Risk and Rate of Return
CHAPTER 8
Introduction
1. Central Ideal
2. While doing ‘risk analysis’ it can help if we
keep in mind the following points:
1. Business Assets and Cash Flows
2. Classification of Assets
3. Measurement of stock risk
•
•
Difference between stand-alone and portfolio risk
Importance of stand-alone risk analysis
4. Stock’s risk in a portfolio context
5. Investor’s aversion to risk
Return
• Whenever an individual makes an investment
he does so with the expectation of earning
more money in the future.
• Return can be defined as ‘something over and
above my basic investment’.
• Return can be expressed both in dollar terms
and percentage return terms.
Return in dollar terms
• Definition
• Calculation
• Two main problems arise when expressing
returns in dollar/rupee terms:
– Size/scale of investment
– Timing of return
Return in percentage terms
• The scale of investment and timing of return problem
associated with expressing returns in $ form can be resolved if
we express investment result in rate of return or percentage
return form.
• Calculation
– ROR solves the problem of knowing scale of investment for forming a
meaningful judgment about value of return.
– ROR solves timing of return problem as returns normally are expressed
on an annual basis.
Risk
• Definition of Risk
• Risk is defined as:
– A hazard, exposure to loss or injury.
– Chance of occurrence of an unfavorable event.
– Risk is the dispersion of returns around mean, or
expected mean.
• In this chapter our focus will be to carry out
risk analysis for stocks.
Investment in stocks and the associated riskiness
•
•
•
•
Goal behind investment of funds
Low risk securities offer low return
Stocks offer higher returns, but are more risky
It is important to remember that,
“No investment should be undertaken unless the expected rate of return
is high enough to compensate for the perceived risk.”
• Investment in risky assets
– Investment in risky assets generally results in returns that is less or more than
what was originally expected of them, rarely if ever do they produce their
expected returns.
• Investment risk
– Investment risk, then, is related to the probability of earning a low or negative
actual return on an investment. The greater the chance of earning lower than
expected or negative returns, the riskier the investment.
Methods for analyzing riskiness of an asset
• As discussed before an asset’s risk can be
analyzed in two ways:
– Stand-alone basis: On a stand alone basis the
riskiness of the asset is considered all by itself.
– Portfolio basis: When analyzing the riskiness of a
stock in a portfolio, the riskiness of the stock is
considered in relation to other stocks held in the
portfolio.
• We’ll first analyze how to measure a stock’s
risk on a stand-alone basis.
1. Stand-alone Risk and Return
1. Calculation of stand-alone risk
– Standard deviation
1. For standard deviation calculation we need to
know expected rate of return
2. Calculation of stand-alone risk involves discussion
on following important concepts:
•
•
•
•
Probability distribution
Expected Return
Standard deviation
Coefficient of Variation (CV)
i. Probability Distribution
• Definition for an event’s probability
• Example
• Definition of probability distribution
i. Probability Distribution (contd.)
• After discussion on probability and describing
what exactly probability distribution is we will
concentrate our attention on determining how
to measure:
– Expected return on individual stock
– Standard deviation or risk on an individual stock
Calculation of Expected Return
(Example # 1)
• Example
– Two companies, Martin Company and US Water
Company.
– Martin Company: In an industry where there is
intense competition.
• Arrival of a better rival can wipe out MC from industry.
• Cash flows of MC are not known with certainty.
• Risky stock means future earnings are not known with
certainty.
Calculation of Expected Return
• We will now set up probability distribution for
MC where outcomes are represented by state
of the economy.
• Probabilities are set by experts according to
expected economic conditions.
Economy
Probability of this
reflecting demand demand occurring
of the product
ROR if this
demand occurs
Strong
0.30
80%
Normal
0.40
10%
Weak
0.30
-60%
Product
(2*3)
Calculation of Expected Return
Formula for calculating rate of return on Martin
Company’s stock
rm  expected rate of return on Martin Company' s stock
^
N
r   ri Pi
i 1
Calculation of Expected Return
• After determining expected return on Martin
company’s stock we now turn our attention to US
water company.
• US water company operates in a stable industry
as it is supplying an essential service so its sale
and profits are relatively more stable and
predictable.
– Even if the economy fluctuates people will not stop drinking water.
– It means USWC has stable future earnings which can be predicted
with much more accuracy hence it is a less risky stock.
Calculation of Expected Return
• US Water Company
Economy
Probability of this
reflecting demand demand occurring
of the product
ROR if this
demand occurs
Strong
0.30
15%
Normal
0.40
10%
Weak
0.30
5%
Product
(2*3)
• Note that USWC’s ROR vary much less than
Martin Company’s because its future earnings
can be predicted with much more certainty.
Graphing the ROR of Martin Co. and
USWC
• Done to obtain a picture of the variability of
possible ROR of each.
• The height of the bar signifies the probability
that an outcome will occur.
Summary
Expected Rate of Return for MC and USWC Stock
Expected return
MC
10%
USWC
10%
• Both the investments give equal expected return.
• Looking at this data only, investment can be made in either
of the two stocks as they give equal rate of return.
• Have we failed to take account of risk? Yes.
• Let’s now calculate take into account riskiness of these
stocks.
Stand-alone risk
• Risk is the dispersion of returns around mean.
• Standard deviation (si) measures stand-alone
risk.
• The larger the si , the lower the probability that
actual returns will be close to the expected return.
Calculation of Risk for Individual Stock
through Standard Deviation Calculation
s  Standard deviation
s  Variance  s
2
σ
N
 (r  r)
i 1
i
2
Pi
Summary
Standard deviation for MC and USWC Stock
Standard Deviation
MC
54.22%
USWC
3.873%
• Standard deviation (si) measures stand-alone, risk.
• The larger the si , the lower the probability that actual
returns will be close to the expected return.
• From the data above, MC has a higher standard deviation
than USWC which means MC’s stock is riskier than USWC’s
and is not as safe a investment as MC.
Decision Time
Stand-alone risk and return for HT and USR stocks
Martin Company
• r= 10%
• s= 54.22%
USWC Co.
• r = 10%
• s= 3.873%
22
Decision Time: Coefficient of Variation
• The Coefficient of Variation (CV) measures risk per unit of
expected return.
• CV is a standardized measure of dispersion about the
expected return.
• In simple words, CV tells how risky a stock is in relation to its
expected rate of return.
• The higher the expected rate of return as compared to
standard deviation of a stock, lower is the riskiness of the
stock and vice versa.
• Formula
23
Decision Time: Coefficient of Variation
CV (MC)
σ/r
54.22/10
5.42
CV (USWC)
σ/r
3.873/10
0.39
Decision: Investment in USWC stock is a safer option as its
risk per unit of return is relatively less than that of USR
stock.
24
Calculation of Expected Return
(Example # 1)

