bm410-10 theory 3 - capm and apt 29sep05

8/12/2019 Bm410-10 Theory 3 - Capm and Apt 29sep05

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BM410: Investments

Capital Asset PricingTheory and APT

orHow do you value stocks?


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Objectives

A. Review and solve problems using the CAL,

MPT, and the Single Index model

B. Understand the implications of capital asset

pricing theory and the CAPM to computesecurity risk premiums

C. Understand the arbitrage pricing theory and

how it works


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A. Solve problems using the CAL, CML,

MPT and Single Index Models

Capital Market Line Review

You estimate that a passive portfolio invested to

mimic the S&P 500 (an index fund) has an expected

return of 13% with a standard deviation of 25%.Your portfolio has an expected return of 17% with a

standard deviation of 27%. With the risk-free rate at

7%, draw the CML and your funds CAL on an

expected return-standard deviation diagram.

A. What is the slope of the CML? Your CAL?

B. Characterize in one short paragraph the

advantage of your fund over the passive fund.


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Answer

Slope of the CML = (13-7)/25 = .24

Slope of your CAL = (17-7)/27 = .37

b. Your fund allows an investor a higher mean for any

given standard deviation than the passive strategy.

17%

13%

7%10% 20% 30%

Your Fund

Index Fund


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MPT Review

Suppose that for some reason you are required toinvest 50% of your portfolio in bonds (sb= 12%,E(rb) = 10%) and 50% in stocks (ss= 25%, E(rs) =17%).

A. If the standard deviation of your portfolio is15%, what must be the correlation coefficient

between stock and bond returns?

B. What is the expected rate of return on yourportfolio?

C. Now suppose that the correlation between stockand bond returns is 0.22 but that you are free tochoose whatever portfolio proportions you desire.Are you likely to be better or worse off that you

were in part a?


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Answer

A. sp2

= w12s1

2+ w22s2

2+ 2W1W2 (r1,2s1s2 )

(.15) 2 =[(.512.121

2) +(.522.252

2) + 2(.51.52)*(.121.252 )] * r1,2

r1,2 = .2183 or 21.8% (take my word for this)

B. E(rp) = (.5 * .10) + (.5 * .17) = 13.5% C. While the current correlation is slightly lower than 22%,

this implies slightly greater benefits from diversification.

However, the 50% bond constraint represents a cost since

you cannot choose your optimal risk-return tradeoff foryour risk level. Unless you would choose to have 50%

bonds anyway, you are better off with the slightly higher

correlation and the ability to choose your own portfolio

weights.


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Factor Review

Investors expect the market rate of return to be

10%. The expected rate of return on the stock

with a beta of 1.2 is currently 12%.

If the market return this year turns out to be8%, how would you review/change your

expectations of the rate of return on the

stock?


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Answer

The expected return on the stock would be

your beta (1.2) times the market return or:

1.2 * 8% = 9.6%

Likewise, you could also determine how much

the return would decrease by multiplying the

beta times the change in the market return or:

1.2 * (8%-10%) = -2.4% + 12% =9.6%


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Questions

Any questions of Capital Allocation Lines,

Modern Portfolio Theory, or Single Index

Models?


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B. Implications of Capital Market

Theory and CAPM

What have we done this far?

We have been concerned with how an individual or

institution would select an optimum portfolio.

If investors act as we think, we should be able todetermine how investors will behave, and how

prices at which markets will clear are set

This market clearing of prices and returns has

resulted in the development of so-called generalequilibrium models

These models allow us to determine the risk for

any asset and the relationship between expected

return and riskfor any asset when the markets

are in equilibrium, i.e. balance or constant state


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Capital Asset Pricing Theory

What is capital asset pricing theory?

It is the theory behind the pricing of assets which

takes into account the risk and return characteristics

of the asset and the market

What is the Capital Asset Pricing Model?

It is an equilibrium model (i.e., a constant state

model) that underlies all modern financial theory

It provides a precise prediction between therelationship between the risk of an asset and its

expected return when the market is in

equilibrium

With this model, we can identify mis-pricing of

securities (in the long-run)


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CAPM(continued)

Why is it important?

It provides a benchmark rate of return for

evaluating possible investments, and identifying

potential mis-pricing of investments For example, an analyst might want to know

whether the expected return she forecast is more

or less than its fair market return.

It helps us make an educated guess as to theexpected return on assets that have not yet been

traded in the marketplace

For example, how do we price an initial public

offering?


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CAPM(continued)

How was it derived?

Derived using principles of diversification with

very simplified (i.e. somewhat unrealistic)

assumptions Does it work, i.e. withstand empirical tests in real life?

Not totally

But it does offer insights that are important and

its accuracy may be sufficient for someapplications

Do we use it?

Yes, but with knowledge of its limitations


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CAPM Assumptions

What does the model assume (some are unrealistic)?

