The Efficient Set for Two Assets
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%
Portfolo Risk and Return Combinations
12.0%
Portfolio Return
% in stocks
11.0%
10.0%
100%
stocks
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk.
These comprise the efficient frontier.
return
Two-Security Portfolios with Various
Correlations
100%
stocks
ρ = -1.0
100%
bonds
ρ = 1.0
ρ = 0.2
σ
Relationship depends on correlation coefficient
-1.0 < ρ < +1.0
If ρ = +1.0, no risk reduction is possible
If ρ = –1.0, complete risk reduction is possible
Perfect Positive Correlation
Corr(RA, RB) = 1
Perfect Negative Correlation
Corr(RA, RB) = −1
Zero Correlation
Corr(RA, RB) = 0
Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
σ
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
Thus diversification can eliminate some,
but not all of the risk of individual securities.
return
The Efficient Set for Many Securities
σP
Consider a world with many risky assets; we can still
identify the opportunity set of risk-return
combinations of various portfolios.
return
The Efficient Set for Many Securities
(contd.)
minimum
variance
portfolio
σP
Given the opportunity set we can identify the
minimum variance portfolio.
return
The Efficient Set for Many Securities
(contd.)
c
effi
ie
t i er
ron
nt f
minimum
variance
portfolio
σP
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
return
Optimal Risky Portfolio with a
Risk-Free Asset
rf
σ
In addition to stocks and bonds, consider a world that
also has risk-free securities like T-bills
return
Riskless Borrowing and Lending
CM
L
rf
σ
Now investors can allocate their money across
the T-bills and a balanced mutual fund
return
Choice of Capital Market Line
CM
L
efficient frontier
rf
σP
With a risk-free asset available and the efficient
frontier identified, we choose the capital market line
with the steepest slope
return
Market Equilibrium
CM
L
efficient frontier
M
rf
σP
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.
return
Market Equilibrium (contd.)
CM
L
rf
σ
Just where the investor chooses along the capital allocation
line depends on his risk tolerance. The big point though is
that all investors have the same CML (Capital Market Line).
return
Market Equilibrium (contd.)
CM
L
Optimal
Risky
Portfolio
rf
σ
All investors have the same CML because they all
have the same optimal risky portfolio given the riskfree rate.
return
The Separation Property
CM
L
efficient frontier
M
rf
σP
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.
return
The Separation Property (contd.)
CM
L
efficient frontier
M
rf
σP
Investor risk aversion is revealed in their choice of where to
stay along the capital market line—not in their choice of the
line.
return
The Separation Property (contd.)
CM
L
Optimal
Risky
Portfolio
rf
σ
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.
Dependence of Optimal Risky
return
Portfolio on Risk-Free Asset
1
f
0
f
r
r
L 0 CML 1
CM
First
Optimal
Risky
Portfolio
Second Optimal
Risky Portfolio
σ
The optimal risky portfolio depends on the riskfree rate as well as the risky assets.