In order to see how a stock’s stand-alone risk and
return are calculated let’s proceed with an example
where we calculate stand-alone return and risk of
two stocks, HT and USR.
Probability Distributions and Expected Returns
Economy
Pi
HT
USR
Recession
0.1
-60%
-8%
Below avg
0.2
-20%
-2%
Average
0.4
30%
5%
Above avg
0.2
60%
12%
Boom
0.1
90%
20%
Expected Rate of Return for HT Stock
^
k HT  expected rate of return on HT stock
^
N
k   ri Pi
i 1
^
k HT  (-60%) (0.1)  (-20%) (0.2)  (30%) (0.4)  (60%) (0.2)  (90%) (0.1)
^
k HT  23%
Expected Rate of Return for USR Stock
^
k USR  expected rate of return on USR stock
^
N
k   ri Pi
i 1
^
k USR  (-8%) (0.1)  (-2%) (0.2)  (5%) (0.4)  (12%) (0.2)  (20%) (0.1)
^
k USR  5.20%
Summary
Expected Rate of Return for HT and USR Stock
Expected return
HT
23%
USR
5.20%
• HT has the highest expected return, and appears to be the
better investment alternative than USR, but is it really?
• Looking at this data only investment should be made in USR’s
stock only.
• Have we failed to take account of risk? Yes.
• Let’s now calculate take into account riskiness of these stocks.
Calculating Standard Deviation
s  Standard deviation
s  Variance  s
2
σ
N
 (K
i 1
 k̂ ) Pi
2
i
Probability Distributions and Expected Returns
Economy
Pi
HT
USR
Recession
0.1
-60%
-8%
Below avg
0.2
-20%
-2%
Average
0.4
30%
5%
Above avg
0.2
60%
12%
Boom
0.1
90%
20%
Standard Deviation for HT
s
N

i 1
(K i  K̂) 2 Pi
(-60 - 23) (0.1)  (-20 - 23) (0.2)

  (30 - 23) 2 (0.4)  (60 - 23) 2 (0.2)
 (90 - 23) 2 (0.1)

2
s HT
s HT
 42%
2





1
2
Standard Deviation for USR
s
N

i 1
(K i  K)2 Pi
(-8 - 5.20) (0.1)  (-2 - 5.20) (0.2)

sUSR   (5 - 5.20) 2 (0.4)  (12 - 5.20) 2 (0.2)
 (20 - 5.20) 2 (0.1)

