CHAPTER 4Risk and Return- The Basics

Stand-alone risk Portfolio risk Risk & return: CAPM / SML

4-2

Risk

The chance of variability of returns associated with an asset

The risk can be considered in two ways:Stand-alone risk (risk of a single asset)

Portfolio risk (risk of an asset is combined with other assets)

ERR should compensate the investors’ perceived risk for the investment

4-3

Investment returns

The rate of return on an investment:

(Amount received – Amount invested)

Return = ________________________

Amount invested

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:

($1,100 - $1,000) ÷ $1,000 = 10%.

4-4

Return: Calculating the expected return

i

n

1=iinn2211 rp=rp+...+rp+rp=r̂ ∑

Demand Probability Rate of Return

Strong 0.3 100%

Normal 0.4 15%

Weak 0.3 (70%)

Total 1.00= (0.3)(100%)+(0.4)(15%)+(0.3)(-70%)=15%

r̂

4-5

Risk: Calculating the SD for expected return

deviation Standard

2Variance

i2

n

1=ii P)r̂r(=σ ∑ -

%66=(0.3)}]15)–(–70{

+(0.4)}15)–(15{+0.3)}()51–100[{(=σ

21

2

22

4-6

Probability distributions

A listing of all possible outcomes, and the probability of each occurrence

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y

The tighter the probability distribution, the smaller the risk of a given investment

4-7

Comparing standard deviations

SR Investment

Prob.T - bill

LR Investment

0 8 13.8 17.4 Rate of Return (%)

•

4-8

Expected return and SD for historical data

Average return for the historical data is simply the average value of the returns over time

SD is calculated by applying the following formula:

1–n

)r–r(=σ

n

1=t

2Avgt∑

4-9

Calculating SD for historical data

Year Return

2002 15%

2003 -5%

2004 20%

%23.13=1–3

)10–20(+)10–5–(+)10–15(=σ

222

Average return: 10%

4-10

Comments on SD as a measure of risk SD (σi) measures total risk. The larger the σi, the lower the probability

that actual returns will be closer to expected returns.

The larger σi is associated with a wider probability distribution of returns.

For a one asset portfolio, the appropriate measure of risk is σi.

Difficult to compare SDs, because return has not been accounted for.

4-11

Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return?

Return is 8%, regardless of the economy.

No. They are still exposed to inflation, although very little inflation is likely in a short time period.

T-bills are risky in terms of reinvestment rate risk but risk-free in the default sense.

4-12

Comparing risk and return

Security Expected return

Risk, σ

T-bills 8.0% 0.0%

HT 17.4% 20.0%

Coll* 1.7% 13.4%

USR* 13.8% 18.8%

Market 15.0% 15.3%

* Seem out of place.

4-13

Coefficient of Variation (CV) A standardized measure of dispersion

about the expected value It shows the risk per unit of return A meaningful basis for comparison when:

The expected returns on two alternatives varyThe returns are expressed in different units

r̂σ

=μσ

= MeanSD

=CV

4-14

Risk rankings by CV CV

T-bill 00/8.00 =0.00HT 20/17.4 =1.15Coll. 13.4/1.7 =7.88USR 18.8/13.8=1.36Market 15.3/15 =1.020

Coll. has the highest amount of risk per unit of return.HT, despite having the highest standard deviation of returns, has a relatively average CV.

4-15

Illustrating the CV as a measure of relative risk

A is riskier that B, despite the same amount of risk

0

A B

Rate of Return (%)

Prob.

Project A: ERR=8% and σ= 9%; CV=9/8=1.125

Project B: ERR =20% and σ= 9%; CV=9/20=0.45

4-16

Investor attitude towards risk

Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.

Risk premium – the difference between the returns on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

4-17

Calculating portfolio expected return

rw = r =Return Expected Portfolio n

1=ii

^

ip

^

∑

Companies Investment Expected ReturnMicrosoft $25,000 12%

General Electric $25,000 11.5%

Pfizer $25,000 10.0%

Coca-Cola $25,000 9.5%

%75.10=

%)5.9(25.0+%)10(25.0+%)5.11(25.0+%)12(25.0=r̂p

4-18

Risk in a Portfolio Context

The risk and return of an individual security should be analyzed in terms of how that security affects the risk and return of the portfolio in which it is held

4-19

Risk in a Portfolio Context

If two assets have the same risk, rational investors will prefer the asset with the higher expected return

If two assets have the same expected return, rational investors will prefer the asset with the smaller risk

4-20

Risk in a Portfolio Context

Portfolio risk is always smaller than the weighted average of the assets’ σs

ρ (roe) is the correlation coefficient which indicates the tendency of two variables to move together

