7/30/2019 Problems Covered in Make-up Class on February 16 2013 (Saturday) From Chapter 7

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FIN 435 (Faculty- SfR)

CHAPTER - 7 (Problems covered during Make-up class on Saturday, February 16, 2013)

(1) An investor can design a risky portfolio with two stocks, A and B. Stock A has an expectedreturn of 18% and a standard deviation of 22.5%. Stock B has an expected return of 15% and a standard

deviation of 15%. The correlation coefficient between the two stocks is 0.75 and the risk-free T-bill rate is9%.

(a) what are the weights of stocks A and B in the optimal risky portfolio (P)?

%6060.04.01

%404.000127.0

00051.0

75.0*15.0*225.0*]09.015.009.018.0[)225.0(*]09.015.0[)15.0(*]09.018.0[

75.0*15.0*225.0*]09.015.0[)15.0(*]09.018.0[

])()([])([])([

])([])([

22

2

22

2

B

A

A

ABBAfBfAAfBBfA

ABBAfBBfAA

w

w

w

rrErrErrErrE

rrErrEw

(b) what is the expected return on the portfolio P?

%20.1615.0*6.018.0*4.0)( PrE

(c) what is the standard deviation of the return on portfolio P?

%8375.16168375.002835.0

75.0*)225.0(*)15.0(*)6.0(*)4.0(*2)15.0(*)6.0()225.0(*)4.0( 2222

p

p

Part (a), (b) and (c) was covered during Problem Class of Chapter 6. Part (d) to (f) is based

on Chapter 7. We will use the expected return and standard deviation we got for Optimal

Risky portfolio in part (b) and (c)

(d) if the investor has a coefficient of risk aversion, A, equal to 3, what is y*, the optimalproportion of the complete portfolio to be invested in the risky portfolio (P)?

%656.8484656.0)163758.0(*3

09.0162.0

)(

2

*

2

*

y

A

rrEy

p

fp

* If in Midterm 1, you have to solve for Minimum variance portfolio, the process will

remain same. You will first find weight of stocks inside minimum variance portfolio, use

those weights to find Expected return and Standard Deviation of Minimum Portfolio using

Markowitz. Use those data to find y* for minimum variance portfolio as we are doing here

for optimal portfolio.

7/30/2019 Problems Covered in Make-up Class on February 16 2013 (Saturday) From Chapter 7

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FIN 435 (Faculty- SfR)

(e) what are the weights of T-bills, stock A and stock B in the complete portfolio?

50794.084656.0*6.0

33862.084656.0*4.0

15344.084656.01*1

B

A

billsT

w

w

yw

Since WA= 40% and WB= 60% (from part a)

(f) what is the maximum level of utility that an investor can achieve? or What is the Utility of

Complete Portfolio?

120477.0020318.0*3*2

1150952.0

2

1)(

020318.0)142540.0()168375.0*84656.0(

150952.009.0*)84656.01(1620.0*84656.0)1()()(

2

2222

CC

PC

fPC

ArEU

y

ryryErE

Since Risk-free rate has 0 Standard Deviation, for calculating 2

c, we only consider weightof optimal portfolio in complete portfolio (y*) and p (Standard Deviation of RiskyPortfolio)

(2)Stock A and B are perfectly negatively correlated. Stock A has expected return of 20% andstandard deviation of 10% while the stock B has expected return of 8% and standard deviation of7.5%. What is the expected return of Minimum Variance Portfolio?

First we need to find-out weights of Minimum variance portfolio using following formula.

f

AB

A

A

ABBABA

ABBAB

BABA

BAB

A

rrE

ww

w

w

rr

rr

w

%143.1313143.008.0*57143.020.0*42857.0)(

%143.5757143.042857.011

%857.4203063.0

01313.0

)1(*)075.0(*)10.0(*2)10.0()075.0(

)1(*)075.0(*)10.0()075.0(

2),cov(2

),cov(

22

2

22

2

22

2

Since the two stocks are perfectly negatively correlated, the minimum variance portfolio of thesetwo stocks will have a zero variance. The return on this portfolio should be equal to the risk-free

rate. Since stocks are perfectively negatively correlated, we can also use WA = B/A+BandWB = A/A+B to calculate Minimum Variance Weights (only for perfectly negativelycorrelated stocks)

But for any other case, we will use Markowitz formula for standard deviation to find-outstandard deviation of a Minimum Variance Portfolio.