MONASH
BUSINESS
SCHOOL
BFF2140
CORPORATE FINANCE I
Joshua Shemesh
MONASH
BUSINESS
SCHOOL
Teaching Week Nine
Risk and Return II: Portfolio Theory and
Capital Asset Pricing Model
Readings
Chapter 12, pp. 351 – 376
Additional Readings: Investments - Zvi Bodie, Alex Kane, Alan J. Marcus 2014
Please read Chapter 7 pp. 206-229
MONASH
BUSINESS
SCHOOL
Learning Objectives
Calculate expected returns and volatility of a portfolio.
Explain the relation between systematic risk and the market
portfolio
Know how to calculate systematic risk
Understand the Capital Asset Pricing Model and use it to
compute the cost of equity capital for stocks.
Risk and Return
Last Week (Recap):
1. Single asset with probability distribution:
2. Single asset with time-series data:
3. Portfolio risk and return (2 risky assets)
Diversification with Multiple Assets
For a diversified portfolio, the variance of the
individual assets contributes little to the risk of
the portfolio.
The risk depends largely on the covariance
between the returns on the assets.
Portfolio Risk: The case of 3 risky assets
2 3 3
Portfolio Variance P wi w j rij i j
i 1 j 1
2 (w ) 2 (w ) 2 (w ) 2
Portfolio Variance P 1 1 2 2 3 3
(2 w1 w2 1 2 r1,2 )
(2 w1 w3 1 3 r1,3 )
(2 w2 w3 2 3 r2,3 )
Portfolio Risk: The case of n risky assets
For example 4 assets
E(RP) = weighted average = W1E(R1) + W2E(R2) + W3E(R3) + W4E(R4)
σ2 P = (W1s1)2 + (W2s2)2 + (W3s3)2 + (W4s4)2
+ 2W1W2σ1σ2r1,2 + 2W1W3σ1σ3r1,3 + 2W1W4σ1σ4r1,4 + 2W2W3σ2σ3r2,3 + 2W2W4σ2σ4r2,4
+ 2W3W4σ3σ4r3,4
Recall σP is the square root of σ2P
For example 5 assets
E(RP) = weighted average = W1E(R1) + W2E(R2) + W3E(R3) + W4E(R4) + W5E(R5)
σ2P = (W1s1)2 + (W2s2)2 + (W3s3)2 + (W4s4)2 + (W5s5)2
+ 2W1W2σ1σ2r1,2 + 2W1W3σ1σ3r1,3 + 2W1W4σ1σ4r1,4 + 2W1W5σ1σ5r1,5 + 2W2W3σ2σ3r2,3
+ 2W2W4σ2σ4r2,4 + 2W2W5σ2σ5r2,5 + 2W3W4σ3σ4r3,4 + 2W3W5σ3σ5r3,5 + 2W4W5σ4σ5r4,5
Recall σP is the square root of σ2P
ETC. ETC. ETC.
Quiz - discussion
• What would happen to your portfolio risk
equation when you invest in 3 assets, A, B and C
(the risk free rate)?
2 (w ) 2 (w ) 2 (w ) 2
Portfolio Variance P 1 1 2 2 3 3
(2 w1 w2 1 2 r1,2 )
(2 w1 w3 1 3 r1,3 )
(2 w2 w3 2 3 r2,3 )
Note: σRF = 0
Diversification and Portfolio Risk
As more shares are added, each new share has a
smaller risk-reducing impact
By forming portfolios, we can eliminate some of the
riskiness of individual shares
However, there is a minimum level of risk that
cannot be diversified away and that is known as
systematic risk
Eugine Fama (1976)
In practice, p falls very slowly after about 12-16 shares are included (Fama, 1976)
In theory the greater the number of assets
In a portfolio the greater the reduction in
Non-systematic risk
Std Dev of portfolio return
Total Non-systematic risk
Risk
Systematic risk
Number of shares in portfolio
MPT (Markowtiz Portfolio Theory)
The idea of diversification is the basic idea behind modern
portfolio theory (also referred to as Markowtiz Portfolio theory
after it’s founder).
MPT is theory of selecting an optimal combination of assets that
are expected to provide the highest possible expected return for
a given level of RISK (or least risk for a given level of return)
Markowitz Portfolio Theory (1952)
MPT:
Quantifies risk
Derives the expected rate of return for a portfolio of
assets and an expected risk measure
Shows that the variance of the rate of return is a
meaningful measure of portfolio risk
Derives the formula for computing the variance of a
portfolio, showing how to effectively diversify a portfolio
Includes only risky assets in MPT
Markowitz Portfolio Theory
MPT can be extended to
1) two stock case (introduced last week)
2) multiple stock case
3) What happens if we add in a riskless asset?
