7. capm beta apt

8/3/2019 7. Capm Beta Apt

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CAPM, BETA AND APT


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CAPITAL MARKET THEORY: AN

OVERVIEW

Capital market theory extends portfolio theory and

seeks to develops a model for pricing all risky assets

based on their relevant risks

Asset Pricing Models

Capital asset pricing model (CAPM) allows for the calculation

of the required rate of return for any risky asset based on

the securitys beta

Arbitrage Pricing Theory (APT) is a multi-factor model for

determining the required rate of return


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ASSUMPTIONS OF CAPITAL MARKET

THEORY

All investors are Markowitz efficient investors (rational)who invest on the efficient frontier.

Investors can borrow or lend any amount of money atthe risk-free rate of return (RFR).

Investors have homogeneous expectations

All investors are risk averse and efficiently diversified.Only the systematic risk is of concern in determiningestimated return.


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ASSUMPTIONS OF CAPITAL MARKET

THEORY All investments are infinitely divisible, which means

that it is possible to buy or sell fractional shares of any

asset or portfolio.

There are no taxes or transaction costs involved in

buying or selling assets.

Capital markets are efficient and no single investor

can influence price (Perfect competition).


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MAKING ASSUMPTIONS

Some of these assumptions are clearly unrealistic

Relaxing many of these assumptions would have only

minor influence on the model and would not change

its main implications or conclusions.

The primary way to judge a theory is on how well itexplains and helps predict behavior, not on its

assumptions.


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THE CAPITAL ASSET PRICING MODEL

An assets covariance with the market portfolio is the

relevant risk measure (helps in beta calculation)

This can be used to determine an appropriate requiredrate of return on a risky asset

CAPM indicates what should be the expected or

required rates of return on risky assets

This helps to value an asset by providing an appropriate

discount rate to use in dividend valuation models


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DETERMINING THE EXPECTED

RETURN

The expected rate of return of a risk asset isdetermined by the RFR plus a risk premium

for the individual asset

The risk premium is determined by the

systematic risk of the asset (beta) and the

prevailing market risk premium (RM-RFR)

RFR)-(RRFR)E(R Mi iF!


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WHAT IS THE MARKET RISK PREMIUM

[RM - RRF]

The additional return over the risk-free rate

needed to compensate investors for assuming an

average amount of risk.


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CAPM AND BETA

)Cov(RFR-R

RFR)E(R Mi,2M

Mi

W

!

RFR)-R(Cov

RFR M2M

Mi,

W!

2

M

Mi,CovW

We then define as beta

RFR)-(RRFR)E(R Mi iF!

)( iF


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CAPM AND REQUIRED RETURN FOR IBMS

COMMON STOCK

Let rRF=6%, rM=12%, and the Beta of IBM commonstock is IBM=1.3

rIBM = rRF + IBM[rM - rRF]

rIBM = 0.06 + 1.3[0.12 0.06]

rIBM = 0.06 + 1.3[0.06]

rIBM = 0.06 + 0.078 =0.138 or 13.8%

Note the following items:

The market risk premium is [12% - 6%] = 6%

IBMs risk premium is 1.3[12% 6%] = 7.8%.


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THE SECURITY MARKET LINE (SML)

The CAPM model is linear and when plotted ona graph paper gives a straight line called SML.

The graphical version of CAPM is SecurityMarket Line.


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GRAPH OF SML

)E(Ri

)Beta(Cov 2Mim/W

0.1

mR

SML

0

Negative Beta

RFR


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OVER AND UNDER PRICED

In equilibrium, all assets and all portfolios of assets

should plot on the SML

The SML gives the market going rate of return or what you

should earn as a return for a security

Any security with an expected return that plots above the

SML is underpriced

Any security with an expected return that plots below theSML is overpriced


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WHAT IF THE SML CHANGES?

Two possible SML changes can definitely occur. Eachhas important effects.

