3.risk n return

5/26/2018 3.risk n return

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Chap ter 5

Risk andReturn

Pearson Education Limited 2004Fundamentals of Financial Management, 12/e

Created by: Gregory A. Kuhlemeyer, Ph.D.

Carroll College, Waukesha, WI


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After s tudy ing Chapter 5,

you should be able to:

1. Understand the relationship (or trade-off) between risk and return.

2. Define risk and return and show how to measure them by calculatingexpected return, standard deviation, and coefficient of variation.

3. Discuss the different types of investor attitudes toward risk.

4. Explain risk and return in a portfolio context, and distinguish betweenindividual security and portfolio risk.

5. Distinguish between avoidable (unsystematic) risk and unavoidable(systematic) risk and explain how proper diversification can eliminateone of these risks.

6. Define and explain the capital-asset pricing model (CAPM), beta, and

the characteristic line.7. Calculate a required rate of return using the capital-asset pricing model

(CAPM).

8. Demonstrate how the Security Market Line (SML) can be used todescribe this relationship between expected rate of return andsystematic risk.

9. Explain what is meant by an efficient financial market and describethe three levels (or forms) to market efficiency.


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Risk and Return

Defining Risk and Return

Using Probability Distributions to

Measure RiskAttitudes Toward Risk

Risk and Return in a Portfolio Context

DiversificationThe Capital Asset Pricing Model (CAPM)

Efficient Financial Markets


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Defin ing Return

Income received on an investmentplus any change in market price,

usually expressed as a percent ofthe beginning market price of the

investment.

Dt+ (Pt- Pt-1)

Pt-1R =


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Return Example

The stock price for Stock A was $10pershare 1 year ago. The stock is currently

trading at $9.50per share and shareholdersjust received a $1 dividend. What return

was earned over the past year?


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Return Example

The stock price for Stock A was $10pershare 1 year ago. The stock is currently

trading at $9.50per share and shareholdersjust received a $1 dividend. What return

was earned over the past year?

$1.00 + ($9.50- $10.00)

$10.00R== 5%


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Def in ing Risk

What rate of return do you expect on yourinvestment (savings) this year?

What rate will you actually earn?Does it matter if it is a bank CD or a share

of stock?

The var iabi l i ty of returns from

those that are expected .


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Determ ining Expected

Retu rn (Discrete Dis t.)

R= ( Ri)( Pi)

Ris the expected return for the asset,

Riis the return for the ithpossibility,

Piis the probability of that returnoccurring,

nis the total number of possibilities.

n

i=1


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How to Determ ine the Expected

Return and Standard Deviat ion

Stock BW

Ri Pi (Ri)(Pi)

-.15 .10 -.015

-.03 .20 -.006

.09 .40 .036

.21 .20 .042

.33 .10 .033

Sum 1.00 .090

Theexpectedreturn, R,for Stock

BW is .09or 9%


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Determ ining Standard

Dev iat ion (Risk Measure)

s= ( Ri- R)2( Pi)Standard Deviation, s, is a statistical

measure of the variability of a distributionaround its mean.

It is the square root of variance.

Note, this is for a discrete distribution.

n

i=1


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How to Determ ine the Expected

Return and Standard Deviat ion

Stock BW

Ri Pi (Ri)(Pi) (Ri - R )2(Pi)

-.15 .10 -.015 .00576-.03 .20 -.006 .00288

.09 .40 .036 .00000

.21 .20 .042 .00288

.33 .10 .033 .00576

Sum 1.00 .090 .01728


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s= ( Ri- R)2( Pi)s= .01728

s= .1315or 13.15%

n

i=1


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Coeff ic ien t o f Variat ion

The ratio of the standard deviat ion ofa distribution to the mean of that

distribution.

It is a measure of RELATIVErisk.

CV = s/ RCV of BW = .1315/ .09= 1.46


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5-14

Disc rete vs . Con t inuous

Dist r ibut ions

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-15% -3% 9% 21% 33%

Discrete Continuous

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

-50%

-41%

-32%

-23%

-14%

-5%

4%

13%

22%

31%

40%

49%

58%

67%


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Determ ining Expected

Retu rn (Con tinuous Dist.)