(1) Calculating Portfolio Risk/Return –
Two Stock Case
The risk and return for a portfolio made up of
AGL and FOA (WAGL = proportion of stock
AGL in portfolio) can be calculated as:
Expected return
E(Rp) = wAGLE(RAGL)+ wFOA E(R FOA) Recall σAGL,FOA is equal
to σAGLσFOArAGL,FOA
Standard deviation
p = (wAGL2σ2AGL + wFOA2σ2FOA
+ 2wAGLwFOAσAGL,FOA)1/2
AGL, FOA Risk/Return
Summary Statistics
AGL FOA
Mean return, E(Ri) 0.0016 0.0025
Standard deviation, i 0.0154 0.0187
Covariance, AGL,FOA 0.00011
To illustrate let us examine the mean and std dev of all possible portfolios, assuming the
weights range from 0 to 1 in increments of 5% (using the formula from slide 14)
Calculating Portfolio Risk/Return –
Two Stock Case A copy of the excel file to construct this frontier is available on Moodle
Portfolio Proportion of portfolio in Standard Mean
Number AGL FOA Deviation Return
(percent) (percent)
1 100% 0%
0.01540 0.0016
2 95% 5%
0.01501 0.0016
3 90% 10%
0.01468 0.0017
4 85% 15%
0.01440 0.0017
5 80% 20%
0.01418 0.0018
6 75% 25%
0.01402 0.0018
7 70% 30%
0.01392 0.0019
8 65% 35%
0.01390 0.0019
9 60% 40%
0.01393 0.0020
10 55% 45%
0.01404 0.0020
11 50% 50%
0.01420 0.0021
12 45% 55%
0.01443 0.0021
13 40% 60%
0.01472 0.0021
14 35% 65%
0.01506 0.0022
15 30% 70%
0.01546 0.0022
16 25% 75%
0.01590 0.0023
17 20% 80%
0.01639 0.0023
18 15% 85%
0.01691 0.0024
19 10% 90%
0.01748 0.0024
20 5% 95%
0.01807 0.0025
21 0% 100%
0.01870 0.0025
Risk/Return Tradeoff –
Different Combinations of AGL and FOA
0.0027
Minimum Variance Frontier Portfolio 21
(100% in FOA)
0.0025
Portfolio 8
Minimum
0.0023
Variance
Portfolio
Return (%) 0.0021
0.0019
Portfolio 1
0.0017
(100% in AGL)
0.0015
0.01300 0.01400 0.01500 0.01600 0.01700 0.01800 0.01900
Std Deviation (Risk)
Quiz discussion
Why in the previous graph would you not invest in
portfolio 4?
Because portfolio 4 (85% AGL, 15% FOA) is
inefficient.
Portfolio 4 does not lie on the efficient frontier.
You can achieve a higher return at the same
level of risk by selecting a portfolio that lies on
the efficient frontier (refer to slide 17)
(2) The Efficient Set for Many Securities
expected
return
Individual Assets/Portfolios
Consider a world with many risky assets; we can still
identify the opportunity set of risk-return combinations
of various portfolios much like we did in the case of 2
risky assets.
The Efficient Set for Many Securities
expected
return
minimum
variance
portfolio
Individual Assets/Portfolios
Given the opportunity set we can identify the
minimum variance portfolio.
Construction of a Portfolio
The efficient frontier
Given risk-aversion, each investor will try to secure
a portfolio on the efficient frontier. The efficient
frontier is determined on the basis of dominance.
Dominance implies that a portfolio is efficient if:
No other portfolio has a higher return for the same risk,
or
No other portfolio has a lower risk for the same return.
The Efficient Set for Many Securities
expected
return
minimum
variance
portfolio
Individual Assets/Portfolios
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
3) The Capital Market Line
Introducing a risk-free asset, the opportunity set for
investors is expanded and results in a new efficient
frontier: the Capital Market Line (CML).
