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

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∑

4-7

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%

4-8

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.

4-9

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.

4-10

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

4-11

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.

4-12

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

4-13

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

4-14

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

4-15

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%

4-16

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%

4-17

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

4-18

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