chp6 techniques
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Techniques for Calculating the Efficient Frontier
Chapter 6
c⃝Kilenthong 2011
() Techniques for Calculating the Efficient Frontier Chapter 6
1 23
8102019 Chp6 Techniques
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Main Issues
Finding an efficient frontier when short sales are allowed and riskless borrowing and lending is possible short sales are allowed but riskless borrowing and lending is not
possible short sales are not allowed but riskless borrowing and lending is
possible neither short sales nor riskless borrowing and lending are possible
Roles of a riskless asset (or riskless lending and borrowing)
Roles of short sales
() Techniques for Calculating the Efficient Frontier Chapter 6
2 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 323
Short Sales and Riskless implies Maximum Slope
The existence of riskless asset implies that the efficient frontier in themean-standard deviation space is the line between the riskless asset
and the portfolio of risky assets that gives the maximum slope of the lineThe efficient frontier is the line passing through R F and B
() Techniques for Calculating the Efficient Frontier Chapter 6
3 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 423
Optimal Portfolio Problem with Short Sales and Riskless
Asset
Mathematically we can find the efficient frontier by solving thefollowing problem
The problem is to find a portfolio of risky assets P whose meanreturn and standard deviation are R P and σP respectively that
maximize the slope
maxX i
R P minus R F
σP
(1)
subject to
N 991761i =1
X i = 1 (2)
() Techniques for Calculating the Efficient Frontier Chapter 6 4 23
8102019 Chp6 Techniques
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Mean and Standard Deviation
The mean return of portfolio P is given by
R P =N 991761i =1
X i R i (3)
where R i is the mean return of asset i
The standard deviation of portfolio P is given by
σP = N 991761i =1
X 2i σ2i +
N 991761i =1
N 991761 j =i
X i X j σij 12
(4)
() Techniques for Calculating the Efficient Frontier Chapter 6 5 23
8102019 Chp6 Techniques
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Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
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Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
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Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
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System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
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System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
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Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
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Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
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Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
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Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
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Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
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Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
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Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
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Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
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Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
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Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
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More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 223
Main Issues
Finding an efficient frontier when short sales are allowed and riskless borrowing and lending is possible short sales are allowed but riskless borrowing and lending is not
possible short sales are not allowed but riskless borrowing and lending is
possible neither short sales nor riskless borrowing and lending are possible
Roles of a riskless asset (or riskless lending and borrowing)
Roles of short sales
() Techniques for Calculating the Efficient Frontier Chapter 6
2 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 323
Short Sales and Riskless implies Maximum Slope
The existence of riskless asset implies that the efficient frontier in themean-standard deviation space is the line between the riskless asset
and the portfolio of risky assets that gives the maximum slope of the lineThe efficient frontier is the line passing through R F and B
() Techniques for Calculating the Efficient Frontier Chapter 6
3 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 423
Optimal Portfolio Problem with Short Sales and Riskless
Asset
Mathematically we can find the efficient frontier by solving thefollowing problem
The problem is to find a portfolio of risky assets P whose meanreturn and standard deviation are R P and σP respectively that
maximize the slope
maxX i
R P minus R F
σP
(1)
subject to
N 991761i =1
X i = 1 (2)
() Techniques for Calculating the Efficient Frontier Chapter 6 4 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 523
Mean and Standard Deviation
The mean return of portfolio P is given by
R P =N 991761i =1
X i R i (3)
where R i is the mean return of asset i
The standard deviation of portfolio P is given by
σP = N 991761i =1
X 2i σ2i +
N 991761i =1
N 991761 j =i
X i X j σij 12
(4)
() Techniques for Calculating the Efficient Frontier Chapter 6 5 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 623
Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
8102019 Chp6 Techniques
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Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
8102019 Chp6 Techniques
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Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
8102019 Chp6 Techniques
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System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
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System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 323
Short Sales and Riskless implies Maximum Slope
The existence of riskless asset implies that the efficient frontier in themean-standard deviation space is the line between the riskless asset
and the portfolio of risky assets that gives the maximum slope of the lineThe efficient frontier is the line passing through R F and B
() Techniques for Calculating the Efficient Frontier Chapter 6
3 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 423