2
sUSR  10.49%
2





1
2
Summary
Expected Rate of Return for HT and USR Stock
Standard Deviation
HT
42%
USR
10.49%
• Standard deviation (si) measures stand-alone, risk.
• The larger the si , the lower the probability that actual returns will
be close to the expected return.
• From the data above, HT has a higher standard deviation than USR
which means HT’s stock is riskier than HT’s and is not as safe a
investment as HT.
• Looking at this data only investment should be made in USR’s stock
only.
• On the other hand, if investment was made looking solely at
expected return, HT was a better choice as it returned 23% against
USR’s 5.20%.
Decision Time: Coefficient of Variation
CV (HT)
σ/k
42/23
1.83
CV (USR)
σ/k
10.49/5.20
2.00
Decision: Investment in HT stock is a safer option as its risk
per unit of return is relatively less than that of USR stock.
35
Stand-alone risk and return for HT
and USR stocks
HT Company
• K= 23%
• s= 42%
• CV = 1.83
USR Co.
• K = 5.20%
• s= 10.49%
• CV = 2.00
36
Risk Aversion and Required Return
• Investors, generally, are risk averse and
require high rate of return in order to induce
them to take the risk of putting their money in
a high risk investment.
2. Portfolio Risk and Return
• Discussion will focus on:
1. Portfolio Returns and Portfolio Risk
– Calculate the expected rate of return and volatility for a portfolio of
investments and describe how diversification affects the returns of a
portfolio of investments.
2. Types of Risk
– A stock’s risk consists of diversifiable and market risk. The only
relevant risk that a stock in a portfolio faces is its market risk. Beta
coefficient is a measure of stock’s market risk and it is measured by
the extent to which the return on a stock moves with the overall stock
market.
3. The CAPM and the SML Equation
– Estimate an investor’s required rate of return.
38
Portfolio Return
Assume a three-stock portfolio, total investment
capital of $150,000 with $50,000 investment in
each of the three stocks A, B and C.
^
Calculate kp and sp.
^
Portfolio Return, rp
^
rp is a weighted average of the expected return of each asset in the
portfolio :
n
rp = S wiri
^
i=1
^
Types of Risk
A stock’s risk consists of diversifiable and market risk.
1. Diversifiable risk
•
Firm-specific, industry specific risk is that part of a
security’s stand-alone risk that can be eliminated by
proper diversification.
2. Market risk.
•
Non-diversifiable, relevant risk is that part of a
security’s stand-alone risk that cannot be eliminated by
diversification, and is measured by beta.
Types of Risk (contd.)
• The more an investor diversify into different
companies located across different industries the
less risky his portfolio becomes.
• If a stock is held in a reasonably well-diversified
portfolio that is containing 40 or more stocks,
almost 50% of its risk can be reduced.
• Total risk = Diversifiable risk + Market risk
• As number of stocks increases, total risk
decreases as diversifiable risk decreases.
i. Diversifiable Risk
• Industry specific factors
• Company specific factors
ii. Non-diversifiable, Market Risk
• Refers to the type of risk that cannot be eliminated through
diversification.
• For example factors such as war, inflation, recession etc.
effects all the companies and industries in an economy.
• All stocks will bear the negative effect of these factors and will
decrease in value.
• For example, after the 9/11 incident all stocks decreased in
value.
• Market risk is beyond control of investors.
• Aim of making portfolio is to reduce diversifiable risk,
remaining risk is called market risk.
44
Figure: Diversifiable and Market Risk
Riskiness of Assets held in portfolios
(Effect of diversification)
• Asset held as part of portfolio is much less riskier than an
asset held alone.
• Investors like to hold their stocks in form of portfolios. (Why?)
• Example to explain the importance of holding assets in the
form of portfolios.
• The process of reducing risk by investing across different
sectors/industries is called ‘diversification’.
• As investors hold portfolio of stocks, the risk and return of a
security should be analyzed in terms of how that stock affects
the portfolio in which it is held.
46
Effect of Portfolio Size on Portfolio
Risk for Average Stocks
• As more stocks are added, each new stock has
a smaller risk-reducing impact.
• sp falls very slowly after about 10 stocks are
included, and after 40 stocks, there is little, if
any, effect.
