Exercises
Financial Economics
Maths Exercises
Problem Set 1
Problem 1
a)
8000 ∗ 1,053 = 9261€
8000 ∗ 1,0513 = 15085,15€
32000
= 1,05𝑛
8000
32000
𝑛𝑙𝑛1,05 = ln (
)
8000
𝑛 = 28,4𝑦𝑒𝑎𝑟
b)
𝑥 ∗ 1,055 = 50000 => 𝑥 = 39176,3€
c)
1,053 = 1,1576
1,0256 = 1,1597
1,0752 = 1,1556
1,00536 = 1,1967 => better
d)
0,1 12
(1 +
) − 1 = 10,47% 𝐸𝐴𝑅
12
10%
0,1 365
(1 +
)
− 1 = 10,52% 𝐸𝐴𝑅
365
e)
𝑥 2
5% = (1 + ) − 1 => 𝑥 = 4,939%
2
𝑥 12
5% = (1 + ) − 1 => 𝑥 = 4,889%
12
f)
1000 ∗ 1,0510 = 1628,89€
0,05 12
) − 1 = 5,116%
12
1000 ∗ 1,0511610 = 1646,98€
(1 +
0,05 𝑛
lim (1 +
) − 1 = 𝑒 0,05 − 1 = 5,127%
𝑛→∞
𝑛
1000 ∗ 1,0512710 = 1648,7€
Problem 2
a)
𝑁𝑃𝑉 = −10000 +
500 1500 10000
+
+
= −2609,36 𝑙𝑎𝑡𝑠
1,06 1,062 1,0610
Maths
Exercises
𝑁𝑃𝑉 = −10000 +
500 1500 10000
+
+
= 135,43 𝑙𝑎𝑡𝑠
1,02 1,022 1,0210
b)
𝐴: 𝑁𝑃𝑉 = −10 +
20
= 8,18
1,1
5
= 9,54
1,1
10
𝐶: 𝑁𝑃𝑉 = 20 −
= 10,9
1,1
=>C
=>B,C
𝐵: 𝑁𝑃𝑉 = 5 +
Comparaison B, C
Projet C
CF today ; 20
CF one year : -10
−15/1,1 = 13,6364
+15
Comparaison B
6,3664 > 5
5
Regardless of pour preferences for cash today vs cash in the future we should always max NPV
first, we can then borrow/lend to shift cash flows through time and find out most preferred patter of
cash flow
Problem 3
30000
= 375000€
0,08 − 0
If we invest 375000€ at 8%, we can withdraw 30000€/ year
30000
= 750000€
0,08 − 0,04
If we invest 750000€ at 8%, with 4% inflation, we can withdraw 30000€/ year
Problem 4
a)
1 − 1,06−18
1200 ∗
= 12 993,1242€
0,06
We are at t=12, there are 18 annuities left.
IF we are at t=0, we multiply by 1,0612
we have : 1200 ∗
1,06−12 −1,06−30
0,06
∗ 1,0612 = 12 993,1242€
b)
1,0517 − 1,117
∗ 1,1−17
1,05 − 1,1
𝑥 = 21 861 455,8€
𝑥 = 2 000 000 ∗
Problem 5
a)
350 000 − 50 000 = 𝑎 ∗
𝑎=
1 − 1,07−36
0,07
300 000 ∗ 0,07
=> 𝑎 = 24 175,92€
1 − 1,07−30
300 000 = 23 500 ∗
1 − 1,07−30
𝑥
+
0,07
1,0730
𝑥 = 63 848,0342€
2
Maths
Exercises
b)
2 000 000 = 𝑎 ∗
1,0536 − 1
0,05
𝑎 = 20 868,91€
We have 36, because in 35 year there are 36 annuities !
In fact one annuity at t=0 and one at t=35.
2 000 000 = 𝑎 ∗
1,0736 − 1,0536
=> 𝑎 = 7102,1138€
1,07 − 1,05
c)
1 − 1,05−𝑛
> 200000
0,05
200000
1,05−𝑛 < − [
∗ 0,05 − 1]
25000
200000
ln(− [
∗ 0,05 − 1])
25000
−𝑛 <
ln 1,05
𝑛 > 10,47 𝑦𝑒𝑎𝑟 (11)
25000 ∗
d)
(1,02𝑛 − 1,08𝑛 )
1,35 − 1,085
∗ 1,08−𝑛 ∗ 1,08−5 + 1000000 ∗ 1,3 ∗
∗ 1,08−5
1,02 − 1,08
1,3 − 1,08
= 42 958 282 + 9 022 932,276 =
1000000 ∗ 1,3 because at t=0, there is 1M so at the end of first year we have 1000000 ∗ 1,3.
At the 6th year we have then 1000000 ∗ 1,35 ∗ 1,02
1000000 ∗ 1,35 ∗ 1,02 ∗
Problem 6
1 − 1,07−35
100 000 ∗
= 1 294 767,23€
0,07
1,0735 − 1
1 294 767,23 = 𝑎 ∗
0,07
𝑎 = 9366,29€
𝑥 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑤𝑎𝑔𝑒
1,0235 − 1,0735
𝑥 ∗ 75000 ∗
= 1 294 767,23€
1,02 − 1,07
𝑥 = 9,948%
Must find n
1000000 − 50000 ∗ (1,05𝑛 ) > 0
𝑛 = 62
𝑁𝑃𝑉 = −10 000 000 − 50 000 ∗
1,05𝑛 − 1,06𝑛
1 − 1,06−𝑛
∗ 1,06−𝑛 + 1 000 000 ∗
1,05 − 1,06
0,06
With 𝑛 = 62
𝑁𝑃𝑉 = 3 995 073,97
3
Maths
Exercises
Problem Set 2
Problem 1
a)
First find 𝑖1
100
= 94
1 + 𝑖1
𝑖1 = 6,3830%
𝑖2
100
= 85
(1 + 𝑖2 )2
𝑖2 = 8,4652%
100
100
+
1,063830
1,0846522

𝑃=

We can also do 94+85…
100
500
𝑃 = 1,063830 + 1,0846522 = 519

𝑃 = 1,063830 + 1,0846522 = 132
50
= 179
100
There an arbitrage opportunity (>130) you buy.
b)
Case 1
Asset 1
𝑃1 = 0,5 = 1 − 0,5
𝑃2 = 3 > 5 − 2,5
Asset 2 is too expensive
We sell it / buy Asset 1
Case 2
Asset 1
𝑃1 = 0,5 =
1
2
+
1 + 𝑖 (1 + 𝑖)2
𝑖 = 224% ?
𝑃2 = 2,5 =
3
10
+
1 + 𝑖 (1 + 𝑖)2
𝑖 = 169% ?
Or *5=>2,5=5+10 : better !
Asset 1 is better : buy, sell asset 2
Problem 2
a)
In the 2 cases it’s 600
b)
market price : 577€
return of 600
3,9861%
c)


3 asset A, 1 asset B
3 ∗ 231 + 1 ∗ 346 = 1039
1800+600
= 1200
2
4
Maths
Exercises
1200
= 1039
1+𝑖
𝑖 = 15,4957%

10%
15,4957 − 3,9861 = 11,50%
Sell asset C
Problem 3
a)
Even if the standard deviation is higher and the expected return lesser, some people would invest
in it to diversify their portfolio, because they are uncorrelated.
b)
the expected return is : 20% ∗ 0,6 + 15% ∗ 0,4 = 18%
the volatility
𝜎𝑝 = √0,6² ∗ 0,2² + 0,4² ∗ 0,25² − 2 ∗ 0,4 ∗ 0,6 ∗ 0,4 ∗ 0,2 ∗ 0,25 = 12,17%
it’s good.