The CML represents the efficient set of all portfolios
that provides the investor with the best possible
investment opportunities when a risk-free asset is
available
Riskless Borrowing and Lending
expected
return
Rf
Now investors can allocate their money across
the Risk free asset and a balanced managed fund
Three Important Facts
• Risk free asset (Rf) has a standard deviation of zero
• The minimum variance portfolio (MV) lies on the boundary of the
feasible set at a point where the variance (std dev) is at a minimum
• The optimal (market) portfolio (M) lies on the feasible set and on a
tangent from the risk-free asset
(note: the optimal portfolio will have the highest Sharpe measure)
Risk-Return Possibilities with Leverage
To attain a higher expected return than is available at
point M (the tradeoff is an exchange for accepting
higher risk)
Either
1) Invest along the efficient frontier beyond point M,
such as point D, or
2) Add leverage to the portfolio by borrowing money at
the risk-free rate and investing in the risky portfolio
at point M
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient Frontier
D
Implications of the Market Portfolio
The CML describes the linear relationship
between expected returns and standard deviations
for different combinations of the market portfolio
and the risk-free security
Because portfolio M lies at the point of tangency,
it has the highest portfolio possibility line
Everybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CML
The market portfolio is the portfolio of all risky
investments, held in proportion to their value
The sum of all investors’ portfolios must equal the
portfolio of all risky securities in the market.
Example 1
• Assume that the risk-free rate of return is 8% per
annum. The risky market portfolio provides an
expected return of 16% per annum and has a standard
deviation of 22%. Mr Pink has chosen a complete
portfolio composed of risk-free assets and the risky
market portfolio which provides an expected return of
10% per annum with a standard deviation of 5.50%
• Graph the Capital Market Line.
Example 1: Solution
Expected
Return (%)
CML
RISKY
MARKET
16%
10%
PINK
Rf =8%
5.5% 22%
Standard
Deviation (%)
Example 1: Solution continued
• Mr Black thinks he can do better than the market by restricting his
investments to three risky stocks which he thinks will provide the best returns.
He chooses a portfolio which provides an expected return of 18% per annum
and 28% standard deviation. Making use of the Sharpe Measure demonstrate
that Mr Black has not found a risky portfolio which is better than the market.
• SOLUTION
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.
• A measure of a security’s systematic risk
• Measures the responsiveness of a security to
movements in the market portfolio.
Cov ( Ri , RM )
i
2 ( RM )
Interpreting Beta
• If beta = 1.0, stock is as risky as the market.
• If beta > 1.0, stock is riskier than the market.
• If beta < 1.0, stock is less risky than the market.
Estimating β with Regression
Security Returns
Slope = i
Return on
market %
Ri = a i + biRm + ei
The Formula for Beta
( Ri )
Cov( Ri , RM )
i
( RM )
2
( RM )
Clearly, your estimate of beta will depend upon
your choice of a proxy for the market portfolio.
Relationship between Risk and
Expected Return (CAPM)
• Expected return on an individual security:
R i RF β i ( R M RF )
Market Risk Premium
This applies to individual securities held within well-
diversified portfolios.
Expected Return on a Security
• This formula is called the Capital Asset Pricing Model
(CAPM):
E ( Ri ) RF β i ( R M RF )
Expected
Risk- Beta of the Market risk
return on = + ×
free rate security premium
a security
• When i = 0, the expected return is RF. Thus, adding risk-free assets
does not change the risk of a risky portfolio.
• Assume i = 1, then R i R M , it’s the market portfolio.
Relationship Between Risk & Return
Expected return
R i RF β i ( R M RF )
RM
RF
1.0
Security Market Line
SML is a graphic representation of the CAPM
R i RF β i ( R M RF )
It is the line which describes the linear
relationship between expected returns for
individual securities (and portfolios) and
systematic risk, as measured by beta
Security Market Line
In equilibrium, all securities must be priced such that their returns lie on the Security Market
Line (SML)
R i RF β i ( R M RF )
Expected return, ki
Km Risk
Market risk premium
premium
Rf
Risk-free
return
m = 1.0
Systematic risk (i)
Security Market Line
SML
Share X (under priced)
Expected return
Direction of
movement Direction of
movement
Rf
Share Y (over priced)
Systematic risk (i)
Security Market Line and inflation
If investors raise inflationary expectations by 3%, what
would happen to the Security Market Line?
Security Market Line and risk
Suppose inflation did not change, but risk aversion
increased enough to cause the market risk premium to
increase by 3 percentage points. What would happen to the
Security Market Line?
Portfolio Beta
Calculated as a weighted average of the betas of the
individual assets in the portfolio:
n
p wi i
i 1
where:
n = number of assets in the portfolio
wi = proportion of the current market value
of portfolio p constituted by the i th asset
Example 2
Share Percentage of Beta Expected
Portfolio % Return %
1 40 1.00 12
2 25 0.75 11
3 35 1.30 15
A) What is the expected return and beta for the portfolio.
B) Given the information above, draw the security market line
and show where the securities fit on the graph. Assume that the
risk-free rate is 8% and that the expected return on the market
portfolio is 12%. How would you interpret these findings?
