National Tsing Hua University

Financial Management
Instructor: Tak-Yuen Wong
Assignment 3

Problem 1. Are the following statements True or False? Justify your answer.
(a) Diversification can reduce risk only when asset returns are negatively correlated.
(b) Stock A has a 10% expected return and 18% volatility, and stock B has a 13% expected
return and 12% volatility. Then, no investor will buy stock A.

Problem 2. You are considering how to invest part of your retirement savings. You have
decided to put $300,000 into three stocks: 60% of the money in GoldFinger (currently
$23/share), 30% of the money in Moosehead (currently $71/share), and the remainder in
Venture Associates (currently $4/share). If GoldFinger stock goes up to $40/share, Moose-
head stock drops to $53/share, and Venture Associates stock rises to $14 per share,
(a.) What is the new value of the portfolio?
(b) What return did the portfolio earn?
(c) If you don’t buy or sell any shares after the price change, what are your new portfolio
weights?

Problem 3. You own three stocks: 600 shares of Apple Computer, 10,000 shares of Cisco
Systems, and 5000 shares of Colgate-Palmolive. The current share prices and expected
returns of Apple, Cisco, and Colgate-Palmolive are, respectively, $547, $18, $95 and 12%,
10%, 8%.
(a) What are the portfolio weights of the three stocks in your portfolio?
(b) What is the expected return of your portfolio?
(c) Suppose the price of Apple stock goes up by $20, Cisco rises by $7, and Colgate-Palmolive
falls by $14. What are the new portfolio weights?
(d) Assuming the stocks’ expected returns remain the same, what is the expected return of
the portfolio at the new prices?

Problem 4. You are given the return data for stock A and B in the following table:
Year 2010 2011 2012 2013 2014 2015
Stock A −10% 20% 5% −5% 2% 9%
Stock B 21% 7% 30% −3% −8% 25%
(a) Estimate (i) the average return and volatility for each stock, (ii) the covariance between
the stocks, and (iii) the correlation between these two stocks. (Hint: note that these are
1
 2

historical returns. Sample estimators for average and volatility are required. Using the lecture
note’s notations, sample covariance between two assets i = 1, 2 is given by
T
1 !
σ̂12 = (r1t − r̄1 )(r2t − r̄2 ).
T − 1 t=1

Consider a portfolio that maintains a 50% weight on stock A and 50% weight on stock B.
(b) What is the return each year of this portfolio?
(c) Based on your results from part (b), compute the average return and volatility of the
portfolio.
(d) Show that the average return of the portfolio is equal to the average of the average
returns of the two stocks.
(e) Now, calculate the volatility of the portfolio directly using the definition of sample stan-
dard deviation, and then verify your result using “a2 + b2 + 2ab” (the blue equation on page
16 of lecture note FM.05pdf).
(f) Explain why the portfolio has a lower volatility than the average volatility of the two
stocks.

Problem 5. Consider an equally weighted portfolio of stocks in which each stock has a
volatility of 40%, and the correlation between each pair of stocks is 27%.
(a) What is the volatility of the portfolio as the number of stocks becomes arbitrarily large?
(b) What is the average correlation of each stock with this large portfolio?

Problem 6. Suppose Ford Motor stock has an expected return of 15% and a volatility of
38%, and Molson-Coors Brewing has an expected return of 12% and a volatility of 28%. If
the two stocks are uncorrelated,
(a) What is the expected return and volatility of a portfolio consisting of 70% Ford Motor
stock and 30% of Molson-Coors Brewing stock?
(b) Given your answer to part (a), is investing all of your money in Molson-Coors stock an
efficient portfolio of these two stocks?
(c) Is investing all of your money in Ford Motor an efficient portfolio of these two stocks?

Problem 7. Suppose Johnson & Johnson and Walgreens Boots Alliance have expected
returns and volatilities shown below, with a correlation of 21%.
Expected return Volatility
Johnson & Johnson 6.9% 17.9%
Walgreens Boots Alliance 9.6% 21.6%
(a) Calculate the expected return and the volatility of a portfolio that is equally invested in
Johnson & Johnson’s and Walgreens’ stock.
(b) If the correlation between Johnson & Johnson’s and Walgreens’ stock were to increase,
would the expected return of the equally invested portfolio rise or fall? What about the
volatility of the portfolio?
 3

(c) Calculate the expected return and the volatility of a portfolio that consists of a long
position of $8,500 in Johnson & Johnson and a short position of $1,500 in Walgreens’ stock.
(d) Plot the expected return as a function of the portfolio volatility. Using your graph,
identify the range of Johnson & Johnson’s portfolio weights that yield efficient combinations
of the two stocks, rounded to the nearest percentage point. (You may consider short-selling
but it suffices to consider portfolio weight below 120%.)
- End -
 I
Ex4 :

(a)

-Average return :

2 9
5
10 20 + 5 + +
3 5%
+
-

Stock A
-

+ : = .