(1) Risk-free rates rRF can change. This can be due to theInflation Premium (IP) or the real rate of interest changing.Note that in this case, the RM must also increase by thesame amount so that the market risk premium [rM-rRF] isunchanged.

(2) The market risk premium [rM-rRF] can change. Thiswould typically be due to changing attitudes toward riskaversion by investors.


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WHAT IF THE SML CHANGES?

Following slides allow for two separate SML changes:

In the first case, we let the risk-free rate rRF increase by 2%

(from 5% to 7%). The market required return rM mustincrease by the same amount (from 10% to 12%), so thatthe market risk premium [rM-rRF] remains at 5%.

In the second case, we allow for a 2% increase in the

market risk premium (from 5% to 7%). The risk-free rateremains the same at 5%, however, rM must increase by 2%(from 10% to 12%) so that the market risk premium cannow be [rM-rRF] = 7%.


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THE SECURITY MARKET LINE

Risk-free rate increases from 5% to 7%. The

new SML is ri = 0.07 + 0.05i

SML, old

ri (%)

0 0.5 1.0 1.5

12

10

7

5

Risk, i

Mkt. risk prem. = 5%

rRF=7%

SML, new


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THE SECURITY MARKET LINE

The market risk premium increases from 5 to

7%. The new SML is ri = 0.05 + 0.07i

SML, old

ri (%)

0 0.5 1.0 1.5

12

10

5

Risk, i

Mkt. risk prem. = 7%

rRF=5%

SML, new


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HOW TO INTERPRET BETA ()

Beta measures a stocks degree of systematic ormarket risk. It can also be thought of as the stockscontribution to the risk of a well-diversified portfolio.

= 1: the stock has average market risk. The stockgenerally tends to go up (down) by the same percentageamount as the market.

= 1.5: The stock generally tends to go up (down) by 50%

(1.5x) more than the market.

= 0.5: The stock generally tends to go up (down) by half asmuch as the market.


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HOW TO INTERPRET BETAS (),

CONTINUED = 0: the stock has no correlation with movements in the

overall stock market. All of this firms risk would actually befirm-specific risk.

< 0: The stock generally tends move in a directionopposite that of the market (very rare).

Firms that supply basic consumer goods (Proctor &Gamble) and utilities (phone, cable, gas, or electric)

tend to have low Betas (lower than 1.0, often around0.4 to 0.6).

Firms that are in economically cyclical industrieswould have higher Betas (greater then 1.0).


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EXPECTED RETURN AND BETA OF A

PORTFOLIO OF STOCKS

Both portfolio expected returns and portfolio Betas are

always a weighted average of the stocks that comprise

the portfolio.

You invest $10,000: $6000 in Apple Computer and

$4000 in Proctor & Gamble (PG).

Let rRF=5% and rM=10%. Let APPLE=1.3 and PG=0.6

The investment weights are (6000/10000)=0.6 for Apple

and (4000/10000)=0.4 for PG. The weights must always

sum up to 1.


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EXPECTED RETURN AND BETA OF A

PORTFOLIO OF STOCKS, CONTINUED

The portfolio Beta is always the weighted

average of each of the stocks betas.

P

= wapple

apple

+ wpg

pg

P = 0.6(1.30) + 0.4(0.6)

P = 1.02

Now find the portfolio required return.

rP = 0.05 + 1.02[0.10 0.05] = 0.101 or 10.1%


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THE CONCEPT OF BETA

Beta(b) measures how the return of an individual asset (or evena portfolio) varies with the market.

b = 1.0 : same risk as the market

b < 1.0 : less risky than the market

b > 1.0 : more risky than the market

Beta is the slope of the regression line (y = a + bx) between astocks return(y) and the market return(x) over time, b fromsimple linear regression.


