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Course
385
Subject
Business
Date
Feb 20, 2024
Type
docx
Pages
10
Uploaded by BailiffLobsterMaster112
Concordia University
John Molson School of Business
FINA 385: Theory of Finance I
Fall 2023
Final review solutions
1.
Using 61 monthly observations you have just estimated that the covariance between return on
stock A and the market index is 0.075 and that the standard deviation of the market index is
0.25. Also, the standard error of the regression is 0.258795.
(a)
what is the beta of stock A?
β
=
n
n
−
1
∗
cov
(
r
A
,r
M
)
σ
M
2
=
61
60
∗
0.075
(
0.25
)
2
=
1.22
(b)
what is the systematic risk of stock A?
β
2
σ
M
2
=(
1.22
)
2
∗
0.0625
=
0.093025
(c)
what is the firm-specific risk of stock A?
σ
2
(
e
i
)=
s
2
=(
0.258795
)
2
=
0.066975
(d)
calculate and interpret the coefficient of determination (R
2
).
R
2
=
β
A
2
σ
M
2
σ
A
2
=
0.093025
0.093025
+
0.066975
=
0.093025
0.16
=
0.581406
58.1406% of the variance of stock A is systematic or market-related
2.
There are three well-diversified portfolios, X, Y, and Z. Portfolio X has a beta of 0.5 and the
expected return of 7%, while the portfolio Y has beta of 1.1 and the expected return of 11.7%.
Portfolio Z has a beta of 1.25 and the expected return of 11.5%. The risk-free rate is 4%. Is
there an arbitrage opportunity? If so, how would you go about exploiting it and what profit
would you expect to make?
for there to be no arbitrage opportunities, the risk premiums have to be proportional to
betas:
E
(
r
X
)−
r
f
β
X
=
E
(
r
Y
)−
r
f
β
Y
=
E
(
r
Z
)−
r
f
β
Z
0.07
−
0.04
0.5
=
0.117
−
0.04
1.1
=
0.115
−
0.04
1.25
0.06
≠
0.07
≠
0.06
since the slope of portfolio Y is greater than that of the other two portfolios, it must be
above the SML and therefore is underpriced.
1
so we need to buy portfolio Y, sell portfolio P with the same beta as that on portfolio Y.
There are three ways to create such a portfolio:
(
1
)
β
P
=
w
X
β
X
+
w
rf
r
f
w
X
=
β
P
β
X
=
1.1
0.5
=
2.2
w
rf
=−
1.2
(
2
)
β
P
=
w
Z
β
Z
+
w
rf
r
f
w
Z
=
β
P
β
Z
=
1.1
1.25
=
0.88
w
rf
=
0.12
(
3
)
β
P
=
w
Z
β
Z
+
w
X
β
X
w
X
=
β
P
−
β
Z
β
X
−
β
Z
=
1.1
−
1.25
0.5
−
1.25
=
0.2
w
Z
=
0.8
any of the above three portfolios will yield you the expected return of 10.6%.
your expected profit from this arbitrage strategy (in percent terms) is buy portfolio Y
+ 11.7%
sell portfolio P
– 10.6% profit
+ 1.1% 3.
An investor can design a risky portfolio with two stocks, A and B. Stock A has an expected
return 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 is 9%. What portfolio of the risk-free investment as well as stocks A and B
will give you the expected return of 12.5%? What is the standard deviation of this portfolio?
