Exercise Sheet 4

Exercise 1The table provides data on the return and standard deviation for di¤erent

compositions of a two-asset portfolio. Plot the data to obtain the portfoliofrontier. Where is the minimum variance portfolio located?

X 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1rp .08 .076 .072 .068 .064 .060 .056 .052 .048 .044 .04�p .5 .44 .38 .33 .29 .26 .24 .25 .27 .30 .35

Solution 1The plot is obtained by putting the data into Excel.

0.035

0.045

0.055

0.065

0.075

0.085

0.2 0.3 0.4 0.5

0=X2.0=X

4.0=X

8.0=X1=X

The minimum variance portfolio is located around X = 0:6.

Exercise 2Assuming that the returns are uncorrelated, plot the portfolio frontier with-

out short sales when the two available assets have expected returns 2 and 5 andvariances 9 and 25.Solution 2The expected return and standard deviation are

rp = XArA +XBrB ; �p =�X2A�

2A +X

2B�

2B + 2XAXB�AB

�1=2:

Using the data in the exercise

rp = XA2 +XB5; �p =�X2A9 +X

2B25 + 2XAXB0

�1=2;

or

rp = XA2 + [1�XA] 5; �p =hX2A9 + [1�XA]

225i1=2

:

Computing these expressions gives the table below.

X 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1rp 5 4.7 4.4 4.1 3.8 3.5 3.2 2.9 2.6 2.3 2.0�p 5 4.5 4.0 3.6 3.2 2.9 2.7 2.58 2.6 2.7 3

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Dell IntelYear Nov Price Dividend Return Year Nov Price Dividend Return

98 30.41 98 24.3799 43 0 0.414009 99 34.79 0.02 0.4283960 19.25 0 ­0.552326 0 34.58 0.08 ­0.0037371 27.93 0 0.450909 1 29.75 0.08 ­0.1373632 28.6 0 0.023989 2 19.09 0.08 ­0.355633 34.57 0 0.208741 3 30.77 0.08 0.6160294 40.52 0 0.172115 4 20.66 0.16 ­0.3233675 30.15 0 ­0.255923 5 24.96 0.32 0.2236216 27.24 0 ­0.096517 6 20.43 0.4 ­0.1654657 24.54 0 ­0.099119 7 25.39 0.44 0.2643178 12.77 0 ­0.479625 8 15.06 0.55 ­0.385191

Mean Return ­0.021375 Mean Return 0.016161

Variance of Return 0.118185 Variance of Return 0.122883

cov(D,I) 0.030263

This data generates the plot.

1.82.32.83.33.84.34.8

2 3 4 5

0=X

2.0=X

4.0=X

8.0=X1=X

Exercise 3Using 10 years of data from Yahoo, construct the portfolio frontier without

short selling for Intel and Dell stock.Solution 3Analyzing the data from Yahoo gives the following results.Plotting this data gives the portfolio frontier in the �gure.

Exercise 4Given the standard deviations of two assets, what is smallest value of the

correlation coe¢ cient for which the portfolio frontier bends backward? (Hint:assuming asset A has the lower return, �nd the gradient of the frontier atXA = 1.)Solution 4

2

­0.025

­0.02

­0.015

­0.01

­0.005

0

0.005

0.01

0.015

0.02

0.25 0.27 0.29 0.31 0.33 0.35 0.37σp

rp

_

Dell = 1, Intel = 0

Dell = 0, Intel = 1

The frontier is described by

rp = XArA+[1�XA] rB ; �p =hX2A�

2A + [1�XA]

2�2B + 2XA [1�XA]�A�B�AB

i1=2:

The gradient of the frontier is de�ned by

drpd�p

=drp=dXAd�p=dXA

:

It is vertical when d�p=dXA = 0. Calculating this derivative

d�pdXA

=1

2[�p]

�1=2 �2XA�

2A � 2 [1�XA]�2B + 2 [1�XA]�A�B�AB � 2XA�A�B�AB

�:

Evaluating this at XA = 1 and setting equal to 0

1

2[�p]

�1=2 �2�2A � 2�A�B�AB

�= 0:

Hence the e¢ cient frontier is vertical at XA = 1 when

�AB =�A�B:

Since �A > 0 and �B > 0 the minimum value of �AB is positive. For any valueless than �A

�Bthe e¢ cient frontier will bend backward.

