portfolio theory - capm

7/30/2019 Portfolio Theory - CAPM

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Portfolio Theory

Capital Asset Pricing Model


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Chapter 5: Portfolio Theory &

Asset Pricing Model

Dont put all your eggs in one basket.- Ancient Chinese Proverb


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Modern portfolio theory

How to reduce risk (diversification) How to pricerisk (beta)

Asset pricing model (CAPM)

We will Cover


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Required

rate of

return =

Risk-free

rate ofreturn

Since Treasurys are essentially, free of defaultrisk, the rate of return on a Treasury security is

considered the rate of return.

What is the Required Return

for a Treasury Security?


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Required

rate ofreturn

=

Risk-free

rate ofreturn +

Risk

Premium

How large of a risk premium should we

require to buy a corporate security?

For a corporate stock or bond, what

is the required rate of return?


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Expected Return - the return that an

investor expects to earn on an asset,

given its price, growth potential, etc.

Required Return - the return that an

investor requires on an asset given itsrisk.

Returns


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Returns may be historical orprospective (anticipated).

Returns can be expressed in:

Dollar terms.

Percentage terms.

Returns


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The possibility that an actual return will

differ from our expected return.

Uncertainty in the distribution of

possible outcomes.

What is Risk?


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Uncertainty in the distribution of possible outcomes.

Assume normal distribution

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

-10 -5 0 5 10 15 20 25 30

Company B

return

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

4 8 12

Company A

return

What is Risk?


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To get a general idea of a stocks

price variability, we could look at

the stocks price range over the pastyear.

A more scientific approach is to

examine the stocks STANDARD

DEVIATION of returns.

How do we Measure Risk?


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Standard deviation is a measure of the

dispersion of possible outcomes. The greater the standard deviation, thegreater the uncertainty, and therefore , thegreater the RISK.

Coefficient of variation (CV) is a relativerisk measure of stand-alone risk.

How do we Measure Risk?


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Combining several securities in a

portfolio can actually reduce overall

risk.

How does this work?

Portfolios


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rate

ofreturn

time

Given stock A & B, the returns on these

stocks do not move together over time(they are not perfectly correlated)


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rate

ofreturn

time

rA


stocks do not move together over time

(they are not perfectly correlated)


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rate

ofreturn

time

rA

rB


stocks do not move together over time

(they are not perfectly correlated)


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rate

ofreturn

time

rp

rA

rB


stocks do not move together over time(they are not perfectly correlated


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rate

ofreturn

time

rp

rA

rB

What has happened to the variability

of returns for the portfolio?


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Calculation of Return and Risk for a Portfolio

of Stocks

Expected return on a portfolio is the weighted

average of individual stock returns:

n

iiiP

REwRE

1

)()(

If the portfolio has two stocks, the portfolio mean is:

)()()(2211

REwREwREP

where w1 = weight for stock 1

w2 = weight for stock 2 &

w1 + w2 = 1

Note: the weights can be negative short selling.


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Calculation of Risk for a Portfolio of Stocks

Standard deviation on a portfolio of N-assets:

If the portfolio has two stocks, the portfolio standard

deviation is:

where w1 = weight for stock 1

w2 = weight for stock 2 & w1 + w2 = 1.0

N

i

N

jij

ijjiii

N

i

port Covwww1 ,1

22

1

BAABAA

2

B

2

A

2

A

2

Ap r)w1(w2)w1(w


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Portfolio return is the weighted average of individualstocks rates of return

Portfolio risk depends on the weighted average of: Variance of each stock

More importantly, covariance between individual stocks

Which dominates (or more important?)

N # of Variance # of COV

2 2 23 3 6

N N N2-1

ReCap:


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Investing in more than one security toreduce risk.

If two stocks are perfectly positively

correlated, diversification has noeffect on risk.

If two stocks are perfectly negatively

correlated, the portfolio is perfectlydiversified.

Diversification


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If you owned a share of every stock

traded on the NYSE and NASDAQ,

would you be diversified?

YES!

Would you have eliminated all of yourrisk?

NO! Common stock portfolios still

have risk. (Remember the October1987 stock market crash?)


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Some risk can be diversifiedaway and some can not

Market Riskis also calledNondiversifiable risk, systematic risk.This type of risk can not be

diversified away. Firm-Specific riskis also called

diversifiable risk, unsystematic risk.

This type of risk can be reducedthrough diversification.


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Unexpected changes in interest rates.