c)
the lowest risk:
𝑥 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝐴,
𝑦 = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝐵
min 𝜎𝑝 = 𝑥² ∗ 0,2² + 𝑦² ∗ 0,25² − 2 ∗ 0,4 ∗ 𝑥 ∗ 𝑦 ∗ 0,2 ∗ 0,25
𝑥+𝑦 =1
So
min 𝜎𝑝 = 𝑥² ∗ 0,2² + (1 − 𝑥)2 ∗ 0,252 − 2 ∗ 0,4 ∗ 𝑥 ∗ (1 − 𝑥) ∗ 0,2 ∗ 0,25
= 0,1425𝑥² − 0,165𝑥 + 0,0625
min′ 𝜎𝑝 = 0,285𝑥 − 0,165
Minimum for 𝑥 = 0,5789
𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 ∶ 𝑥 = 0,5789, 𝑦 = 0,4211,
𝐸 = 17,9%, 𝜎 = 12,14%
d)
𝐴
∑4 × 4 𝐵
𝐶
[𝐷
𝐴
𝐵
𝐶
𝐷
10 −10 5 12
−10 15 10 −5
5
10 20 0
12
−5
0 12 ]
𝑉𝑎𝑟𝑃 = 0,25 ∗ (0,25 ∗ 10 + 0,40 ∗ −10 + 0,2 ∗ 5 + 0,15 ∗ 12) + 0,40
∗ (0,25 ∗ −10 + 0,40 ∗ 15 + 0,2 ∗ 10 + 0,15 ∗ −5) + 0,20
∗ (0,25 ∗ 5 + 0,40 ∗ 10 + 0,2 ∗ 20 + 0,15 ∗ 0) + 0,15
∗ (0,25 ∗ 12 + 0,40 ∗ −5 + 0,2 ∗ 0 + 0,15 ∗ 12) = 4,495
𝜎 = 2,12%
1,3
𝛽𝐴 =
= 0,2892
4,495
4,75
𝛽𝐵 =
= 1,0567
4,495
9,25
𝛽𝐶 =
= 2,0578
4,495
2,8
𝛽𝐷 =
= 0,6229
4,495
The proportion of 𝐴 is 0,2892 ∗ 0,25 = 7,23%, 𝑒𝑡𝑐 …
5
Maths
Exercises
Problem 4
a)
Maximize expected returns : security 2
Minimize risk : security 1
b)
𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 = 1
So 100% security 1
c)
𝑉𝑎𝑟 = 𝑥² ∗ 0,05² + 𝑦² ∗ 0,08² − 2 ∗ 𝑥𝑦 ∗ 0,05 ∗ 0,08 = 𝑥² ∗ 0,0025 + 𝑦² ∗ 0,0064 − 0,0080 ∗ 𝑥𝑦
𝑥+𝑦 =1
𝑀𝑖𝑛𝑉𝑎𝑟 = 25𝑥² + 64(𝑥 2 − 2𝑥 + 1) − 80 ∗ (𝑥 − 𝑥 2 ) = 169𝑥² − 208𝑥 + 64 = 0
2082 − 4 ∗ 169 ∗ 64 = 0
𝑏
208
𝑥=−
=
= 0,6154
2𝑎 2 ∗ 169
𝑥 = 61,54%
𝑦 = 38,46%
d)
𝐸 = 0,6154 ∗ 0,1 + 0,3846 ∗ 0,16 = 12,3%
Not invest on Tbill, because it’s risk free in all the cases,
Risk premium Portfolio=2,3%
Problem 5
a)
100 variance,
𝑆𝑛 = 1 + 2 + 3 … + 99
𝑆𝑛 = 99 + 98 + ⋯ + 1
2 ∗ 𝑆𝑛 = 100 ∗ 99
9900
𝑆𝑛 =
= 4950
2
b)
1
2
…
𝑛
1
0,3² = 0,09
0,036
…
0,036
2 0,3² ∗ 0,4 = 0,036 0,09
…
0,036
…
…
…
…
0,036
[𝑛
0,036
0,036 0,036 0,09 ]
𝑥1 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 1 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 = 𝑥2 = ⋯ 𝑥100 = 0,01
𝑉𝑎𝑟 𝑃 = 𝑥1 ∗ (𝑥1 ∗ 0,09 + 𝑥2 ∗ 0,036 … + 𝑥100 ∗ 0,036) + 𝑥2
∗ (𝑥1 ∗ 0,036 + 𝑥2 ∗ 0,09 … + 𝑥100 ∗ 0,036) + ⋯ + 𝑥100 ∗ (𝑥1 ∗ 0,036 + 𝑥2 ∗ 0,036 …
+ 𝑥100 ∗ 0,09)
𝑉𝑎𝑟 𝑃 = 0,01 ∗ 0,09 + 0,01 ∗ 0,036 … + 0,01 ∗ 0,036
𝑉𝑎𝑟 𝑃 = 0,01 ∗ 0,09 + 0,99 ∗ 0,036 = 0,03654
𝜎𝑃 = 19,1154%
c)
𝑉𝑎𝑟 𝑃 =
1
1
∗ 0,09 + (1 − ) ∗ 0,036
𝑛
𝑛
1
∗ 0,09 + (1 − ) ∗ 0,036 = 0,036
𝑛
𝜎𝑃 = 18,9737%
lim
1
𝑛→∞ 𝑛
6
Maths
Exercises
Problem 6
a)
𝑉𝑎𝑟𝑃 = 0,5² ∗ 0,627² + 0,5² ∗ 0,507² + 2 ∗ 0,5 ∗ 0,5 ∗ 0,66 ∗ 0,627 ∗ 0,507 = 0,267448
𝜎𝑃 = 51,7154%
b)
Correlation TBill with asset =0
1 1
1
1
1
𝜎𝑃 ² = ∗ ( ∗ 0,6272 + ∗ 0,627 ∗ 0,507 ∗ 0,66 + ∗ 0,627 ∗ 0 ∗? ) +
3 3
3
3
3
1
1
1
1
∗ ( ∗ 0,627 ∗ 0,507 ∗ 0,66 + ∗ 0,507² + ∗ 0,507 ∗ 0 ∗? ) +
3
3
3
3
1
1
1
∗ ( ∗ 0,627 ∗ 0 ∗? + ∗ 0,507 ∗ 0 ∗? + ∗ 0²)
3
3
3
1 1
1
1 1
1
2
𝜎𝑃 ² = ∗ ( ∗ 0,627 + ∗ 0,627 ∗ 0,507 ∗ 0,66) + ∗ ( ∗ 0,627 ∗ 0,507 ∗ 0,66 + ∗ 0,507²)
3 3
3
3 3
3
= 0,11886572
𝜎𝑃 = 34,4769%
c)
0,517 ∗ 2 = 1,034
103,4%
d)
𝛽𝐷𝑒𝑙𝑙 = 2,21
𝛽𝑀𝑖𝑐𝑟𝑜 = 1,81
𝜎𝑃 ² = 15%
𝜎𝑝𝐷𝑒𝑙𝑙 = 0,15 ∗ 2,21 = 33,15%
𝜎𝑝𝑀𝑖𝑐𝑟𝑜 = 0,15 ∗ 1,81 = 27,15%
Problem 7
a)
𝐸(𝐴1) = 0,5 ∗ 0,08 + 0,3 ∗ −0,02 + 0,2 ∗ 0,12 = 5,8%
𝐸(𝐴2) = 0,5 ∗ −0,05 + 0,3 ∗ 0,14 + 0,2 ∗ 0,09 = 3,5%
b)
𝑉𝑎𝑟(𝐴1) = (8 − 5,8)2 ∗ 0,5 + (−2 − 5,8)2 ∗ 0,3 + (12 − 5,8)2 ∗ 0,2 = 28,36
𝑉𝑎𝑟(𝐴2) = (−5 − 3,5)² ∗ 0,5 + (14 − 3,5)2 ∗ 0,3 + (9 − 3,5)2 ∗ 0,2 = 75,25
c)
𝑐𝑜𝑣𝐴1𝐴2 = (8 − 5,8) ∗ (−5 − 3,5) ∗ 0,5 + (−2 − 5,8) ∗ (14 − 3,5) ∗ 0,3 + (12 − 5,8) ∗ (9 − 3,5) ∗ 0,2
= −27,1
27,1
𝜌=−
= −0,586628
√28,36 ∗ √75,25
d)
5,8 + 3,5
= 4,65%
2
2
2
𝜎𝑃 = 0,5 ∗ 28,36 + 0,52 ∗ 75,25 + 2 ∗ 0,52 ∗ (−27,1) = 12,3525%
𝜎𝑝 = 3,515%
7
Maths
Exercises
Problem Set 3
Problem 1
a)
b)
Find 𝐸(3)
𝑤𝑒 ℎ𝑎𝑣𝑒:
𝛽 = 1,8 ∶ 𝑥 = 12%
𝛽 = 0,8 ∶ 𝑥 = 7%
12−7
So the slope is :
= 0,05
1,8−0,8
Assume 𝛽 = 0,8 𝑖𝑠 𝛽 ∗ = 0
Then 𝛽 = 1,2 𝑖𝑠 𝛽 ∗ = 0,4
So
𝐸 = 0,05𝛽 ∗ + 0,07
𝐸(3) = 9%
OR
𝑟𝑖 = 𝑟𝑓 + 𝛽𝑖 ∗ (𝑟𝑚 − 𝑟𝑓 )
𝑠𝑡𝑜𝑐𝑘 1 = 𝑟𝑓 + 1.8 ∗ (𝑟𝑚 − 𝑟𝑓 ) = 12
𝑠𝑡𝑜𝑐𝑘 2 = 𝑟𝑓 + 0.8 ∗ (𝑟𝑚 − 𝑟𝑓 ) = 8
𝑟𝑚 = 8%, 𝑟𝑓 = 3%
𝑠𝑡𝑜𝑐𝑘 3 = 3% + 1,2(8% − 3%) = 9%
c)
𝑠𝑡𝑜𝑐𝑘 4 = 3% + 2(8% − 3%) = 13%
Real is 16%, price is underpriced (too low = too much return)
𝑠𝑡𝑜𝑐𝑘 5 = 3% + 1,05 ∗ (8% − 3%) = 8,25%
Real is 7%, price is overpriced
If stocks are not on the SML, then the market portfolio is inefficient and we can improve upon the
market portfolio by buying underpriced and selling overpriced. We can beat the market.