2
w
A
=
[
E
(
r
A
)−
r
f
]
σ
B
2
−[
E
(
r
B
)−
r
f
]
σ
A
σ
B
ρ
AB
[
E
(
r
A
)−
r
f
]
σ
B
2
+[
E
(
r
B
)−
r
f
]
σ
A
2
−[
E
(
r
A
)−
r
f
+
E
(
r
B
)−
r
f
]
σ
A
σ
B
ρ
AB
w
A
=
[
0.18
−
0.09
]∗(
0.15
)
2
−[
0.15
−
0.09
]∗
0.225
∗
0.15
∗
0.75
[
0.18
−
0.09
]∗(
0.15
)
2
+[
0.15
−
0.09
]∗(
0.225
)
2
−[
0.18
−
0.09
+
0.15
−
0.09
]∗
0.225
∗
0.15
∗
0.75
w
A
=
0.00051
0.00127
=
0.40
w
B
=
1
−
0.4
=
0.60
E
(
r
P
)=
0.4
∗
0.18
+(
0.6
)∗
0.15
=
16.2%
σ
p
=
√
(
0.4
)
2
∗(
0.225
)
2
+(
0.6
)
2
∗(
0.15
)
2
+
2
∗(
0.4
)∗(
0.6
)∗(
0.15
)∗(
0.225
)∗
0.75
σ
p
=
√
0.02835
=
0.16837
=
16 .837%
S
=
E
(
r
P
)−
r
f
σ
p
=
0.162
−
0.09
0.16837
=
0.42762
E
(
r
C
)=
y
∗
E
(
r
P
)+(
1
−
y
)
r
f
y
=
E
(
r
C
)−
r
f
E
(
r
P
)−
r
f
=
0.125
−
0.09
0.162
−
0.09
=
0.4861
w
RF
=
1
−
y
=
1
−
0.4861
=
0.5139
w
A
=
0.4
∗
0.4861
=
0.19444
w
B
=
0.6
∗
0.4861
=
0.29167
the standard deviation of the above portfolio is
E
(
r
C
)=
r
f
+
Sσ
C
σ
C
=
E
(
r
C
)−
r
f
S
=
0.125
−
0.09
0.42762
=
0.08185
=
8.185%
4.
Please refer to the following diagram (bear spread).
Payoff
$60
$75
Stock price
(a)
what combination of puts will give you the above payoff?
buy a put with exercise price of $75 and sell a put with exercise price of $60.
3
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(b) suppose the underlying stock currently sells at $75, the risk-free rate is 5% and the
standard deviation of return on the stock is 25%. If you were to use 3-month (0.25 years)
put options to replicate the above payoff, how much would it cost you to establish this
position?
C
0
=
S
0
N
(
d
1
)−
Xe
−
rT
N
(
d
2
)
where
d
1
=
ln
(
S
0
/
X
)+(
r
+
σ
2
/
2
)
T
σ
√
T
and
d
2
=
d
1
−
σ
√
T
d
1
=
ln
(
75
/
75
)+(
0.05
+(
0.25
)
2
/
2
)∗
0.25
0.25
∗
√
0.25
d
1
=
0.02031
0.125
=
0.1625
and
d
2
=
0.1625
−
0.125
=
0.0375
N
(
d
1
)=
N
(
0.16
)=
0.5636
and
N
(
d
2
)=
N
(
0.04
)=
0.5160
C
0
=
75
∗
0.5636
−
75
∗
e
−
0.05
∗
0 .25
∗
0.5160
C
0
=
42.27
−
38.22
C
0
=
4.05
P
=
C
+
Xe
−
r
∗
T
−
S
0
P
=
4.05
+
75
e
−
0.05
∗
0.25
−
75
P
=
4.05
+
74.06834
−
75
=
3.12
For the put with exercise price of $60:
C
0
=
S
0
N
(
d
1
)−
Xe
−
rT
N
(
d
2
)
where
d
1
=
ln
(
S
0
/
X
)+(
r
+
σ
2
/
2
)
T
σ
√
T
and
d
2
=
d
1
−
σ
√
T
d
1
=
ln
(
75
/
60
)+(
0.05
+(
0.25
)
2
/
2
)∗
0.25
0.25
∗
√
0.25
d
1
=
0.22314
+
0.02031
0.125
=
1.94763
and
d
2
=
1.94763
−
0.125
=
1.82263
4
N
(
d
1
)=
N
(
1.95
)=
0.9738
+
0.9750
2
=
0.9744
and
N
(
d
2
)=
N
(
1.82
)=
0.9556
C
0
=
75
∗
0.9744
−
60
∗
e
−
0.05
∗
0.25
∗
0.9556
C
0
=
73.08
−
56.62
C
0
=
16.46
P
=
C
+
Xe
−
r
∗
T
−
S
0
P
=
16.46
+
60
e
−
0 .05
∗
0.25
−
75
P
=
16.46
+
59.25
−
75
=
0.71
total cost
to establish the position = $3.12 – $0.71 = $2.41
5.
