Lecture 1 - Nov. 3, 2011 (1)

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

1/18

11/3/20

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


WHY?

MICEX + constituents: levels


MICEX + constituents: returns


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

2/18

11/3/20

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)


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


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


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)


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


This class


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

3/18

11/3/20

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


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


Syllabus

40% of class 3 projects

Find one partner

E-mail me if you have [email protected]

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!


Portfolio Selection


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


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


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

4/18

11/3/20

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


We can show our preferences (Utility)

E[r]

Q

S

Higher

Return

2

2

1][ ArEU

Indifference

Curve


P

R

More

Risk

Critical Assumption

Variance (Standard Deviation) is onlycharacteristic ofrisk important to us

For example Skewness doesnt matter


Normal Distribution


rr

Symmetric distributionSymmetric distribution

s.d. s.d.

Skewed Distribution: Large Negative Returns Possible


rrNegativeNegative PositivePositive

Median

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

5/18

11/3/20

Skewed Distribution: Large Positive Returns Possible


rrNegativeNegative PositivePositive

Median

We will assume returns follow a Normal Distribution


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


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


-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


-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%


rf=4%

Reward-to-risk 0.4796 0.1450 0.4944 0.3988 0.4862 0.6223

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

6/18

11/3/20

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


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%


-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


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


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)


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


rrff = 7%= 7%

22%22%00

FF

) S = 8/22) S = 8/22

E(rE(rpp)) -- rrff = 8%= 8%

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

7/18

11/3/20

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


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)


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


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)


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%


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%


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

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

8/18

11/3/20

How does risk reduction depend on ?

Asset 1: E(r1) = 10% 1 = 12%

Asset 2: E(r2) = 17% 2 = 25%


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


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


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?


15%

20%

25%

Return

B2


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


15%

20%

25%

Re

turn

There is a unique optimal portfolio

MMarket


0%

5%

10%

0% 5% 10% 15% 20% 25% 30%

Standard Deviation

Expected


15%

20%

25%

Re

turn

There is a unique optimal portfolio

MMarket

Systematic/Covariance Risk Idiosyncratic Risk (ni)


0%

5%

10%

0% 5% 10% 15% 20% 25% 30%

Standard Deviation

Expected

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

9/18

11/3/20

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


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.


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.


Simulated Efficient Frontiers (Sentana, 2009)


Sampling Distribution of the GMM estimator


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


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

10/18

11/3/20

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


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


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


A relude

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

11/18

11/3/20

Last Time: a Steeper CAL is better

E(r)E(r)

E(rE(rpp) = 15%) = 15%

aa

CALCAL(Capital(CapitalAllocationAllocation

Line)Line)


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


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


Lower risk

Diversify with Risky Assets


15%

20%

25%

Return

What Happens When We Add More Assets?


15%

20%

25%

Return

B2


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


15%

20%

25%

Re

turn

Which portfolio does everyone choose?

MMarket


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


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

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

12/18

11/3/20

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?


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


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?


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.


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.


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)


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

13/18

11/3/20

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



20%

25%

n

Capital Market Line (CML)

MM rk t

CML


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.


,

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:


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)


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


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

14/18

11/3/20

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


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)


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


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.


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


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


8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

15/18

11/3/20

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


Rolls Critique (1977) continued

Only true test if the true market portfolio is ex-postefficient

True market portfolios is unobservable. So, good luck.


Testing CAPM

an t er sset r c ng o e s


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:


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 ][


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 ][


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

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

16/18

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

17/18

11/3/20

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


Predicting Betas: Correcting for Mean Reversion

In order to correct for mean reversion

Because the average beta is 1

We calculate the following:


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

()


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


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)


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

8/3/2019 Lecture 1 - Nov. 3, 2011 (1)

18/18

11/3/20

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


Related findings- Fama French 1992 - Size and Beta


Size and Book to Market


Lecture 1 - Nov. 3, 2011 (1)

Documents

Transcript of Lecture 1 - Nov. 3, 2011 (1)