Risk and Return: Portfolio Theory
and Assets Pricing Models
Chapter Objectives
• Discuss the concepts of portfolio risk and
return.
• Determine the relationship between risk and
return of portfolios.
• Highlight the difference between systematic
and unsystematic risks.
• Examine the logic of portfolio theory .
• Show the use of capital asset pricing model
(CAPM) in the valuation of securities.
Introduction
• A portfolio is a bundle or a combination of
individual assets or securities.
• The portfolio theory provides a normative
approach to investors to make decisions to invest
their wealth in assets or securities under risk.
– It is based on the assumption that investors are risk-averse.
– An investor may want to maximize the returns from his
investments for a given level of risk.
– The third assumption of the portfolio theory is that the returns
of assets are normally distributed.
Markowitz Portfolio Theory
• The basic portfolio model was developed by
Harry Markowitz, who derived the expected rate
of return for a portfolio of assets and an expected
risk measure. The Markowitz model is based on
several assumptions regarding investor behavior:
1. Investors consider each investment alternative
as being represented by a probability
distribution of expected returns over some
holding period.
2. Investors maximize one-period expected utility,
and their utility curves demonstrate diminishing
marginal utility of wealth
Markowitz Portfolio Theory
Investors estimate the risk of the portfolio on the
basis of the variability of expected returns.
4. Investors base decisions solely on expected
return and risk, so their utility curves are a
function of expected return and the expected
variance (or standard deviation) of returns only.
5. For a given risk level, investors prefer higher
returns to lower returns. Similarly, for a given
level of expected return, investors prefer less risk
to more risk
Markowitz Portfolio Theory
• Under these assumptions, a single asset or
portfolio of assets is considered to be efficient
if no other asset or portfolio of assets offers
higher expected return with the same (or
lower) risk, or lower risk with the same (or
higher) expected return
Portfolio Return: Two-Asset Case
• The return of a portfolio is equal to the
weighted average of the returns of individual
assets (or securities) in the portfolio with
weights being equal to the proportion of
investment value in each asset.
Expected return on portfolio  weight of security X × expected return on security X
 weight of security Y × expected return on security Y
Portfolio Return
• Consider the following stock returns
Stocks
A
B
C
D
Weights Expected Return
0.2
0.1
0.3
0.11
0.3
0.12
0.2
0.13
Calculate the portfolio return
Portfolio Risk: Two-Asset Case
• The portfolio variance or standard deviation
depends on the co-movement of returns on
two assets. Covariance of returns on two
assets measures their co-movement.
• The formula for calculating covariance of
returns of the two securities X and Y is as
follows:
Portfolio Risk: Two-Asset Case
• The portfolio variance or standard deviation depends on the
co-movement of returns on two assets. Covariance of returns
on two assets measures their co-movement.
• The formula for calculating covariance of returns of the two
securities X and Y is as follows:
Covariance XY = Standard deviation X ´ Standard deviation Y ´
Correlation XY
• The variance of two-security portfolio is given by the following
equation:
 p2   x2 wx2   y2 wy2  2wx wy Co varxy
  x2 wx2   y2 wy2  2wx wy x y Corxy
Portfolio Risk Depends on Correlation
between Assets
• When correlation coefficient of returns on
individual securities is perfectly positive (i.e.,
cor = 1.0), then there is no advantage of
diversification.
• The weighted standard deviation of returns on
individual securities is equal to the standard
deviation of the portfolio.
• We
may
therefore
conclude
that
diversification always reduces risk provided
the correlation coefficient is less than 1.