When perfectly negatively correlated, ρ=–1.0

When perfectly positively correlated, ρ=+1.0

ABBABA

2

B

2

B

2

A

2

AP ρσσww2+σw+σw=σ

4-21

Risk in a Portfolio ContextDiversification can not reduce risk if

the portfolio consists of perfectly positively correlated stocks

∑ ∑n

1=t

n

1=t

2Avg,Bt,B

2Avg,At,A

AvgB,t,BAvgA,t,A

n

1=t

AB

)r–r()r–r(

)r–r)(r–r(=ρ

∑BA

n

1=iiBBiAAi

AB σσ

p)r̂–r)(r̂–r(=ρ

∑

For h

istor

ical d

ata

(see

pag

e 14

1)

For e

xpec

ted

data

,

pro.

dis.

is n

eede

d

(see

pag

e 17

4,ta

b-5-

1

4-22

Risk in a Portfolio Context

For expected data formula: rAi is the return on stock A under the ith

state of economy and is the expected return on stock A

For historical data formula: rA,t is the actual return on stock A in

period t, and rA, Avg is the average return on stock A during the period

Ar̂

4-23

Distribution of returns for 2 perfectly negatively correlated stocks (ρ = -1.0)

-10

15 15

25 2525

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM

4-24

Year Stock W Stock M Portfolio WM

2007 40% (10%) 15%

2008 (10%) 40% 15%

2009 35% (5%) 15%

2010 (5%) 35% 15%

2011 15% 15% 15%

Average Return

15% 15% 15%

SD (σ) 22.6% 22.6% 0.0%

)r( P

Distribution of returns for 2 perfectly negatively correlated stocks (ρ = -1.0)

)r( M)r( W

4-25

Distribution of returns for 2 perfectly positively correlated stocks (ρ=+1.0)

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

4-26

Year Stock M Stock M Portfolio MM’

2007 (10%) (10%) (10%)

2008 40% 40% 40%

2009 (5%) (5%) (5%)

2010 35% 35% 35%

2011 15% 15% 15%

Average Return 15% 15% 15%

SD (σ) 22.6% 22.6% 22.6%

)r( P

Distribution of returns for 2 perfectly positively correlated stocks (ρ=+1.0)

)r( M′)r( M

4-27

Distribution of returns for 2 partially correlated stocks (ρ=+0.67)

201120112011

4-28

Year Stock W Stock Y Portfolio WY

2007 40% 28% 34%

2008 (10%) 20.0% 5%

2009 35% 41% 38%

2010 (5%) (17%) (11%)

2011 15% 3% 9%

Average Return 15% 15% 15%

SD (σ) 22.6% 22.6% 20.6%

)r( P

Returns distribution for two partially positively correlated stocks (ρ=+0.67)

)r( Y)r( W

4-29

Comments on Risk in a Portfolio Context The portfolio risk will decline as the

number of stocks in the portfolio increases In the real world, no two stocks are

perfectly positively or negatively correlated; most stocks are positively correlated

It is impossible to form completely riskless stock portfolios

Diversification can reduce risk, but it cannot eliminate risk

4-30

Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio10 20 30 40 2,000+

Company-Specific/diversifiable Risk

Market Risk/Non-diversifiable Risk

20

0

Total Security Risk, p

p (%)35

4-31

Breaking down sources of risk

Total risk= Market risk + Diversifiable risk

Market risk– portion of a security’s total risk that cannot be eliminated through diversification

Also called systematic or non-diversifiable risk (beta) is inherent in the marketCaused by war, inflation, recession, high interest rates etc that systematically affect most firms

4-32

Diversifiable risk– portion of a security’s total risk that can be eliminated through proper diversification.

Also called company-specific or unsystematic riskCaused by random events like lawsuits, strikes, unsuccessful marketing programs and other events that are unique to a particular firm

Breaking down sources of risk

4-33

If an investor chooses to hold a one-stock portfolio, would the investor be compensated for the risk he bears?

NO!

Rational, risk-averse investors are concerned with σp, which is based upon market risk.

There can be only one price (the market return) for a given security.

No compensation should be earned for holding unnecessary, diversifiable risk.

4-34

CAPM

A basic model that links non-diversifiable risk and return for all assetsAssumes: A stock’s ERR is equal to the risk-free rate plus a risk premium that reflects the riskiness of the stock after diversification.

Primary Conclusion: The relevant risk of an individual stock is the amount of risk the stock contributes to a well-diversified portfolio.

4-35

Beta (β) or Market Risk

The extent to which the returns on a given stock move with the stock marketA relative and most relevant measure of a stock’s non-diversifiable risk

Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

A stock with a high β will move more than the market on average and vice-versa

4-36

Calculating β from historical data

Run a regression of past returns of a security against past returns on the market.

The slope of the regression line, sometimes called security’s characteristic line, is defined as the β coefficient for the security.

4-37

Illustrating the calculation of β

.

.

