risk and return : stocks

7/30/2019 risk and return : stocks

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Prentice Hall, 2004

9Corporate Financial Management 2e

Emery Finnerty Stowe

Risk and Return:

Stocks


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Learning Objectives

Calculate average realized returns for a security.

Estimate expected returns from securities and

portfolios.Estimate the standard deviation of returns on

securities and for portfolios.

Explain why diversification is beneficial.

Describe the efficient frontier and the Capital

Market Line.


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Chapter Outline

9.1 Historical Security Returns in the United

States

9.2 Probability and Statistics

9.3 Expected Return and Specific Risk

9.4 Investment Portfolios

9.5 A Prescription for Investing

9.6 Some Practical Advice


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Risk and Return and the

Principles of Finance

Diversification

Risk-Return Trade-OffEfficient Capital Markets

Incremental Benefits

Two-Sided TransactionsTime-Value-of-Money


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Realized Rates of Return

Compute Peters realized return from his

investment in Iomega common shares.

Three months ago, Peter Lynch purchased 100

shares of Iomega Corp. at $50 per share. Last

month, he received dividends of $0.25 pershare from Iomega. These shares are worth

$56 each today.


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Dollar Returns

Total amount invested

$50(100) = $5,000

Total dividends received

$0.25(100) = $25

Total proceeds from sale of stock

$56 (100) = $5,600

Capital gain

$5,600$5,000 = $600


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Dollar Returns

Total Dollar Return =

Dividends + Capital Gain (or Loss)

= $25 + $600 = $625

Capital gain is part of the total dollar return

even if it is not yet realized.


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Holding Period Return

The Holding Periodis defined as the length

of time over which the assets percentage

return is computed.In Peter Lynchs case, the holding period is

3 months long.


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The Holding Period Return(HPR) is defined

as:

where Ptis the price at the end of period t,

Pt1 is the price at the end of period t1,

and Dtis the dividend received during period t.

HPRP D P

Ptt t t

t=

+ -

-1

1


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In Peter Lynchs case,

Pt1 = $50

Pt= $56Dt= $0.25

or=+ -

=25

0 125 12 50%$56 $0 . $50

$50. .

HPRP D P

Pt

t t t

t

=+ -

-

1

1


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The total return of 12.50% consists of:

Dividend Yield

and

Capital Gains Yield

= =

=-

=

$0 .

$50 .

$56 $50

$50.

25

0 50%

12 00%


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Reinvestment of Interim Cash

Flows

In the single period example, we assumed that the

dividend of $0.25 was received at the end of the

holding period.

When the holding period is long, and there are

interim cash flows during this period, we will

assume that these cash flows are re-invested in

additional units of the same asset.In the case of common stocks, dividends received

are used to purchase additional shares.


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Reinvestment of Cash Flows

Fifteen months ago, John Vinick purchased

100 shares of Iomega Corp.s common stock at

$50 each. At the end of every quarter, Iomegahas paid a dividend of $0.25 per share. John

has reinvested these dividends back into

Iomegas common stock. The end-of-quarter

share prices are given to you.

Compute the HPR and APY for each

quarter.


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Time Event SharePrice

0.000.250.500.75

1.001.25

Purchase 100 sharesDPS of $0.25 per shareDPS of $0.25 per shareDPS of $0.25 per share

DPS of $0.25 per shareDPS of $0.25 per share

$50$56$54$50

$56$60


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1.00 $56 $25.35 0.446 101.855

0.25 $56 $25.00 0.446 100.446 $5,625.000.50 $54 $25.11 0.463 100.909 $5,449.220.75 $50 $25.23 0.500 101.409 $5,070.80

$5,704.651.25 $60 $25.47 0.417 102.272 $6,137.59

Time SharePrice

TotalDivs.

SharesPurchased Owned

TotalValue

0.00 $50 100.000 100.000 $5,000.00

Shares


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HPR and APY for 2 Quarters

After 2 quarters, John owns 100.911 shares worth $54

each. The HPR is thus

The APY is (1.0898)2 - 1 or 18.78%

HPR q2449 22 000

0008 98%=

-=

$5 , . $5 ,

$5 ,.


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HPR and APY for Various

Holding Periods

Holding Period HPR APY

0.250.500.751.00

1.25

12.50%8.98%1.42%

14.09%

22.75%

60.18%18.78%1.89%

14.09%

17.82%


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Average Annual Returns, 1926-2000

Class of Security ArithmeticMean

GeometricMean

StandardDeviation

Large firm common stock

Small firm common stockLong term corp. bondsLong term govt. bondsIntermediate term govt.

bondsU.S. Treasury bills

13.0%

17.3%6.0%5.7%

5.5%3.7%

11.0%

12.4%5.7%5.3%

5.3%3.8%

20.2%

33.4%8.7%9.4%

5.8%3.2%


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Probability Concepts

Random variable Something whose value in the future is subject to

uncertainty.