Optimal Portfolio Problem with Short Sales and Riskless
Asset
Mathematically we can find the efficient frontier by solving thefollowing problem
The problem is to find a portfolio of risky assets P whose meanreturn and standard deviation are R P and σP respectively that
maximize the slope
maxX i
R P minus R F
σP
(1)
subject to
N 991761i =1
X i = 1 (2)
() Techniques for Calculating the Efficient Frontier Chapter 6 4 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 523
Mean and Standard Deviation
The mean return of portfolio P is given by
R P =N 991761i =1
X i R i (3)
where R i is the mean return of asset i
The standard deviation of portfolio P is given by
σP = N 991761i =1
X 2i σ2i +
N 991761i =1
N 991761 j =i
X i X j σij 12
(4)
() Techniques for Calculating the Efficient Frontier Chapter 6 5 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 623
Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 723
Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 823
Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
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System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
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System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
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Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
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Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
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Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
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Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
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Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
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Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
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Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
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Optimal Portfolio Problem with Short Sales and Riskless
Asset
Mathematically we can find the efficient frontier by solving thefollowing problem
The problem is to find a portfolio of risky assets P whose meanreturn and standard deviation are R P and σP respectively that
maximize the slope
maxX i
R P minus R F
σP
(1)
subject to
N 991761i =1
X i = 1 (2)
() Techniques for Calculating the Efficient Frontier Chapter 6 4 23
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Mean and Standard Deviation
The mean return of portfolio P is given by
R P =N 991761i =1
X i R i (3)
where R i is the mean return of asset i
The standard deviation of portfolio P is given by
σP = N 991761i =1
X 2i σ2i +
N 991761i =1
N 991761 j =i
X i X j σij 12
(4)
() Techniques for Calculating the Efficient Frontier Chapter 6 5 23
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Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
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Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
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Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
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System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
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System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 523
Mean and Standard Deviation
The mean return of portfolio P is given by
R P =N 991761i =1
X i R i (3)
where R i is the mean return of asset i
The standard deviation of portfolio P is given by
σP = N 991761i =1
X 2i σ2i +
N 991761i =1
N 991761 j =i
X i X j σij 12
(4)
() Techniques for Calculating the Efficient Frontier Chapter 6 5 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 623
Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
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httpslidepdfcomreaderfullchp6-techniques 723
Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 823
Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1023
System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
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Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 623
Solving for Optimal Portfolio
The problem can be written as
maxX i
983131 N 991761i =1
X i
1048616R i minus R F
1048617983133
F 1(X)
N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus
12
F 2(
X)
(5)
Solving this using first order conditions (FOCs) differentiating theobjective function with respect to a choice variable and take it equalto zero
The FOC wrt X k is
part F 1 times F 2
part X k = 0 =rArr F 1
part F 2
part X k + F 2
part F 1
part X k = 0 (6)
() Techniques for Calculating the Efficient Frontier Chapter 6 6 23
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Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
8102019 Chp6 Techniques
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Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
8102019 Chp6 Techniques
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Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
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System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
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Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 723
Solving for Optimal Portfolio More Details
Each derivative is
part F 1
part X k = R k minus R F
and
part F 2
part X k = minus
1
2
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 3
2
2X k σ2k + 2
991761 j =k
X j σ jk
() Techniques for Calculating the Efficient Frontier Chapter 6 7 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 823
Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 923
Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1023
System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 823
Solving for Optimal Portfolio More Details
Putting these together
0 =1048667R P minus R F
1048669983080minus
1
2
983081 N 991761i =1
X 2i σ2
i +N 991761i =1
N 991761 j =i
X i X j σij
minus 32
times 2X k σ2k + 2991761 j =k
X j σ jk
+ 1048667R k minus R F 1048669
N
991761i =1
X 2i σ2
i +N
991761i =1
N
991761 j =i
X i X j σij
minus 1
2
= minus1048667R P minus R F
1048669σminus3P
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669σminus1P
() Techniques for Calculating the Efficient Frontier Chapter 6 8 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 923
Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1023
System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 923
Solving for Optimal Portfolio More Details
Multiplying this equation by σP and rearranging the terms give
