CHAPTER 4 Risk and Return- The Basics Stand-alone risk Portfolio risk Risk & return: CAPM / SML.
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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
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