Figure: Effect of Portfolio Size on Portfolio Risk for Average Stocks
Beta Coefficient
For measuring relevant risk of a stock
• As more and more assets are added to a portfolio,
portfolio risk reduces.
• However, we could put every conceivable asset in the
world into our portfolio and still have risk remaining.
• This remaining risk is called Market Risk and is measured
by Beta.
• When we hold a stock in the portfolio we reduce its
diversifiable risk, the only risk remaining is ‘market risk’.
• Beta is a measure stock’s market risk and is measured by
the extent to which return on a stock moves with the
overall stock market.
49
Beta Coefficient
For measuring relevant risk of a stock
• For a stock having a beta of 1 it means it is as risky as market.
If market moves up by 10% stock also moves up by 10%
similarly if market moves down by 10% stock also moves
down by 10%.
• For a stock having a beta of 2 it means it is more volatile than
the market. If market moves up by 10% stock moves up by
20% similarly if market moves down by 10% stock moves
down by 20%.
• For a stock having a beta of 0.5 it means it is only half as
volatile as the market. If market moves up by 10% stock
moves up by 5% similarly if market moves down by 10% stock
moves down by 5%.
Beta Coefficient
For measuring relevant risk of a stock
•
•
•
•
If beta = 1.0, average stock.
If beta > 1.0, stock riskier than average.
If beta < 1.0, stock less risky than average.
Most stocks have betas in the range of 0.5 to 1.5.
Use of Beta Coefficient to measure
Return on a Stock
• After calculating the riskiness of a stock using
the beta coefficient, we now measure the ROR
on a stock using the SML equation.
Use of Beta Coefficient to measure
Return on a Stock
• Suppose there are two companies, AA and BB.
The beta coefficient for company AA is 0.85
and the beta coefficient for company BB is
1.50.
• Keeping this data in mind we can calculate the
required returns on the stocks of AA and BB
with the help of SML equation.
Security
Beta
AA
0.85
BB
1.50
SML Equation
• RPM = market risk premium = rM - rRF
• RPi = stock risk premium = (RPM)bi
• ri = rRF + (rM - rRF )bi
ri = rRF + (RPM)bi
56
Using the SML to calculate the required returns
SML: ri = rRF + (rM – rRF)bi .
•
•
•
•
Assume :
rRF = 8%.
rM = 15%.
RPM = rM – rRF = 15% – 8% = 7%.
Using the SML to calculate the required return on
stock of AA and BB companies.
SML: ri = rRF + (rM – rRF)bi .
• ri = rRF + (rM – rRF)bi
• rAA = 8% + (7 * 0.85)
• rAA = 13.95%
• ri = rRF + (rM – rRF)bi
• rBB = 8% + (7 * 1.50)
• rBB= 18.5%
SML: ri = 8% + (15% – 8%) bi
ri (%)
SML
.
BB
rM = 15
rRF = 8
-1
0
. .
. T-bills
1
AA
2
Risk, bi
CAPM Example
• What is Intel’s required return if its B = 1.2
(from ValueLine Investment Survey), the
current 3-month T-bill rate is 5%, and the
historical US market risk premium of 8.6% is
expected?
• Solve
60
We now calculate beta for a portfolio with 50%
investment in AA stock and 50% in BB stock
bp= Weighted average
= 0.5(bAA) + 0.5(bBB)
= 0.5(0.85) + 0.5(1.50)
= 1.175
Changes to SML Equation
• What happens if inflation increases?
• What happens if investors become more risk
averse about the stock market?
• What happens if beta increases?
62
i. Impact of increase in inflation
• If investors raise inflation expectations by 3%, what would happen
to the SML?
• The risk-free rate as measured by the rate on U.S. Treasury
securities is called the nominal, or quoted, rate; and it consists of
two elements: rRF = r* + IP.
• Therefore, the 6% rRF mentioned in the previous equation (ri = 6% +
(5%) bi ) might be thought of as consisting of a 3% real risk-free rate
of return plus a 3% inflation premium:
rRF = r* + IP = 3% + 3%
• However, now assume that expected inflation rate rose by 2% i.e.
IP = 3% + 2% = 5%
• Because of increase in inflation premium, risk free rate increases to
8%.
rRF = k* + IP = 3% + 5% = 8%
Impact of increase in inflation
• In simple words,
– As inflation increases, kRF increases.
• ki = kRF + (kM – kRF)bi
kRF = k* + IP

Under CAPM an increase in kRF leads to an
equal increase in ROR of all risky assets.
ii. Impact of increase in investor’s risk
aversion
• The slope of the SML reflects the extent to which
investors are averse to risk—the steeper the slope of
the line, the more the average investor requires as
compensation for bearing risk.
iii. Impact of increase in beta
• An increase in beta means an increase in
market riskiness of the stock which leads to
increased required rate of return on stocks.