Session 1
Portfolio Theory and the Capital Asset Pricing
Model (CAPM)
FIN 625: Corporate Finance
Learning Objectives
LO 1: Explain and calculate expected return and
variance/standard deviation of an individual
security.
LO 2: Explain and calculate covariance and
correlation.
LO 3: Distinguish between systematic and
unsystematic risks.
LO 4: Explain the idea of diversification and its
importance for portfolio theory.
LO 5: Explain the implications of Capital Market
Line (CML)
LO 6: Explain the Security Market Line and the
Capital Asset Pricing Model (CAPM)
Outline
1.
2.
3.
4.
5.
6.
Return and Risk of Individual Securities
Covariance or Correlation between Two
Securities
Return and Risk of Portfolios
Systematic versus Unsystematic Risk
Capital Asset Pricing Model (CAPM)
Expected Return and Cost of Capital
1. The Return and Risk of Individual
Securities


We calculate the expected return
We use variance or standard deviation to
measure (total) risk
Example
Consider the following two risky-asset
world. There is a 1/3 chance of each
state of the economy, and the only
assets are a stock fund and a bond fund.
Scenario
Recession
Normal
Boom
Rate of Return
Probability Stock Fund Bond Fund
33.3%
-7%
17%
33.3%
12%
7%
33.3%
28%
-3%
Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond
Rate of
Return
17%
7%
-3%
7.00%
0.0067
8.2%
Fund
Squared
Deviation
0.0100
0.0000
0.0100
Deviation compares return in each state to the expected
return.
Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
1
1
1
E ( rS )  ( 7%)  (12%)  (28%)
3
3
3
E ( rS )  11%
Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
2
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
(-7%-11%) =.0324
Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
1
0.0205 = (0.0324 + 0.0001 + 0.0289)
3
Standard Deviation
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
14.3% = 0.0205
Note that stocks have a higher expected return than
bonds and higher risk.
Risk & Return:
Historical Perspective (1926-2014)
2. Covariance or Correlation Between
Two Securities

Covariance or correlation measure the comovement of two securities


Positive covariance or correlation indicates
that the two securities move in similar
directions
Negative covariance or correlation indicates
that the two securities move in different
directions
Covariance
Scenario
Recession
Normal
Boom
Sum
Covariance
Stock
Bond
Deviation Deviation
-18%
10%
1%
0%
17%
-10%
Product
-0.0180
0.0000
-0.0170
Weighted
-0.0060
0.0000
-0.0057
-0.0117
-0.0117
Weighting takes the product of the deviations multiplied
by the probability of that state.
Correlation
 ab 
Cov (a, b)
 a b
 .0117
 ab 
 0.998
(.143)(.082)
3. Return and Risk of Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
Let us turn now to the risk-return tradeoff of a portfolio
that is 50% invested in bonds and 50% invested in
stocks.
Portfolios
Rate of Return
Stock fund Bond fund Portfolio squared deviation
-7%
17%
5.0%
0.0016
12%
7%
9.5%
0.000025
28%
-3%
12.5%
0.001225
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted
average of the returns on the stocks and bonds in the
portfolio:
r w r w r
P
B B
S S
5% = 50%(-7%) + 50%(17%)
Portfolios
Rate of Return
Stock fund Bond fund Portfolio squared deviation
-7%
17%
5.0%
0.0016
12%
7%
9.5%
0.000025
28%
-3%
12.5%
0.001225
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
9.0%
0.0010
3.08%
The expected rate of return on the portfolio is a
weighted average of the expected returns on the
securities in the portfolio.
E ( rP )  w B E ( rB )  wS E ( rS )
9% = 50%(11%) + 50%(7%)
Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio squared deviation
-7%
17%
5.0%
0.0016
12%
7%
9.5%
0.000025
28%
-3%
12.5%
0.001225
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
9.0%
0.0010
3.08%
The variance of the rate of return on the two risky
assets portfolio is
σ P2  (w B σ B )2  (w S σ S )2  2(w B σ B )(w S σ S )ρBS
where BS is the correlation between the returns on
the stock and bond funds.
Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio squared deviation
-7%
17%
5.0%
0.0016
12%
7%
9.5%
0.000025
28%
-3%
12.5%
0.001225
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
9.0%
0.0010
3.08%
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50% in
bonds) has less risk than either stocks or bonds held in
isolation.
Portfolios


In general, as long as ρ<1, the standard
deviation of portfolio returns < weighted average
of the standard deviations of the securities in the
portfolio.
Note that the expected return of a portfolio is
always equal to the weighted average of the
expected returns of the securities in the portfolio.
This suggests the benefit of diversification.
The Efficient Set
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Risk and Return Combinations
Portfolio Return
% in stocks
100%
stocks
12.0%
11.0%
10.0%
100%
bonds
9.0%
8.0%
7.0%
6.0%
5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
The Efficient Set
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Risk and Return Combinations
Portfolio Return
% in stocks
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
100%
bonds
7.0%
6.0%
5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level
of risk.
Portfolios with Various Correlations
return