Problem 2
𝑆𝑀𝐿 = 𝑟𝑖 = 𝑟𝑓 + 𝛽𝑖 ∗ (𝑟𝑚 − 𝑟𝑓 )
𝑟𝑚 − 𝑟𝑓
𝐶𝑀𝐿 = 𝑟𝑖 = 𝑟𝑓 + 𝜎𝑖 ∗
𝜎𝑚
a)
𝑆𝑀𝐿 = 𝑟𝑖 = 0,06 + 2 ∗ (0,15 − 0,06) = 24%
Because in the 0,15 there is already some diversification.
b)
if efficient : CML, because we can’t do more diversification, it will be on the CML
We would have 𝜎𝑖 = 𝜎𝑀
0,15 − 0,06
𝐶𝑀𝐿 = 𝑟𝑖 = 0,06 + 0,15 ∗
= 0,15
0,15
c)
𝛽=
0,15
=1
0,15
d)
Equation 1 is used for 1 security. (Also for efficient portfolio.)
Equation 2 is not used for inefficient security, but appropriate for efficient portfolio
8
Maths
Exercises
e)
What is the amount of risk of this security in part a) that is diversified away ?
AB diversifiable risk
CB systematic risk
CA total risk
𝐵𝐴 = 𝐶𝐴 − 𝐶𝐵 = 𝜎𝐴 − 𝜎𝐵
CML
𝑟𝑚 − 𝑟𝑓
𝑟𝐵 = 𝑟𝑓 +
∗ 𝜎𝐵
𝜎𝑀
SML
𝑟𝐴 = 𝑟𝑓 + (𝑟𝑚 − 𝑟𝑓 ) ∗ 𝛽𝐴
𝑟𝐴 = 𝑟𝐵 =>
Lecture 4 slide 23
𝐴𝐵 = 𝜎𝐴 − 𝜎𝐵 = 𝜎𝐴 − 𝛽𝐴 𝜎𝑀 = 50 − 2 ∗ 15 = 20%
Problem 3
a)
𝐷
𝐸
+ 𝛽𝑒𝑞𝑢𝑖𝑡𝑦 ∗
𝐷+𝐸
𝐷+𝐸
𝛽𝑑𝑒𝑏𝑡 = 0 because it’s risk free.
So we have
𝐸
𝛽𝑎𝑠𝑠𝑒𝑡 = 𝛽𝑒𝑞𝑢𝑖𝑡𝑦 ∗
𝐷+𝐸
Competitors
Estimated 𝛽𝑒𝑞𝑢𝑖𝑡𝑦
𝛽𝑎𝑠𝑠𝑒𝑡 = 𝛽𝑑𝑒𝑏𝑡 ∗
Rimi foods
Sony Electronics
Dow chemicals
0,8
1,6
1,2
𝐷
𝐷+𝐸
0,3
0,2
0,4
b)
𝐷
𝐸
+ 𝛽𝑒𝑞𝑢𝑖𝑡𝑦 ∗
𝐷+𝐸
𝐷+𝐸
𝛽𝑎𝑠𝑠𝑒𝑡 = 𝛽𝑒𝑞𝑢𝑖𝑡𝑦 ∗ 0,6
𝛽𝑎𝑠𝑠𝑒𝑡 = 0,56 ∗ 0,5 + 1,28 ∗ 0,3 + 0,72 ∗ 0,2 = 0,808
0,808
𝛽𝑒𝑞𝑢𝑖𝑡𝑦 =
= 1,35
0,6
𝛽𝑎𝑠𝑠𝑒𝑡 = 𝛽𝑑𝑒𝑏𝑡 ∗
c)
𝑟𝑓𝑜𝑜𝑑 = 0,07 + 0,56 ∗ (0,15 − 0,07) = 11,48%
𝑟𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐𝑠 = 0,07 + 1,28 ∗ (0,15 − 0,07) = 17,24%
𝑟𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙𝑠 = 0,07 + 0,72 ∗ (0,15 − 0,07) = 12,76%
𝑟𝐿 = 0,07 + 0,808 ∗ (0,15 − 0,07) = 13,464%
d)
𝛽𝐴𝐹 = 0,2 ∗ 0,3 + 0,8 ∗ 0,7 = 0,62
𝛽𝐴𝐸 = 0,2 ∗ 0,2 + 1,6 ∗ 0,8 = 1,32
𝛽𝐴𝐶 = 0,2 ∗ 0,4 + 1,2 ∗ 0,6 = 0,8
𝛽𝑎𝑠𝑠𝑒𝑡 = 0,62 ∗ 0,5 + 1,32 ∗ 0,3 + 0,8 ∗ 0,2 = 0,866
𝑟𝑓𝑜𝑜𝑑 = 0,07 + 0,62 ∗ (0,15 − 0,07) = 11,96%
9
𝐸
𝐷+𝐸
0,7
0,8
0,6
𝛽𝑎𝑠𝑠𝑒𝑡
0,8 ∗ 0,7 = 0,56
1,6 ∗ 0,8 = 1,28
1,2 ∗ 0,6 = 0,72
Maths
Exercises
𝑟𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑖𝑐𝑠 = 0,07 + 1,32 ∗ (0,15 − 0,07) = 17,56%
𝑟𝑐ℎ𝑒𝑚𝑖𝑐𝑎𝑙𝑠 = 0,07 + 0,8 ∗ (0,15 − 0,07) = 13,4%
𝑟𝐿 = 0,07 + 0,866 ∗ (0,15 − 0,07) = 13,298%
Problem 4
𝑟𝑓 = 2%
𝑟𝑚 = 8% + 2% = 10%
𝜎𝑚 = 𝜎𝑆𝑃 = 20%
a)
(𝑟𝑡 − 𝑟𝑓 ) = 𝛼 + 𝛽(𝑟𝑚𝑡 − 𝑟𝑓 ) + 𝜀
𝑟𝑡 = 0 + 1 ∗ 0,08 + 0,02 + 0 = 0,10 = 10%
b)
0,10 − 0,02
= 0,4
0,2
10 − 2
=
= 0,2828
√800
𝑆𝑆&𝑃 =
𝑆𝐼𝑈𝐸𝐹
𝜎𝑖 ² =
2
2
2
2
𝛽𝑖𝑚
𝜎𝑚
𝛽𝑖𝑚
𝜎𝑚
202
=
=
1
∗
= 800
2
0,5
𝑅²
𝜌𝑖𝑚
c)
The IUEF is inefficient so we don’t take it.
