Homework Exercises
NB:
These exercises are similar to those that you will encounter on the exam and more difficult than
those in Berk and DeMarzo book. Making these exercises is not obligatory but of course if you do not
make them, most likely you will not be able to pass the exam. During AWVs these exercises will be
discussed. Needless to say, additionally you should be able to solve all the problems at the ends of
Chapters 1013; these problems are similar to your MyFinanceLab tests
.
1.
It is wellknown that the volatility of gold prices has been very low historically: around 12% per
annum. Lately, the annualized gold volatility has reached 32%. For both historical and current
gold volatilities, compute also the daily, weekly and monthly gold volatilities.
Historic:
σ
day
= 12%/√(251) = 0.757%
Current:
σ
day
= 32%/√(251) = 2.020%
σ
week
= 12%/√(52) = 1.664%
σ
week
= 32%/√(52) = 4.438%
σ
month
= 12%/√(12)= 3.464%
σ
month
= 32%/√(12)= 9.238%
2.
Historical annualized return on stocks is approximately 8% and the annualized volatility is 35%.
Compute daily, weekly and monthly
volatilities.
σ
day
= 35%/√(251) = 2.209%
, σ
week
= 35%/√(52) = 4.854% , σ
month
= 35%/√(12)= 10.104%
3.
The following table gives monthly closing prices of two stocks for the last 12 months.
Date
QLogic
Mattel
Dec
27
.
6
22
.
8
Jan
25
.
4
21
.
7
Feb
26
.
4
22
.
6
Mar
28
.
6
23
.
5
Apr
27
.
9
23
.
6
May
25
.
2
22
.
0
Jun
25
.
8
22
.
8
Jul
26
.
1
23
.
2
Aug
21
.
5
19
.
9
Sep
19
.
2
17
.
2
Oct
19.1
16
.
3
Nov
19
.
9
16
.
4
a)
Compute monthly stock price returns. Assume no dividends. Compute average monthly returns.
b)
Compute variances, volatilities and annualized volatilities of both stock returns.
c)
Compute the covariance between these stock returns and the correlation coefficient.
d)
What are the average return, variance and annualized volatility of the portfolio consisting for
50% of QLogic and 50% of Mattel?
e)
Now suppose you have 10 000 $ to invest. You borrow and short sell 5 000 $ worth of Mattel
stock and invest the resulting 15 000 $ into QLogic. What is now the expected return and the
monthly and yearly volatility
of your portfolio?
f)
Explain what happens to the portfolio volatility in e) and in f) if the correlation between the
stock increases? Why this happens?
a)
Returns(t+1): (Rt+1

Rt)/Rt
Qlogic:
Return(jan): (25.4

27.6)/27.6=

0.0797 or

7.97%
Return(feb): (26.4

25.4)/25.4= 0.0394 or 3.94%
etc.
Mattel:
Return(jan):
(21.7

22.8)/22.8=

4.82%
Return(feb):
(22.6

21.7)/21.7= 4.15%
etc.
Average monthly returns:
:
Qlogic= 
2.631% , Mattel=

2.734%
b)
Var(R
Qlogic
, monthly)=( (

7.97%
 
2.631%)2 + (3.94% + 2.631%)2 + ……)/(11

1)= 0.00622,
σ
Qlogic
(monthly)= √(0.00622)= 7.89%,
σ
Qlogic
(yearly)= √(12)*7.89%= 27.32%
Var(R
Mattel
,monthly)
=
( (

4.82%
 
2.734%)2 + (4.15% + 2.734%)2 + ……)/(11

1) = 0.00451,
σ
Mattel
(monthly)= √(0.00451)= 6.71%,
σ
Mattel
(yearly)= √(12)*6.71%= 23.26%
c
)
((

7.97%
 
2.631%)
*
(

4.82%
 
2.734%)+(3.94% + 2.631%)
*
(4.15% + 2.734%)+…
… … …)/(11

1)=
0.004823
Measurement of the variance between stocks
Positive: returns tend to move in the same direction, Negative: returns tend to move in the
opposite direction
Correlation
=ρ
i,j
=
0.004823/(7.89%*6.71%)=
0.9165
‘’
normalized covariance’’, always between

1 and +1. If Ri on average changes with x% then Rj
will on average change with
ρ
i,j
*x%
d)
Rp=0.5*

2.631%+0.5*

2.734%=

2.682%
Var(Rp)= 0.5^2*0.00622 +0.5^2*0.00451 + 2*0.5*0.5*0.004823 = 0.00511
σ(monthly)= √(0.00511)= 7.147%,
σ(yearly)= √(12)* 7.147%= 24.76%
e)
Weights
: Qlogic =
15000/10000= 1.5(150%)
, Mattel=

5000/10000=

0.5(

50%)
Rp=1.5*

2.631%+

0.5*

2.734%=

2.579%
Var(Rp)= 1.5^2*0.00622 +(

0.5)^2*0.00451 + 2*1.5*0.5*0.004823 = 0.00784
σ(monthly)= √(0.00784)= 8.855%,
σ(yearly)= √(12)* 8.855
%
= 30.67
%
f)
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