+ Stock B : 21 + 7 + 30 -
3 -
8 + 25
(a) False = 12 %
6
not
Justification : Diversification can reduce risk as
long as asset returns are
-

Volatility :

↑
correlated, 319-35]
perfectly positively
correlated. Even returns are uncorrelated or
slightly positively StockA GA (20-3 5) (5-3 5)+ -S-3 5)(2-3
= 12-10-3
+
5
.

: .
.

+
.

diversification can still lower overall risk.

~ 10 6 %
False
.

(b) + Stock B

/ 52(21
Stock A because it could improve : SB ( 3 12) + ( 8 12)7 123 12)]
buy (7 12) + 130 127 + ( 3 12)
2
+
-

still
-

12)
-
-

Some investors
-

might
- -

+
-

Justification
= - -

if its risk-
their portfolio by reducing overall risk or
fitting their specific goals, even
~ 15 6 % .

less favorable its own

-return profile seem on .

- Covariance :

12)+ (5 3 5)(30 12) + ( 5

-

3 5)( 3 12) - -
+
3 5)(7
&
-

Ex2
-

(20
- .

260 3 5) (21-12)
- -

+
Giz
.

: -
.

= .

0
, 6x 300, 000
The number of shares purchased + Gold finger :
= 7, 826 09 = 7 , 826
3 5)( 8 12) + (9
-

3 5)(25
-

12)]
(2
: . - -
.
-
-

23
.

+ Moosehead 0 , 3x300 , 000

1 267 1268 0 00104
61 -
: .

-
.

, . =
71
Correlation :

+ Venture 0 , 1x300, 000

-

: 7 500
,00104
=
0
p
,

Associate 4 = = 0 .
0627
.

0 106 0 156
. . .

7 816 09x40 313 , 043 48 (D)

The new value of each stock : + Gold finger :
,
.
= .

b) Portfolio Returns
-

CL) 0 Sx( 3) 4 0%
0 5x( 5)
-
+
67 , 185 33
-

1 267 61x53 2013 :

.
-

5 5%
.

0 5x2)
+
Moosehead
=
. .

+ : ,
.

2010 : 0 Sx( .
-
10) + .
= .

0 5(-z) 3 0%
-

+ 2014 : 0S x 2 + =
=

13 5 %
. .
.

(3) 2011 :
0 5 x 20 + 0 5 x7 = .

105 , 000
.
.

17 0%
-

+ Venture 7 500 x 14 =
0 5x9 + 0 3x25 =

0 Sx 5 + 0 5x30 17 5 % 2015
.

, : .

2012
.

: - : . . = . +

Associate
(3) 485 , 226 58 .
(2)
c) Portfolio Metrics :
(1)
= .

+ +
value of the portfolio is :
a) The new

5 5 + 13 5 + 17 5 40 3 0 17 0
portfolio :
38
-

OS6
+
7 75 %1)
-
. . . . . .
.

b) The return of
-1 = 61 , 74 % o Portfolio Return :
p =
= .

- Average 6

c) The portfolio weight: Goldfinger 313 043 48

Volatility = (15 5 -7 73) (13 5 7 75)2+ (17 5 - 7 75)2 + ( 4 0 7 75) + 1 3 0
-

Portfolio
-
-

new
Sp
-

:
. . .
.

64 51 %
.

,
.
. .

:
.

~
.

.
-

485 , 226 58 .

-
7 .
737 + (17 0 .
- 7 .
75)"]
67 , 185 33
Moosehead 13 85 %
.

+ :
= .

485 , 226 58 .
= 9 72% .
(*]

105 , 000 A
(d) -verage return of 2 stocks : = 0 5(3 5%) + 05 (12 0 %)
21 64 %
Ventureaerate
.

p
. . .
.

:
+
.

485 , 226 58 .
= 7 75%
. (2)

(1) (2) E of 2 stocks ( 7 75 % )

Ex3 : portfolio return
average return
=

average
.

The 600 x 347 5 000 95 $953

a) portfolio value : + 10 000 ,
x 18 + , x = ,
200 00 .

(e)
of the three stocks
The weight Op W W 2wAWBPEPS
-

= + +
600 x 547
+ Apple :
x100% = 33 38 % .

983 200 00
,

64 + 12 0 3 0 5 0 06
.