R= ( Ri) / ( n)

Ris the expected return for the asset,

Riis the return for the ith observation,

nis the total number of observations.

n

i=1


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n

i=1s= ( Ri- R)2

( n)

Note, this is for a continuous

distributionwhere the distribution isfor a populat ion. Rrepresents thepopulation mean in this example.


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Cont inuous

Distr ibu t ion Prob lem

Assume that the following list represents thecontinuous distribution of population returnsfor a particular investment (even thoughthere are only 10 returns).

9.6%, -15.4%, 26.7%, -0.2%, 20.9%,28.3%, -5.9%, 3.3%, 12.2%, 10.5%

Calculate the Expected Return andStandard Deviation for the populat ionassuming a continuous distribution.


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Lets Use the Calculator!

Enter Data first. Press:

2nd Data

2nd CLR Work

9.6 ENTER

-15.4 ENTER

26.7 ENTER Note, we are inputting data

only for the X variable andignoring entries for the Yvariable in this case.


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Enter Data first. Press:

-0.2 ENTER

20.9 ENTER

28.3 ENTER

-5.9 ENTER

3.3 ENTER

12.2 ENTER

10.5 ENTER


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Examine Results! Press:

2nd Stat

through the results.

Expected return is 9% forthe 10 observations.Population standard

deviation is 13.32%. This canbe much quicker

than calculating by hand,but slower than using aspreadsheet.


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Certainty Equivalent (CE) is theamount of cash someone would

require with certainty at a point intime to make the individual

indifferent between that certain

amount and an amount expected tobe received with risk at the same

point in time.

Risk A tt itudes


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Certainty equivalent > Expected value

Risk Preference

Certainty equivalent = Expected value

Risk Indifference

Certainty equivalent < Expected valueRisk Aversion

Mostindividuals are Risk Averse.

Risk A tt itudes


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Risk Attitude Example

You have the choice between (1) a guaranteeddollar reward or (2) a coin-flip gamble of

$100,000 (50% chance) or $0 (50% chance).The expected value of the gamble is $50,000.

Mary requires a guaranteed $25,000, or more, tocall off the gamble.

Raleigh is just as happy to take $50,000 or takethe risky gamble.

Shannon requires at least $52,000 to call off thegamble.


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What are the Risk Attitude tendencies of each?

Risk A tt itude Examp le

Maryshows risk aversionbecause her

certainty equivalent < the expected value ofthe gamble.

Raleighexhibits risk indifferencebecause hercertainty equivalent equals the expected value

of the gamble.

Shannonreveals a risk preferencebecause hercertainty equivalent > the expected value ofthe gamble.


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RP= ( Wj)( Rj)

RPis the expected return for the portfolio,

Wjis the weight (investment proportion)for thejthasset in the portfolio,

Rjis the expected return of the jthasset,

mis the total number of assets in the

portfolio.

Determ ining Port fo l io

Expec ted Return

m

j=1


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Standard Dev iat ion

m

j=1

m

k=1sP= WjWk jk

Wjis the weight (investment proportion)for thejthasset in the portfolio,

Wkis the weight (investment proportion)

for the kthasset in the portfolio,

jkis the covariance between returns forthejthand kthassets in the portfolio.


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Tip Sl ide: Appendix A

Slides 5-28 through 5-30

and 5-33 through 5-36assume that the student

has read Appendix A inChapter 5


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What is Covar iance?

sjk= j k rjkj is the standard deviation of thej

th

asset in the portfolio,

kis the standard deviation of the kth

asset in the portfolio,

rjkis the correlation coefficient between thejthand kthassets in the portfolio.


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Correlat ion Coeff ic ien t

A standard ized stat ist ical measu re

o f the l inear relat ionsh ip between

two variables.

Its range is from -1.0 (perfect

negative correlation), through 0(no correlation), to +1.0 (perfect

positive correlation).


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Variance - Covariance Matrix

A three asset portfolio:

Col 1 Col 2 Col 3

Row 1 W1W1s1,1 W1W2s1,2 W1W3s1,3Row 2 W2W1s2,1 W2W2s2,2 W2W3s2,3Row 3 W3W1s3,1 W3W2s3,2 W3W3s3,3sj,k= is the covariance between returns for

thejthand kthassets in the portfolio.


