Chapter 04 Risk, Return, and The Portfolio Theory
Chapter 04 Risk, Return, and The Portfolio Theory
Chapter 04 Risk, Return, and The Portfolio Theory
And
the Portfolio Theory
Definition of Return and Risk
Meaning of Return
• Returns measure the financial results of an
investment
• Returns may be historical or prospective
(anticipated)
• Returns can be expressed in:
• Dollar/Birr terms
• Percentage terms
2
Example:
What is the return on an investment that costs Br. 1,000
and is sold after 1 year for Br. 1,100?
• Dollar/Birr Return =
Birr Received - Birr Invested
Br. 1,100 - Br. 1,000 = Br. 100.
• Percentage Return =
Birr Return Birr Invested
Br. 100 Br. 1,000 = 0.10 = 10%.
3
Meaning of Risk
4
Measuring Risk and Return of a
Single Asset
• The characteristics of individual asset
that are of interest are the:
• Expected Return
• Variance or Standard Deviation
• Coefficient of Variation
5
Measuring Expected Return
n
k̂ = k i Pi
i =1
Where,
ḱ = expected rate of return (Central Tendency)
ki = possible future returns
Pi = probability of the return occurring
n = total number of possibilities 6
Measuring Risk
Probability
Distribution
Stock X
Stock Y
Rate of
-20 0 15 50 return (%)
7
• Which stock is riskier? Why?
Measuring Risk: A Closer Look
Measures of Dispersion
• spreading or scattering of the possible outcomes in
the probability distribution
• measures how likely it is that an outcome will vary
from the central tendency
• Two of the widely used measures of dispersion are
variance and standard deviation
Variance
2
k
n
2
i k Pi . 8
i 1
Measuring Risk: A Closer...
• Standard deviation measures the stand-alone risk
of an investment.
• The larger the standard deviation, the higher the
probability that returns will be far below the
expected return.
• Coefficient of variation (CV) is an alternative
measure of stand-alone risk.
• CV = σ/ḱ
9
Example:
Given: Investments in X and Y.
Rate of Return
Scenario Probability Stock X Bond Y
Recession 33.3% -7% 17%
Normal 33.3% 12% 7%
Boom 33.3% 28% -3%
11
Risk Attitudes (Behaviors)
Return/
Reward
Risk Averse
Risk
Indifference
Risk Seeking
Risk
Portfolio Return
Stock X Bond Y
Expected Returns (ḱ) 11% 7%
Risk (σ), 14.31% 8.16%
0 15% Return
1 35% ; 2 20%.
16
Exercise 1:
Security
A B
Expected Returns (ḱ) 14% 11.5%
Risk (σ), 10.7% 1.5%
Diversifiable Risk;
Nonsystematic Risk; Firm
Specific Risk; Unique
Risk
Portfolio risk
Non-diversifiable risk;
Systematic Risk; Market
Risk
No. of Assets
Thus, diversification can eliminate some, but not all
of the risk of individual securities.
Stand-alone Market Diversifiable
Risk = Risk + Risk
• Market risk:
• Is defined as the contribution of a security to the
overall riskiness of the portfolio.
• Is relevant for stocks held in well-diversified
portfolios.
• Is measured by a stock‟s beta coefficient (β), which
measures a stock‟s volatility relative to the market.
• What is the relevant risk for a stock held in
isolation? 24
Definition of Risk When Investors Hold
the Market Portfolio
• Researchers have shown that the best measure
of the risk of a security in a large portfolio is
the beta (β) of the security.
• Beta measures the responsiveness of a security
to movements in the market portfolio.
Cov(ki , k M )
i
(k M )
2
25
How are betas calculated?
ki = a i + ikm + ei 27
How is beta interpreted?
k i k F β i (k M k F )
Expected return
kM
kF
33
1.0
Challenges to the CAPM
1. Market Imperfections
2. Anomalies:
• Size Effect
• Price-Earnings (P/E) & Market to Book
Ratios
• The January Effect
3. Multifactor Models
• Arbitrage Pricing Model (APM)
34
The Arbitrage Pricing Model (APM)
• The Arbitrage Pricing Theory: 1976, Stephen Ross
• Assumes:
• several factors affect Expected Return
• does not specify factors
• Implications
• Expected Return is a function of several factors,
Fi, each with its own β.