100%
stocks
 = -1.0
100%
bonds
 = 1.0
 = 0.2



Relationship
depends on
correlation
coefficient
-1.0 <  < +1.0
If  = +1.0, no risk
reduction is
possible
If  = –1.0,
complete risk
reduction is
possible
return
The Efficient Set for Many Securities
Individual Assets
P
Consider a world with many risky assets; we
can still identify the opportunity set of riskreturn combinations of various portfolios.
return
The Efficient Set for Many Securities
minimum
variance
portfolio
Individual Assets
P
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
return
Optimal Portfolio with a Risk-Free Asset
100%
stocks
rf
100%
bonds

In addition to stocks and bonds, consider a world that
also has risk-free securities like treasury bills.
return
Riskless Borrowing and Lending
rf
P
With a risk-free asset available and the efficient
frontier identified, we choose the capital allocation
line with the steepest slope.
return
Market Equilibrium
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
tangency (market) portfolio, M. In a world with homogeneous
expectations, M is the same for all investors.
return
Market Equilibrium
100%
stocks
Balanced
fund
rf
100%
bonds

Just where the investor chooses along the Capital Market
Line depends on his risk tolerance. The central point is that
all investors have the same CML (Separation Principle).
4. Systematic versus Unsystematic Risk

Systematic Risk
Risk factors that affect a large number of
assets
 Also known as non-diversifiable risk or market
risk
 Includes such things as tax change, inflation,
war, etc.

Systematic versus Unsystematic Risk

Unsystematic Risk
Risk factors that affect a limited number of
assets
 Also known as unique risk, diversifiable risk,
or asset-specific risk
 Includes such things as CEO change, labor
strike, etc.

Principle of Diversification



Diversification can substantially reduce the
variability of returns without an equivalent
reduction in expected returns.
Specifically, diversification can alleviate or
eliminate the unsystematic portion of the total
risk.
However, there is a minimum level of risk that
cannot be diversified away, and that is the
systematic portion.
Risk for a Well-Diversified Portfolio

In a large portfolio the unsystematic risk is
effectively diversified away, but the systematic
risk is not.
Diversifiable Risk;
Unsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable
risk; Systematic
Risk; Market Risk
n
Risk For an Individual Security in a
Well-Diversified Portfolio
Total risk =
systematic risk + unsystematic risk
 The standard deviation of returns is a
measure of total risk.
 For well-diversified portfolios, unsystematic
risk is very small.
 Consequently, the risk for a single security in
a well diversified portfolio that matters is
essentially equivalent to the systematic risk.

5. Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model uses
beta to capture the systematic risk of an
asset and then provides an equation that
prices an asset with respect to its beta.
Beta (b)

Beta measures the responsiveness of a
security to movements in the market
portfolio (i.e., systematic risk).
bi 
Cov ( Ri , RM )
 ( RM )
2
Security Returns
Estimating b with Regression
Slope = bi
Return on
market %
Ri = a i + biRm + ei
Proxy for Market Return
Clearly, your estimate of beta will depend
upon your choice of a proxy for the market
portfolio. Generally, we use the return of a
market index as a proxy for the market
return. The market index may be Down
Jones Industrial Average (DJIA), S&P 500,
Russell 3000, etc.
Portfolio Beta



The Beta of a portfolio is the weighted
average of the betas of individual
securities in that portfolio.
Example: Calculate the beta of a portfolio
consisting of 40% of General Electric
stock with beta equal to 1.22, and 60% of
Walmart stock with beta equal to 0.04.
Beta(portfolio) =
40%*1.22+60%*0.04=0.512
Capital Asset Pricing Model Equation

Expected Return on the Market:
R M  RF  Market Risk Premium
• Expected return on an individual security:
R i  RF  β i  ( R M  RF )
Market Risk Premium
This applies only to individual securities held within
well-diversified portfolios!
Expected Return on a Security

This formula is called the Capital
Asset Pricing Model (CAPM):
R i  RF  β i  ( R M  RF )
Expected
return on
a security
RiskBeta of the
=
+
×
free rate
security
• Assume
• Assume
bi  0 , then R i  RF
bi  1, then R i  R M
Market risk
premium
Expected return
Relationship Between Risk & Return
R i  RF  β i  ( R M  RF )
RM
RF
1.0
b
Expected return
Relationship Between Risk & Return
13.5%
3%
β i  1.5
RF  3%
1.5
R M  10%
b
𝑅𝑖 = 3% + 1.5 10% − 3% = 13.5%
6. Expected Return and Cost of Capital



If the return of a security cannot meet
investors’ expectations based on its
(systematic) risk, no one will buy and hold
such a security.
Therefore, the expected return serves as
a hurdle that each security must
overcome given its (systematic) risk.
This is why expected return for investors
becomes the cost of capital for the firm.
Cost of Capital and Capital Budgeting

We should use the cost of capital to
discount future cash flows in capital
budgeting, as we will learn in more details
in Session 4.
Readings

Chapter 11