Chapter Four
Risk and Return
• Return is the total gain or loss
experienced on behalf of the owner
of an investment over a given period
of time.
K = actual, expected, or required
rate of return during period t.
Pt = price (value) of asset at time
t.
Pt-1 = price (value) of asset at t-1
Ct = cash (flow) received from the
asset investment in time period t-1
to t.
Example
Alpha Co. wishes to determine the actual rate of
return on two of its video machines, X and Y. X
was purchased exactly one year ago for
$20,000 and currently has a market value of
$21,500. During the year it generated $800 of
after-tax cash receipts. Y was purchased four
years ago, and its value at the beginning and
end of the year just ended declined from
$12,000 to $11,800. During the year it
generated $1700 of after-tax cash receipts.
What is the annual rate of return on asset X
and asset Y?
RISK
• The probability or likelihood that actual
results (rates of returns) deviate from
expected returns.
• Attitudes towards risk
–Indifferent
–Risk averse
–Risk seeking
• Managers are assumed to be risk averse
Implications
1. If two projects have the same expected
return, the manager will prefer the one
with the lesser amount of risk.
2. If two projects have the same degree of
risk, manager will prefer the one with the
higher expected return.
Measures of risk for a single asset
• A probability distribution is used to
measure risk.
• Probability distribution lists the set
of possible returns that can occur at
a specific time and their associated
probabilities of occurrence.
• Using probability distribution, we will
compute:
1. expected rate of return
2. standard deviation or variance
3. coefficient of variation
1. Expected rate of return ( k )
• The return expected to be realized
from an investment.
• The mean value of the probability
distribution of possible returns.
where, ki = possible return in year i
pi =probability of occurrence of ki
n = number of possible returns
Example 1
Mr. Max is considering the possible
rates of return (dividend yield plus
capital gain or loss) that he might
earn next year on a $10,000
investment in the stock of either
Alpha Co. or Beta Co. The rate-ofreturn probability distributions for
the two companies are shown below.
Standard Deviation (σ)
• Measures the average dispersion of
the probability distribution around its
expected value.
• The most common statistical
indicator of an asset’s risk (stand
alone risk)
• Alpha is a riskier investment than
Beta.
Coefficient of variation (CV)
• Shows the risk per unit of return
• Provides a more meaningful basis for
comparison when the expected
returns on two alternatives are not
the same.
Portfolio Risk and Diversification
• Portfolio is a collection or a group of
investment assets.
• Through diversification, risk can be
reduced
• The risk and return of individual assets
should be analyzed in terms of how
they affect the risk and return of the
portfolio in which they are included.
• The goal of the financial manager
should be to create an efficient
portfolio.
• An efficient portfolio maximizes
return for a given level of risk or
minimizes risk for a given level of
return.
Where:
wi = the proportion of the portfolio
devoted to the ith asset
Ki = expected rates or return on the
th
i asset
n = number of assets composing
the portfolio
Example 4: Consider a portfolio of
three stocks A, B, and C, with
expected returns of 16%, 12%, and
20%, respectively. The portfolio
consists of 50% stock A, 25% stock B,
and 25% stock C. What is the
expected return on this portfolio?
Portfolio risk
• Depends not only on the risk of
the individual securities, or
assets, but also on the
correlations between their
returns.
Correlation (co-movement)
• Measures the degree of linear
relationship to which two variables,
such as returns on two assets, move
together.
• Takes on numerical values that range
from +1 to -1.
• The positive or negative sign indicates
the direction of the co movement.
• The absolute value of the correlation
indicates the relative strength of the
association.
• The closer the correlation coefficient
is to +1 or -1, the stronger the
association.
+Ve Correlation coefficient:
The variables move together.
-Ve Correlation coefficient:
 The variables move in opposite
direction.
A correlation of 0.0:
No relationship between the
variables( unrelated).
Correlation coefficient of +1.0:
The relative magnitude of the
movements is exactly the same
(perfect positive correlation)
Correlation of -1.0:
 The variables move exactly opposite
to each other( perfect negative
correlation)
• The closer the correlation is to 0.0,
the lesser the two sets of returns
move together.
Two-security portfolio risk
wx, wy= the proportion of the total portfolio
devoted to asset X and to asset Y, respectively.
Corrxy = correlation of asset X with asset Y.
