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Essentials ofInvestments

2001 The McGraw-Hill Com anies, Inc. All ri hts reserved.

FourthEdition

Irwin / McGraw-Hill

Bodie Kane Marcus1

Chapter6

Risk and Return: Pastand Prologue

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Rates of Return: Single Period

HPR P P DP

1 0 1

0

HPR = Holding Period Return

P1= Ending price

P0= Beginning price

D1= Dividend during period one

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Rates of Return: Single Period

Example

Ending Price = 24

Beginning Price = 20Dividend = 1

HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

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Data from Text Example p. 154

1 2 3 4

Assets(Beg.) 1.0 1.2 2.0 .8

HPR .10 .25 (.20) .25TA (Before

Net Flows 1.1 1.5 1.6 1.0

Net Flows 0.1 0.5 (0.8) 0.0

End Assets 1.2 2.0 .8 1.0

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FourthEdition

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Returns Using Arithmetic and

Geometric AveragingArithmetic

ra = (r1+ r2+ r3+ ... rn) / n

ra = (.10 + .25 - .20 + .25) / 4= .10 or 10%

Geometric

rg= {[(1+r1) (1+r2) .... (1+rn)]}1/n - 1

rg= {[(1.1) (1.25) (.8) (1.25)]}1/4 - 1

= (1.5150)1/4

-1 = .0829 = 8.29%

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Dollar Weighted Returns

Internal Rate of Return (IRR) - thediscount rate that results present value

of the future cash flows being equal tothe investment amount

Considers changes in investment

Initial Investment is an outflow

Ending value is considered as an inflow

Additional investment is a negative flow

Reduced investment is a positive flow

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FourthEdition

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Dollar Weighted Average Using TextExample

Net CFs 1 2 3 4

$ (mil) - .1 - .5 .8 1.0

Solving for IRR

1.0 = -.1/(1+r)1

+ -.5/(1+r)2

+ .8/(1+r)3

+1.0/(1+r)4

r = .0417 or 4.17%

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Quoting Conventions

APR = annual percentage rate

(periods in year) X (rate for period)

EAR = effective annual rate( 1+ rate for period)Periods per yr- 1

Example: monthly return of 1%

APR = 1% X 12 = 12%

EAR = (1.01)12- 1 = 12.68%

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Characteristics of ProbabilityDistributions

1) Mean: most likely value

2) Variance or standard deviation

3) Skewness

* If a distribution is approximately normal,the distribution is described bycharacteristics 1 and 2

F h

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r

Symmetric distribution

Normal Distribution

s.d. s.d.

F h

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rNegative Positive

Skewed Distribution: Large Negative

Returns Possible

Median

F th

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rNegative Positive

Skewed Distribution: Large Positive

Returns Possible

Median

F th13

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Subjective returns

p(s) = probability of a state

r(s) = return if a state occurs

1 to s states

Measuring Mean: Scenario orSubjective Returns

E(r) = p(s) r(s)Ss

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Numerical Example: Subjective or

Scenario DistributionsState Prob. of State rinState

1 .1 -.05

2 .2 .053 .4 .15

4 .2 .25

5 .1 .35

E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)

E(r) = .15

i l fFourthdi15

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Standard deviation = [variance]1/2

Measuring Variance or Dispersion of

ReturnsSubjective or Scenario

Variance = Ss

p(s) [rs - E(r)]2

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]Var= .01199

S.D.= [ .01199] 1/2 = .1095

Using Our Example:

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FourthEdition

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Real vs. Nominal Rates

Fisher effect: Approximation

nominal rate = real rate + inflation premium

R = r + i or r = R - i

Example r = 3%, i = 6%

R = 9% = 3% + 6% or 3% = 9% - 6%

Fisher effect: Exact

r = (R - i) / (1 + i)2.83% = (9%-6%) / (1.06)

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FourthEdition

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Annual Holding Period ReturnsFrom Figure 6.1 of Text

Geom Arith Stan.