Individual investors are price takers (cannot affectprices)

Single-period investment horizon (an its identical for all)

Investments are limited to traded financial assets

No taxes, and no transaction costs (costless trading)

Information is costless and available to all investors

Investors are rational mean-variance optimizers Investors analyze information in the same way, and

have the same view, i.e., homogeneous expectations


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Resulting Equilibrium Conditions

Based on the previous assumptions:

All investors will hold the same portfolio for risky

assetsthe market portfolio (M)

The market portfolio (M) contains all securities andthe proportion of each security is its market value as

a percentage of total market value

The risk premium on the market depends on the

average risk aversion of all market participants The risk premium on an individual security is a

function of its covariance (correlation and sssm)

with the market


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E(r)

E(rM)

rf

MCML

sm

Capital Market Line

s

M = Market portfolio rf = Risk free rateE(rM) - rf= Market risk premium

[E(rM) - rf]/sM= Market price of risk

The efficient frontier without

lending or borrowing


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Expected Return and Risk

of Individual Securities

What does this imply?

The risk premium on individual securities is

a function of the individual securitys

contribution to the risk of the marketportfolio

Individual securitys risk premium is a

function of the covariance of returns with

the assets that make up the market portfolio


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CAPM Key Thoughts

Key statements:

Portfolio risk is what matters to investors, and

portfolio risk is what governs the risk premiums

they demand

Non-systematic, or diversifiable risk can be reduced

through diversification.

Investors need to be compensated for bearing only

non-systematic risk (risk that cannot be diversified

away)

The contribution of a security to the risk of a

portfolio depends only on its systematic risk, as

measured by beta. So the risk premium of the asset

is proportional to its beta. ( = [COV(ri,rm)] / sm2)


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Expected ReturnBeta Relationship

Expected return - beta relationship of CAPM:

E(rM) - rf = E(rs) - rf

1.0 bsIn other words, the expected rate of return of an asset

exceeds the risk-free rate by a risk premium equal to the

assets systematic risk (its beta) times the risk premium

of the market portfolio. This leads to the familiar re-arrangement of terms to give (memorize this):

E(rs) = rf + bs [E(rM) - rf ]


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E(r)

E(rM)

rf

SML

M

= 1.0

The Security Market Line

Notice that instead of using standarddeviation, the Security Market Line uses Beta

SML Relationships

= [COV(ri,rm)] / sm2

Slope SML = E(rm)rf = market riskpremium

SML = rf + [E(rm) - rf]


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Differences Between the SML and CML

What are the differences?

The CML graphs risk premiums of efficient

portfolios , i.e. complete portfolios made up of the

risk portfolio and risk-free asset, as a function of

standard deviation

The SML graphs individual asset risk premiums as

a function of asset risk.

The relevant measure of risk for individual

assets is not standard deviation; rather, it is beta

The SML is also valid for portfolios


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Example: SML Calculations

Put the following data on the SML. Are

they in equilibrium?

Market data: E(rm) - rf= .08 rf = .03

Asset data: bx = 1.25 by= .60

Calculations:

bx = 1.25 so E(r) on x =

E(rx) = .03 + 1.25(.08) = .13 or 13%

by= .60 so E(r) on y =

E(ry) = .03 + .6(.08) = .078 or 7.8%


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E(r)

Rx=13%

SML

m

1.0

Rm=11%

Ry=7.8%

3%

x

1.25

y.6

.08

Graph of Sample Calculations

They are in equilibrium


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Disequilibrium Example

Suppose a security with a beta of 1.25 is

offering expected return of 15%

According to SML, it should be 13%

Under priced: offering too high of a rate of

return for its level of risk. Investors

therefore would:

Buy the security, which would increasedemand, which would increase the price,

which would decrease the return until it

came back into line.


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E(r)

15%

SML

1.0

Rm=11%

rf=3%

1.25

Disequilibrium Example

The return is above the

SML, so you would buy it

As more people bought

the security, it would

push the price up,

which would bring the

return down to the line.


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CAPM and Index Models

CAPM Problems

It relies on a theoretical market portfolio which

includes all assets

It deals with expected returns To get away from these problems and make it testable,

we change it and use an Index model which:

Uses an actual index, i.e. the S&P 500 for

measurement Uses realized, not expected returns

Now the Index model is testable


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The Index Model

With the Index model, we can:

Specify a way to measure the factor that affectsreturns (the return of the Index)

Separate the rate of return on a security into its

macro (systematic) and micro (firm-specific)components

Components

= excess return if market factor is zero

iRm= component of returns due to movements in theoverall market

ei = component attributable to company specificevents

Ri

= ai

+ i

Rm

+ ei

(Notice the similarity to the Single Index model discussed earlier)


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Security Characteristic Line

Excess Returns (i)

SCL

.

.

...

.

. .

. ..

. . .. .

. ..

..

.

. .

. ..

.

..

. .

..

.

. . .. .

.

. ... .. .. .

