Principles of Finance
Grzegorz Trojanowski
Lecture 4:
Calculating efficient portfolios
October 26, 2004
Principles of Finance - Lecture 4
1
Lecture 4 material
• Required reading:
9 Elton et al., Chapters 5, 6
• Supplementary reading:
9 Luenberger, Chapter 6
9 Sharpe et al., Chapters 7, 8
9 Alexander et al., Chapters 7-9
October 26, 2004
Principles of Finance - Lecture 4
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1
Lecture 4: Checklist
• By the end of this lecture you should:
9 Understand
the concepts of an efficient
portfolio
9 Be able to compute two efficient portfolios
when short sales are allowed
9 Be able to compute the efficient frontier when
short sales are allowed
9 Be able to compute the capital market line
when there is a risk free asset
9 Be able to compute the efficient frontier when
short sales are not allowed
October 26, 2004
Principles of Finance - Lecture 4
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Last lectures recap (1)
•
Consider N assets with the expected return and variance-covariance matrix
given by:
⎡ σ 12 σ 12
⎡ Ε(r1 ) ⎤
⎢
⎢ Ε( r ) ⎥
σ
σ 22
Ε( R ) = ⎢ 2 ⎥ and Ω = ⎢ 21
⎢ M
⎢ M ⎥
M
⎢
⎢Ε ( r ) ⎥
⎣ N ⎦
⎣σ N 1 σ N 2
•
L σ 1N ⎤
⎥
L σ 2N ⎥
O M ⎥
⎥
L σ N2 ⎦
The expected return of the portfolio described by the (column) vector of
weights W equals to E(rP) = W T E(R)
•
The portfolio variance is given by σ P2 = W Τ ΩW
•
Finally, the covariance between two portfolios whose portfolio weight
vectors are W1 and W2 is given by σ = W Τ ΩW
12
1
2
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2
Last lectures recap (2)
• The feasible set is the set that contains all possible
portfolios made up of different combinations of the N
assets
• The minimum variance set, or the envelope, comprises
the portfolios that have the lowest standard deviation (or
variance) for any given expected return
• Today: how to construct optimal portfolios given
investor’s preferences?
October 26, 2004
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Efficient portfolios (1)
• Portfolios that have the lowest standard deviation
given their expected return, and the highest expected
return given their standard deviation, are known as
efficient portfolios
• The problem faced by the investor is to combine
individual assets into a portfolio that is efficient, and
given his or her risk preferences, to choose the
efficient portfolio that is optimal
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3
Efficient portfolios (2)
• Formally, the problem is to choose portfolio weights
{wi: i = 1, …, N} to solve the following minimisation
problem
N
N N
i =1
i =1 j =1
j ≠i
min σ P = ∑ wi2σ i2 + ∑ ∑ wi w jσ ij
N
N
i =1
i =1
subject to ∑ wi = 1 and ∑ wi Ε(ri ) = Ε(rP )
• Solving the minimisation problem for each possible
expected return E(rP) yields the minimum variance
frontier or the envelope
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Efficient portfolios (3a)
• The efficient frontier is the segment of the minimum
variance frontier that lies above the GMV portfolio
GMV Portfolio
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4
Efficient portfolios (3b)
• The efficient frontier is the segment of the minimum
variance frontier that lies above the GMV portfolio
GMV Portfolio
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Efficient portfolios: Example (1)
• Consider the four assets and two portfolios used in the
example last week
A
1
B
C
D
E
Variance-covariance matrix
F
Mean return vector
0.10
0.01
0.03
0.05
H
I
Portfolio weights
W1
W2
6%
0.2
0.2
2
3
G
4
0.01
0.30
0.06
-0.04
8%
0.3
0.1
5
0.03
0.06
0.40
0.02
10%
0.4
0.1
6
0.05
-0.04
0.02
0.50
15%
0.1
0.6
• The four assets and all the possible combinations of
the two portfolios can be plotted in mean-standard
deviation space
October 26, 2004
Principles of Finance - Lecture 4
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5
Efficient portfolios: Example (2)
16.00%
15.00%
Stock D
14.00%
Expected return
13.00%
12.00%
11.00%
10.00%
Stock C
9.00%
8.00%
Stock B
7.00%
6.00%
5.00%
0.00%
Stock A
10.00% 20.00% 30.00% 40.00%
50.00% 60.00% 70.00% 80.00% 90.00%
Standard deviation
October 26, 2004
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Efficient portfolios: Example (3)
• It can be seen that stocks B and C are inefficient: there
exist combinations of the two portfolios that have the
same standard deviation but offer higher expected return
• Moreover, not all the combinations of the two portfolios
are minimum variance portfolios since if they were, then