𝐸(𝑃) = 8% = 𝑥 ∗ 0,02 + 𝑦 ∗ 0,10
𝑥+𝑦 =1
𝑥 = 0,25
𝑦 = 0,75
d)
Now we take IUEF
𝑟𝑡 = 0,02 + 1 ∗ 0,08 + 0,02 = 12%
𝐸(𝑃) = 𝑥 ∗ 0,02 + 𝑦 ∗ 0,10 + 𝑧 ∗ 0,12 = 0,08
𝑥+𝑦+𝑧 =1
We’ve
𝑥 ∗ 2 + 𝑦 ∗ 10 + (1 − 𝑥 − 𝑦) ∗ 12 = 8
{
𝑧 =1−𝑥−𝑦
−𝑥 ∗ 10 − 𝑦 ∗ 2 + 12 = 8
{
𝑧 =1−𝑥−𝑦
𝑥 = 0,4 − 0,2𝑦
{
𝑧 = 0,6 − 0,8𝑦
202
20
+ 𝑦𝑧 ∗ √0,5 ∗ 20 ∗
= 400𝑦² + 800𝑧² + 800𝑦𝑧
0,5
√0,5
𝑉𝑎𝑟 = 400𝑦² + 800 ∗ (0,6 − 0,8𝑦)2 + 800 ∗ 𝑦 ∗ (0,6 − 0,8𝑦)
= 400𝑦² + 800 ∗ (0,82 𝑦 2 − 0,8 ∗ 2 ∗ 0,6𝑦 + 0,62 ) + 800 ∗ 0,6𝑦 − 800 ∗ 0,8𝑦²
= 272𝑦² − 288𝑦 + 288
𝑀𝑖𝑛𝑉𝑎𝑟 = 544𝑦 − 288
𝑉𝑎𝑟 = 𝑦² ∗ 20² + 𝑧² ∗
𝑥 = 29,40%
𝑦 = 52,95%
𝑧 = 17,65%
10
Maths
Exercises
𝑟𝑚∗ = 10,4998%
Other method
2
2
𝜎𝑝2 = 𝑦² ∗ 𝜎𝑀
+ 𝑧² ∗ 𝜎𝐼𝑈
+ 2 ∗ 𝑦 ∗ 𝑧 ∗ 𝜎𝐼𝑈,𝑀 = 𝑦² ∗ 400 + 𝑧² ∗ 800 + 2 ∗ 𝑦 ∗ 𝑧 ∗ 400
𝜎𝑖𝑚
2
(𝛽𝑖𝑚 = 𝜎2 => 𝜎𝐼𝑈,𝑚 = 𝛽𝐼𝑈𝑚 𝜎𝑚
= 400)
𝑚
𝐿(𝑦, 𝑧, 𝛾) = 𝑦² ∗ 400 + 𝑧² ∗ 800 + 𝑦 ∗ 𝑧 ∗ 400 + 𝛾(8 ∗ 𝑦 + 10𝑧 − 6)
𝜕𝐿(𝑦, 𝑧, 𝛾)
= 800𝑦 + 800𝑧 + 8𝛾
𝜕𝑦
𝜕𝐿(𝑦, 𝑧, 𝛾)
= 1600𝑧 + 800𝑦 + 10𝛾
𝜕𝑧
𝜕𝐿(𝑦, 𝑧, 𝛾)
= 8𝑦 + 10𝑧 − 6
𝜕𝛾
Then 𝑦 = 0,5294
𝑧 = 0,1765
𝑠𝑜 𝑥 = 0,2941
Problem 5
a)
1,3 ∗ 𝑥 + 0,9 ∗ 𝑦 = 1
𝑥+𝑦 =1
1,3𝑥 + 0,9 − 0,9𝑥 = 1
𝑥 = 0,25
𝑦 = 0,75
b)
on a
𝑉𝑎𝑟 𝜀 = 𝑥1 (𝑥1 ∗ 𝑉𝑎𝑟 𝜀1 + 𝑥2 ∗ 𝐶𝑜𝑣 𝜀1 𝜀2 … + 𝑥𝑛 ∗ 𝐶𝑜𝑣 𝜀1 𝜀𝑛 )
+ 𝑥2 (𝑥1 ∗ 𝐶𝑜𝑣 𝜀2 𝜀1 + 𝑥2 ∗ 𝑉𝑎𝑟 𝜀2 … + 𝑥𝑛 ∗ 𝐶𝑜𝑣 𝜀2 𝜀𝑛 ) + ⋯
+ 𝑥𝑛 (𝑥1 ∗ 𝐶𝑜𝑣 𝜀𝑛 𝜀1 + 𝑥2 ∗ 𝐶𝑜𝑣 𝜀𝑛 𝜀2 … + 𝑥𝑛 ∗ 𝑉𝑎𝑟 𝜀𝑛 )
1
𝑥1 = 𝑥2 = ⋯ = 𝑥𝑛 = 𝑥 =
𝑛
𝜌=0
So
1
𝑉𝑎𝑟 𝜀 = ∗ 𝑉𝑎𝑟 𝜀𝑖
𝑛
1
lim ∗ 𝑉𝑎𝑟 𝜀𝑖 = 0
𝑛→∞ 𝑛
c)
∝≠ 0 so the CAPM doesn’t hold
d)
1
1
∗ 𝑉𝑎𝑟 𝜀𝑖 + (1 − ) ∗ 𝐶𝑜𝑣 𝜀𝑖 𝜀𝑗
𝑛
𝑛
1
1
lim ∗ 𝑉𝑎𝑟 𝜀𝑖 + (1 − ) ∗ 𝐶𝑜𝑣 𝜀𝑖 𝜀𝑗 = 𝐶𝑜𝑣 𝜀𝑖 𝜀𝑗
𝑛→∞ 𝑛
𝑛
𝑉𝑎𝑟 𝜀 =
(= 𝜌𝑉𝑎𝑟 𝜀𝑖 )
Problem 6
a)
𝑋 = 1,75 ∗ 0,04 + 0,25 ∗ 0,08 = 0,09 = 9%
𝑌 = −1 ∗ 0,04 + 2 ∗ 0,08 = 12%
𝑍 = 2 ∗ 0,04 + 1 ∗ 0,08 = 16%
11
Maths
Exercises
b)
80 + 60 − 40 = 100
80
60
40
∗ 1,75 −
−2∗
=0
100
100
100
80
60
40
∗ 0,25 +
∗2−1∗
=1
100
100
100
𝑟𝑝 = 0 ∗ 0,04 + 1 ∗ 0,08 = 0,08
c)
1600 + 20 − 80 = 1540
1600
20
80
∗1−
−2∗
=1
1540
1540
1540
1600
20
80
∗ 0,25 +
∗2−1∗
=0
1540
1540
1540
𝑟𝑝 = 1 ∗ 0,04 + 0 ∗ 0,08 = 0,04
d)
Problem Set 4
Problem 1
Protective put/portfolio insurance : you can also achieve this return, by buying a call and a bond.
160
Problem
1
A]
140
120
100
Asset Price
80
Buy Put return
60
40
Portfolia
Return
20
0
0 20 40 60 80 100 120 140
Perfect immunization
300
Problem
1 Price
Asset
B]
250
200
150
Buy Put Return
100
50
0
-50
-100
0
40
80 120 160 200 240
Write Call
Return
Portfolio Return
-150
-200
12
Maths
Exercises
Straddle : (careful you don’t own the stock in this diagram)
250
Problem
1
C]
200
150
Buy Put
Return
Buy Call
Return
Portfolio
Return
Price Asset
100
50
0
20
40
60
80
100
120
140
160
180
200
0
200
Problem
1
D] Price Asset
150
100
50
160
140
120
100
80
60
40
-50
20
0
0
Stock Short
Price :
Return of
portfolio
-100
-150
-200
250
Problem 1
E] a)
200
150
Received by
Equityholder
100
Firm Asset
50
0
Equity
0
50
100
150
200
Equity can be viewed as a call option (buy) at price exercice = price of
debt. (here 100)
Below this price, the equity holder receive nothing (because the debt
is paid before)
13
Maths
Exercises
250
Problem 1
E] b)
200
150
Debt Received by
Debtholder
100
Firm Asset
50
0
0
50
100
150
200
Debt
Debt can be viewed as a put option (write) at price exercice = price of debt
(here 100) + buy a bond
Below, the debtholder receive less that the debt, until firm asset = 0 => debt
can't be paid.
Strangle
140
Problem 1
F] a)
125
120
100
100
80
75
60
Price Asset
Buy Call
50
40
Buy Put
25
20
0 0
-20 0
25
50
75
100
125
-40
Bull spread
200
Problem 1
F] b)
150
Price Asset
100
Long Call E1
50
Short Call E2
0
0
25
50
75
100
125
Portfolio return
-50
-100
Long condor
14
Maths
Exercises
Problem 1
F] c)
200
150
Price Asset
100
Long Call E1
50
Long Call E2
0
Short Call E3
0
25
50
75
100
125
-50
Short Call E4
Portfolio Return
-100
-150
Problem 2
ASSUME always take 𝑆 + 𝑃 = 𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶, 𝑖𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
a)
𝐶 = 10, 𝐸 = 110, 𝑆 = 120, 𝑡 = 1, 𝑟 = 10%
(𝑆 + 𝑃 = 𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶
𝑠𝑜 𝑡ℎ𝑎𝑡 𝑆 ≤ 𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶)
Real price
𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶 = 110 ∗ 𝑒 −0,10∗1 + 10 = 109,5321
The stock at 120 is too expensive, so you short sell it, and you buy a bond.
Strategy
CF today
Short sell 1 stock
Buy 1 call
Buy a risk free bond
+120
-10
−110 ∗ 𝑒 −0,10∗1
= 99,53
+10.4679
Risk free profit today
CF at the date of
maturity
S1<E
-S1
0
110
S1>E
-S1
S1-110
110
110-S1
P=max[0,110-S1]
0
b)
(𝑆 + 𝑃 ≥ 𝐸 ∗ 𝑒 −𝑟𝑡 )
1
𝐸 ∗ 𝑒 −𝑟𝑡 = 165 ∗ 𝑒 −0,02∗6 = 164,4505
So you buy a stock and sell a put.