Unexpected changes in cash flows due to

tax rate changes, foreign competition, andthe overall business cycle.

Market Risk


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A companys labor force goes on strike.

A companys top management dies in a

plane crash. A huge oil tank bursts and floods a

companys production area.

Firm-Specific Risk


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portfolio

risk

number of stocks

As you add stocks to your

portfolio, firm-specific risk isreduced.


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portfolio

risk

number of stocks

Market risk

As you add stocks to your

portfolio, firm-specific risk isreduced.


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Large

0 15

Prob.

2

1

1 35% ; Large 20%.

Return


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As more stocks are added, each new stock has asmaller risk-reducing impact on the portfolio.

p falls very slowly after about 40 stocks areincluded. The lower limit for p is about 20% =M .

By forming well-diversified portfolios, investors

can eliminate about half the riskiness of owning asingle stock.

Foundation of Modern Portfolio Theory

Conclusion


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Developed by Harry Markowitz in 1958.

Based on diversification effect: If we combine

stocks into portfolios, depending upon thecorrelation between stocks in the portfolio andthe weighting of each stock, some portfolios willdominate others, e.g. some portfolios will have

higher rates of return given the same risk(standard deviation) or some portfolios will havelower standard deviations given the same return.These portfolios will form the EFFICIENTFRONTIER.

Modern Portfolio Theory


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Investors all think in terms of a single holding period.

All investors have identical expectations.

Investors can borrow or lend unlimited amounts at the risk-free rate.

All assets are perfectly divisible.

There are no taxesand no transactions costs.

All investors are price takers, that is, investors buying andselling wont influence stock prices.

Quantitiesof all assets are given and fixed.

What are the assumptions

of the MPT and Asset Pricing Models


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ExpectedPortfolioReturn, r

p

Risk, p

Efficient Set

Feasible Set

Feasible and Efficient Portfolios

with Risky Assets


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The feasible set of portfolios represents all

portfolios that can be constructed from a givenset of stocks.

An efficient portfolio is one that offers:

the most return for a given amount of risk, or

the least risk for a give amount of return.

The collection of efficient portfolios is calledthe efficient set or efficient frontier.


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IB2 IB1

IA2IA1

Optimal PortfolioInvestor A

Optimal Portfolio

Investor B

Risk p

ExpectedReturn, rp

Optimal Portfolios


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Indifference curvesreflect an investorsattitude toward risk as reflected in his orher risk/return tradeoff function. Theydiffer among investors because of

differences in risk aversion.An investors optimal portfoliois defined

by the tangency point between theefficient set and the investors

indifference curve.


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IB2 IB1

IA2IA1

Optimal PortfolioInvestor A

Optimal Portfolio

Investor B

Risk p

ExpectedReturn, rp

Optimal Portfolios


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When a risk-free asset is added to thefeasible set, investors can create portfoliosthat combine this asset with a portfolio ofrisky assets.

The straight line connecting rfwith M, thetangency point between the line and the oldefficient set, becomes the new efficientfrontier.

What impact does rF have on

the efficient frontier?


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M

Z

.Arf

M Risk, p

Efficient Set with a Risk-Free Asset

The Capital MarketLine (CML):

New EfficientFrontier

.

.B

rM

ExpectedReturn, rp

Optimal Portfolio Choice With a


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Optimal Portfolio Choice With a

Risk-Free Asset

G

Standard deviation

Expected

return M

C

rF

E

Capital market line


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The Capital Market Line (CML)is all linearcombinations of the risk-free asset andPortfolio M.

Portfolios below the CML are inferior.

The CML defines the new efficient frontier.

All investors will choose a portfolio on the

CML. Optimal portfolio choice is M and

combinations of risk-free asset and MTobins Separation Theorem

What is the Capital Market Line?


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rf

MRisk, p

I1I2

CML

P = OptimalPortfolio

.P

.M

rP

rM

P

ExpectedReturn, rp


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What is the implication of CAPM andmodern portfolio theory on individualsecurities?

From modern portfolio theory, welearned: diversification reduces riskOnly systematic risk is important

From capital market line, we learnedthe efficient portfolio is the marketportfolio M


Security Market Line (SML)


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Yes. For example:

Interest rate changes affect all firms, but

which would be more affected:

a) Retail food chain

b) Commercial bank

Do some firms have more

market risk than others?


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Yes. For example:

Interest rate changes affect all firms, but

which would be more affected:

a) Retail food chain

b) Commercial bank

Do some firms have more

market risk than others?