Portfolio Risk
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Returns
Coca-Cola
Pepsi
0.12
0.24
0.23
0.08
0.15
0.21
0.07
0.15
0.25
0.03
0.32
-0.09
0.11
-0.35
-0.09
0.21
0.26
-0.15
-0.12
0.2
0.29
0.06
0.03
0.31
Portfolio Risk
• Calculate the covariance between the two
assets
• Calculate the portfolio risk and return
assuming the assets are equally weighted in
the portfolio
Portfolio Risk
• Covariance = -0.0123
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Returns
Coca-Cola
Pepsi
0.12
0.24
0.23
0.08
0.15
0.21
0.07
0.15
0.25
0.03
0.32
-0.09
0.11
-0.35
-0.09
0.21
0.26
-0.15
-0.12
0.2
0.29
0.06
0.03
0.31
0.135
0.075
Coca-Cola
Pepsi
(R-Rbar)C* (R-Rbar)P
(R-Rbar) (R-Rbar)
-0.015
0.165
-0.002475
0.095
0.005
0.000475
0.015
0.135
0.002025
-0.065
0.075
-0.004875
0.115
-0.045
-0.005175
0.185
-0.165
-0.030525
-0.025
-0.425
0.010625
-0.225
0.135
-0.030375
0.125
-0.225
-0.028125
-0.255
0.125
-0.031875
0.155
-0.015
-0.002325
-0.105
0.235
-0.024675
-0.1473
Portfolio Risk
• Portfolio return = (0.5*0.135) + (0.5*0.075)
=0.105
• Coca-Cola
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Returns
0.12
0.23
0.15
0.07
0.25
0.32
0.11
-0.09
0.26
-0.12
0.29
0.03
0.135
Portfolio Risk
(R-Rbar)
-0.015
0.095
0.015
-0.065
0.115
0.185
-0.025
-0.225
0.125
-0.255
0.155
-0.105
(R-Rbar)^2
0.000225
0.009025
0.000225
0.004225
0.013225
0.034225
0.000625
0.050625
0.015625
0.065025
0.024025
0.011025
0.2281
variance= 0.0207
Std = 0.1440
Portfolio Risk
• Pepsi
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Returns
0.24
0.08
0.21
0.15
0.03
-0.09
-0.35
0.21
-0.15
0.2
0.06
0.31
0.075
(R-Rbar)
0.165
0.005
0.135
0.075
-0.045
-0.165
-0.425
0.135
-0.225
0.125
-0.015
0.235
(R-Rbar)^2
0.027225
2.5E-05
0.018225
0.005625 variance= 0.0364
0.002025
0.027225 std= 0.1909
0.180625
0.018225
0.050625
0.015625
0.000225
0.055225
0.4009
Portfolio Risk
𝟐 𝟐 𝟐 𝟐 𝟐
𝝈𝒑 = 𝝈𝒙 𝒘𝒙 +𝝈𝒚 𝒘𝒚
+𝟐𝒘𝒙𝒘𝒚 𝑪𝒐𝒗𝒂𝒓𝒙𝒚
𝝈𝟐𝒑 = (𝟎. 𝟎𝟐𝟎𝟕 ∗ 𝟎. 𝟓𝟐) + (𝟎. 𝟎𝟑𝟔𝟒 ∗ 𝟎. 𝟓𝟐) + 𝟐(𝟎. 𝟓 ∗ 𝟎. 𝟓 ∗ −𝟎. 𝟎𝟏𝟐𝟑) =
𝟎. 𝟎𝟎𝟖𝟐
𝝈𝒑 = 𝟎.𝟎𝟎𝟖𝟐 = 𝟎.𝟎𝟗𝟎𝟔
Example
• Suppose you invest 60% of your portfolio in
Exxon Mobil and 40% in Coca Cola. The
expected dollar return on your Exxon Mobil
stock is 10% and on Coca Cola is 15%. The
expected return on your portfolio is:
Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola.
The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is
15%. The standard deviation of their annualized daily returns are 18.2% and
27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the
portfolio variance.
Portfolio standard deviation
Risk Diversification: Systematic and
Unsystematic Risk
Unique
risk
Market risk
0
5
10
Number of Securities
15
Beta and Unique Risk
1. Total risk =
diversifiable risk +
market risk
2. Market risk is
measured by beta,
the sensitivity to
market changes
Expected
stock
return
beta
+10%
-10%
- 10%
+10%
-10%
Expected
market
return
Beta and Unique Risk
• Market Portfolio - Portfolio of all assets in the
economy. In practice a broad stock market
index, such as the GSE All Share Index, S&P
Composite are used to represent the market.
• Beta - Sensitivity of a stock’s return to the
return on the market portfolio.
