variance (Standard Deviation) is onlycharacteristic ofrisk important to us
For example Skewness doesnt matter
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Normal Distribution
2008 Patrick J. Kelly 23 2008 Patrick J. Kelly 23
rr
Symmetric distributionSymmetric distribution
s.d. s.d.
Skewed Distribution: Large Negative Returns Possible
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rrNegativeNegative PositivePositive
Median
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Skewed Distribution: Large Positive Returns Possible
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rrNegativeNegative PositivePositive
Median
We will assume returns follow a Normal Distribution
2008 Patrick J. Kelly 26 2008 Patrick J. Kelly 26
rr
s.d. s.d.
M52: Investors care about the risk and return of their portfolio (of Many
Risky Assets)
For a portfolio ofNrisky securities, variance is:
The first term sumsNvariances; the second term
p2 wi
2 var( i) w iwj cov( i, j)j1i j
N
i 1
N
i1
N
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capturesNx(N-1) covariances.
Diversification does not eliminate all risk.
Several Stocks
Yearly Returnsfor 5 Stocks
40.00%
60.00%
80.00%
100.00%
120.00%
GE
S.Cal.Edison
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-60.00%
-40.00%
-20.00%
0.00%
20.00%
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Return
Year
McGrawHill
ConAgra
CitiGroup
Portfolio of Several Stocks
YearlyReturns for 5 Stocks
40.00%
60.00%
80.00%
100.00%
120.00%
GE
S.Cal.Edison
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-60.00%
-40.00%
-20.00%
0.00%
20.00%
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Return
Year
McGrawHill
ConAgra
CitiGroup
Returns and Standard Deviations
GE S.Cal.Edison McGrawHill ConAgra CitiGroup eqAvg
Mean 16.60% 8.11% 13.34% 12.97% 22.34% 16.17%
SD 26.27% 28.34% 18.89% 22.49% 37.72% 19.53%
2008 Patrick J. Kelly 30 2008 Patrick J. Kelly 30
rf=4%
Reward-to-risk 0.4796 0.1450 0.4944 0.3988 0.4862 0.6223
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Possible to split investment funds between safe andrisky assets
Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio)
Controlling Risk: Allocating Between Risky & Risk-Free Assets
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Ultimately we will see that under some assumptionsa risk-free asset and a particular risky portfolio spanthe mean-variance return space.
Entire Stock Market vs. T-Bills
Entire Market Return vs. T-Bill Rates
10.00%
20.00%
30.00%
40.00%
50.00%
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-40.00%
-30.00%
-20.00%
-10.00%
0.00%
1958
1960
1962
1964
1966
1968
1970
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
1996
1998
2000
2002
T-B i ll Ra t e En ti re Mark e t
rf= 7%rf= 7% rf= 0%rf= 0%
E[ra] = 15%E[ra] = 15% a = 22%a = 22%
Example
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w = % in aw = % in a (1-w) = % in rf(1-w) = % in rf
E[rp] = wE[ra] + (1 - w)rfE[rp] = wE[ra] + (1 - w)rf
rp = combined portfoliorp = combined portfolio
For example, w = .75For example, w = .75
Expected Returns for Combinations
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E[rp] = .75(.15) + .25(.07) = .13 or 13%E[rp] = .75(.15) + .25(.07) = .13 or 13%
p = .75(.22) + .25(0) = .165 or 16.5%p = .75(.22) + .25(0) = .165 or 16.5%
What Return for What Risk?