all other assets would lie within the graph (note that now
stocks A and D lie outside the graph
October 26, 2004
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6
Identifying efficient portfolios (1)
• Consider a rate of return δ and the line originating from
the point (0, δ) and tangent to the efficient set
• The tangency point is known as an envelope portfolio
E(rP)
P
δ
October 26, 2004
σP
Principles of Finance - Lecture 4
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Identifying efficient portfolios (2)
• In practice, the efficient frontier can be found by solving
the following optimisation problem for every value of δ
max Θ =
Ε(rP − δ )
σP
N
subject to ∑ wi = 1
i =1
N
N N
N
i =1
i =1 j =1
j ≠i
i =1
where σ P = ∑ wi2σ i2 + ∑ ∑ wi w jσ ij and ∑ wi Ε(ri ) = Ε(rP )
October 26, 2004
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7
Identifying efficient portfolios (3)
• Proposition 1: The weights of the envelope portfolio P
are proportional to the vector z that solves the set of
simultaneous equations E(R) - δ = Ω z where E(R) - δ
is the ‘excess’ return vector given by
⎡ Ε(r1 ) − δ ⎤
⎢ Ε( r ) − δ ⎥
⎥
Ε( R ) − δ = ⎢ 2
M
⎥
⎢
⎢Ε ( r ) − δ ⎥
⎣ N
⎦
and Ω is the variance-covariance matrix
October 26, 2004
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Identifying efficient portfolios (4)
• This system of equations is solved by z = Ω-1 [E(R) - δ]
• Once we have solved this system for z, we can recover
the actual weights of the envelope portfolio by re-scaling
the elements of z so that they sum up to unity
∀i = 1,K, N : wi =
zi
N
∑zj
j =1
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8
Identifying efficient portfolios (5)
• Proposition 2: Given any two envelope portfolios, x and
y, any combination of these two portfolios α x + (1 - α) y
will also be an envelope portfolio
• Proposition 1 gives a method of computing an envelope
portfolio given some arbitrarily chosen constant δ
• Proposition 2 says that once we have computed two such
envelope portfolios, we can trace out the entire envelope
simply by combining the two portfolios that we have
computed in varying proportions
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2 envelope portfolios: Example (1)
• Recall the four-assets example discussed so far
• For δ = 0, we can use Proposition 1 to solve for the first
envelope portfolio
A
1
B
C
D
E
F
Variance-covariance matrix
E[R]
2
0.10
0.01
0.03
0.05
3
0.01
0.30
0.06
-0.04
8.0%
4
0.03
0.06
0.40
0.02
10.0%
5
0.05
-0.04
0.02
0.50
15.0%
6.0%
6
=MMULT(MINVERSE(A2:D5),F2:F5)
7
8
z_x
x
9
0.3861
0.3553
10
0.2567
0.2362
11
0.1688
0.1553
12
0.2752
0.2532
October 26, 2004
Cell C9 contains the function
=B9/SUM(B$9:B$12)
This function is copied down
to cells C10:C12
Principles of Finance - Lecture 4
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9
2 envelope portfolios: Example (2)
• Arbitrarily choosing the value of δ (e.g. δ = 0.04), we
can solve for the second envelope portfolio Compute ‘excess’
1
Variance-covariance matrix
E[R]
E[R]- δ
2
0.10
0.01
0.03
0.05
6.0%
2.0%
returns
for
δ
chosen, e.g. cell
G2
contains
formula
3
0.01
0.30
0.06
-0.04
8.0%
4.0%
=F2-F$7
0.03
0.06
0.40
0.02
10.0%
6.0%
0.02
0.50
15.0%
11.0%
A
4
5
6
7
B
C
D
Cell G9 contains the function
=F9/SUM(F$9:F$12)
0.05
-0.04
This function is copied down
to cells G10:G12
E
δ
F
G
4.0%
8
z_x
x
z_y
y
9
0.3861
0.3553
0.0404
0.0782
10
0.2567
0.2362
0.1386
0.2684
11
0.1688
0.1553
0.1151
0.2227
12
0.2752
0.2532
0.2224
0.4307
Enter here an
arbitrarily chosen
value of δ
=MMULT(MINVERSE(A2:D5),G2:G5)
October 26, 2004
Principles of Finance - Lecture 4
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2 envelope portfolios: Example (3)
• We can then compute the expected returns, variances
and standard deviations as well as the covariance for the
two envelope portfolios
=MMULT(I4:L4,F2:F5)
I
1
2
3
4
J
K
L
0.1553
0.2532
Transpose x
0.3553
0.2362
=MMULT(I4:L4,MMULT
(A2:D5,G9:G12))
Transpose y
0.0782
0.2684
0.2227
0.4307
6
Mean(x)
9.37%
Mean(y)
11.30%
7
Var(x)
0.0862
Var(y)
0.1414
8
Sigma(x)
29.37%
Sigma(y)
37.60%
5
=SQRT(L7)
=MMULT(I2:L2,MMULT
(A2:D5,G9:G12))
9
10
Cov(x,y)
0.1040
11
Corr(x,y)
0.9419
October 26, 2004
=J10/(J8*L8)
Principles of Finance - Lecture 4
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10
2 envelope portfolios: Example (4)
• From Proposition 2 we can now trace the entire envelope
by combining the two envelope portfolios that we have
computed in varying proportions
I
J
K
L
14
W_x
W_y
Std. dev.