.ri

_

rM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM^ ^

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

4-38

Comments on β

If β = 1.0, the security is just as risky as the average stock.

If β > 1.0, the security is riskier than average.

If β < 1.0, the security is less risky than average.

Most stocks have βs in the range of 0.5 to 1.5.

4-39

Can β of a security be negative?

Yes, if the correlation between the return on stock i and market return is negative (ρi,m < 0).

If the correlation is negative, the regression line would slope downward, and the β would be negative.

However, a negative β is highly unlikely.

4-40

β coefficients for HT, Coll, and T-Bills

ki

_

kM

_

-20 0 20 40

40

20

-20

HT: β = 1.30

T-bills: β = 0

Coll: β = -0.87

4-41

Comparing expected return and β coefficients

Security Exp. Ret. Beta HT 17.4% 1.30Market 15.0 1.00USR 13.8 0.89T-Bills 8.0 0.00Coll. 1.7 -0.87

Riskier securities have higher returns, so the rank order is OK.

4-42

Calculation of β coefficients

Year2009 10% 10% 10% 10%10%

2010 30 20 15 2020

2011 (30) (10) 0 (10)(10)

Hr Ar Lr Mr

4-43

Explanation of relative volatility of stocks H, A, and L

The tendency of a stock to move up and down with the market is reflected in its β coefficient

Stock HHigh Risk: β=2

Stock AAverage Risk: β=1.0

Stock LLow Risk: β=0.5

Return on Stock i, (%)ir

Return on the Market i, (%)

Mr

4-44

Some additional comments on risk

1. Most investors do diversify, either by holding large portfolios or by purchasing shares in a mutual fund

2. Investors must be compensated for bearing market risk only

3. A portfolio consisting of low-β securities will itself have a low β because

nn2211p bw+....+bw+bw=b

4-45

Security market line (SML) or Characteristic line

A part of CAPM which shows the linear relationship between systematic risk () and expected return at a given time

Shows all risky marketable securities

determines the risk factors of the SML

The slope of the SML is the reward-to-risk ratio: (rM – rRF) / M = (rM – rRF)

If the security's risk versus ERR is plotted above the SML, it is undervalued because the investor can expect a greater return for the inherent risk

4-46

SML

SML: ri=rRF+(RPM)bi

=6%+(5%) biERR (%)

0 0.5 1.0 1.5 2.0

rRF=6

Risk, βi

Relatively Risky Stock’s Risk Premium:10%

rH=16

rM =rA=11

rL=8.5 Safe Stock’s RP:2.5%

4-47

SML formula

ri = rRF + (rM – rRF) bi

is the correlation between the ith stock’s return and the return on the market

is the standard deviation of the ith stock’s return

is the standard deviation of the market’s return

M,iM

ii ρ

σσ

=b

M,iρ

iσ

Mσ

4-48

What is the market risk premium?

Additional return over the risk-free rate needed to compensate investors

Its size depends on the perceived risk of the stock market and investors’ risk averse attitude

Varies from year to year, but mostly it ranges from 4% to 8% per year

4-49

Factors that change the SML What if investors raise inflation expectations

by 3%, what would happen to the SML?

SML1

ri (%)SML2

0 0.5 1.0 1.5

1815

11 8

I = 3%

Risk, βi

4-50

Factors that change the SML What if investors’ risk aversion increased,

causing the MRP to increase by 3%, what would happen to the SML?

SML1

ri (%) SML2

0 0.5 1.0 1.5

1815

11 8

RPM = 3%

Risk, βi

4-51

The SML: Calculating RRR

Assume rRF = 8% and rM = 15%.

The market (or equity) risk premium is RPM = rM – rRF = 15% – 8% = 7%.

When bi=1.5, SML=8%+(7%)(1.5)=18.5%

4-52

Calculating required rates of return

rHT = 8.0% + (15.0% - 8.0%)(1.30)

= 8.0% + (7.0%)(1.30)= 8.0% + 9.1% = 17.10%

rM = 8.0% + (7.0%)(1.00) = 15.00% rUSR = 8.0% + (7.0%)(0.89) = 14.23% rT-bill = 8.0% + (7.0%)(0.00) = 8.00% rColl = 8.0% + (7.0%)(-0.87)= 1.91%

4-53

Expected vs. Required returns

r) <r( Overvalued 1.9 1.7 Coll.

r) =r( uedFairly val 8.0 8.0 bills-T

r) <r( Overvalued 14.2 13.8 USR

r) =r( uedFairly val 15.0 15.0 Market

r) >r( dUndervalue 17.1% 17.4% HT

r r

^

^

^

^

^

^

4-54

Illustrating the SML

..Coll.

.HT

T-bills

.USR

SML

rM = 15

rRF = 8

-1 0 1 2

.

