La Frontera Eficiente de Inversión
Dra. Gabriela Siller Pagaza
Departamento de Economía
[email protected]
Factores relevantes
• A los inversionistas les interesa conocer sobre
los activos:
– Rendimiento
– Riesgo
E R  
n
R
i
pi  
i 1
Var  R   
n
2

 p R
i
i
 
2
i 1
• Además, invierten en una canasta de activos,
para obtener beneficios por diversificación.
Expected Return, Variance, and Covariance
Rate of Return
Scenario Probability Stock fund Bond fund
Recession
33.3%
-7%
17%
Normal
33.3%
12%
7%
Boom
33.3%
28%
-3%
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.
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E (rS )  1  (7%)  1  (12%)  1  (28%)
3
3
3
E (rS )  11%
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E (rB )  1  (17%)  1  (7%)  1  (3%)
3
3
3
E (rB )  7%
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(11%  7%)  3.24%
2
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(11%  12%)  .01%
2
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(11%  28%)  2.89%
2
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
1
.0205  (3.24%  0.01%  2.89%)
3
Expected Return, Variance, and Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
14.3%  0.0205
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Note that stocks have a higher expected return than bonds and higher risk.
Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested
in bonds and 50% invested in stocks.
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
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 )  wB E (rB )  wS E (rS )
9%  50%  (11%)  50%  (7%)
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The variance of the rate of return on the two risky assets portfolio is
σ P2  (wB σ B )2  (wS σ S )2  2(wB σ B )(wS σ S )ρBS
σ P2  (wB σ B )2  (wS σ S )2  wB wS 2Cov
where BS is the correlation coefficient between the returns on the stock and
bond funds.
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
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 stocks or bonds held in isolation.
% in stocks
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 Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
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)
We can consider other portfolio
weights besides 50% in stocks and 50%
in bonds …
% in stocks
Risk
Return
0%
0%
5%
5%
10%
10%
15%
15%
20%
20%
25%
25%
30%
30%
35%
35%
40%
40%
45%
45%
50%
50%
55%
55%
60%
60%
65%
65%
70%
70%
75%
75%
80%
80%
85%
85%
90%
90%
95%
95%
100%
100%
8.2%
8.2%
7.0%
7.0%
5.9%
5.9%
4.8%
4.8%
3.7%
3.7%
2.6%
2.6%
1.4%
1.4%
0.4%
0.4%
0.9%
0.9%
2.0%
2.0%
3.1%
3.1%
4.2%
4.2%
5.3%
5.3%
6.4%
6.4%
7.6%
7.6%
8.7%
8.7%
9.8%
9.8%
10.9%
10.9%
12.1%
12.1%
13.2%
13.2%
14.3%
14.3%
7.0%
7.0%
7.2%
7.2%
7.4%
7.4%
7.6%
7.6%
7.8%
7.8%
8.0%
8.0%
8.2%
8.2%
8.4%
8.4%
8.6%
8.6%
8.8%
8.8%
9.0%
9.0%
9.2%
9.2%
9.4%
9.4%
9.6%
9.6%
9.8%
9.8%
10.0%
10.0%
10.2%
10.2%
10.4%
10.4%
10.6%
10.6%
10.8%
10.8%
11.0%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
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)
We can consider other portfolio
weights besides 50% in stocks and 50%
in bonds …
The Efficient Set for Two Assets
% in stocks
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%
100%
activo A
100%
Activo B
Note that some portfolios are “better” than
others. They have higher returns for the same
level of risk or less.
These compromise the efficient
frontier.
Minimum Variance Portfolio
σ P2  (wB σ B )2  ( 1  wB σ S )2  wB 1  wB 2Cov
σ
 2 wB B2  2 wB S2  2 S2  2Cov  4 wB Cov  0
wB
2
P
WB 
  Cov BS
2
S
    2 Cov BS
2
S
2
B
Portfolio Risk/Return Two Securities:
Correlation Effects
• Relationship depends on correlation
coefficient
• -1.0 <  < +1.0
• The smaller the correlation, the greater the
risk reduction potential
• If  = +1.0, no risk reduction is possible
Portfolio Risk as a Function of the
Number of Stocks in the Portfolio

In a large portfolio the variance terms are effectively diversified away, but the
covariance terms are not.
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
Individual Assets
P
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
minimum
variance
portfolio
Individual Assets
P
Given the opportunity set we can identify the
minimum variance portfolio.
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 Risky 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 T-bills
return
Riskless Borrowing and Lending
100%
stocks
Balanced
fund
rf
100%
bonds

Now investors can allocate their money across
the T-bills and a balanced mutual fund
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 market portfolio M. In a world with
homogeneous expectations, M is the same for all investors.
return
The Separation Property
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.
return
The Separation Property
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.
return
Market Equilibrium
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.
return
Market Equilibrium
100%
stocks
Optimal
Risky
Porfolio
rf
100%
bonds

All investors have the same CML because they all have the
same optimal risky portfolio given the risk-free rate.
return
The Separation Property
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.
return
Optimal Risky Portfolio with a Risk-Free Asset
r
1
f
0
rf
100%
stocks
First
Optimal
Risky
Portfolio
Second Optimal Risky
Portfolio
100%
bonds

By the way, the optimal risky portfolio depends
on the risk-free rate as well as the risky assets.
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 (b)of the security.
• Beta measures the responsiveness of a
security to movements in the market
portfolio.
bi 
Cov( Ri , RM )
 ( RM )
2
Security Returns
Estimating b with regression
Slope = bi
Return on
market %
Ri = a i + biRm + ei
The Formula for Beta
bi 
Cov( Ri , RM )
 ( RM )
2
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 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 to individual securities held within welldiversified portfolios.
Expected Return on an Individual Security
• This formula is called the Capital Asset Pricing
Model (CAPM)
Ri  RF  βi  ( R M  RF )
Expected
return on a
security
=
Risk-free
rate
+
Beta of the
security
×
Market risk
premium
• Assume bi = 0, then the expected return is RF.
• Assume bi = 1, then R i  R M
Expected return
Relationship Between Risk & Expected Return
Ri  RF  βi  ( R M  RF )
RM
RF
1.0
Ri  RF  βi  ( R M  RF )
b
Expected return
Relationship Between Risk & Expected Return
13.5%
3%
1.5
β i  1.5
RF  3%
b
R M  10%
R i  3%  1.5  (10%  3%)  13.5%
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