Series Mean% Mean% Dev.%

Lg Stk 11.01 13.00 20.33Sm Stk 12.46 18.77 39.95

LT Gov 5.26 5.54 7.99

T-Bills 3.75 3.80 3.31

Inflation 3.08 3.18 4.49

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Annual Holding Period Risk

Premiums and Real ReturnsRisk RealSeries Premiums% Returns%

Lg Stk 9.2 9.82Sm Stk 14.97 15.59

LT Gov 1.74 2.36

T-Bills --- 0.62

Inflation --- ---

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Possible to split investment fundsbetween safe and risky assets

Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio)

Allocating Capital Between Risky &Risk-Free Assets

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Allocating Capital Between Risky &

Risk-Free Assets (cont.) Issues Examine risk/ return tradeoff

Demonstrate how different degrees of riskaversion will affect allocations betweenrisky and risk free assets

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ou thEdition

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rf= 7% srf = 0%E(rp) = 15% sp= 22%

y = % in p (1-y) = % in rf

Example Using the Numbers in

Chapter 6 (pp 171-173)

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Edition

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E(rc) = yE(rp) + (1 - y)rf

rc= complete or combined portfolio

For example, y = .75

E(rc) = .75(.15) + .25(.07)= .13 or 13%

Expected Returns for Combinations


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Edition

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E(r)

E(rp) = 15%

rf= 7%

22%0

P

F

Possible Combinations

s


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Edition

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Bodie Kane Marcus24

pc =

Sincerf

y

Variance on the Possible CombinedPortfolios

= 0, then

s


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Edition

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Bodie Kane Marcus25

c = .75(.22) = .165 or 16.5%

If y = .75, then

c = 1(.22) = .22 or 22%

If y = 1

c = 0(.22) = .00 or 0%

If y = 0

Combinations Without Leverage

sss

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Edition

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Using Leverage with CapitalAllocation Line

Borrow at the Risk-Free Rate and investin stock

Using 50% Leveragerc= (-.5) (.07) + (1.5) (.15) = .19

sc= (1.5) (.22) = .33

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Edition

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E(r)

E(rp) = 15%

rf= 7%

= 22%0

P

F

P

) S = 8/22

E(rp) - rf= 8%

CAL

(Capital

AllocationLine)

s

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Edition

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Risk Aversion and Allocation

Greater levels of risk aversion lead tolarger proportions of the risk free rate

Lower levels of risk aversion lead tolarger proportions of the portfolio ofrisky assets

Willingness to accept high levels of riskfor high levels of returns would result inleveraged combinations

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Edition

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Quantifying Risk Aversion

pfp ArrE s 005.E(rp) = Expected return on portfolio p

rf = the risk free rate

.005 = Scale factor

A x sp= Proportional risk premium

The larger A is, the larger will be the

addedreturn required for risk

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Edition

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Quantifying Risk Aversion

rrEA

p

fp

2

.005

)(

Rearranging the equation and solving for A

Many studies have concluded that

investors average risk aversion isbetween 2 and 4


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Chapter7

EfficientDiversification


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rp = W1r1 +W2r2W1= Proportion of funds in Security 1

W2= Proportion of funds in Security 2r1 = Expected return on Security 1

r2 = Expected return on Security 2

Two-Security Portfolio: Return

WiSi=1

n

= 1


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Edition

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sp2= w12s12+ w22s22+ 2W1W2 Cov(r1r2)s12 = Variance of Security 1s22 = Variance of Security 2

Cov(r1r2) = Covariance of returns for

Security 1 and Security 2

Two-Security Portfolio: Risk


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Edition

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Covariance

r1,2

= Correlation coefficient of

returns

Cov(r1r2) = r1 2s1s2

s1= Standard deviation ofreturns for Security 1s2= Standard deviation ofreturns for Security 2

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f


Edition

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Correlation Coefficients: Possible

Values

If r = 1.0, the securities would beperfectly positively correlated

If r = - 1.0, the securities would beperfectly negatively correlated

Range of values for r1,2

-1.0 < r < 1.0


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Edition

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s2p = W12s12+ W22s22+ 2W1W2

rp = W1r1 +W2r2 + W3r3

Cov(r1r2)

+ W32s32

Cov(r1r3)+ 2W1W3

Cov(r2r3)+ 2W2W3

Three-Security Portfolio


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f


dition

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rp= Weighted average of the

n securities

sp2 = (Consider all pair-wisecovariance measures)

In General, For an n-SecurityPortfolio:


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f

2001 The McGraw-Hill Com anies, Inc. All ri hts reserved.Irwin / McGraw-Hill

E(rp) = W1r1 +W2r2

Two-Security Portfolio

sp2= w12s12+ w22s22+ 2W1W2 Cov(r1r2)sp= [w12s12+ w22s22+ 2W1W2Cov(r1r2)]1/2


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f


r= 0

E(r)

r= 1r= -1

r= -1r= .3

13%

8%

12% 20% St. Dev

TWO-SECURITY PORTFOLIOS WITH

DIFFERENT CORRELATIONS


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f


Portfolio Risk/Return Two Securities:Correlation Effects

Relationship depends on correlationcoefficient

-1.0 < r< +1.0 The smaller the correlation, the greater

the risk reduction potential

Ifr = +1.0, no risk reduction is possible


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1

1 2

- Cov(r1r2)

W1

=

+ - 2Cov(r1r2)

2

W2 = (1 - W1)

Minimum Variance Combination

s 2s 2

2E(r2) = .14 = .20Sec 212 = .2

E(r1) = .10 = .15Sec 1 ss r

s 2


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W1 =(.2)2- (.2)(.15)(.2)

(.15)2

+ (.2)2

- 2(.2)(.15)(.2)

W1 = .6733

W2 = (1 - .6733) = .3267

Minimum VarianceCombination: r= .2


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rp= .6733(.10) + .3267(.14) = .1131

p = [(.6733)2(.15)2 + (.3267)2(.2)2 +

2(.6733)(.3267)(.2)(.15)(.2)]1/2

p= [.0171]1/2

= .1308

Minimum Variance: Return and Riskwith r= .2

s

s


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W1 =(.2)2- (.2)(.15)(.2)

(.15)2

+ (.2)2

- 2(.2)(.15)(-.3)

W1 = .6087

W2 = (1 - .6087) = .3913

Minimum VarianceCombination: r= -.3


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rp= .6087(.10) + .3913(.14) = .1157

p = [(.6087)2(.15)2 + (.3913)2(.2)2 +

2(.6087)(.3913)(.2)(.15)(-.3)]1/2

p= [.0102]

1/2= .1009

Minimum Variance: Return and Risk

with r= -.3

s

s


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Extending Concepts to All Securities

The optimal combinations result inlowest level of risk for a given return

The optimal trade-off is described as theefficient frontier

These portfolios are dominant


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E(r)The minimum-variance frontier of

risky assets

Efficientfrontier

Globalminimum

variance

portfolio Minimum

variancefrontier

Individual

assets

St. Dev.


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Extending to Include Riskless Asset

The optimal combination becomeslinear

A single combination of risky andriskless assets will dominate


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E(r)

CAL (Global

minimum variance)

CAL (A)CAL (P)

M

P

A

F

P P&F A&FM

A

G

P

M

s

ALTERNATIVE CALS


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Dominant CAL with a Risk-FreeInvestment (F)

CAL(P) dominates other lines -- it has thebest risk/return or the largest slope

Slope = (E(R) - Rf) / s[ E(RP) - Rf) / s P] > [E(RA) - Rf) / sA]

Regardless of risk preferences

combinations of P & F dominate


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Single Factor Model

ri= E(Ri) + iF + e

i= index of a securities particular returnto the factor

F= some macro factor; in this case F isunanticipated movement; F is commonlyrelated to security returns

Assumption: a broad market index like theS&P500 is the common factor


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Single Index Model

Risk PremMarket Risk Prem

or Index Risk Prem

i = the stocks expected return if the

markets excess return is zero

i(rm- rf) = the component of return due tomovements in the market index

(rm- rf) = 0

ei = firm specific component, not due to market

movements

a

errrr ifmiifi a


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Let: Ri = (ri- rf)

Rm = (rm- rf)

Risk premium

format

Ri = ai+ i(Rm) + ei

Risk Premium Format


E ti ti th I d M d l

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Estimating the Index ModelExcess Returns (i)

Security

Characteristic

Line

. ..

..

.

. .

. ..

. .

.

. .

. ..

..

.

. .

. ..

.

..

..

..

.

. ..

. .

.

. ... .. .

. .