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CALCULATING BETA: THE

CHARACTERISTIC LINE

The systematic risk input of an individual asset is derived from a

regression model, referred to as the assets characteristic line

with the model portfolio:

IFE ! tM,iiti, RRwhere:Ri,t = the rate of return for asset i during period t

RM,t = the rate of return for the market portfolio M during t

miii R-R FE !

2M

Mi,CovW

F !i

error termrandomthe!I


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ESTIMATING THE BETA COEFFICIENT

Generally, these quantities are not known.

We usually rely on their historical values to provide uswith an estimate of beta.

If we know the securitys correlation with the market, its

standard deviation, and the standard deviation of the

market, we can use the definition of beta:

M

jMj

jW

WVF

,!


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INTERPRETING THE BETA

COEFFICIENTThe beta of the market portfolio is always equal to

1.0:

1,

!!

M

MMM

M

W

WVF

1, !MMVSince

The beta of the risk-free asset is always equal to 0:

0,

!!M

rMr

r

ff

fW

WV

F

0!frWSince


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PROJECT, UN-GEARED & GEARED

BETA Beta Geared: The Beta attaching to the ordinary

shares of a geared firm. These bear a risk higherthan the firms basic activity.

Beta Un-geared: The geared Beta stripped of theeffect of gearing. Corresponds to the activity Beta inan equivalent un-geared firm.

An indication of the systematic riskiness attachingto the returns on ordinary shares. It equates to theasset Beta for an un-geared firm, or is adjustedupwards to reflect the extra riskiness of shares in a

geared firm., i.e. the Geared Beta


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ARBITRAGE PRICING THEORY (APT)

CAPM is criticized because of the difficulties in

selecting a proxy for the market portfolio as a

benchmark

An alternative pricing theory with fewer

assumptions was developed:

Arbitrage Pricing Theory


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ASSUMPTIONS OF ARBITRAGE PRICING

THEORY

Capital markets are perfectly competitive

Investors always prefer more wealth to lesswealth with certainty

The stochastic process generating assetreturns can be presented as a k-factor model


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ARBITRAGE PRICING THEORY

Multiple factors expected to have an impact on allassets:

Inflation

Growth in GNP Major political upheavals

Changes in interest rates

And many more.

Contrast with CAPM assumption that only beta isrelevant


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ARBITRAGE PRICING THEORY (APT)

where:

= the expected return on an asset with zerosystematic risk where

ikkiii bbbE PPPP ! ...22110

0P

1P = the risk premium related to each of the common factors - for

example the risk premium related to interest rate risk

bik = the pricing relationship between the risk premium and asset i - that is how

responsive asset i is to this common factorK


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EXAMPLE OF TWO STOCKS

AND A TWO-FACTOR MODEL

= changes in the rate of inflation. The risk premium

related to this factor is 1 percent for every 1 percent

change in the rate

1P

)01.(1

!P

= percent growth in real GNP. The average risk premium related to this

factor is 2 percent for every 1 percent change in the rate

= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3

percent

2P

)02.( 2 !P

)03.(3

!P3P


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= the response of asset Xto changes in the rate of

inflation is 0.501x

b)50.(

1!

xb

= the response of asset Y to changes in the rate of inflation is 2.00

1yb

= the response of asset X to changes in the growth rate of real GNP is

1.50

= the response of asset Y to changes in the growth rate of real GNP is

1.75

2xb

2yb

)50.1( 2 !xb

)75.1( 2 !yb


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= .03 + (.01)bi1 + (.02)bi2

Ex = .03 + (.01)(0.50) + (.02)(1.50)

= .065 = 6.5%

Ey = .03 + (.01)(2.00) + (.02)(1.75)

= .085 = 8.5%

22110 iii bbE PPP !


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EMPIRICAL TESTS OF THE APT

Studies by Roll and Ross and by Chen support APT by

explaining different rates of return with some better

results than CAPM

Dhrymes and Shanken question the usefulness of APT

because it was not possible to identify the factors and

therefore may not be testable

7. capm beta apt

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