A six-month call option with an exercise price of $55 on a stock that currently trades at $50 a
share is selling for $4.00. If the risk-free rate is 5% and the put option with the same time to
expiration and the same exercise price is selling for $6, is there an arbitrage opportunity? If so,
how would you go about making an arbitrage profit?
P
+
S
0
=
C
+
Xe
−
rT
P
=
C
+
Xe
−
rT
−
S
0
P
=
$
4 .00
+
$
55
e
−
0 .05
∗
0 .5
−
50.00
P
=
$
4 .00
+
$
53.64205
−
$
50
=
$
7.64205
yes, there is an arbitrage opportunity since the put is undervalued
you want to buy the put and sell the right hand side of the second equation above:
CF now
S < 55
S > 55
buy put
– 6.00 55 – S
0
borrow PV(X)
+ 53.64205 – 55
– 55
buy stock
– 50.00
S
S
sell/write call + 4.00
0
– (S – 55)
1.64205
0
0
6.
You are asked to estimate beta of Alcan’s stock. Using S&P 500 as a proxy for the market
index you obtain the regression output below. SUMMARY OUTPUT
Regression Statistics
Multiple R
0.57700
R Square
0.33292
Standard Error
0.06838
Observations
67
Coefficients
Standard Error
t Stat
P-value
Intercept
0.00758
0.00835
0.90752
0.36748
S&P 500 Index
1.02818
0.18052
(a)
is your estimate of beta statistically significantly different from zero?
5
yes, it is statistically significant at the 1% level because the t
-statistic is 5.69562,
which is higher than the critical value for the 1% level of significance (2.576)
(b)
what is the standard deviation of the firm specific component of Alcan’s stock return?
standard deviation of the firm specific component is also called standard error of
regression
and in this case is 0.06838
(c)
what is the total risk of Alcan?
we know that R
2 tells us the fraction of the total risk that is systematic. Since we
know both R
2
and the firm-specific risk, we can figure out the total risk:
R
2
=
1
−
σ
2
(
e
i
)
σ
A
2
σ
A
2
=
σ
2
(
e
i
)
1
−
R
2
=
(
0.06838
)
2
1
−
0.33292
=
0.0046758
0.66708
=
0.0070094
(d)
what is Alcan’s systematic risk?
systematic risk is the difference between the total risk and firm-specific risk:
σ
A
2
=
β
2
σ
M
2
+
σ
2
(
e
i
)
β
2
σ
M
2
=
σ
A
2
−
σ
2
(
e
i
)=
0.0070094
−
0.0046758
=
0.0023336
(e)
what is the standard deviation of the return on S&P 500 index?
since systematic risk is
β
2
σ
M
2
, it follows that σ
M
2
=
0.0023336
(
1.02818
)
2
=
0.0022074
σ
M
=
√
0.0022074
=
4.69834%
7.
A 6% annual coupon bond has a modified duration of 10 years and is currently selling at
$800. Its yield to maturity is 8%. If the yield to maturity increases to 9%, what is the
expected change in the bond’s price?
the expected change in the price can be expressed as a product of modified duration
and the change in the yield to maturity:
ΔP
P
=−
D
¿
×
Δy
ΔP
=[−
D
¿
×
Δy
]∗
P
ΔP
=−
10
∗
0.01*$800
=−
$
80
8.
Two years ago you purchased a bond that paid 10% annual coupon and had 6 years to maturity.