SML: ri = 8% + (15% – 8%) βi

ri (%)

Risk, βi

Theoretically, every

security sh

ould lie on

the SML

If a se

curity is

below

the SML, it is

overpriced

and vice-versa.

4-55

CML (Capital Market Line) Vs. SML

CML SML

Definition CML is a line that plots the return vs. total risk (SD)

SML is a line that plots the return vs. market risk (ß)

Risk Measurement

CML uses SD as the measure of risk

SML uses ß  as the measure of risk

Equation ri = rRF + [(rM –

rRF)/σM] σi

ri = rRF + (rM – rRF) bi

Efficient and Non-efficient

CML graph defines efficient portfolio

SML graph defines both efficient and non-efficient portfolios

4-56

An example:Equally-weighted two-stock portfolio

Create a portfolio with 50% invested in HT and 50% invested in Collections.

The beta of a portfolio is the weighted average of each of the stock’s betas.

βP = wHT βHT + wColl βColl

βP = 0.5 (1.30) + 0.5 (-0.87)

βP = 0.215

4-57

Calculating portfolio RRR The RRR of a portfolio is the weighted

average of each of the stock’s RRR.

rP = wHT rHT + wColl rColl

rP = 0.5 (17.1%) + 0.5 (1.9%)

rP = 9.5%

Or, using the portfolio’s beta, CAPM can be used to solve for expected return.

rP = rRF + (rM – rRF) βP

rP = 8.0% + (15.0% – 8.0%) (0.215)

rP = 9.5%

4-58

More thoughts on the CAPM

Difficult to test the validity of CAPM statistically.

Betas do not remain stable over time.

Betas are calculated using historical data, but historical data may not reflect investors’ expectations about future riskiness.

4-59

More thoughts on the CAPM

Investors are concerned with both market risk and total risk and thus ri

should be:

ri = rRF + (rM – rRF) βi + ???

Two variables are consistently related to stock returns: (i) the firm’s size and (ii) its market/book ratio. After adjusting for other factors, smaller firms

and stocks with low market/book ratios have relatively high returns

4-60

Problems 4-1: A stock’s return has the following distribution

Demands for Products

P(Demand)

Rate of Return if Demand Occurs

Weak 0.1 (50%)

Below average 0.2 (5)

Average 0.4 16

Above average

0.2 25

Strong 0.1 60

Total Weight 1.00 Calculate the stock’s expected return,

standard deviation, and coefficient of variation

4-61

Solutions 4-1

Demands

Prob. Rate of Return

Weak 0.1 (50%) -0.05 0.376996

Below Avg.

0.2 (5) -0.01 0.0053792

Average 0.4 16 0.064 0.0008464

Above Avg.

0.2 25 0.05 0.0036992

Strong 0.1 60 0.06 0.0236196

1.00 = 0.114 0.071244

∑ )r(p=r̂ iiAvg ∑ i2

Avgi p)r̂r(

Avgr̂

267.0=0.071244=p)r̂r(=σ i2

Avgi∑ 34.2=114.0267.0

=CV

4-62

Problems and Solution 4-2

An individual has $35,000 invested in a stock which has a beta of 0.8 and $40,000 invested in a stock with a beta of 1.4. If these are the only two investments in her portfolio, what is her portfolio beta?

Solution:

12.1=)4.1)(000,75000,40

(+)8.0)(000,75000,35

(=bP

4-63

Problems and Solution 4-3

Assume that the risk-free rate is 5% and the market risk premium is 6%. a) What is the expected return for the overall stock market? b) What is the required rate of return on a stock that has a beta of 1.2?

Solution: a) Expected return = 5%+(6%)(1.0)=11% b) RRR= 5%+ (6%)(1.2)=12.2%

4-64

Problems and Solution 4-4

Assume that the risk-free rate is 6% and the expected return on the market is 13%. What is the required rate of return on a stock that has a beta of 0.7?

Solution:

RRR= 6%+ (13% – 6%)(0.7)=10.9%

4-65

Problem 4-7

Suppose, rRF=9%, rM=14% and bi=1.3.a) What is ri, the required rate of return on Stock i?b) Now suppose rRF (i) increases to 10% or (ii) decreases to 8%. The slope of the SML remains constant. How would this affect c) Now assume rRF remains at 9% but rM (i) increases to 16% or (ii) falls to 13%. The slope of the SML does not remain constant. How would these changes affect

4-66

Solution 4-7

a) Given

b-i)

b-ii)

c-i)

c-ii)

%,9=rRF %,14=rM .3.1=bi

%5.15=)3.1%)(9–%14(+%9=ri

%5.16=)3.1%)(10–%15(+%10=r%;15=r iM

%5.14=)3.1%)(8–%13(+%8=r%;13=r iM

%1.18=)3.1%)(9–%16(+%9=ri

%2.14=)3.1%)(9–%13(+%9=ri