Strategy
CF today
Buy 1 stock
Buy 1 put
Short sell a risk free
bond (borrow at risk
free rate)
Risk free profit today
-160
-1
0,02
165 ∗ 𝑒 − 6
= 164,4509
+3,4509
CF at the date of
maturity
S1<E
+S1
165-S1
-165
S1>E
+S1
0
-165
0
S1-165
C=Max[0,S1-165]
15
Maths
Exercises
c)
Buy a call
100 ∗ 𝑒 −0,5∗0,10 = 95,1229$
95,1229 − 90 = 5,1229$
+8$ = 13,123$
Problem 3
a)
300
250
235
200
200
Firm Asset
Loan AA
150
Loan BB
100
100
Equity 35
100
50
35
0
0
50
100
150
200
250
300
350
𝐿𝑜𝑎𝑛 𝐴𝐴 = 𝐵𝑜𝑛𝑑 (𝐹𝑉 100) − 𝑃𝑢𝑡 (𝐸 = 100)
𝐿𝑜𝑎𝑛 𝐵𝐵 = 𝐶𝑎𝑙𝑙 (𝐸 = 100) − 𝐶𝑎𝑙𝑙 (𝐸 = 200)
𝑆𝐸𝑂 = 𝐶𝑎𝑙𝑙 (𝐸 = 200)
b)
300
250
Problem 3
B)
235
200
200
150
Firm Asset
100
100
Debt AA
Risk Free 100
50
Put Write 100
0
0
50
100
150
200
250
-50
-100
-150
16
300
350
Maths
Exercises
The seller is the company, the buyer is the bank (who sell the credit)
𝑃𝑉(𝑙𝑜𝑎𝑛𝐴𝐴) = 𝑃𝑉(𝐹𝑉 = 100) − 𝑃(𝐸 = 100)
100
100
𝑃(𝐸 = 100) = 𝑃𝑉(𝐹𝑉 = 100) − 𝑃𝑉(𝐿𝑜𝑎𝑛 𝐴𝐴) =
−
= 0,4267
1,08 1,085
More simpler, the loan AA is the addition of a risk free asset and a put (write), so the put (write) is
the difference between the risk free asset and the loan, and we must actualize them
(The price of the put is 0,4267 ∗ 100000000 = 426693,98)
c)
300
250
Problem 3
C)
235
200
200
150
Firm Asset
100
100
Call buy 100
Debt Loan BB
50
Call Write 200
0
0
50
100
150
200
250
300
350
-50
-100
-150
𝐶(𝐸 = 200) = 35
𝐸
𝑆+𝑃 =
+𝐶
1+𝑖
𝑆 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡𝑠
𝑆 + 𝑃(𝐸 = 100) = 𝑃𝑉(𝐹𝑉 = 100) + 𝐶(𝐸 = 100)
𝐶(𝐸 = 100) = −𝑃𝑉(𝐹𝑉 = 100) + 𝑆 + 𝑃(𝐸 = 100) = 200 + 0,4267 −
𝑃𝑉(𝐿𝑜𝑎𝑛𝐵𝐵) = 107,8341 − 35 = 72,8341
Problem 4
a)
𝑢 = 1,5
𝑑 = 0,75
𝑆0 = 100
𝑆𝑢 = 150
𝑆𝑑 = 75
𝐸 = 105
𝐶𝑢 = 150 − 105 = 45
𝐶𝑑 = 0
∝∗ =
𝐶𝑢 − 𝐶𝑑
45
=
= 0,6
𝑆(𝑢 − 𝑑) 100 ∗ 0,75
17
100
= 107,8341
1,08
Maths
Exercises
b)
𝛽∗ =
𝑢𝐶𝑑 − 𝑑𝐶𝑢 1,5 ∗ 0 − 0,75 ∗ 45
=
= −42,857
𝑟(𝑢 − 𝑑)
1,05 ∗ 0,75
c)
𝐶 = ∝∗ ∗ 𝑆 + 𝛽 ∗ = 0,6 ∗ 100 − 42,857 = 17,143
Or
1,05 − 0,75
𝑝=
= 0,4
1,5 − 0,75
0,4 ∗ 45 + 0,6 ∗ 0
𝐶=
= 17,143
1,05
Problem Set 5
Problem 1
𝐶𝑢 − 𝐶𝑑
∝∗ =
𝑆(𝑢 − 𝑑)
𝑢𝐶𝑑 − 𝑑𝐶𝑢
𝛽∗ =
𝑟(𝑢 − 𝑑)
𝑡=0
𝑡=1
𝑡=2
𝑡=3
86,4
𝐶𝑢𝑢𝑢 = 46,4
72
50 ∗ 1,2 = 60
64,8
𝐶𝑢𝑢𝑑 = 24,8
50
54
50 ∗ 0,9 = 45
48,6
𝐶𝑢𝑑𝑑 = 8,6
40,5
𝑒 𝑟𝑡 −𝑑
𝑝=
in case t=1 => 𝑝 =
𝑢−𝑑
So that
1,05 − 0,9
𝑝=
= 0,5
1,2 − 0,9
1 − 𝑝 = 0,5
𝑒 𝑟𝑡 −𝑑
𝑢−𝑑
36,45
𝐶𝑑𝑑𝑑 = 0
=
𝑟−𝑑
𝑢−𝑑
𝑝 ∗ 𝐶𝑢𝑢𝑢 + (1 − 𝑝) ∗ 𝐶𝑢𝑢𝑑 0,5 ∗ (46,4 + 24,8)
=
= 33,9048
𝑟
1,05
𝑝 ∗ 𝐶𝑢𝑢𝑑 + (1 − 𝑝) ∗ 𝐶𝑢𝑑𝑑 0,5 ∗ (24,8 + 8,6)
𝐶𝑢𝑑 =
=
= 15,9048
𝑟
1,05
𝑝 ∗ 𝐶𝑢𝑑𝑑 + (1 − 𝑝) ∗ 𝐶𝑑𝑑𝑑 0,5 ∗ (8,6)
𝐶𝑑𝑑 =
=
= 4,0952
𝑟
1,05
𝑝 ∗ 𝐶𝑢𝑢 + (1 − 𝑝) ∗ 𝐶𝑢𝑑 0,5 ∗ (33,9048 + 15,9048)
𝐶𝑢 =
=
= 23,7188
𝑟
1,05
𝑝 ∗ 𝐶𝑢𝑑 + (1 − 𝑝) ∗ 𝐶𝑑𝑑 0,5 ∗ (15,9048 + 4,0952)
𝐶𝑑 =
=
= 9,5238
𝑟
1,05
𝑝 ∗ 𝐶𝑢 + (1 − 𝑝) ∗ 𝐶𝑑 0,5 ∗ (23,7188 + 9,5238)
𝐶=
=
= 15,8298
𝑟
1,05
𝐶𝑢𝑢 =
OR
𝐶=
0,53 ∗ (86,4 − 40) + 3 ∗ 0,52 ∗ 0,5 ∗ (64,8 − 40) + 3 ∗ 0,5 ∗ 0,52 ∗ (48,6 − 40)
= 15,829
1,053
18
Maths
Exercises
b)
This method is the dynamic replication, less precise.