Probability The relative likelihood of each possible outcome (or

value) of a random variable.

Probabilities of individual outcomes cannot be negativenor greater than 1.0.

Sum of the probabilities of all possible outcomes mustequal 1.0.


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Mean The long run average of the random variable.

Equals the expected value of the random variable.

Variance (and Standard Deviation) Measure the dispersion in the possible outcomes.

Standard deviation is the square-root of the variance.

Higher variance implies greater dispersion in the

possible outcomes.


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Covariance Measures how two random variables vary

together (or co-vary).

Covariance can be negative, positive or zero.

Its magnitude has no bounds.

Correlation Coefficient

A standardized measure of co-variationbetween two random variables.

Always lies between -1.0 and +1.0.


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Positive Covariance (or correlation)

When one random variables outcome is above the

mean, the other is also likely to be above its mean.

Negative Covariance (or correlation)

When one random variables outcome is above the

mean, the other is likely to be below its mean.

Zero Covariance (or correlation) There is no relationship between the outcomes of the

two random variables.


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Computing the Basic Statistics

A security analyst has prepared the following

probability distribution of the possible returns on the

common stock shares of two companies: Compu-

Graphics Inc. (CGI) and Data Switch Corp. (DSC).

Probability Return onCGI

Return onDSC

0.300.500.20

10%14%20%

40%16%20%


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The Mean

LetNrepresent the number of possible

outcomes,

pn represent the probability of the nthoutcome,

xn represent the value of the nth outcome.

The mean of the distribution (mx) is computedas:

mx n

n

N

p xn=

=

1


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The Variance and the Standard

Deviation

The variance of the distribution of returns for the stock

is computed as:

2

1

2 )( xxpN

n

nn


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Variance and Standard Deviation

The variance of the distribution of a random variablex

is computed as:

The standard deviation is the square-root of thevariance.

2

1

2 )( xxpN

n

nnx

2

xx


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Variance and Standard Deviation

The variance of CGIs returns is:

2

1

2

)( xxp

N

nnnCGI

00.12

)1420(20.0)1414(50.0)1410(30.0 222


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The Variance and the Standard

Deviation

The Standard Deviation of CGIs return is:

Similarly, the variance of DSCs returns is 112.00,and its standard deviation is 10.58%

%46.300.12


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The Covariance

The Covariance of two random variablesx andy is

computed as:

))((),(1

yyxxpYXCov n

N

n

nn


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The Covariance

The covariance of the returns on CGI and DSC is

thus:

))((),(1

, yyxxpDSCCGICov n

N

n

nnyx

00.24

)2420)(1420(20.0

)2416)(1414(50.0

)2440)(1410(30.0


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The Correlation Coefficient

The Correlation Coefficient between the returns on

two random variables (x andy) is computed as:

r

x,yx.y

x y


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The Correlation Coefficient

The correlation coefficient between CGI and DSC is

thus:

YX

YX

YXCov

r

),(,

58.1046.3

00.24,

YXr

655.0, YXr


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Summary of Results for CGI and DSC

CGI DSC

MeanStandard Deviation

14.00%3.46%

24.00%10.58%

Correlation Coefficient -0.655


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Summary of Results for CGI and DSC

The mean return is a measure of the expected

returnfrom the security.

The expected return on DSC is 1.7 times higherthan the expected return on CGI.

The standard deviation is a measure of the specific

riskof the security.

The specific risk of DSC is 3 times higher thanthe specific risk of CGI.

The returns on DSC and CGI are negatively

correlated.


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Portfolio Expected Return and Risk

100% in CGI

100% in DSC

10.00%

12.00%

14.00%

16.00%

18.00%

20.00%

22.00%

24.00%

26.00%

0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00%

Risk

Return


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Portfolios of Securities

A portfolio is a combination of two or more

securities.

Combining securities into a portfolio reduces risk.An efficient portfoliois one that has the highest

expected return for a given level of risk.

We will look at two-asset portfolios in fair detail.

Our results will hold for n-asset portfolios.


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Notation

Let the return to asset i beRiwith expected

return ri (i = 1,2).

Let i represent the standard deviation ofthe returns on asset i (i = 1,2).

Let rij represent the correlation coefficient

between two assets i andj.Let wi represent the proportion invested in

asset i (i = 1,2).


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Portfolio Weights

Suppose you have $600 to invest.

You buy $400 worth of CGI stock and $200

worth of DSC stock.Let CGI be stock no. 1 and DSC be stock

no. 2.

w and w1 20 667 0 333= = = =$400

$600.

$200

$600.