Excess returns

on market index

Ri= a i+ iRm+ ei

Plot of a companys excess return as a

function of the excess return of the market


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Does the CAPM hold?

There is much evidence that supports the

CAPM

There is also evidence that does not support the

CAPM Is the CAPM useful?

Yes. Return and risk are linearly related for

securities and portfolios over long periods of time

Yes. Investors are compensated for taking onadded market risk, but not diversifiable risk

Perhaps instead of determining whether the CAPM is

true or not, we might ask: Are there better models?


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Questions

Any questions on capital asset pricing

and the Capital Asset Pricing Model?


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CAPM Problem

Suppose the risk premium on the market portfolio is

9%, and we estimate the beta of Dell as bs = 1.3. The

risk premium predicted for the stock is therefore 1.3

times the market risk premium of 9% or 11.7%. The

expected return on Dell is the risk-free rate plus therisk premium. For example, if the T-bill rate were

5%, the expected return of Dell would be 5% +(1.3 *

9%) = 16.7%.

a. If the estimate of the beta of Dell were only 1.2,what would be Dells required risk premium?

b. If the market risk premium were only 8% and

Dells beta was 1.3, what would be Dells risk

premium?


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Answer

a. If Dells beta was 1.2 the required risk premium

would be (remember the risk premium is the

expected return less the risk-free rate):

E(rs

) = rf

+ bs

[E(rM

) - rf

] or the expected return on

Dell = 5% + 1.2 (9%) = 15.8%

Dells risk premium (over the risk free rate) =

15.8% - 5% = 10.8%

b. If the market risk premium was 8%:

E(rs) = rf + bs [E(rM) - rf ]

E(r) of Dell = 5% + 1.3 (8%) = 15.4%

Dells new risk premium is 15.4 5% = 10.4%


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C. Understand Arbitrage Pricing Theory

(APT) and How it Works

What is arbitrage? The exploitation of security mis-pricing to earn

risk-free economic profits

It rises if an investor can construct a zeroinvestment portfolio (with a zero net investment

position netting out buys and sells) with a sure

profit

Should arbitrage exist? In efficient markets (and in CAPM theory),

profitable arbitrage opportunities will quickly

disappear as more investors try to take advantage of

them


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Arbitrage Pricing Theory (APT) (continued)

What is APT based on?

It is a variant of the CAPM, and is an attempt to

move away from the mean-variance efficient

portfolios (the calculation problem) Ross instead calculated relationships among

expected returns that would rule out riskless profits

by any investor in a well-functioning capital market

What is it? It is a another theory of risk and return similar to the

CAPM.

It is based on the law of one price: two items that

are the same cant sell at different prices


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APT (continued)

In its simplest form, it is:

Ri= a i+ iRm+ ei the same as CAPM

The only value for a which rules out arbitrage

opportunities is zero. So set a to zero and you get:Ri= iRmSubtract the risk-free rate and you get the

well-known equation:

E(rs) = rf + bs [E(rM) - rf ] from CAPM


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APT and CAPM Compared

Differences:

APT applies to well diversified portfolios and not

necessarily to individual stocks

With APT it is possible for some individual stocks

to be mispricedto not lie on the SML

APT is more general in that it gets to an expected

return and beta relationship without the assumption

of the market portfolio

APT can be extended to multifactor models, such

as:

Ri= a i+ 1R1+ 2R2+ 3R3+ nRn + ei


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APT and Investment Decisions

Roll and Ross argue that APT offers an approach tostrategic portfolio planning

Investors need to recognize that a few systematicfactors affect long-term average returns

Investors should understand those factors and setup their portfolios to take those factors intoaccount

Key Factors:

Changes in expected inflation Unanticipated changes in inflation

Unanticipated changes in industrial production

Unanticipated changes in default-risk premium

Unanticipated changes in the term structure ofinterest rates


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Questions

Any questions on Arbitrage Pricing Theory

and how it differs from CAPM?


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Problem

Suppose two factors are identified for the U.S.

economy: the growth rate of industrial

production (IP) and the inflation rate (IR). IP

is expected to be 4% and IR 6% this year. Astock with a beta of 1.0 on IP and 0.4 on IR

currently is expected to provide a rate of return

of 14%. If industrial production actually

grows by 5% while the inflation rate turns outto be 7%, what is your best guess on the rate

of return on the stock?


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Answer

The revised estimate on the rate of return on

the stock would be:

Before

14% = a+[4%*1] + [6%*.4]

a= 7.6%

With the changes:

7.6% + [5%*1] + [7%*.4]The new rate of return is 15.4%


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Review of Objectives

A. Can you solve problems using the CAL,

MPT, and the Single Index model?

B. Do you understand the implications of

capital asset pricing theory and the CAPM tocompute security risk premiums?

C. Do you understand arbitrage pricing theory

and how it works?

bm410-10 theory 3 - capm and apt 29sep05

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