ki k f 1F1 2 F2 3 F3 .... N FN 35
Chapter
Ends
36
Appendix to the Chapter
Portfolio Return
5% 7.0% 7.2%
10% 5.9% 7.4%
15% 4.8% 7.6% 12.0%
20% 3.7% 7.8% 11.0% 100%
25% 2.6% 8.0% 10.0% stocks
30% 1.4% 8.2% 9.0%
35% 0.4% 8.4%
8.0%
40% 0.9% 8.6% 100%
7.0%
45% 2.0% 8.8% bonds
50.00% 3.08% 9.00% 6.0%
55% 4.2% 9.2% 5.0%
60% 5.3% 9.4% 0.0% 5.0% 10.0% 15.0% 20.0%
65% 6.4% 9.6%
70% 7.6% 9.8% Portfolio Risk (standard deviation)
75% 8.7% 10.0%
80% 9.8% 10.2% We can consider other portfolio
85% 10.9% 10.4%
90% 12.1% 10.6% weights besides 50% in stocks &
95% 13.2% 10.8% 50% in bonds …
100% 14.3% 11.0%
The Efficient Set for Two Assets
%%ininstocks
stocks Risk
Risk Return
Return Portfolo Risk and Return Combinations
Portfolio Return
0%0%
0% 8.2%
8.2%
8.2% 7.0%
7.0%
7.0%
5%5%
5% 7.0%
7.0%
7.0% 7.2%
7.2%
7.2% 12.0%
10%
10%
10% 5.9%
5.9%
5.9% 7.4%
7.4%
7.4% 11.0%
15%
15%
15% 4.8%
4.8%
4.8% 7.6%
7.6%
7.6% 10.0% 100%
20%
20%
20% 3.7%
3.7%
3.7% 7.8%
7.8%
7.8% 9.0% stocks
25%
25%
25% 2.6%
2.6%
2.6% 8.0%
8.0%
8.0% 8.0%
30%
30%
30% 1.4%
1.4%
1.4% 8.2%
8.2%
8.2% 7.0%
100%
35%
35%
35% 0.4%
0.4%
0.4% 8.4%
8.4%
8.4% 6.0%
bonds
40%
40%
40% 0.9%
0.9%
0.9% 8.6%
8.6%
8.6% 5.0%
45%
45%
45% 2.0%
2.0%
2.0% 8.8%
8.8%
8.8% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
50%
50%
50% 3.1%
3.1%
3.1% 9.0%
9.0%
9.0%
55%
55%
55% 4.2%
4.2%
4.2% 9.2%
9.2%
9.2% Portfolio Risk (standard deviation)
60%
60%
60% 5.3%
5.3%
5.3% 9.4%
9.4%
9.4%
65%
65%
65% 6.4%
6.4%
6.4% 9.6%
9.6%
9.6%
70%
70%
70% 7.6%
7.6%
7.6% 9.8%
9.8%
9.8%
75%
75%
75% 8.7%
8.7%
8.7% 10.0%
10.0%
10.0% We can consider other
80%
80%
80% 9.8%
9.8%
9.8% 10.2%
10.2%
10.2%
85%
85%
85% 10.9%
10.9%
10.9% 10.4%
10.4%
10.4% portfolio weights besides 50%
90%
90%
90% 12.1%
12.1%
12.1% 10.6%
10.6%
10.6% in stocks & 50% in bonds …
95%
95%
95% 13.2%
13.2%
13.2% 10.8%
10.8%
10.8%
100%
100% 14.3%
14.3% 11.0%
11.0%
The Efficient Set for Two Assets
% in stocks Risk Return
0% 8.2% 7.0%
5% 7.0% 7.2% Portfolo Risk and Return Combinations
Portfolio Return
10% 5.9% 7.4%
12.0%
15% 4.8% 7.6%
11.0%
20% 3.7% 7.8%
10.0% 100%
25% 2.6% 8.0% stocks
9.0%
30% 1.4% 8.2% 8.0%
35% 0.4% 8.4% 7.0% 100%
40% 0.9% 8.6% 6.0% bonds
45% 2.0% 8.8% 5.0%
50% 3.1% 9.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
55% 4.2% 9.2% Portfolio Risk (standard deviation)
60% 5.3% 9.4%
65% 6.4% 9.6%
70% 7.6% 9.8% Note that some portfolios are “better”
75% 8.7% 10.0% than others. They have higher returns
80% 9.8% 10.2% for the same level of risk or less. These
85% 10.9% 10.4%
90% 12.1% 10.6% compromise the efficient frontier.
95% 13.2% 10.8%
100% 14.3% 11.0%
Two-Security Portfolios with Various
Correlations
return
100%
= -1.0 stocks
= 1.0
100%
= 0.2
bonds
41
The Efficient Set for Many
Securities
return
Individual Assets
P
Consider a world with many risky assets; we can still
identify the opportunity set of risk-return combinations 42
of various portfolios.
The Efficient Set for Many
Securities
return
minimum
variance
portfolio
Individual Assets
P
return
minimum
variance
portfolio
Individual Assets
P
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
Optimal Risky Portfolio with a
Risk-Free Asset
return
100%
stocks
rf
100%
bonds
return
100%
stocks
Balanced
fund
rf
100%
bonds
return
rf
P
With a risk-free asset available and the efficient
frontier identified, we choose the capital allocation
line with the steepest slope
Market Equilibrium
return
M
rf
P
With the capital allocation line identified, all investors choose
a point along the line—some combination of the risk-free
asset and the market portfolio M. In a world with
homogeneous expectations, M is the same for all investors.
The Separation Property
return
M
rf
P
The Separation Property states that the market portfolio,
M, is the same for all investors—they can separate their
risk aversion from their choice of the market portfolio.
The Separation Property
return
M
rf
P
Investor risk aversion is revealed in their choice of
where to stay along the capital allocation line—not in
their choice of the line.
Market Equilibrium
return
100%
stocks
Balanced
fund
rf
100%
bonds
Just where the investor chooses along the Capital Asset
Line depends on his risk tolerance. The big point though
is that all investors have the same CML.
Market Equilibrium
return
100%
stocks
Optimal
Risky
Portfolio
rf
100%
bonds
All investors have the same CML because they all have
the same optimal risky portfolio given the risk-free rate.
The Separation Property
return
100%
stocks
Optimal
Risky
Porfolio
rf
100%
bonds
The separation property implies that portfolio choice
can be separated into two tasks: (1) determine the
optimal risky portfolio, and (2) selecting a point on the
CML.
Optimal Risky Portfolio with a Risk-
Free Asset
return
100%
stocks
55