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Note

As we know, the market compensates

investors for accepting risk - but only

for market risk. Firm-specific risk can

and should be diversified away.

So - we need to be able to measure

market risk.


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Beta: a measure of market risk.

Specifically, it is a measure of how an

individual stocks returns vary withmarket returns.

Its a measure of the sensitivity of an

individual stocks returns to changes in

the market.

This is why we have BETA.


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A firm that has a beta = 1 has average market

risk. The stock is no more or less volatile than

the market.

A firm with a beta > 1 is more volatile than the

market (ex: computer firms)

Aggressive stocks

A firm with a beta < 1 is less volatile than the

market (ex: utilities). Defensive stocks.

The markets beta is 1


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Run a regression line ofpast returns on

Stock iversus returns on the market.

The regression line is called thecharacteristic line.

The slope coefficient of the characteristic

line is defined as the beta coefficient.

How are betas calculated?


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Method of Calculation

Analysts use a computer with statistical or

spreadsheet software to perform the regression.

At least 3 years of monthly returns or 1

years of weekly returns are used.

Many analysts use 5 years of monthly

returns.

Most stocks have betas in the range of 0.5 to

1.5.

C l l ti B t


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-5-15 5 10 15

-15

-10

-10

-5

5

10

15

XYZ Co. returns

S&P 500

returns

. . . .

. . . .. . . .

. . . .

. . . .

. . . .

. . . .. . . .

. . .

. . . .

. . . .

Calculating Beta

C l l ti B t


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-5-15 5 10 15

-15

-10

-10

-5

5

10

15

XYZ Co. returns

S&P 500

returns

. . . .

. . . .. . . .

. . . .

. . . .

. . . .

. . . .. . . .

. . .

. . . .

. . . .

Beta = slope

= 1.20

This is the

characteristic line

of XYZ stock

Calculating Beta


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Interpreting Regression Results

The R2measures the percent of a stocksvariance that is explained by the market.The typical R2 is:

0.3 for an individual stock over 0.9 for a well diversified portfolio

The 95% confidence interval shows therange in which we are 95% sure that the

true value of beta lies. The typical range is: from about 0.5 to 1.5 for an individual stock

from about .92 to 1.08 for a well diversified portfolio


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Beta Equation [11.17], p 351

Mathematically, beta is calculated as:

)(

),(2

M

Mii

R

RRCOV


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We know how to measure risk, using

standard deviation for overall risk and beta

for market risk.

We know how to reduce overall risk to only

market risk through diversification.

We need to know how to price risk so we will

know how much extra return we should

require for accepting extra risk.

Summary:


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The return on an investment

required by an investor given theinvestments risk.

What is the Required Rate of

Return?


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Required

rate of

return

=

Risk-free

rate of

return+

Risk

Premium


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Required

rate of

return

=

Risk-free

rate of

return+

Risk

Premium

Market

Risk


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Required

rate of

return

=

Risk-free

rate of

return+

Risk

Premium

Market

Risk

Firm-specific

Risk

i


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= +

Required

rate of

return

Risk-free

rate of

return

Risk

Premium

Market

Risk

Firm-specific

Risk

can be diversified

away

Required


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Required

rate of

return

Beta

Lets try to graph this

relationship!

Required


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Required

rate of

return

Risk-free

rate ofreturn

(6%)

Beta

12% .

1

Required


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q

rate of

return

Risk-free

rate ofreturn

(6%)

Beta

12% .

1

security

marketline

(SML)


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This linear relationship between risk

and required return is known asthe Capital Asset Pricing Model

(CAPM)

The CAPM is an equilibrium model thatspecifies the relationship between riskandrequired rate of returnfor assets held in well-diversified portfolios.

It is based on the premise that only one factoraffects risk - the market portfolio.


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Basic Messages of CAPM

If you want to earn higher returns, you must

be prepared to bear higher risk.

If you are not fully diversified, you are

bearing risk without being compensated.

Required

SML


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rate of

return

Risk-free

rate ofreturn

(6%)

Beta

12% .

1

SML

0

Is there a riskless

(zero beta) security?

Required

SML


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rate of

return

Beta

12% .

1

SML

0

Is there a riskless

(zero beta) security?

Treasury

securities are

as close to riskless

as possible.Risk-free

rate ofreturn

(6%)

Required

SMLWh d s th S&P 500


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rate of

return

Beta

12% .

1

SMLWhere does the S&P 500

fall on the SML?