Return
15
-2.00
3.00
60.78%
15.17%
16
-1.75
2.75
57.65%
14.68%
:
:
:
:
:
23
0.00
1.00
37.60%
11.30%
+J16^2*$L$7
24
0.25
0.75
35.20%
10.82%
+2*I16*J16*$J$10)
25
0.50
0.50
33.00%
10.34%
26
0.75
0.25
31.04%
9.86%
27
1.00
0.00
29.37%
9.37%
:
:
:
:
:
34
2.75
-1.75
29.00%
5.99%
35
3.00
-2.00
30.60%
5.51%
October 26, 2004
=1-I15
=I15*$J$6+J15*$L$6
=SQRT(I16^2*$J$7
Portfolio y
Portfolio x
Principles of Finance - Lecture 4
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2 envelope portfolios: Example (5)
• Plotting the portfolios from the data table yields the
envelope and the efficient frontier
16.00%
Asset D
Portfolio y
14.00%
Portfolio x
12.00%
Return
10.00%
Asset C
8.00%
Asset B
6.00%
Asset A
4.00%
2.00%
0.00%
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
Std. deviation
October 26, 2004
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11
Capital market line (1)
• Suppose there exist a risk-free asset that offers the
return rf
• Consider a solution, z, to the simultaneous equations
E(R) - rf = Ω z
• This system is solved by z = Ω-1 [E(R) - rf], and the
weights, z, re-scaled so that sum to unity
• This yields the portfolio, M
• If all the investors face the same optimisation problem,
then M is the market portfolio
October 26, 2004
Principles of Finance - Lecture 4
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Capital market line (2)
• Now, consider combinations of the market portfolio M
and the risk-free asset
• The expected return and standard deviation of such
combinations are given by
Ε(rP ) = αr f + (1 − α )Ε(rM )
σ P = α 2σ 2f + (1 − α ) 2 σ M2 + 2α (1 − α )σ fM = (1 − α )σ M
• The locus of all such combinations is the capital
market line
October 26, 2004
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12
Capital market line (3)
Portfolio expected return
Capital market line (CML)
Market portfolio (M)
Risk-free rate (rf)
Portfolio standard deviation
October 26, 2004
Principles of Finance - Lecture 4
25
Capital market line: Example
• For the four-asset case discussed so far, suppose that the
risk-free asset earns a rate of return rf = 3%
• We can then compute the market portfolio
=MMULT(TRANSPOSE(C41:44),F2:F5)
38
A
B
rf
3.00%
C
D
39
=F2-$B$38
40
E[R]- rf
z
M
41
3.0%
0.1268
0.1924
42
5.0%
0.1682
0.2551
43
7.0%
0.1285
0.1949
44
12.0%
0.2356
0.3575
=MMULT(MINVERSE
(A2:D5),G2:G5)
October 26, 2004
E
F
rM
10.51%
σM
33.75%
=MMULT(TRANSPOSE
(C41:44),MMULT
(A2:D5,C41:C44))
Cell C41 contains the function
=B41/SUM(B$41:B$44)
This function is copied down
to cells C42:C44
Principles of Finance - Lecture 4
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13
Efficient frontier – recap (1)
• The efficient frontier is found by solving the following
optimisation problem for every value of δ
max Θ =
Ε(rP − δ )
σP
N
subject to ∑ wi = 1
i =1
N
N N
N
i =1
i =1 j =1
j ≠i
i =1
where σ P = ∑ wi2σ i2 + ∑ ∑ wi w jσ ij and ∑ wi Ε(ri ) = Ε(rP )
• The weights of tangency portfolio, P, are proportional to
the vector z that solves the set of simultaneous equations
E(R) - δ = Ω z
October 26, 2004
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Efficient frontier – recap (2)
• This system can be solved by z = Ω-1 [E(R) - δ], and the
weights of the portfolio can be recovered by xi =
zi
N
∑zj
j =1
• However, the solution may involve negative weights
• A negative weight implies that the asset is short sold by
the investor, and that the proceeds from the short sale
can be used immediately to purchase another asset
• In practice, many investors are prohibited from selling
an asset short, and those are not likely to be able to use
all of the short sale proceeds
October 26, 2004
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14
Efficient frontier when