The par value of the bond was $1,000. At the time of purchase the yield to maturity of the bond
was 8%. Calculate your realized compound yield over the last two years if:
(a)
the yield to maturity remained the same
if you reinvest the coupons at the yield to maturity, then your realized compound
yield will be equal to the yield to maturity:
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P
buy
=
$
1
,
000
(
1
+
0.08
)
6
+
∑
t
=
1
n
$
100
(
1
+
0.08
)
6
=
$
1
,
092. 46
P
sell
=
$
1
,
000
(
1
+
0.08
)
4
+
∑
t
=
1
n
$
100
(
1
+
0.08
)
4
=
$
1
,
066.24
capital loss =
$1,066.24 - $1,092.46
=
$26.22
total dollar return =(
1.08
)
*$100
+
$100 −
$26.22 =
$181.78
percentage annual return =
√
1
+
$181.78
$1,092 .46
−
1
=
8%
(b)
the yield to maturity increased to 9% after right before you received your first coupon
payment
P
buy
=
$
1
,
000
(
1
+
0.08
)
6
+
∑
t
=
1
n
$
100
(
1
+
0.08
)
6
=
$
1
,
092. 46
P
sell
=
$
1
,
000
(
1
+
0.09
)
4
+
∑
t
=
1
n
$
100
(
1
+
0.09
)
4
=
$
1
,
032.40
capital loss =
$1,032.40 - $1,092.46
=
$60.06
total dollar return =(
1.09
)
*$100
+
$100 −
$60.06 =
$148.94
percentage annual return =
√
1
+
$148.94
$1,092.46
−
1
=
6.599%
9.
Calculate the effective annual yield of a bond that is currently selling at $98,039 and has 2
months to maturity.
yield to maturity, YTM, can be calculated as ($100,000 – $98,039) / $98,039 – 1 = 0.02.
Effective annual yield, EAY, is (1 + YTM)
6
– 1 = 12.618%
10.
A stock is currently trading at $100 per share. The stock price is equally likely to increase to
$120 or to decline to $80 per share over the next year. What must be the price of a one year
put option with exercise price equal to $100 if the risk-free rate is 10%?
Stock Price
Put Payoff
120
0
80
20
The hedge ratio is – 0.5. A portfolio comprised of one share and two puts would provide a guaranteed payoff of $120, with present value $120/1.10 = $109.09. Therefore, S + 2P = 109.09
100 + 2P = 109.09
P = $4.545
7
11. Current prices of call options on Furniture City Inc stock are given in the table below.
Identify two apparent pricing discrepancies in the table. Identify which of the basic option
pricing relationships each discrepancy violates.
Expiration month
Stock price
Exercise price
June
July
August
September
$119.50
$110
$9 $12.50
$15
$18
$119.50
$120
$1.50
$3.75
$3
$4.50
$119.50
$130
$1
$2.25
$2.75
$5
first, look at the calls with X = $120. The price of a call option should be increasing with
the time to maturity, but this is not the case. The August call with X = $120 seems to be
mispriced.
second, look at September calls. The price of a call option should be decreasing with the
exercise price. This, clearly, is not the case. The September call with X = $130 seems to
mispriced.
12.
The following is a list of prices of zero coupon bonds with different maturities and par value
of $1,000. Time to Maturity Price
1
$943.40
2
$881.68
3
$808.88
4
$742.09
Calculate the following:
(a)
yield to maturity for each zero coupon bond
(b)
forward rates for each of the next 4 years
yields to maturity and forward rates are given in the table below.
Time to
maturity
Price
YTM
forward
rates
1
943.4
6.000%
6.00%
2
881.68
6.499%
7.00%
3
808.88
7.326%
9.00%
4
742.09
7.742%
9.00%
Forward rates are calculated using the following formula:
(
1
+
f
n
)=
(
1
+
y
n
)
n
(
1
+
y
n
−
1
)
n
−
1
f
2
=
(
1
+
y
2
)
2
(
1
+
y
1
)
−
1
f
2
=
(
1
+
0.06499
)
2
(
1
+
0.06
)
−
1
=
7%
13.