𝐶𝑢𝑢𝑢 − 𝐶𝑢𝑢𝑑
46,4 − 24,8
∝∗𝑢𝑢 =
=
=1
𝑆𝑢𝑢 (𝑢 − 𝑑)
72 ∗ (1,2 − 0,9)
𝐶𝑢𝑢𝑢 −∝∗𝑢𝑢 𝑆𝑢𝑢𝑢 46,4 − 1 ∗ 86,4
𝛽𝑢𝑢 =
=
= −38,0952
𝑟
1,05
𝐶𝑢𝑢 =∝∗𝑢𝑢 ∗ 𝑆𝑢𝑢 + 𝛽𝑢𝑢 = 1 ∗ 72 − 38,0952 = 33,9048
𝐶𝑢𝑢𝑑 − 𝐶𝑢𝑑𝑑
=1
𝑆𝑢𝑑 (𝑢 − 𝑑)
𝐶𝑢𝑢𝑑 −∝∗𝑢𝑑 𝑆𝑢𝑢𝑑 24,8 − 1 ∗ 64,8
=
=
= −38,0952
𝑟
1,05
=∝∗𝑢𝑑 ∗ 𝑆𝑢𝑑 + 𝛽𝑢𝑑 = 1 ∗ 54 − 38,0952 = 15,9048
∝∗𝑢𝑑 =
𝛽𝑢𝑑
𝐶𝑢𝑑
𝐶𝑢𝑑𝑑 − 𝐶𝑑𝑑𝑑
8,6
=
= 0,7078
𝑆𝑑𝑑 (𝑢 − 𝑑)
40,5 ∗ (1,2 − 0,9)
𝐶𝑢𝑑𝑑 −∝∗𝑑𝑑 𝑆𝑢𝑑𝑑 8,6 − 0,7078 ∗ 40,5
=
=
= −24,5706
𝑟
1,05
= ∝∗𝑑𝑑 ∗ 𝑆𝑑𝑑 + 𝛽𝑑𝑑 = 0,7078 ∗ 40,5 − 24,5706 = 4,0953
∝∗𝑑𝑑 =
𝛽𝑑𝑑
𝐶𝑑𝑑
𝐶𝑢𝑢 − 𝐶𝑢𝑑
=1
𝑆𝑢 (𝑢 − 𝑑)
𝐶𝑢𝑢 −∝∗𝑢 𝑆𝑢𝑢 33,9048 − 1 ∗ 72
𝛽𝑢 =
=
= −36,2811
𝑟
1,05
𝐶𝑢 =∝∗𝑢 ∗ 𝑆𝑢 + 𝛽𝑢 = 1 ∗ 60 − 36,2811 = 23,7189
∝∗𝑢 =
𝐶𝑢𝑑 − 𝐶𝑑𝑑 15,9048 − 4,0953
=
= 0,8748
𝑆𝑢 (𝑢 − 𝑑)
45 ∗ (1,2 − 0,9)
𝐶𝑢𝑢 −∝∗𝑢 𝑆𝑢𝑢 15,9048 − 0,8748 ∗ 54
𝛽𝑑 =
=
= −29,8423
𝑟
1,05
∗
𝐶𝑑 =∝𝑑 ∗ 𝑆𝑑 + 𝛽𝑑 = 0,8748 ∗ 45 − 29,8423 = 9,5237
∝∗𝑑 =
𝐶𝑢 − 𝐶𝑑
23,7189 − 9,5237
=
= 0,9463
𝑆(𝑢 − 𝑑)
50 ∗ (1,2 − 0,9)
∗
𝐶𝑢 −∝ ∗ 𝑆𝑢
𝛽=
= −31,4848
𝑟
∗
𝐶 =∝ ∗ 𝑆 + 13 = 0,9463 − 31,4848 = 15,8302
∝∗ =
𝑡=0
𝑡=1
𝑡=2
𝑡=3
𝑆𝑢𝑢𝑢 = 86,4
𝐶𝑢𝑢𝑢 = 46,4
𝑆𝑢𝑢 = 72
∝∗𝑢𝑢 = 1
𝛽𝑢𝑢 = −38,0952
𝐶𝑢𝑢 = 33,9048
𝑆𝑢 = 50 ∗ 1,2 = 60
∝∗𝑢 = 1
𝛽𝑢 = −33,2811
𝐶𝑢 = 23,7189
𝑆 = 50
𝑆𝑢𝑢𝑑 = 64,8
𝐶𝑢𝑢𝑑 = 24,8
𝑆𝑢𝑑 = 54
∝∗𝑢𝑑 = 1
𝛽𝑢𝑑 = −38,0952
𝐶𝑢𝑑 = 15,9048
50 ∗ 0,9 = 45
𝑆𝑢𝑑𝑑 = 48,6
19
Maths
Exercises
∝∗𝑑 = 0,8748
𝛽𝑑 = −29,8423
𝐶𝑑 = 9,5237
𝐶𝑢𝑑𝑑 = 8,6
40,5
∝∗𝑑𝑑 = 0,7078
𝛽𝑑𝑑 = −24,5706
𝐶𝑑𝑑 = 4,0953
𝑆𝑑𝑑𝑑 = 36,45
𝐶𝑑𝑑𝑑 = 0
Dynamic riskfree arbitrage
We look in a risk free profit today
We dynamically adjust the composition of the portfolio to have a zero value at the date of maturity
Adjustments are self financing (no net cost)
𝐶𝑀 = 20, 𝐶𝑇 = 15,8302
𝑃𝑎𝑡ℎ 1 ∶ 𝑆 => 𝑑𝑠 => 𝑢𝑑𝑠 => 𝑢𝑢𝑑𝑠
𝑃𝑎𝑡ℎ 2 ∶ 𝑆 => 𝑑𝑠 => 𝑑𝑑𝑠 => 𝑑𝑑𝑑𝑠
Period/node
𝑡=0
CF in period t
+20
+31,4848
−0,9463 ∗ 50 = −47,315
(−20 − 15,8302) = −4,1698
(total 0)
𝑡=1
Sell (0,9463 − 0,8748) 𝑎𝑡 45
+(0,9463 − 0,8748) ∗ 45
Use the proceeds to reduce the debt
= +3,2175
−3,2175
Debt
outstanding
:
31,4848 ∗ 1,05 − (total 0)
3,2175 = 29,84154
Buy (1 − 0,8748) 𝑎𝑡 𝑆𝑢𝑑 = 54
𝑡=2
−(1 − 0,8748) ∗ 54 = −6,7608
Borrow 6,7608
+6,7608
(total 0)
Debt outstanding : 29,84154 ∗ 1,05 +
6,7608 = 38,0944
The call is in the money and will be −24,8
𝑡=3
exercised
+64,8
Sell a full stock at 𝑆𝑢𝑢𝑑 = 64,8
−38,0944 ∗ 1,05 = −39,99912
Repay the debt off
𝑇𝑜𝑡𝑎𝑙 = 0!
At the end we have (𝐶𝑀 − 𝐶𝑇 ) ∗ (1 + 𝑟)𝑛 = (20 − 15,8302) ∗ 1,053 = 4,8271
Period/node
𝑡=0
𝑡=1
𝑡=2
Strategy
Write 1 call
Borrow 31,4848
Buy ∝∗ of a stock at 𝑆 = 50
Deposit
Strategy
Write 1 call
Borrow 31,4848
Buy ∝∗ of a stock at 𝑆 = 50
Deposit
CF in period t
+20
+31,4848
−0,9463 ∗ 50 = −47315
(−20 − 15,8302) = −4,1698
(total 0)
Sell (0,9463 − 0,8748) 𝑎𝑡 45
+(0,9463 − 0,8748) ∗ 45
Use the proceeds to reduce the debt
= +3,2175
−3,2175
Debt outstanding : 31,4848 ∗ 1,05 − 3,2175 = (total 0)
29,84154
(0,8748 − 0,7078) ∗ 40,5
Sell (0,8748 − 0,7078) 𝑎𝑡 𝑆𝑑𝑑 = 40,5
Repaid 6,6,7635
= 6,7635
−6,7635
Debt outstanding : 29,84154 ∗ 1,05 − 6,7635 = (total 0)
20
Maths
Exercises
𝑡=3
24,570117
The call is out of the money, won’t be exercised
Sell 0,7077 at 𝑆𝑑𝑑𝑑 = 36,45
Repay the debt
Problem 2
a)
𝑡=0
𝑡=1
0,7073 ∗ 36,45 = 25,79931
−24,570117 ∗ 1,05
= −25,79862
𝑇𝑜𝑡𝑎𝑙 = 0!
𝑡=2
𝑆𝑢𝑢 = 138,11
𝐶𝑢𝑢 = 48,11
𝑆𝑢 = 117,52
𝑆 = 100
𝑆𝑢𝑑 = 104,09
𝐶𝑢𝑑 = 14,09
𝑆𝑑 = 88,57
𝑆𝑑𝑑 = 78,45
𝐶𝑑𝑑 = 0
𝑢 = 1,1752
𝑑 = 0,8857
Risk neutral valuation
𝑡
1
𝑒 𝑟𝑛 − 𝑑 𝑒 0,06∗2 − 𝑑
𝑝=
=
𝑢−𝑑
𝑢−𝑑
1
𝑒 0,06∗2 = 1,0304
1,0304 − 0,8857
= 0,4998
1,1752 − 0,8857
0,49982 ∗ 48,11 + 2 ∗ 0,5002 ∗ 0,4998 ∗ 14,09
𝐶=
=
1,03042
0,4998 ∗ 48,11 + 0,5002 ∗ 14,0875
𝐶𝑢 =
= 30,1744
1,0304
0,4998 ∗ 14,0875 + 0,5002 ∗ 0
𝐶𝑑 =
= 6,8332
1,0304
0,4998 ∗ 30,1744 + 0,5002 ∗ 6,8332
𝐶=
= 17,9533
1,0304
𝑝=
b)
𝑡=0
𝑡=1
𝐷𝐼𝑉 = 5
𝑡=2
𝑆𝑢𝑢 = 132,2452
𝐶𝑢𝑢 = 42,2452
𝑆𝑢 = 117,52
𝑆𝑢𝑑𝑖𝑣 = 117,52 − 5 = 112,52
100
𝛼 = 0,8130
𝛽 = −66,0168
𝐶 = 15,2832
𝑆𝑢𝑑 = 99,659
𝐶𝑢𝑑 = 9,6590
𝑆𝑑𝑢 = 98,2115
𝐶𝑑𝑢 = 8,2115
∗
𝑆𝑑 = 88,57 − 5 = 83,57
𝛼𝑑∗ = 0,3354
𝛽𝑑 = −24,3803
𝐶𝑑 = 3,9833
𝑆𝑑𝑑 = 74,0149
𝐶𝑑𝑑 = 0
21
Maths
Exercises
0,4998 ∗ 42,2452 + 0,5002 ∗ 9,6590
= 25,1801
1,0304
𝐶𝑢Exercise = 𝑚𝑎𝑥{0, 𝑆𝑢 − 𝐸} = 117,52 − 90 = 27,52
𝐶𝑢 = 𝑚𝑎𝑥 = 27,52
0,4998 ∗ 8,2115
𝐶𝑑𝑁𝑜 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 =
= 3,9830
1,0304
𝐶𝑑Exercise = 0
𝐶𝑑 = 𝑚𝑎𝑥 = 3,9830
0,4998 ∗ 27,52 + 0,5002 ∗ 3,9830
𝐶 𝑁𝑜 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 =
= 15,2822