The covariance is a measure of the
degree of linear relationship
between two variables.
• The covariance may take on any value
(positive or negative), whereas the
correlation coefficient can take on
values only from +1.0 through zero to 1.0.
Where, n = the number of possible
states
Pi= the probability associated with the
ith state
Example 6:
The portfolio is composed of 50%
investment in each asset.
Required: calculate:
a) expected return of the portfolio
b) portfolio risk
Correlation, return and risk for two-asset
portfolio
• The risk of any single proposed asset
investment should not be viewed
independent of other assets.
• New investments must be considered
in light of their impact on the risk and
return of the portfolio of assets.
• To reduce overall risk (to diversify
risk), it is best to combine or add to
the portfolio assets that have a
negative or a low positive
correlation
• A negative or a lower positive
correlation reduces risk below the risk o
the least risky asset.
• Negatively correlated assets reduce ris
more effectively than uncorrelated
assets.
• Uncorrelated assets reduce risk
more effectively than positively
correlated assets.
• Perfectly correlated assets can not
reduce portfolio risk (no
diversification) below the risk
of the least risky asset. i.e. it is equal
to the mean risk.
Types of risk
 Diversifiable risk (unsystematic risk
or Company-specific risk)
 Non-diversifiable risk (systematic or
market risk)
Diversifiable risk( unsystematic risk or
Company-specific risk)
• The portion of an asset’s risk that
is attributable to firm-specific,
random causes, such as strikes,
lawsuits, product development.
Non-diversifiable risk (systematic or
market risk)
• Is attributable to market factors that
affect all firms.
E.g. Inflation, war, monetary and fiscal
policies
• This risk can not be eliminated through
diversification.
Total security risk
= non-diversifiable risk + diversifiable
risk
Risk and Required Rate of Return
• Required Rate of return is the
minimum expected rate of return
that would induce an investor to
acquire it.
The capital asset pricing model (CAPM)
• The required rate of return of a
security is influenced by the risk free
rate and a premium to compensate
for the security risk.
• Beta is a measure of how much non-diversifiable
risk (market risk) asset j has relative to the average
asset in the market.
• KM - KRF = the market risk premium
• βj(KM - KRF) = the risk premium for
asset j.
• Example7: Alem is considering two
investment opportunities, X &Y. She
estimates the standard deviation for X is
10.56% and its correlation coefficient
with the market is +0.45. Y has a standard
deviation of 12.15% and a correlation
coefficient with the market of +0.85%.
The standard deviation of the market is
8.67%. A local brokerage firm gives. Alem
estimates of the expected risk-free rate
of return and required market return to
be 6% & 11% respectively.
Required: what is the required rate
of return on: a) asset X? b) Asset
Y?
• The reward for bearing risk, the
expected or required risk premium,
depends only on the systematic risk
of an investment.
• The market does not reward risks
born unnecessarily.
Portfolio Betas
• Portfolio beta is a weighted
average of the betas of individual
assets.
Example 8: suppose you
have $10,000 invested in
each of 10 stocks. The
amount invested in each
stock is 10% and all the
stocks have a beta of 1.2
each.
Required:
a)what is the portfolio beta?
b) What is the portfolio beta if you sell one
of the stocks and reinvest in another
stock of the same price but of a beta of i)
0.60 ii) 2.00
c) What is the required return of the
portfolio in “b” above if KRF and KM are
6% and 11% , respectively?
The Security Market Line (SML)
• The depiction of the CAPM as a
graph that reflects the required
return for each level of nondiversifiable risk (Beta).
Shifts in the SML
1. Change in inflationary
expectations
2. Changes in Risk Aversion
Change in inflationary
expectations
Example 10
Assume in example 9 above that the KRF
includes 2% real rate of interest and 5%
inflation premium. Assume also that
recent economic events have resulted in
an increase of 3% inflationary
expectations, raising the inflation
premium from 5% to 8%. In this case the
new returns are KRF = 2% + 8% = 10%
and Km = 11% + 3% = 14%.
Changes in Risk Aversion
• The slope of the SML, KM – KRF,
reflects the general risk preference
of investors in the market place.
• The steeper the slope, the greater
the degree of risk aversion
Example 11: Assume in example 9 above
that as a result of a recent economic
events investors have become more risk
averse, causing a new higher market
return of 14%.