E[r]E[r]
E[rE[raa] = 15%] = 15%
aa
CALCAL(Capital(CapitalAllocationAllocationLine)Line)
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rrff = 7%= 7%
22%22%00
ff
) S = 8/22) S = 8/22
E[rE[raa]] -- rrff = 8%= 8%
E[rE[rpp] = 13%] = 13%
16.5%16.5%
Extending the CAL
E(r)
E(rE(rpp) = 15%) = 15%
aa
2008 Patrick J. Kelly 36 2008 Patrick J. Kelly 36
rrff = 7%= 7%
22%22%00
FF
) S = 8/22) S = 8/22
E(rE(rpp)) -- rrff = 8%= 8%
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The Capital Allocation Line
CAL depicts the possible asset allocations, i.e. risk-return combinations
Slope S of CAL =
Measures the increase in expected return an investor obtains
364.022822715 PFP
rrrES
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or ta ng on one a t o na un t o stan ar ev at on r s
Also called the
reward-to-variability ratio
Sharpe Ratio
Extending the CAL
The portfolio has the most risk if all monies areinvested in a(w=1)
Q: How can we assume even more risk?
A: Since risk increases linearly in w, we need to increasew > 1 (borrow money to invest in a)
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Borrowing at the risk-free rate is not available forindividual investors, need to borrow at a rate rB > rF
CAL with Borrowing
E(r)
E(rE(rpp) = 15%) = 15%
aa
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rrff = 7%= 7%
22%22%00
FF
) S = 8/22) S = 8/22
E(rE(rpp)) -- rrff = 8%= 8%
Is there an optimal portfolio choice?
E(r)E(r)
E(rE(rpp) = 15%) = 15%
aa
CALCAL(Capital(CapitalAllocationAllocationLine)Line)
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rrff = 7%= 7%
22%22%00
FF
) S = 8/22) S = 8/22
E(rE(rpp)) -- rrff = 8%= 8%
Portfolio Example
Consider an example where we can invest into riskyassets (stocks, funds) 1 and 2.
Asset 1: E(r1) = 10% 1 = 12%
Asset 2: E(r2) = 17% 2 = 25%
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What is the expected portfolio return and standarddeviation?
Benefits from Diversification
2211 rEwrEwrE p
2,12121
2
2
2
2
2
1
2
1 2ww wwp
Asset 1: E(r1) = 10% 1 = 12%
Asset 2: E(r2) = 17% 2 = 25%
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Portfolio Standard Deviation (%) for Given Correlation
Weight in 1 E(r) portf. 1,2 = - 1 1,2 = 0 1,2 = 0.2 1,2 = 0.5 1,2 = 1
0
0.2
0.4
0.6
0.8
1
Portfolio Standard Deviation (%) for Given Correlation
Weight in 1 E(r) portf. 1,2 = - 1 1,2 = 0 1,2 = 0.2 1,2 = 0.5 1,2 = 1
0 17.0 25.0 25.0 25.0 25.0 25.0
0.2 15.6 17.6 20.1 20.6 21.3 22.4
0.4 14.2 10.2 15.7 16.6 17.9 19.8
0.6 12.8 2.8 12.3 13.4 15.0 17.2
0.8 11.4 4.6 10.8 11.7 12.9 14.6
1 10.0 12.0 12.0 12.0 12.0 12.0The lower the correlation the
lower the portfolio variance
Even though the
expected return
is the same
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How does risk reduction depend on ?
Asset 1: E(r1) = 10% 1 = 12%
Asset 2: E(r2) = 17% 2 = 25%
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Portfolios of More the Two Risky Assets
We are looking for the lowest variance portfolio for agiven return
MINp2 wi
2var(ri ) wiwj cov(ri,rj )
j1ij
N
i1
N
i1
N
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subjectto : E[rp] a given return
and wi
i1
N
1
Portfolios of More the Two Risky Assets
We are looking for the lowest variance portfolio for agiven return
2
1
2 wwMIN p
2008 Patrick J. Kelly 45
1and
:
w
Ewtosubject
0and w
with short-sale constraints
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Return
What Happens When We Add More Assets?
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Return
B2
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0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
A1
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Re
turn
There is a unique optimal portfolio
MMarket
2008 Patrick J. Kelly 47
0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Re
turn
There is a unique optimal portfolio
MMarket
Systematic/Covariance Risk Idiosyncratic Risk (ni)
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0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
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How can we tell?