1,0304
𝐶 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 = 10
𝐶𝑢No exercise =
c)
𝛼𝑢∗ =
𝐶𝑢𝑢 − 𝐶𝑢𝑑
=
𝑑𝑖𝑣 (𝑢
𝑆𝑢 ∗ − 𝑑)
𝐶𝑑𝑢 − 𝐶𝑑𝑑
=
𝑑𝑖𝑣 (𝑢
𝑆𝑑 ∗ − 𝑑)
𝐶𝑑𝑢 − 𝛼𝑑∗ ∗ 𝑆𝑑𝑢
𝑛𝑜𝑡ℎ𝑖𝑛𝑔 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑤𝑒 ′ 𝑣𝑒 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒𝑑 ‼
8,2215
= 0,3394
83,57 ∗ (1,1152 − 0,8857)
8,2115 − 0,3394 ∗ 98,2215
𝛽𝑑 =
=
= −24,3803
𝑟
1,0304
𝐶𝑑 = 𝛼𝑑∗ ∗ 𝑆𝑑𝑑𝑖𝑣 + 𝛽𝑑 = 0,3394 ∗ 83,57 − 24,3803 = 3,9833
𝛼𝑑∗ =
𝐶𝑢 − 𝐶𝑑
27,52 − 3,9833
=
= 0,8130
𝑆 ∗ (𝑢 − 𝑑) 100 ∗ (1,1752 − 0,8857)
𝐶𝑢 − 𝛼 ∗ ∗ 𝑆𝑢 27,52 − 0,8130 ∗ 117,52
𝛽=
=
= −66,0168
𝑟
1,0304
𝐶 = 𝛼 ∗ ∗ 𝑆 + 𝛽 = 0,8150 ∗ 100 − 66,0168 = 15,2832
𝛼∗ =
𝐶𝑀 = 10
𝐶𝑇 = 15,2832
Period/node
𝑡=0
𝑡=1
𝑡=2
Strategy
Buy 1 call
Short sell 𝛼 ∗ at 𝑆 = 100
Lend at risk free rate
Deposit (15,2832-10) at 3,04% for 2 periods
CF in period t
−10
+0,8130 ∗ 100 = 81,30
−66,0168
−5,2832
(𝑇𝑜𝑡𝑎𝑙 = 0)
Return (0,8130-0,3354) at 83,57
−39,57875
Receive interest on lending : 66,0168 ∗ 0,0304
+2,00691
Decrease lending by : (66,0168 − 24,3803)
+42,6365
Compensate the owner of the stock !! −5 ∗ −4,065
0,8130
(𝑇𝑜𝑡𝑎𝑙 ∶ 0)
Exercise the call
+8,2115
Receive interest and withdraw the lending : +25,1215
24,3803 ∗ 1,0304
−0,3394 ∗ 98,2215
Return 0,3394 of a stock at 𝑆𝑑𝑢
= −33,33298
(𝑇𝑜𝑡𝑎𝑙 = 0)
d)
𝐸 = 110
𝑃𝑢𝑢 = 0
𝑃𝑢𝑑 = 5,9125
𝑃𝑑𝑑 = 31,5536
22
Maths
Exercises
𝑝 ∗ 𝑃𝑢𝑢 + (1 − 𝑝) ∗ 𝑃𝑢𝑑 (0,4998 ∗ 0 + (1 − 0,4998) ∗ 5,1125)
=
= 2,8702
𝑟
1,0304
𝑃𝑢𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 = max{0, 𝐸 − 𝑆𝑢 } = 0
𝑝 ∗ 𝑃𝑢𝑑 + (1 − 𝑝) ∗ 𝑃𝑑𝑑 (0,4998 ∗ 5,1125 + (1 − 0,4998) ∗ 31,5536)
𝑃𝑑𝑛𝑜 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 =
=
= 18,1853
𝑟
1,0304
𝑃𝑑𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 = max{0, 𝐸 − 𝑆𝑢 } = 21,43
𝑃𝑢𝑛𝑜 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 =
𝑝 ∗ 𝑃𝑢 + (1 − 𝑝) ∗ 𝑃𝑑
= 11,7952
𝑟
𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒
𝑃
= 10
𝑃 = 𝑚𝑎𝑥 = 11,7952
𝑃𝑛𝑜 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 =
If E=90
It’s wrong to do that :
𝑆 + 𝑃 = 𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶
𝑃 = 𝐸 ∗ 𝑒 −𝑟𝑡 + 𝐶 − 𝑆 = 5,293398 wrong!!
Problem 3
𝑆𝑢𝑢 = 196
𝑆𝑢 = 140
𝑆 = 100
𝑆𝑢𝑑 = 105
𝐶𝑢𝑑 = 5
𝐶𝑑𝑢 = 25
𝑆𝑑 = 75
𝑆𝑑𝑑 = 56,25
1,04 − 0,75
𝑝=
= 0,4462
1,4 − 0,75
1 − 𝑝 = 0,5538
0,4462 ∗ 96 + 0,5538 ∗ 5
𝐶𝑢 =
= 43,486
1,04
0,4462 ∗ 25
𝐶𝑑 =
= 10,73
1,04
43,846 ∗ 0,4462 + 10,73 ∗ 0,5538
𝐶=
= 24,52
1,04
𝐶𝑑𝑢 − 𝐶𝑑𝑑
25
=
= 0,513
𝑆𝑑 ∗ (𝑢 − 𝑑) 75 ∗ (1,4 − 0,75)
𝐶𝑢𝑢 − 𝐶𝑢𝑑
96 − 5
𝛼𝑢 =
=
=1
𝑆𝑢 ∗ (𝑢 − 𝑑) 140 ∗ (1,4 − 0,75)
𝐶𝑢𝑢 − 𝛼𝑢 ∗ 𝑆𝑢𝑢 96 − 1 ∗ 196
𝛽𝑢 =
=
= −96,1538
𝑟
1,04
𝐶𝑢𝑑 − 𝛼𝑑 ∗ 𝑆𝑢𝑑 25 − 0,513 ∗ 105
𝛽𝑑 =
=
= −27,75
𝑟
1,04
𝐶𝑢 = 1 ∗ 140 − 96,1538 = 43,8462
𝐶𝑑 = 0,513 ∗ 75 − 27,75 = 10,73
43,8462 − 10,73
𝛼=
= 0,5096
100 ∗ (1,4 − 0,75)
43,8462 − 0,5096 ∗ 140
𝛽=
= −26,4402
1,04
𝐶 = 0,5096 ∗ 100 − 26,4402 = 24,5192
𝛼𝑑 =
b)
23
Maths
Exercises
Problem Set 6
Problem 1
a)
We should use 365 days in the year… we have 23,3221 for the call and 1,8098 for the put
(It’s deep in the money, so high call, low put)
𝐸 = 100
𝑆 = 120
𝜎 = 0,4
𝑟𝑓 = 0,0618
(𝑢 = 𝑒 0,4√0,25 = 1,2214
𝑑 = 𝑒 −0,4√0,25 = 0,81873)
𝑑1 =
𝑆
𝜎2
ln (𝐸 ) + (𝑟 + 2 ) ∗ 𝑡
𝜎√𝑡
=
120
0,42
ln (100) + (0,0618 + 2 ) ∗ 0,25
0,4 ∗ √0,25
120
0,42
ln (
) + (0,0618 −
) ∗ 0,25
100
2
𝑆
𝜎2
ln ( ) + (𝑟 − ) ∗ 𝑡
𝐸
2
𝑑2 =
=
𝜎√𝑡
0,4 ∗ √0,25
𝑁(𝑑1 ) = 0,8599 + 0,89 ∗ (0,8621 − 0,8599) = 0,8619
𝑁(𝑑2 ) = 0,8106 + 0,89 ∗ (0,8133 − 0,8106) = 0,8130
= 1,0889
= 0,88886
𝐶 = 𝑆 ∗ 𝑁(𝑑1 ) − 𝐸 ∗ 𝑒 −𝑟𝑡 ∗ 𝑁(𝑑2 ) = 120 ∗ 0,8619 − 100 ∗ 𝑒 −0,25∗0,0618 ∗ 0,8130
𝐶 = 23,3744
𝑃 = −𝑆 ∗ (1 − 𝑁(𝑑1 )) + 𝐸 ∗ 𝑒 −𝑟𝑡 ∗ (1 − 𝑁(𝑑2 ))
= −120 ∗ (1 − 0,8619) + 100 ∗ 𝑒 −0,25∗0,0618 ∗ (1 − 0,8130)
𝑃 = 1,8413
b)
𝐸 = 25
𝑆 = 25
𝜎 = 0,3
𝑟𝑓 = 0,0618
40
35
30
25
20
15
10
5
0
Price Share
Return
0
5
10
15
20
25
30
35
Price of the put
Maturity = 1 year
V(offer)=N(S+P)
24
Maths
Exercises
𝑆
𝜎2
ln (𝐸 ) + (𝑟 + 2 ) ∗ 𝑡
25
0,32
ln ( ) + (0,0618 + 2 ) ∗ 1
25
𝑑1 =
=
= 0,356
𝜎√𝑡
0,3 ∗ √1
𝑆
𝜎2
25
0,32
ln ( ) + (𝑟 − ) ∗ 𝑡 ln ( ) + (0,0618 −
)∗1
𝐸
2
2
25
𝑑2 =
=
= 0,056
𝜎√𝑡
0,3 ∗ √1
𝑁(𝑑1 ) = 0,6368 + 0,6 ∗ (0,6406 − 0,6368) = 0,639
𝑁(𝑑2 ) = 0,5199 + 0,6 ∗ (0,5239 − 0,5199) = 0,522
𝑃 = −𝑆 ∗ (1 − 𝑁(𝑑1 )) + 𝐸 ∗ 𝑒 −𝑟𝑡 ∗ (1 − 𝑁(𝑑2 )) = −25 ∗ (1 − 0,639) + 25 ∗ 𝑒 −0,0618 ∗ (1 − 0,522)
𝑃 = 2,2043
We can also have the same result with a long bond and risk free.