What is the modified duration of a seven-year bond which has a coupon rate of 9% and sells at
par?
8
D
=
1
+
y
y
−
(
1
+
y
)+
T
(
c
−
y
)
c
[(
1
+
y
)
T
−
1
]+
y
D
=
1
+
0.09
0.09
−
(
1
+
0.09
)+
7
(
0.09
−
0.09
)
0.09
[(
1
+
0.09
)
7
−
1
]+
0.09
D
=
12.11
−
1.09
0.1645235
=
12.11
−
6.6251
=
5.4848075
D
¿
=
D
(
1
+
y
)
=
5.4848075
1.09
=
5.03
14. Bond A is an 8% coupon bond with 20 years to maturity selling at par (face) value. Bond B is
an 8% coupon bond, with 20 years to maturity selling below par (face) value. Which of the two
bonds has a longer duration?
bond B has a higher yield to maturity than bond A since its coupon payments and
maturity are equal to those of A, while its price is lower. Therefore, its duration must
be shorter.
15.
The common stock of CALL Corporation has been trading in a narrow range around $50 per
share for months, and you are convinced that it is going to stay in that range for the next three
months. The price of a 3-month put option with an exercise price of $50 is $4. The stock will
pay no dividends for the next three months. (a)
if the risk-free rate is 10% per year, what must be the price of a 3-month call option on
CALL stock with an exercise price of $50 if it is in the money?
from put-call parity, C = P + S
0
– X*e
-rT
C = 4 + 50 – 50e
-0.1*0.25
= $5.23
(b)
what would be a simple options strategy using a put and a call to exploit your conviction
about the stock price’s future movement? What is the most money you can make on this
position? How far can the stock price move in either direction before you lose money?
sell a straddle, i.e., sell a call and
a put to realize premium income of $4 + $5.23 =
$9.32. If the stock ends up at $50, both the options will be worthless and your profit
will be $9.23. This is your maximum possible profit since at any other stock price,
you will need to pay off on either the call or the put. The stock price can move by
$9.23 in either direction before your profits become negative. (c)
how can you create a position using a put, a call, and risk-free lending that would have the same payoff structure as the stock at expiration? How much will it cost you to establish this position?
buy the call, sell (write) the put, lend $50e
-0.1*0.25
. The payoff is as follows:
Position
Immediate CF
CF in 3 months
S
T
X
S
T
> X
Call (long)
C = 5.23
0
S
T
– 50
Put (short)
– P = – 4.00
–(50 – S
T
)
0
Lending position 50/(1.10)
1/4
= 48.77
50 50
––––––––––––––––––
–––––––
–––––––
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TOTAL
C – P + X*e
-rT
= $50.00
S
T
S
T
by the put-call parity theorem, the initial outlay equals the stock price, S
0
, or $50. In
either scenario, you end up with the same payoff as you would if you bought the
stock itself. 16.
A vertical combination is the purchase of a call with exercise price X
2
and a put with exercise
price X
1
, with X
2
greater than X
1
. Graph the payoff to this strategy.
Position
S
T
< X
1
X
1
S
T
X
2
X
2 <
S
T
Buy call (X
2
)
0
0
S
T
– X
2
Buy put (X
1
)
X
1 – S
T
0
0
Total
X
1 – S
T
0
S
T
– X
2
17. Prove that the portfolio variance formula in the case of two perfectly negatively correlated
assets simplifies to σ
P
2
=(
w
1
σ
1
−
w
2
σ
2
)
2
.
σ
P
2
=
w
1
2
σ
1
2
+
w
2
2
σ
2
2
+
2
w
1
w
2
σ
1
σ
2
ρ
12
σ
P
2
=
w
1
2
σ
1
2
+
w
2
2
σ
2
2
−
2
w
1
w
2
σ
1
σ
2
σ
P
2
=(
w
1
σ
1
−
w
2
σ
2
)
2
10
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