Beta and Unique Risk
 im
Bi  2
m
Investment Opportunity Sets (2 Assets)
given Different Correlations
20
Cor = - 1.0
R
Cor = - 0.25
15
Portfolio return, %
Cor = + 0.50
Cor = + 1.0
10
Cor = - 1.0
L
5
0
0
5
10
15
20
Porfolio risk (Stdev, %)
25
30
Mean-Variance Criterion
• A risk-averse investor will prefer a portfolio
with the highest expected return for a given
level of risk or prefer a portfolio with the
lowest level of risk for a given level of
expected return. In portfolio theory, this is
referred to as the principle of dominance
Investment Opportunity Set: The NAsset Case
• An efficient portfolio is one that has the
highest expected returns for a given level of
risk. The efficient frontier is the frontier
formed by the set of efficient portfolios. All
other portfolios, which lie outside the efficient
frontier, are inefficient portfolios.
Investment Opportunity Set: The NAsset Case
Return
R
D
x
C
x
B
Q
x
P
x
x
x
x
A
Risk, 
Efficient Frontier
Goal is to move up
and left.
Return
WHY?
B
ABN AB
A
N
Risk
Efficient Frontier
Return
Low Risk
High Risk
High Return
High Return
Low Risk
High Risk
Low Return
Low Return
Risk
Efficient Frontier
Return
Low Risk
High Risk
High Return
High Return
Low Risk
High Risk
Low Return
Low Return
Risk
A Risk-Free Asset and a Risky Asset
• A risk-free asset or security has a zero
variance or standard deviation.
• Return and risk when we combine a risk-free
and a risky asset:
E ( Rp )  wE ( R j )  (1  w) R f
 p  w j
Security Market Line
For a given amount of
systematic risk, SML
shows the required
rate of return.
Return
Market Return = rm
Efficient Portfolio
Risk Free
Return
.
=
rf
Risk
Security Market Line
Return
Market Return = rm
.
Efficient Portfolio
Risk Free
Return
=
rf
1.0
BETA
A security market line (SML) is a line that visually represents
the relationship between risk and the expected or the required
rate of return on an asset.
Return
Security Market Line
SML
rf
1.0
BETA
SML Equation = rf + B ( rm - rf )
Capital Asset Pricing Model
R = rf + B ( rm - rf )
CAPM
Capital Asset Pricing Model
Given beta of 2.5 and risk premium on the
market as 5%, if the risk free rate is 19%, use
the CAPM to find the opportunity cost.
Capital Asset Pricing Model
• Ama is considering the following investments.
The current rate on Tbill is 5.5%, and the
expected return for the market is 11%. Using
CAPM, what rates of return should Ama
require for each individual security?
Stock
Beta
A
0.75
B
1.4
C
0.95
D
1.25
Capital Asset Pricing Model (CAPM)
• The capital asset pricing model (CAPM) is a
model that provides a framework to
determine the required rate of return on an
asset and indicates the relationship between
return and risk of the asset.
• Assumptions of CAPM
–
–
–
–
–
Market efficiency
Risk aversion and mean-variance optimisation
Homogeneous expectations
Single time period
Risk-free rate
Implications of CAPM
• Investors will always combine a risk-free asset
with a market portfolio of risky assets. They will
invest in risky assets in proportion to their market
value.
• Investors will be compensated only for that risk
which they cannot diversify. This is the marketrelated (systematic) risk.
• Beta, which is a ratio of the covariance between
the asset returns and the market returns divided
by the market variance, is the most appropriate
measure of an asset’s risk.
• Investors can expect returns from their
investment according to the risk. This implies a
linear relationship between the asset’s expected
return and its beta.
Limitations of CAPM
• It is based on unrealistic assumptions.
• It is difficult to test the validity of CAPM.
• Betas do not remain stable over time.
The Arbitrage Pricing Theory (APT)
• In APT, the return of an asset is assumed to
have two components: predictable (expected)
and unpredictable (uncertain) return. Thus,
return on asset j will be:
E ( R j )  R f + UR
where Rf is the predictable return (risk-free return on a zero-beta asset) and UR
is the unanticipated part of the return. The uncertain return may come from
the firm specific information and the market related information:
E ( R j )  R f  ( 1 F1   2 F2   3 F3 
  n Fn )  URs
Steps in Calculating Expected Return
under APT
• Factors:
–
–
–
–
–
industrial production
changes in default premium
changes in the structure of interest rates
inflation rate
changes in the real rate of return
• Risk premium
• Factor beta