Exercise 5

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Allowing short selling, show that the minimum variance portfolios for �AB =+1 and �AB = �1 have a standard deviation of zero. For the case of a zerocorrelation coe¢ cient, show that it must have a strictly positive variance.Solution 5The portfolio variance is

�2p = X2A�

2A + [1�XA]

2�2B + 2XA [1�XA] �AB�A�B

With perfect positive correlation,

XA =�2B � �A�B

�2A + �2B � 2�A�B

=�B

�B � �A;

Hence

�2p =

��B

�B � �A

�2�2A +

�1� �B

�B � �A

�2�2B + 2

��B

�B � �A

��1� �B

�B � �A

��A�B

= 0:

With perfect negative correlation

XA =�2B + �A�B

�2A + �2B + 2�A�B

=�B

�A + �B:

Hence

�2p =

��B

�A + �B

�2�2A +

�1� �B

�A + �B

�2�2B � 2

��B

�A + �B

��1� �B

�A + �B

��A�B

= 0:

When the assets are uncorrelated

XA =�2B

�2A + �2B

:

Hence

�2p =

��2B

�2A + �2B

�2�2A +

�1� �2B

�2A + �2B

�2�2B

=�2A�

2B

�2A + �2B

> 0:

Exercise 6Using the data in Exercise 2, extend the portfolio frontier to incorporate

short selling.Solution 6

4

Using Excel the returns and standard deviations can be computed. This isshown in the table where the left-hand column is the proportion of asset A, thecentral column the expected return and the �nal column the standard deviation.

-0.5 6.5 7.64852927-0.4 6.2 7.102112362-0.3 5.9 6.562011887-0.2 5.6 6.029925373-0.1 5.3 5.5081757420 5 50.1 4.7 4.5099889140.2 4.4 4.0447496830.3 4.1 3.61386220.4 3.8 3.2310988840.5 3.5 2.9154759470.6 3.2 2.6907248090.7 2.9 2.580697580.8 2.6 2.60.9 2.3 2.7459060441.0 2 31.1 1.7 3.3376638541.2 1.4 4.1785164831.3 1.1 4.651881341.4 0.8 4.651881341.5 0.5 5.14781507The frontier is plotted in the �gure.

01234567

0 2 4 6 8

0=X

1=X

Exercise 7Calculate the minimum variance portfolio for the data in Example 43. Which

asset will never be sold short by an e¢ cient investor?

EXAMPLE: Let asset A have expected return �rA = 2 and standarddeviation �A = 2 and asset B have expected return �rB = 8 andstandard deviation �B = 6: Table 4.3 gives the expected return

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and standard deviation for various portfolios of the two assets when�AB = � 1

2 :

Solution 7In Example 43 asset A has expected return �rA = 2 and standard deviation

�A = 2, and asset B has expected return �rB = 8 and standard deviation �B = 6:The correlation coe¢ cient is �AB = � 1

2 :The proportion of asset A in the minimum variance portfolio is given by

XA =�2B � �A�B�AB

�2A + �2B � 2�A�B�AB

=36� 2� 6� (�0:5)

4 + 36� 2� 2� 6� (�0:5)= 0:80769;

soXB = 0:19231:

Exercise 8For a two-asset portfolio, use (�rP = rf +

h�rp�rf�p

i�P ) to express the risk

and return in terms of the portfolio proportions. Assuming that the assets haveexpected returns of 4 and 7, variances of 9 and 25 and a covariance of �12;graph the gradient of the risk�return trade-o¤ as a function of the proportionheld of the asset with lower return. Hence identify the tangency portfolio andthe e¢ cient frontier.Solution 8The risk�return tradeo¤ is always given by

r� = rf +

�rp � rf�p

���;

where � is a portfolio of the risk free asset and a risky portfolio. The gradientis rp�rf

�pwhich can be evaluated using the data in the exercise as

rp � rf�p

=XArA + [1�XA] rB � rfh

X2A�

2A + [1�XA]

2�2B + 2XA [1�XA]�AB

i1=2=

XA4 + [1�XA] 7� rfhX2A9 + [1�XA]

225� 2XA [1�XA] 12

i1=2=

7� 3XA � rf[58X2

A � 74XA + 25]1=2:

Assume that rf = 1. The gradient can be plotted as below. This shows thatif rf = 1 the tangency portfolio is XA = 0:6202 and XB = 0:3798:

6

X

p

fp rrσ

−

6202.0

Exercise 9Taking the result in Example 61, show the e¤ect on the tangency portfolio

of (a) an increase in the return on the risk-free asset and (b) an increase in theriskiness of asset A. Explain your �ndings.Solution 9Assume that rf = 2. The gradient can be plotted as below.So if rf = 2 the

tangency portfolio is XA = 0:6145 and XB = 0:3855: Comparing with rf = 1the tangency portfolio has less of asset A and more of asset B.The example shows that the proportion of asset A in the tangency portfolio

is given by

XA =�2B [�rA � rf ]

�2A [�rB � rf ] + �2B [�rA � rf ]:

(a) As rf goes up, XA goes down.(b) As �A goes up, XA goes down.

Exercise 10What is the outcome if a risk-free asset is combined with (a) two assets whose

returns are perfectly negatively correlated and (b) two assets whose returns areperfectly positively correlated?Solution 10(a) There is an arbitrage opportunity unless the minimum variance portfolio

has the same return as the risk free asset.

7

X

p

fp rrσ

−

6145.0

pσ

pr

mvpr

fr

fr

Eitherleads toarbitrage

8

pσ

pr

mvpr

fr

fr

Eitherleads toarbitrage

(b) There is also an arbitrage opportunity unless the risk free asset has thesame return as the minimum variance portfolio.

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