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Expected Return of the Portfolio

The portfolios expected return is:

2111 )1( rwrwrp


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Expected Return of the Portfolio

The expected return of the portfolio of CGI

and DSC is:

%2431%14

32 pr

%33.17pr


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Portfolio Risk

The risk of the portfolio (as measured by its

standard deviation) is:

212111

2

2

2

1

2

1

2

1 ),()1(2)1( RRCorrwwwwp

As you can see, p is not a simple weighted

average of1 and 2.


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Portfolio Risk

The risk of the portfolio of $400 worth of CGI stock

and $200 worth of DSC stock is:

)58.10)(46.3)(655.0(3132258.103146.3322222

p

%67.2p


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Diversification of Risk

Note that while the expected return of the portfolio

is between those of CGI and DSC, its risk is less

than either of the two individual securities.

Combining CGI and DSC results in a substantial

reduction of risk - diversification!

This benefit of diversification stems primarily

from the fact that CGI and DSCs returns arenegatively correlated.


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Portfolio Expected Return

The expected return of the portfolio depends on:

The expected return of the securities in the

portfolio. The portfolio weights.

The risk of the portfolio depends on:

The risk of the securities in the portfolio.

The portfolio weights.

The correlation coefficient of the returns on the

securities.

Eff f P f li W i h i


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Effect of Portfolio Weights on its

Expected Return and Risk

Portfolio Weights Portfolios

CGI DSC Expected

Return

Standard

Deviation1.00

0.75

0.67

0.500.25

0.00

0.00

0.25

0.33

0.500.75

1.00

14.00%

16.50%

17.33%

19.00%21.50%

24.00%

3.46%

2.18%

2.64%

4.36%7.40%

10.58%


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C l ti C ffi i t d


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Correlation Coefficient and

Portfolio Risk

All else being the same, the lower the

correlation coefficient, the lower is the risk

of the portfolio. Recall that the expected return of the portfolio

is not affected by the correlation coefficient.

Thus, lower the correlation coefficient,greater is the diversification of risk.


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Perfect Positive Correlation

When the returns on two stocks are

perfectly positively correlated, there is no

diversification of the risk.The risk of the portfolio is then simply the

weighted average of the risk of the

individual assets.2111

2

2

2

1

2

1

2

1 )1(2)1( wwwwp

2111)1( wwp


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Perfect Positive Correlation

1 2

2r

1r

2111 )1( wwp


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Perfect Negative Correlation

When the returns on two stocks are

perfectly negatively correlated, it is possible

to diversify away ALL of the risk byappropriate weighting of the two stocks.

2111

2

2

2

1

2

1

2

1 )1(2)1( wwwwp

There exists a w1 such that:

0)1(2)1( 21112

2

2

1

2

1

2

1

2 wwwwp


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Perfect Negative Correlation

1 2

2r

1r

0)1(2)1(2111

2

2

2

1

2

1

2

1 wwww

21

1*

1

w

C l ti C ffi i t d


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Correlation Coefficient and

Portfolio Risk

Consider stocks of two companies, X and Y. The

table below gives their expected returns and standard

deviations.

Stock X Stock YExpected Return

Standard Deviation

10%

12%

25%

30%

Plot the risk and expected return of portfolios of thesetwo stocks for the following (assumed) correlation

coefficients:

-1.0 0.5 0.0 +0.5 +1.0


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Various Correlations

x y

yr

x

r

r 1 r 0r 1


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Many Asset Portfolios

The above framework can be expanded to

the case of portfolios with a large number of

stocks.In forming each portfolio, we can vary

the number of stocks that make up the portfolio,

the identity of the stocks in the portfolio, and

the weights assigned to each stock.

Look at the plot of the expected returns

versus the risk of these portfolios.

All Combinations of Risky


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All Combinations of Risky

Assets

F

E1N

E2

E

expect

edreturn


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Efficient Portfolios

A portfolio is an efficient portfolioif

no other portfolio with the same expected

return has lower risk, orno other portfolio with the same risk has a

higher expected return.

Investors prefer efficient portfolios over

inefficient ones.

The collection of efficient portfolio is called

an efficient frontier.


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The Efficient Frontier

F

E1N

E2

E

expect

edreturn


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Choosing the Best Risky Asset

Investors prefer efficient portfolios over

inefficient ones.

Which one of the efficient portfolios is best?

We can answer this by introducing a riskless asset.

There is no uncertainty about the future value of this

asset (i.e. the standard deviation of returns is zero). Let

the return on this asset be rf. For practical purposes, 90-day U.S. Treasury Bills are

(almost) risk-free.


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The Best Risky Asset

expected

return

rf

M


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The Capital Market Line

Assume investors can lend andborrow at the risk-free rate of interest.

Borrowing entails a negative investment in the

riskless asset.Because every investor holds a part of the bestrisky assetM, portfolioMis the market portfolio.

The market portfolio consists of all risky assets.

Each assets weight is proportional to its marketvalue.

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risk and return : stocks