How can we tell if adding more assets improves theefficient frontier?
Mathematically adding any less than perfectly correlated
security ought to improve the ex-ante efficient frontier Improvement may not be meaningful
The problem is that the ex-ante efficient frontier willalways lie within the ex-post efficient frontier.
Simply due to luck? consider
2011 Patrick J. Kelly 49
Testing Portfolio Efficiency
Consider the question whether a certain portfolio 1 isstill efficient if one makes available additional assets (2,3 and 4)
If Expected returns and covariances were known, then easy If you cannot form a portfolio with higher mean and/or lower
variance by including assets 2, 3, and 4 then 1 still efficient
Problem:
With a probably of 1, the true ex-ante efficient portfolio iswithin the empirically observed efficient portfolio.
To illustrate the problem, consider.
2011 Patrick J. Kelly 50
Illustration (Sentana, 2009)
Suppose assets/funds 1, 2, 3 and 4 all have E[r-rf]=0and Cov(i,j) =0 for all
In truth, adding assets 2, 3 and 4 will not improve thereward for risk and will not change the efficientport o o.
The next chart simulates two years of daily returns andplots the efficient frontiers from these simulationswhere the TRUE ex-ante Sharpe ratio is ZERO.
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Simulated Efficient Frontiers (Sentana, 2009)
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Sampling Distribution of the GMM estimator
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trueMean of
Simulated
The point
The sampling error is large when using sample means,variances and covariances to estimate expected meansand variances, but
Statistical tests are well suited to the problem
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Intuition: Tests of Mean Variance Efficiency
Cochrane (2001, Chapters 2 and 5) shows that anyexpected return can be related to any mean-varianceefficient portfolio lying on the efficient frontier:
E ri rf i,mv E rmv rf
This structure suggests a natural test for efficiency:
Test whether is non-zero.
This is the basic intuition of most tests of portfolio efficiency
Differences across tests are mostly econometric refinements
2011 Patrick J. Kelly 55
E ri rf i,mv E rmv rf
Test of Mean Variance Efficiency
Gibbons, Ross, Shenken (1989)
T N1N
1 ET f f
2
1
1~ FN,TN1
Wherefis a return based factor or portfolio return on themean-variance efficient frontier, ET(f) is the sample mean ofthe factor, and (f) the sample standard deviation, Tis thenumber of observations,Nis the number of test assets, 1 isthe number of factors, is a vector of the intercepts from theN test-asset regressions, and is the cross test asset residualcovariance matrix, such that
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E t t Mean of
Refinements
The GRS test is reasonably robust to small departuresfrom normality of the return distribution
MacKinlay and Richardson (1991,Journal of Finance)propose refinements to deal with leptokurtosis inreturns in a GMM framework.
Hodgson, Linton, and Vorkink (2002,Journal of AppliedEconometrics) derive semi-parametric tests that possesmore power to reject the null.
See Sentana (2009, Econometrics Journal) for a discussionof further refinements 2011 Patrick J. Kelly 57
Testing whether a portfolio is mean-variance efficient
Useful for:
Mutual Fund performance evaluation
Measuring gains from portfolio diversification
Tests of linear asset pricing models
Not trivial because:
Realized returns are not the same as expected returns
*Ideas borrowed from Sentana (2009)
Homework 1
Discussion of
Liquidity and its definitions
Finding the efficient frontier and the optimal portfolio @Risk and VaR analyses
Bootstrapping
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A relude
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Last Time: a Steeper CAL is better
E(r)E(r)
E(rE(rpp) = 15%) = 15%
aa
CALCAL(Capital(CapitalAllocationAllocation
Line)Line)
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rrff = 7%= 7%
22%22%00
ff
S = 0.36S = 0.36S = 0.72S = 0.72
Risk in Equally-Weighted Portfolios
Avg. Std. Dev. 30%
Avg. Correlation 0.2
# of Assets Portfol io Due to Due to
Std. Dev. Variances Covariances
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2 23.24% 83.33% 16.67%
3 20.49% 71.43% 28.57%
4 18.97% 62.50% 37.50%
100 13.68% 4.81% 95.19%
1000 13.44% 0.50% 99.50%
10000 13.42% 0.05% 99.95%
How Can We Raise the CAL?