Risk-free

rate ofreturn

(6%)

0

Required

SMLWhere does the S&P 500


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rate of

return

Beta

12% .

1

SWhere does the S&P 500

fall on the SML?

The S&P 500 isa good

approximation

for the market

Risk-free

rate ofreturn

(6%)

0 Required

SML


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rate of

return

Beta

12% .

1

UtilityStocks

Risk-free

rate ofreturn

(6%)

0 Required

SMLHigh-tech


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rate of

return

Beta

12% .

1

High tech

stocks

Risk-free

rate ofreturn

(6%)

0


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E(Ri) = RF +

Beta,Risk-freeRate

[E(RMRF)]


SML [Equation 11.9, p 351]

Cov(Ri,RM)

M

Equity(Market) RiskPremium


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Suppose the Treasury bond rate is

6%, the average return on the S&P

500 index is 12%, and Walt Disneyhas a beta of 1.2.

According to the CAPM, what should

be the required rate of return onDisney stock?

Example:


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=

According to the CAPM, Disney stock

should be priced to give a 13.2% return. If the expected return > 13.2%

Underpriced, good investment

If the expected return < 13.2%

Overpriced, bad investment

If the expected return = 13.2%

Properly priced, good investment

E(RDisney) = RF+ [E(RM) RF] Disney

Required

f

SMLTheoretically every


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rate of

return

Beta

12% .

10

Theoretically, every

security should lie

on the SML

Risk-free

rate ofreturn

(6%)

Required

f

SMLTheoretically every


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rate of

return

Beta

12% .

10

Theoretically, every

security should lie

on the SML

If every stock

is on the SML,investors are being fully

compensated for risk.Risk-free

rate ofreturn

(6%)

Required

f

SMLIf a security is above


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rate of

return

Beta

12% .

10

If a security is above

the SML, it is

underpriced.

Risk-free

rate ofreturn

(6%)

Required

t f

SMLIf a security is above


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rate of

return

Beta

12% .

10

If a security is above

the SML, it is

underpriced.

If a security isbelow the SML, it

is overpriced.Risk-free

rate ofreturn

(6%)

The Global Efficient Set


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The Global Efficient Set

B

Standard deviation

Expected

return

A

Efficient frontier-

US and foreign stocks

C

DEfficient frontier-

US stocks only

Gl b l Di ifi i Eff


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portfolio

risk

number of stocks

Domestic Portfolio

International Portfolio

Global Diversification Effect

Gl b l Di ifi i


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Global Diversification

Expands the set of securities available toinvestors.

If security returns are not perfectly

synchronized, investors can achieve greaterrisk reduction through global diversificationrather than limiting their choices in thedomestic markets.

Global diversification pushes out the efficientfrontier.

In Class Problem #6


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You are given the following information on the returns for market portfolio, Mand the returns on common stock A:

Probability RM RA RF.25 .20 .25 .06

.25 -.05 -.15 .06

.25 .30 .55 .06

.25 -.05 -.15 .06

A. Determine beta for stock A.

B. Using SML criterion to determine whether you would invest in the stock.

To answer this, you have to plot the expected SML. Mark your axis clearlyand make sure you plot the risk-free rate, beta, and expected returns andrequired returns properly on the graph.

C. Which asset is more risky, the common stock A or the market portfolio, M?In systematic risk? In total risk? Explain.

In Class Problem #6

In Class Problem #7


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Combine CAPM and Dividend Growth

Model:

Use CAPM to calculate required rate of

return

Use dividend growth model to calculate the

value of the stock

In Class Problem #7

In Class Problem #7


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The risk-free rate of return is 11 percent; the required rate of return on

the market is 14 percent; and UHM Companys stock has a betacoefficient of 1.5.

If the dividend expected during the coming year, D1, is $2.25, and if g =a constant 5%, at what price should UHMs stock sell?

Now, suppose the Federal Reserve Board increases the moneysupply, causing the risk-free rate to drop to 9 percent and marketreturn to fall to 12 percent. What would this do to the price of thestock?

Now, suppose UHM has a change in management. The new groupinstitutes policies that increase the expected constant growth rate to

6 percent. Also, the new management stabilizes sales and profits,and thus causes the beta coefficient to decline from 1.5 to 1.3.Assume that RF and RM are equal to the values in part B. After allthese changes, what is UHMs new equilibrium price? (Note: D1goes to $2.27).

In Class Problem #7

portfolio theory - capm

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