E 10 6 15 6) = 9 72
%
Ciscosystem 10 , 000 18 2
-

+ :
x
x 100% % = 18 31 % .
-

10 5210 6) + 10 5 15 . .
.
.
.
.
.

e
. . .
.
.
. .

983 200 00
. .

, .

+
Colgate-Pamolive : 5 ,000x95
x 100% = 48 31 % .

detrition.
985 ,200 00 . ( 1 : using the
b) Expectedreturn :
Cf ( two stocks' volatilities
the average of the
(0 3338 x0 12) + (0 1831 x 0 1) + 10 4831x0 08) = 0 0978 9 70 % The has lower volatility than
portfolio
=
I
.
. .
.
.
. . .

reduces portfolio's risks

-

portfolio value
c) The because of diversification =>
000x(95
new :

7) 5, 14) 995 200

600x(547 20) 10 000x(18 + + = ,

(p 0 0627(1) is low
-

+ + ,
-
The correlation = .

The new portfolio weight :

600 (547 + 20) Ex Si

34 10 %
x
+ Apple :
x 100 % = ,
of the portfolio :

995 , 200 (a) The volatility

+
Cisco 10 ,000 x (18 + 7) 0 40 x0 40 x 0 27 =
0 2078 - 20 78 %
.
.

100% 25 , 12 %
. .
.

x =

System 995 ,
200
(b) The average correlation :
5, 000 x (95-14)
+ Colgate :
x 100% 40 70%
40
=

0 27 x 0 40x0 0 5196 51 96%

.

Palmolive 995 , 200 = =

.
-
. .
.
.

0 08) 2078
return (0 3418 x 0 12) + (0 2512x0 10) + 10 4070 x 0 40 x 0 .

Expected Portfolia
. .

d)
.
.

:
.

. .

= 0 0987 .

= 9 87% .
 I
Ex6 :

return of the portfolio is (d)

expected
:

(a) -
The
0 141
0 15 + 0 30x0 12
·

. 70
0 x . .
. = .

(1)
= 14 1 %
.

is
The of the
given portfolio
:

-
variance
"
2x0 70 x 0 30 x0 0 38 0 28
0 30
2 x
2
0 38 0 28 + x .

Var (1) =
0 70 .
x .
+ . x .
.
.
.

= 0 0778 .

SD(Rp) = Var(Rp)
of the given portfolios
:

-
The volatility = 0 0778
.

= 27 89 % (2)
.

(b) We got :

Given portfolio
Molson-Coors Stock
10+
%

↓
108 % W
< 14, 1 %

--
Return 12 %
Expected : 9%

%
+

27 89 %
28% 7

T
.

Volatility
n
+ :

would not make an efficien t

in Molson-Coors 100% JJ

=> Investing all money

Molson-Coors terms of
&

in
dominates investing entirely
in
18% %o 18 % igno 20% 21%
portfolio (portfolio 15%

risk-return trade offf

< 60 %, portfolio are efficient
.
c) We got : Given portfolio => With x(j)
ford Motor Stocks

+ Expected Return : 15 % > 14 , 1 % slightly lower

+ Volatility :
38 % 7 27 89 %
.
=> significantly lower
would not make an efficient
all money in ford Motors
= Investing return
reduce riskE
small decrease in

portfolio (for some investors ,

Ex7

(a)The expected return of the portfolio :

ECRp] =
xjE[Rj] + kwE[Rw]
0 069 + 0 50 0 896
= 0 50
.
. .
. .
.

= 0 0825 .

= 8 25 % .

-The volatility of the portfolio

SD(Rp) =
(SD(Rj)"+" SD(Rw) + 2xiXwCorr(R iRw)SD(Rj)
SDCRw)

~
502 (0 179) 0 50 ? (0 216)" (0 50) ? 21) (0 (0 179) (0 216)
:

. .
+ . + 2 . .
.
.
. . .

= 0 1540
.

= 15 4 % .

constant , assuming the correlation

return would only
(b) The expected remain

change (0 5x0 069 +

.
.
0 5 x 0 096
.
. = 5 25. %)

(due to the correlation term in the

of the portfolio would increase
-
The volatility
calculation for the volatility of a portfolio
In this the total investment is
(C) case
,

2 500 -
,1 500 = 7, 000
,

= The portfolio weight are :

xj =
8500
x 100 % = 121 43 %;n
=
, 100x100% 1 43 %
.

7000

R(P) = xjRj + kwRw =

=
1 2143

6 325 % .
.
x 0 .
069 - 0 2143x0 096
.
.

I
SD(RA) = (1 2143)? (0 1795
.
.
-
0 21432 0 2162 + 1 2413 (-0 2143) x 0 21x0 179 0 216
. . . . . . .
.
.
.

= 0 1721
.

= 17 21 % .