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Techniques for Calculating the Efficient Frontier

Chapter 6

c⃝Kilenthong 2011

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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

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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

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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)

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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)

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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)

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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

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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

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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)

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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

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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

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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)

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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

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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

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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

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Example Solution

The slope of the efficient frontier is equal to 166

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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

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C

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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

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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

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Example Continued

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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

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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

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M G l Effi i t P tf li P bl

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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

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