Excess returns

on market index

Ri= ai+ iRm+ ei


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Components of Risk

Market or systematic risk: risk related to themacro economic factor or market index

Unsystematic or firm specific risk: risk not

related to the macro factor or market index Total risk = Systematic + Unsystematic


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Measuring Components of Risk

si2 = i

2sm2 + s2(ei)

where;

si2 = total variance

i2sm

2 = systematic variance

s2(ei) = unsystematic variance


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Total Risk = Systematic Risk +Unsystematic Risk

Systematic Risk/Total Risk = r2

i2 sm

2 / s2= r2

i2sm

2/ i2sm

2 + s2(ei) = r2

Examining Percentage of Variance


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Advantages of the Single Index Model

Reduces the number of inputs fordiversification

Easier for security analysts to specialize


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Chapter8

Capital Asset Pricing

and Arbitrage

Pricing Theory


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Capital Asset Pricing Model (CAPM)

Equilibrium model that underlies all modernfinancial theory

Derived using principles of diversificationwith simplified assumptions

Markowitz, Sharpe, Lintner and Mossin areresearchers credited with its development


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Assumptions

Individual investors are price takers

Single-period investment horizon

Investments are limited to tradedfinancial assets

No taxes, and transaction costs


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Assumptions (cont.)

Information is costless and available toall investors

Investors are rational mean-varianceoptimizers

Homogeneous expectations


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Resulting Equilibrium Conditions

All investors will hold the same portfoliofor risky assetsmarket portfolio

Market portfolio contains all securitiesand the proportion of each security is itsmarket value as a percentage of totalmarket value


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Risk premium on the market dependson the average risk aversion of all

market participants Risk premium on an individual security

is a function of its covariance with the

market

Resulting Equilibrium Conditions

(cont.)


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E(r)

E(rM)

rf

M CML

sm

Capital Market Line

s


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M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium

E(rM) - rf = Market price of risk

= Slope of the CAPM

Slope and Market Risk Premium

Ms


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Expected Return and Risk onIndividual Securities

The risk premium on individualsecurities is a function of the individual

securitys contribution to the risk of themarket portfolio

Individual securitys risk premium is a

function of the covariance of returnswith the assets that make up the marketportfolio


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E(r)

E(rM)

rf

SML

M

= 1.0

Security Market Line


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SML Relationships

= [COV(ri,rm)] / sm2

Slope SML = E(rm) - rf

= market risk premium

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


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Sample Calculations for SML

E(rm) - rf= .08 rf = .03

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

y= .6e(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


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E(r)

15%

SML

1.0

Rm=11%

rf=3%

1.25

Disequilibrium Example


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Disequilibrium Example

Suppose a security with a of 1.25 isoffering expected return of 15%

According to SML, it should be 13%

Underpriced: offering too high of a rateof return for its level of risk


Security Characteristic Line

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Security Characteristic LineExcess Returns (i)

SCL

..

...

.

. .

. ..

. .

.

. .

. ..

. .

.

. .

. ..

.

..

..

.

.

.

. ..

. .

.

. ... .. .

. .

Excess returns

on market index

Ri= ai+ iRm+ ei


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Using the Text Example p. 245, Table

8.5:

Jan.Feb...

DecMeanStd Dev

5.41-3.44..

2.43-.604.97

7.24.93..

3.901.753.32

Excess

Mkt. Ret.

Excess

GM Ret.


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Estimated coefficientStd error of estimate

Variance of residuals = 12.601

Std dev of residuals = 3.550R-SQR = 0.575

-2.590(1.547)

1.1357(0.309)

rGM- rf = + (rm- rf)

Regression Results:

aa


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Arbitrage Pricing Theory

Arbitrage - arises if an investor canconstruct a zero investment portfoliowith a sure profit

Since no investment is required, aninvestor can create large positions tosecure large levels of profit

In efficient markets, profitable arbitrageopportunities will quickly disappear


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Arbitrage Example from Text pp. 255-

257 Current Expected StandardStock Price$ Return% Dev.%

A 10 25.0 29.58

B 10 20.0 33.91

C 10 32.5 48.15

D 10 22.5 8.58


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Arbitrage Portfolio

Mean Stan. Correlation

Return Dev. Of Returns

Portfolio

A,B,C 25.83 6.40 0.94

D 22.25 8.58


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Arbitrage Action and Returns

E. Ret.

St.Dev.

* P* D

Short 3 shares of D and buy 1 of A, B & C to form

P

You earn a higher rate on the investment than

you pay on the short sale


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APT and CAPM Compared

APT applies to well diversified portfolios andnot necessarily to individual stocks

With APT it is possible for some individual

stocks to be mispriced - not lie on the SML APT is more general in that it gets to an

expected return and beta relationship withoutthe assumption of the market portfolio

APT can be extended to multifactor models

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