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You are creating a portfolio of Stock D and StockBW (from earlier). You are investing $2,000in

Stock BW and $3,000in Stock D. Remember that

the expected return and standard deviation ofStock BWis 9%and 13.15%respectively. The

expected return and standard deviation ofStock Dis 8%and 10.65%respectively. The correlation

coefficient between BW and D is 0.75.

What is the expected return and standarddeviation of the portfolio?

Port fo l io Risk and

Expec ted Return Examp le


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Expec ted Return

WBW= $2,000 / $5,000 = .4

WD= $3,000 / $5,000 =.6

RP= (WBW)(RBW) + (WD)(RD)

RP= (.4)(9%) + (.6)(8%)

RP= (3.6%) + (4.8%) = 8.4%


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Two-asset portfolio:

Col 1 Col 2

Row 1 WBW WBWsBW,BW WBW WDsBW,DRow 2 WD WBWsD,BW WD WDsD,DThis represents the variance - covariance

matrix for the two-asset portfolio.




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Col 1 Col 2

Row 1 (.4)(.4)(.0173) (.4)(.6)(.0105)

Row 2 (.6)(.4)(.0105) (.6)(.6)(.0113)

This represents substitution into thevariance - covariance matrix.




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Col 1 Col 2

Row 1 (.0028) (.0025)

Row 2 (.0025) (.0041)

This represents the actual element valuesin the variance - covariance matrix.




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sP= .0028+ (2)(.0025) + .0041sP= SQRT(.0119)

sP= .1091or 10.91%A weighted average of the individualstandard deviations is INCORRECT.


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The WRONG way to calculate is aweighted average like:

sP= .4(13.15%)+.6(10.65%)sP= 5.26+ 6.39= 11.65%

10.91% = 11.65%

This is INCORRECT.


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Stock C Stock D Portfolio

Return 9.00% 8.00% 8.64%

Stand.Dev. 13.15% 10.65% 10.91%

CV 1.46 1.33 1.26

The portfolio has the LOWESTcoefficientof variation due to diversification.

Summary o f the Port fo l io

Return and Risk Calcu lat ion


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Combining securities that are not perfectly,

positively correlated reduces risk.

Divers i f icat ion and the

Cor relat ion Coeff ic ien t

IN

VESTMENTRE

TURN

TIME TIMETIME

SECURITY E SECURITY FCombination

E and F


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Systemat ic Risk is the variability of returnon stocks or portfolios associated with

changes in return on the market as a whole.

Unsys temat ic Risk is the variability of returnon stocks or portfolios not explained bygeneral market movements. It is avoidable

through diversification.

Total Risk = Sys tematic

Risk + Unsystemat ic Risk

Total Risk = SystematicRisk+Unsystemat icRisk


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5-41



Total

Risk

Unsystematic risk

Systematic risk

STD

DEVOFPORTFOLIO

RETURN

NUMBER OF SECURITIES IN THE PORTFOLIO

Factors such as changes in nations

economy, tax reform by the Congress,or a change in the world situation.


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Total

Risk

Unsystematic risk

Systematic risk

STD

DEVOFPORTFOLIO

RETURN

NUMBER OF SECURITIES IN THE PORTFOLIO

Factors unique to a particular companyor industry. For example, the death of akey executive or loss of a governmental

defense contract.


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CAPM is a model that describes therelat ionshipbetween risk and

expected (required) return; in thismodel, a securitys expected

(required) return is the risk-free rate

plus a premium based on thesystemat ic r isk of the security.

Cap ital Asset

Pric ing Model (CAPM)


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1. Capital markets are efficient.

2. Homogeneous investor expectations

over a given period.3. Risk-freeasset return is certain

(use short- to intermediate-term

Treasuries as a proxy).4. Market portfolio contains only

systemat ic r isk (use S&P 500 Indexor similar as a proxy).

CAPM Assumpt ions


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Charac ter ist ic L ine

EXCESS RETURNON STOCK

EXCESS RETURNON MARKET PORTFOLIO

Beta=Rise

Run

Narrower spreadis higher correlation

Characteristic Line


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Calculating Beta

on You r Calculator

Time Pd. Market My Stock

1 9.6% 12%

2 -15.4% -5%

3 26.7% 19%

4 -.2% 3%

5 20.9% 13%

6 28.3% 14%

7 -5.9% -9%

8 3.3% -1%

9 12.2% 12%

10 10.5% 10%

The Marketand My

Stockreturns areexcess

returns andhave the

riskless ratealready

subtracted.