short sales are not allowed
• We can recast the portfolio choice problem with the
added restriction of no short sales by including the
following constraint in the previous optimisation
problem
xi ≥ 0
∀ i = 1, …, N
• This considerably complicates the problem, since the
tangency portfolio weights can no longer be found as the
solution to a set of simultaneous equations
• Instead, we can solve the constrained optimisation
problem numerically (e.g. using Excel’s Solver tool)
October 26, 2004
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29
Efficient frontier with
no short sales: Example (1)
• Let’s come back to the four-asset example discussed
before
A
B
C
D
E
F
1
Variance-covariance matrix
E[R]
2
0.10
0.01
0.03
0.05
6.0%
3
0.01
0.30
0.06
-0.04
8.0%
4
0.03
0.06
0.40
0.02
10.0%
5
0.05
-0.04
0.02
0.50
15.0%
• We can then arbitrarily choose a value for a constant,
say δ = 5%, and set up the optimisation problem in
Excel in the following way
October 26, 2004
Principles of Finance - Lecture 4
30
15
Efficient frontier with
no short sales: Example (2)
A
15
B
C
Delta
5.00%
Enter here an arbitrarily chosen
constant, e.g. 5%
16
17
Optimal portfolio weights
18
x_1
0.2500
19
x_2
0.2500
20
x_3
0.2500
21
x_4
0.2500
22
Total
1.0000
=MMULT(TRANSPOSE(C18:C21),
F2:F5)
24 Portfolio mean
9.75%
=SQRT(MMULT(TRANSPOSE
(C18:C21),MMULT(A2:D5,C18:C21)))
25 Portfolio sigma
31.22%
26 Theta
0.1521
Enter here some starting values
for the portfolio weights
=SUM(C18:C21)
23
October 26, 2004
=(C24-C15)/C25
Principles of Finance - Lecture 4
31
Efficient frontier with
no short sales: Example (3)
• The optimal portfolio weights can be computed using
the Excel’s Solver tool
• Go to the Tools/Solver option which will generate the
following dialogue box (see last lecture)
Note that this time we
choose maximisation
option
Note that this time we
change a number of
cells (four of them)
rather than just one
Slide 33 shows how to
add new constraints
October 26, 2004
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16
Efficient frontier with
no short sales: Example (4)
• The user-defined entries in the dialogue box tell Solver
to change the portfolio weights in a way that maximises
the value of
Ε ( RP − δ )
N
and ∑ xi = 1
σP
subject to constraints ∀i : xi ≥ 0
i =1
October 26, 2004
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33
Efficient frontier with
no short sales: Example (5)
• The constraints can be added by clicking on the Add
button (in the Solver dialogue box), which generates
the following dialogue box (completed here for the first
constraint)
Enter here the addresses of
the cells on which you want
to impose a constraint
Select here
the type of
constraint
Enter here the value
corresponding to the
constraint imposed
Confirm here
clicking ‘OK’
October 26, 2004
Principles of Finance - Lecture 4
by
34
17
Efficient frontier with
no short sales: Example (6)
A
15
B
C
Delta
5.00%
16
In the example discussed, the
constrained
optimisation
procedure (explained on
Slides 30-33) yields the
following result
17
Optimal portfolio weights
18
x_1
0.0000
19
x_2
0.2618
20
x_3
0.2500
21
x_4
0.4981
22
Total
1.0000
23
October 26, 2004
24 Portfolio mean
11.97%
25 Portfolio sigma
41.18%
26 Theta
0.1692
Principles of Finance - Lecture 4
35
Efficient frontier with
no short sales: Example (7)
• By changing the value of δ, we can trace out the entire efficient set
without the short sales
16.00%
Asset D
Expected return
14.00%
12.00%
10.00%
Asset C
8.00%
Asset B
6.00%
4.00%
0.00%
Asset A
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
70.00%
80.00%
Standard deviation
• Note that now having two portfolios is not enough to derive the
whole set!
October 26, 2004
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18