Example 2 Solution
n
(a) R p wi Ri weighted average
i 1
w1R1 w2 R2 w3 R3
0.40(0.12) 0.25(0.11) 0.35(0.15)
0.128 or 12.80%
n
p wi i weighted average
i 1
w11 w2 2 w3 3
0.40(1) 0.25(0.75) 0.35(1.30)
1.04
Example 2 Solution
R i R F i (k m R F )
b)
R1 0.08 1.00(0.12 0.08) 0.12 or 12%
R2 0.08 0.75(0.12 0.08) 0.11 or 11%
R3 0.08 1.30(0.12 0.08) 0.13 or 13%
Example 2 Solution
Ri
SML
3
RM = 12%
1
2
Rf = 8%
0.75 1 1.3 i
Shares 1 and 2 seem to be correctly priced according to the CAPM as they plot
directly on the SML, hence they are fairly priced and provide a fair return, given
their level of systematic risk. However share 3 is earning more than a fair return
(plots above the SML) and so appears to be undervalued.
Example 3: Portfolio Theory
You are given the following variance, covariance matrix for Sifty Sasha’s Burbon
Bar (SSBB), Warren’s Winery (WW) and the market portfolio.
SSBB WW MARKET
SSBB 0.0221
WW 0.0011 0.0165
MARKET 0.0040 0.0020 0.0100
Additionally you are informed that the expected return(standard deviation) for
SSBB and WW are 10.40% (14.87%) and 9.20% (12.85%) respectively. The risk
free rate of interest is 8% per annum. The expected market return is 14% and you
have a total of $500,000 available for investment.
How to read variance, co variance matrix.
SSBB WW MARKET
SSBB variance
WW covariance variance
variance
MARKET covariance covariance
Example 3
a). Justify that the risk and return estimates quoted are
correct.
b). Assume that you are required to create a portfolio
consisting of 40% investment in SSBB and 60%
investment in WW. Calculate the Expected return and
standard deviation for this portfolio.
c). Assume that you are now required to create a new
portfolio that consists of investing 200,000 in SSBB
shares, $200,000 in WW shares with the balance being
invested in the risk free asset. Calculate the expected
return and standard deviation of this portfolio.
d). Which portfolio combination is best? Justify your decision
Example 3 (a)
a). Justify that the risk and return estimates quoted are correct.
i, M 0.0040 0.0020
i SSBB 0.4 WW 0.2
2
M 0.0100 0.0100
Justify that expected returns are correct
RSSBB RF RM RF 0.08 0.40(0.14 0.08) 0.1040
RWW RF RM RF 0.08 0.20(0.14 0.08) 0.0920
Justify that standard deviations are correct
2
SSBB SSBB 0.0221 0.1487
2
WW WW 0.0165 0.1285
Example 3 (b)
b) Assume that you are required to create a portfolio consisting of 40% investment in
SSBB and 60% investment in WW. Calculate the Expected return and standard
deviation for this portfolio.
RPortfolio wSSBB RSSBB wWW RWW
(0.4 0.1040) (0.6 0.092) 0.0968
2 (w
Portfolio Variance P ) 2 (w ) 2
SSBB SSBB WW WW
(2 wSSBB wWW SSBB WW rSSBB ,WW )
P2 (0.4 0.1487 ) 2 (0.6 0.1285) 2 (2 0.4 0.6 0.0011)
P 0.1001
Example 3 (c)
c) Assume that you are now required to create a new portfolio that consists of investing
200,000 in SSBB shares, $200,000 in WW shares with the balance being invested in the
risk free asset. Calculate the expected return and standard deviation of this portfolio.
RPortfolio wSSBB RSSBB wWW RWW wRF RRF
(0.4 0.1040) (0.4 0.092) (0.2 0.08) 0.0944
2 (w
Portfolio Variance P 2 2 2
SSBB SSBB ) ( wWW WW ) ( wRF RF )
(2 wSSBB wWW SSBB WW rSSBB ,WW )
(2 wSSBB wRF SSBB RF rSSBB , RF )
(2 wWW wRF WW RF rWW , RF )
P2 (0.4 0.1487 ) 2 (0.4 0.1285) 2 (2 0.4 0.4 0.0011)
P 0.0808
Example 3 (d)
d) Which portfolio combination is best? Justify your decision
R i - R RF 0.0968 - 0.08
SHARPEPortfolio B 0.1678
i 0.1001
R i - R RF 0.0944 - 0.08
SHARPEPortfolio C 0.1782
i 0.0808
σ 0.1001
CVPortfolio B 1.034
R 0.0968
σ 0.0808
CVPortfolio C 0.856
R 0.0944
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