Raise Return
P
FP
r
rrES
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Lower risk
Diversify with Risky Assets
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Return
What Happens When We Add More Assets?
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Return
B2
2008 Patrick J. Kelly 64
0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
A1
Efficient Trade-off Line & Efficent Frontier Curve
15%
20%
25%
Re
turn
Which portfolio does everyone choose?
MMarket
2008 Patrick J. Kelly 65
0%
5%
10%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
Expected
Optimal Decision Rule
If everyone prefers more return to more risk and everyonesees the same assets there is only one best riskyportfolio. Meaning:
Optimal Portfolio Selection takes 2 steps:
1. Choose Optimal RiskyPortfolio
2008 Patrick J. Kelly 66
2. OptimalAllocationbetween Risky and Riskless
This is called the Separation Property
Notice this means: All rational risk-averse investors will passively index holdings
to some risky fund and account for risk aversion by keepingsome money totally safe
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Asset Pricing Implications
If everyone chooses the same portfolio (by theseparation property), how do we value an individualstock?
Does its return matter?
Does its standard deviation matter?
2008 Patrick J. Kelly 67
Many Risky Assets
For a portfolio ofNrisky securities, variance is:
The first term sumsNvariances; the second term
p2
wi2
var( i) wiwj cov( i, j)j1i j
N
i 1
N
i1
N
2008 Patrick J. Kelly 68
capturesNx(N-1) covariances. AsNgets large, which of the two dominates?
The variances are overwhelmed.
Nx(N-1) gets much larger thanN
What if... ?
If there were enough assets to diversifyaway all firm-specific risk, would you want
to hold any?
2008 Patrick J. Kelly 69
Why?
Rational investors and diversified portfolio
If you are able to complete diversify away firm specificrisk, you will
Because no one will be willing to pay you anything fortaking that risk.
2008 Patrick J. Kelly 70
Capital Asset Pricing Model
William Sharpe & John Lintner (1964) insight was thatwe should only care about how a stock comoves withthe rest of the market because the firm-specificvariance can be diversified away.
2008 Patrick J. Kelly 71
Assumptions of the CAPM
1. Perfect competition: Markets are large and investors areprice takers
2. Frictionless markets: no taxes and no transactions costs
3. Complete markets: All risky assets are publically traded
4. Unlimited borrowing and lendingat the risk-free rate (orunlimited shorting)
5. Homogenous expectations: Everyone sees the sameefficient frontier, because everyone holds the samebeliefs about expect returns and variances
6. Investors follow the mean-variance criterionand have anidentical one holding period over which the mean-variance criterion is applied (utility is maximized)
2008 Patrick J. Kelly 72
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From Assumptions to CAPM
1. Raise the capital allocation line
Mean variance criterion
2. Maximum diversification
Frictionless markets
3. Investors choose the same optimal portfolio
Homogeneous expectations, perfect competition, completemarkets, and unlimited borrowing and lending at the risk-free rate
4. Only systematic (market) risk is important
Only covariance risk matters
Only one price of risk
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Efficient Trade-off Line & Efficent Frontier Curve
20%
25%
n
Capital Market Line (CML)
MM rk t
CML
2008 Patrick J. Kelly 74
0%
5%
10%
15%
0% 5% 10% 15% 20% 25% 30%
Standard Deviation
ExpectedRetur
M
fM rrE
)(
Measuring Systematic Risk
Could use cov(E[rstock], E[rmarket])
BUT:
again, the units are a pain with cov(). Correlation, then?
No; what is the correlation between 2 systematic risks? Doesnt account for how much it moves when correlated
We need a measure of a stocks systematic riskonly.