0 = minusR P minus R F
σ2P
λP
X k σ
2k +
991761 j =k
X j σ jk
+
1048667R k minus R F
1048669
which can be rewritten in a compact form as for each asset k
R k minus R F = λP X k
Z k
σ2k +
991761 j =k
λP X j
Z j
σ jk
R k minus R F = Z k σ2k +
991761 j =k
Z j σ jk (7)
() Techniques for Calculating the Efficient Frontier Chapter 6 9 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1023
System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1023
System of Simultaneous Equations
After collecting all FOC for every asset together we will end up witha system of simultaneous equations
R 1 minus R F = Z 1σ21 + Z 2σ12 + Z 3σ13 + + Z N σ1N
R 2 minus R F = Z 1σ12 + Z 2σ22 + Z 3σ23 + + Z N σ2N
R N minus R F = Z 1σ1N + Z 2σ12 + Z 3σ1N + + Z N σ2N
() Techniques for Calculating the Efficient Frontier Chapter 6 10 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1123
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
Rminus R F 1 = Σtimes Z
where
R =
R 1R N
1 =
11
Z =
Z 1Z N
Σ =
σ21 σ12 σ1N
σ12 σ22 σ2N
σ1N σ2N σ2
N
() Techniques for Calculating the Efficient Frontier Chapter 6 11 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1223
Recovering the Optimal Portfolio
In principle we will be able to solve for Z i using several methodseg
1 inverse matrix
Z = Σminus11048616
Rminus R F 11048617
(8)
2 repetitive substitution (see example)
The solution of this mathematical problem is Z i but what we reallywant is X i How can we get X i
From Z i = λP X i we can show that
991761i Z i = λP 991761i
X i = λP (9)
Hence we can recover the optimal portfolio X from Z using
X k = Z k
λP
= Z k
sumi Z i (10)
() Techniques for Calculating the Efficient Frontier Chapter 6 12 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1323
Example
Suppose there are three risky assets say CP (asset 1) Centrals (asset
2) and PTT (asset 3)Using past information we can calculate mean returns and variancecovariance matrix of these assets as
R =
148
20
Σ = 6times 6 05times 6times 3 02times 6times 15
05times 6times 3 3times 3 04times 3times 1502times 6times 15 04times 3times 15 15times 15
Suppose that the riskless rate is 5
() Techniques for Calculating the Efficient Frontier Chapter 6 13 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1423
Example
Hence a system of equations for this problem is
9 = 36Z 1 + 9Z 2 + 18Z 3
3 = 9Z 1 + 9Z 2 + 18Z 3
15 = 18Z 1 + 18Z 2 + 225Z 3
Students do it on the broad
The solution is
Z = 14
631
633
63
() Techniques for Calculating the Efficient Frontier Chapter 6 14 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1523
Example
We can then find portfolio weight X as
X 1 = 14
18 X 2 =
1
18 X 3 =
3
18
The mean return of the optimal portfolio is
R P = 14
18 times 14 +
1
18 times 8 +
3
18 times 20 = 1467
The variance of the optimal portfolio is
σ2P = XT ΣX = 3383
() Techniques for Calculating the Efficient Frontier Chapter 6 15 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1623
Example Solution
The slope of the efficient frontier is equal to 166
() Techniques for Calculating the Efficient Frontier Chapter 6 16 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1723
Short Sales without Riskless
We will now consider a case where short sales are allowed but there isno riskless asset
We can use the same technique as before but with an assumed rateR F Different assumed rates will lead to different efficient portfolios
() Techniques for Calculating the Efficient Frontier Chapter 6 17 23
C
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1823
Example Continued
Suppose that the riskless rate is now R F = 2
The system of equations now becomes
12 = 36Z 1 + 9Z 2 + 18Z 3
6 = 9Z 1 + 9Z 2 + 18Z 3
18 = 18Z 1 + 18Z 2 + 225Z 3
whose solution is
Z 1 = 42
189
Z 2 = 72
189
Z 3 = 6
189
and X 1 = 7
20
X 2 = 12
20
X 3 = 1
20and
R P = 107 σ2P = 1370
() Techniques for Calculating the Efficient Frontier Chapter 6 18 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 1923
Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
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Example Continued
In principle we need to find only two of them (Two Fund Theorem)That is any combination of these two portfolios (which themselvesare assets) is on the efficient frontier
For example put 50minus 50 weight We can show that σ2P = 21859
Then we can find the covariance between the two portfolios using
σ2P = X
21 σ2
1 + X 22 σ2
2 + 2X 1X 2σ12
This leads to σ12 = 1995
With the information of expected returns variances and covariancebetween the two portfolios we can trace out the whole frontier
() Techniques for Calculating the Efficient Frontier Chapter 6 19 23
E l C i d
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Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
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Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
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Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
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More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2023
Example Continued
() Techniques for Calculating the Efficient Frontier Chapter 6 20 23
Ri kl b t N Sh t S l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2123
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless assetIn principle an efficient portfolio problem is a constrainedmaximization problem In this case we can write
maxX i
R P minus R F
σP (11)
subject to
991761i X i = 1 (12)
X i ge 0foralli (13)
where the last one represents the no short-sales constraint
() Techniques for Calculating the Efficient Frontier Chapter 6 21 23
Ri kl b t N Sh t S l E l
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2223
Riskless but No Short Sales Example
Consider again the example with three assets and risk-free rateR F = 5
Recall that the efficient portfolio in this case is
X 1 = 14
18 X 2 = 1
18 X 3 = 3
18
Remember that this solution is solved under an assumption that shortsales are allowed
What if we now impose the no short-sales constraint should we get adifferent answer
() Techniques for Calculating the Efficient Frontier Chapter 6 22 23
M G l Effi i t P tf li P bl
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23
8102019 Chp6 Techniques
httpslidepdfcomreaderfullchp6-techniques 2323
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959)
maxX 991761i X
2i σ2
i + 991761i 991761 j =i
X i X j σij (14)
subject to
991761i X i = 1 (15)
991761i
X i R i ge R P (16)
X i ge 0 foralli (17)
991761i X i d i ge D (18)
where the last constraint is the so called dividend requirementconstraint
The role of a riskless asset is to simplify the objective function as aslope
() Techniques for Calculating the Efficient Frontier Chapter 6 23 23