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Calculating Beta

on You r Calculator

Assume that the previous continuousdistribution problem represents the excessreturns of the market portfolio (it may still be

in your calculator data worksheet -- 2nd Data ).

Enter the excess market returns as X

observations of: 9.6%, -15.4%, 26.7%, -0.2%,20.9%, 28.3%, -5.9%, 3.3%, 12.2%, and 10.5%.

Enter the excess stock returns as Y observations

of: 12%, -5%, 19%, 3%, 13%, 14%, -9%, -1%,12%, and 10%.


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Calculating Beta

on You r Calculator

Let us examine again the statisticalresults (Press 2ndand then Stat )

The market expected return and standarddeviation is 9% and 13.32%. Your stockexpected return and standard deviation is6.8% and 8.76%.

The regression equation isY=a+bX. Thus, ourcharacteristic line isY= 1.4448+ 0.595Xandindicates that our stock has a betaof 0.595.


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An index of systemat ic r isk.

It measures the sensi t iv i tyof astocks returns to changes inreturns on the market portfolio.

The betafor a portfolio is simply aweighted average of the individual

stock betas in the portfolio.

What is Beta?


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Charac ter ist ic L ines

and Dif feren t Betas

EXCESS RETURNON STOCK

EXCESS RETURNON MARKET PORTFOLIO

Beta < 1(defensive)

Beta = 1

Beta > 1(aggressive)

Each characteristicline has adifferent slope.


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Rjis the required rate of return for stock j,Rfis the risk-free rate of return,

bjis the beta of stock j (measuressystematic risk of stock j),

RMis the expected return for the market

portfolio.

Securi ty Market Line

Rj= Rf+ bj(RM- Rf)


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Rj= Rf+ bj(RM- Rf)

bM= 1.0Systematic Risk (Beta)

Rf

RM

Requ

iredReturn

RiskPremium

Risk-freeReturn


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Obtaining Betas

Can use historical dataif past best represents theexpectations of the future

Can also utilize services like Value Line, IbbotsonAssociates, etc.

Adjusted Beta

Betas have a tendency to revert to the mean of 1.0

Can utilize combination of recent betaand mean

2.22(.7) + 1.00(.3) = 1.554 + 0.300 = 1.854 estimate

D t i t i f th


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Lisa Miller at Basket Wondersisattempting to determine the rate of return

required by their stock investors. Lisa isusing a 6% Rfand a long-term market

expected rate of return of 10%. A stock

analyst following the firm has calculatedthat the firm betais 1.2. What is therequ ired rate o f returnon the stock of

Basket Wonders?

Determ inat ion o f the

Requ ired Rate o f Retu rn

BW R i d


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RBW= Rf+ bj(RM- Rf)RBW= 6%+ 1.2(10%- 6%)

RBW= 10.8%

The required rate of return exceedsthe market rate of return as BWs

beta exceeds the market beta (1.0).

BWs Required

Rate of Retu rn

D t i t i f th


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Lisa Miller at BW is also attempting todetermine the intrinsic value of the stock.She is using the constant growth model.

Lisa estimates that the dividend next periodwill be $0.50and that BW will growat a

constant rate of 5.8%. The stock is currently

selling for $15.

What is the intrinsic value of the stock?Is the stock overor underpriced?


In tr ins ic Value of BW

D t i t i f th


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The stock is OVERVALUEDasthe market price ($15) exceeds

the intrinsic value ($10).


In tr ins ic Value of BW

$0.5010.8%- 5.8%

IntrinsicValue

=

= $10


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Systematic Risk (Beta)

RfR

equiredRet

urn

Direction ofMovement

Direction ofMovement

Stock Y (Overpriced)

Stock X (Underpriced)

D t i t i f th


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Small-firm Effect

Price / Earnings Effect

January Effect

These anomalies have presentedserious challenges to the CAPMtheory.


Requ ired Rate o f Retu rn

3.risk n return

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