2008 Patrick J. Kelly 75
,
Why it isnt this:
Actually, numerically identical
)][var(
)][,][cov(
fm
fmfi
irrE
rrErrE
])[var(
])[],[cov(
m
mii
rE
rErE
Reward for risk
Reward for no systematic risk (=0)
rf
Reward for investing in the market portfolio (=1)
E[rm]-rf
Reward for investing in any asset:
2008 Patrick J. Kelly 76
fmifi rrErrE ][][
Asset Pricing Models in General
Asset pricing models relate expected stock returns tofactors. Typically, these are written as linear models(because easier than non-linear):
These linear models are an approximation of marginalutility growth.
This says: Discounted aggregate marginal utility growthapproximately follows some function of factors (f)
2011 Patrick J. Kelly 77
What Are Factors?
These factors are proxies for marginal utility growth:
Factors signal current (or forecast future) marginal
utility growth. What grows or stunts marginal utility?
States of the economy: consider when the economy goes bad, putextra value on assets/portfolios with high payoffs in these bad states
Such portfolios will have high prices and low returns
In some models factors can be those that forecast future
marginal utility growth, but they shouldnt predict too well(otherwise the models will predict larger than factual interest variation)
This is why we use changes, not levels: returns, not prices
2011 Patrick J. Kelly 78
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Asset Pricing Models:
Different ways to model marginal utility growth
Capital Asset Pricing Model (CAPM)
One factor
Intertemporal CAPM (ICAPM)
CAPM + 1 factor to capture changes in investmentopportunities
Arbitrage Pricing Theory (APT)
Multi-factor
2011 Patrick J. Kelly 79
CAPM
rw is the return to current wealth NOT just themarket. Includes:
Labor income
Real estate
Any private property
All public property (parks, lakes, roads, bridges)
One period model (ignores time)
2011 Patrick J. Kelly 80
Mertons (1973) ICAPM
Multi-period version of CAPM
Factors are state variables that determine how well theinvestor can do his/her optimization. A factor that can beanything that affects
curren wea
the distribution of distribution of future returns
Labor market income, Housing value, Small business
Changes in the investment opportunity set.
Investors with long horizons are unhappy with news that future returnsare lower
High value to assets which are negatively correlated with long termwealth. That is, prefer stocks with high payouts during recessions
Anything that affects the average investor
2011 Patrick J. Kelly 81
Note about the extra ICAPM factor
The ICAPM state variable(s) should affect the averageinvestor.
Consider a risk that in the future makes A better off and Bworse off
B sells the risk
A buys it
Net effect is zero.
This helps explains why LOTS of variables arecorrelated with returns, but do not carry any priced risk
For example: Industry returns comove, but not once you
control for priced risks.
2011 Patrick J. Kelly 82
APT
Using the Law of One Price
Common comovements of stock returns should havethe same price
Complete idiosyncratic price movements are not priced
If well diversified only common factors affect consumption
ntu t on s t e same as prev ous ar trage exampe
No restrictive assumptions about returns or preferences
APT does not suggest factors. It says start statistical
Find comovement in stock returns
ICAPM says start with theory when looking for factors
Variables that affect the distribution of returns
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Weaknesses of CAPM and ICAPM
CAPM is linear and does not price non-normallydistributed returns well
Should price derivatives, but generally cannot, their returnsare too non-normal
Why linear?
en y o s a e var a es s un nown
But, they should forecast something about the economy
Portfolio of wealth is unobservable
It includes are return generating assets (including humancapital and public property)
Transactions costs and illiquidity can delink consumption andreturns at high frequencies
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Rolls Critique (1977)
If a portfolio against which returns are measured is ex-post efficient, then no security will have abnormalreturns
If the portfolio is inefficient then any abnormal returnis possible
a p a s non-zero s ecause e mar e por o o sinefficient or is it because there is mispricing?
If returns are linear in Beta, all that proves is that themarket is ex-post efficient
If they are not linear in beta you do not know if it is becausethe market is
ex-post inefficient or because
there is a missing factor
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Rolls Critique (1977) continued
Only true test if the true market portfolio is ex-postefficient
True market portfolios is unobservable. So, good luck.
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Testing CAPM
an t er sset r c ng o e s
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The Market Model
Alphas and betas are measured statistically usinghistorical returns on the security and the marketportfolio proxy, e.g. S&P 500
Run the regression of theMarket Model:
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Apply this to a particular stock, and you get a SecurityCharacteristic Line...
itftmtiiftit rrrr
Security Characteristic Line
Equation of Line: )( fmfi rrrr
40%
50%
60%
remium
itftmtiiftit rrrr ][
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Beta=Slope
Alpha=Intercept
-20%
-10%
0%
10%
20%
30%
-15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35%Market Risk Premium
StockRisk
How to Calculate Beta
Returns are for what period? Best is annual or quarterly but too few observations
Practice is to use monthly
Make sure that all returns are stated monthly
itftmtiiftit rrrr ][
2008 Patrick J. Kelly 90
Pay attention to the risk free rate, because that is usually stated yearly
Use portfolios to reduce estimation error Run a CAPM type regression on each stock
Sort stocks into Beta portfolios
Calculate portfolio Betas
Attribute portfolio Betas to each stock
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Mean Reversion as Seen with IBMs Betas
When Beta is below the mean (typically 1) betas tend torise.
When Beta is above then mean betas tend to fall
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Predicting Betas: Correcting for Mean Reversion
In order to correct for mean reversion
Because the average beta is 1
We calculate the following:
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13
1
3
2 taHistoricBeBetaAdjusted
CAPM predictions
should be zero.
If not, there may be missing factors.
is the only relevant factor
Relation between and returns is linear
Over long periods the return on the market is greaterthan the risk free return
In general, riskier stocks should earn higher return on average
Market portfolio is mean-variance efficient
If you cross-sectionally regress on risk premia,estimates should equal the average market risk premium
()
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Fama-McBeth (1973)
Tests if high beta is associated with high returns andvice versa.
General format:
Time-series regressions to get betas for test portfolios
Usually beta sorted porttflios
-associated with high return
Cochrane (1999) shows that GMM panel regressionsare identical under some assumptions
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General FM73 Algorithm
Use monthly data from years 1 through 7 to estimate CAPM Beta1-7 for eachcompany.
Use to rank stocks (rank1-7) into 20 equally sized groups (call portfoliosformed on rank1-7 portfolio1-7)
Use monthly data from years 8 through 12 to estimate CAPM Beta8-12 (= ) foreach company.
Find the average Beta8-12 and average (equally weighted) return for each1-7 . .
out each month).
Roll one year forward and repeat steps 1-4 till done.
Example in step 1 use data from years 2 through 8
For each month run a cross-sectional regression using the average Betas andreturns calculated in step 4.
Collect the time series of the coefficients (s)
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Example from FF92
They look at whether CAPM works.
It doesnt.
Here is what they do:
The sort stocks by size and by the stocks pre-ranking beta. re-ranking betas: betas are calculated for each stock over the 5 years ofmonthly data preceding (not including) the current year (from t-5 to t-1).
Stocks are sorted by size and then each size portfolio is sorted by thepre-ranking beta.
For each month in year t, calculate equally weighted portfolio returns
For each size/pre-ranking-beta portfolio calculate Beta for the entire
sample, including one lag of the value weighted market to correct fornon-synchronous trading.
Assign the betas to each stock in the portfolio and run monthly cross-sectional re ressions. 2011 Patrick J. Kelly 102
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Average Slopes (t-Statistics) from Month-by-Month Regressions of Stock Returns on
,B,Size, Book-to-Market Equity, Leverage, and E/P: July 1963 to December 1990
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Related findings- Fama French 1992 - Size and Beta
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Size and Book to Market
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