Also, it’s an American, so the offer is at least 27,2043M
Problem 2
a)
𝑆 = 200
𝐸 = 180
𝜎 = 22,3%
𝑖 = 21%
200
0,2232
ln (
) + (0,21 +
)∗1
180
2
𝑑1 =
= 1,5257
0,223 ∗ √1
200
0,2232
ln (180) + (0,21 − 2 ) ∗ 1
𝑑2 =
= 1,3027
0,223 ∗ √1
𝑁(𝑑1 ) = 0,9357 + 0,57 ∗ (0,9370 − 0,9357) = 0,9364
𝑁(𝑑2 ) = 0,9032 + 0,27 ∗ (0,9049 − 0,9032) = 0,9037
𝐶 = 200 ∗ 0,9364 − 180 ∗ 𝑒 −0,21 ∗ 0,9037
𝐶 = 55,4255
b)
We should use replicating portfolio
𝑢 = 𝑒 𝜎√𝑡 = 𝑒 0,223 = 1,2498
𝑑 = 𝑒 −𝜎√𝑡 = 0,8001
𝑆𝑢 = 1,2498 ∗ 200 = 249,96
𝑆𝑑 = 0,8001 ∗ 200 = 160,02
𝑒 0,21 − 0,8001
𝑝=
= 0,9641
1,2498 − 0,8001
0,9641 ∗ (249,96 − 180) + (1 − 0,9641) ∗ 0
𝐶=
𝑒 0,21
𝐶 = 54,67
c)
We should use replicating portfolio
𝑢 = 𝑒 𝜎√𝑡 = 𝑒 0,223√0,5 = 1,1708
𝑑 = 𝑒 −𝜎√𝑡 = 0,8541
𝑆𝑢𝑢 = 1,1708 ∗ 234,16 = 274,15
𝑆𝑢 = 1,1708 ∗ 200 = 234,16
𝑆 = 200
𝑆𝑑𝑢 = 200
𝑆𝑑 = 0,8541 ∗ 200 = 170,82
25
Maths
Exercises
𝑆𝑑𝑑 = 0,8541 ∗ 170,82 = 145,897
0,21∗0,5
𝑒
− 0,8541
= 0,8103
1,1708 − 0,8541
𝑒 0,21∗0,5 = 1,1107
0,8103 ∗ (274,15 − 180) + (1 − 0,8103) ∗ 20
𝐶𝑢 =
= 72,10
1,1107
0,8103 ∗ 20
𝐶𝑑 =
= 14,59
1,1107
0,8103 ∗ 72,10 + 14,59 ∗ (1 − 0,8103)
𝐶=
1,1107
𝐶 = 55,14
𝑝=
d)
𝐶𝑢 − 𝐶𝑑
72,10 − 14,59
=
= 0,908
𝑆(𝑢 − 𝑑) 200 ∗ (1,1708 − 0,8541)
(274,15 − 180) − 20
𝛼𝑢∗ =
=1
234,16 ∗ (1,1708 − 0,8541)
20
𝛼𝑑∗ =
= 0,3697
170,82 ∗ (1,1708 − 0,8541)
𝛼∗ =
Node
Stock price
BMS model
2 period binomial model
𝐶
∆
𝐵
𝐶
∆
𝐵
𝑆𝑢
200
55,4255
0,9364 131,8515
56,7273
0,9195
127,1727
𝑆𝑑
170,82
15,4166
0,6601 97,3436
14,8996
0,3696
48,2355
𝑆
234,16
72,2031
0,9921 160,1041
73,1428
1
161,0162
If the stock price decreases : very big difference between the two model in 𝛼 𝑎𝑛𝑑 𝛽 even if the
prices of the call are similar.
Problem 3
a)
300
250
200
Commercial Loan
150
Tranche 1
100
Tranche 2
50
Tranche 3
0
0
100
200
300
400
b)
Tranche 1
𝑉𝑎𝑙𝑢𝑒 𝑇1 = 𝑉 − 𝐶(𝐸 = 𝐷1)
We also can do Put write+bond(D1) but not in this problem…
Tranche 2
Call buy (D1)+Call write (D1+D2)
𝑉𝑎𝑙𝑢𝑒 𝑇2 = 𝐶(𝐸 = 𝐷1) − 𝐶(𝐸 = 𝐷1 + 𝐷2)
Tranche 3
Call buy (D2)
Call write (D1+D2)
𝑉𝑎𝑙𝑢𝑒 𝑇3 = 𝐶(𝐸 = 𝐷1 + 𝐷2)
26
Maths
Exercises
c)
∂C
>0
∂σ
1
1
σ2 = σ2 + (1 − ) ρσ2
N
N
1 : =>n=>inf : Lim=0
Minimize the volatility (put)
Λ=
2 : =>n=>1
Maximize the volatility (call)
d)
Momentum : when there is a non random period in the price of an asset, increase or decrease,
then random => increase or decrease etc.
Mean version => when prices don’t vary around 0
27
Maths
Exercises
Problem Set 7
Exercise 1 : bond pricing
a)
1 − 1,04−5
𝑃𝑟𝑖𝑐𝑒 𝑏𝑜𝑛𝑑 = 6 ∗
+ 100 ∗ 1,04−5 = 108,9
0,04
b)
𝑃𝑟𝑖𝑐𝑒 𝑇𝑏𝑖𝑙𝑙 = 30 ∗
1 − 1,02−10
+ 1000 ∗ 1,02−10 = 1089,8
0,02
Exercise 2
a)
1 − 1,0375−8
𝑃𝑟𝑖𝑐𝑒 𝐵𝑜𝑛𝑑 𝑃 = 5000 ∗
+ 100000 ∗ (1,0375−8 ) = 108503,49
0,0375
1 − 1,0375−16
𝑃𝑟𝑖𝑐𝑒 𝐵𝑜𝑛𝑑 𝑄 = 5000 ∗
+ 100000 ∗ (1,0375−16 ) = 114837,7
0,0375
b)
1 − 1,06−8
+ 100000 ∗ (1,06−8 ) = 93790,2
0,06
1 − 1,06−16
𝑃𝑟𝑖𝑐𝑒 𝐵𝑜𝑛𝑑 𝑄 = 5000 ∗
+ 100000 ∗ (1,06−16 ) = 89894,1
0,06
𝑃𝑟𝑖𝑐𝑒 𝐵𝑜𝑛𝑑 𝑃 = 5000 ∗
c)
Maturity up = more variation in price
Higher yield = lower price
Yield higher than coupon = discount and vice versa
Exercise 3
a)
1
100000 = 97645 ∗ (1 + 𝑖)4 => 𝑖 = 10%
b)
10% semi annual
𝑥 2
(1 + ) − 1 = 0,10
2
𝑥 = 2 ∗ (√1,10 − 1) = 9,76%
The first is better
Exercise 4
a)
?
1 − 1,04−6
+ 100 ∗ 1,04−6 = 105,24
0,04
1 − 1,04−5
𝑃𝑟𝑖𝑐𝑒 𝑖𝑛 6 𝑚𝑜𝑛𝑡ℎ𝑠 = 5 ∗
+ 100 ∗ 1,04−5 = 104,452
0,04
𝑃𝑟𝑖𝑐𝑒 𝑡𝑜𝑑𝑎𝑦 = 5 ∗
b)
104,452 + 5 − 105,242
= 4%
105,242
c)
28
Maths
Exercises
𝑃𝑟𝑖𝑐𝑒 𝑖𝑛 6 𝑚𝑜𝑛𝑡ℎ𝑠 = 5 ∗
1 − 1,03−5
+ 100 ∗ 1,03−5 = 109,159
0,03
109,159 + 5 − 105,242
= 8,47%
105,242
Exercise 5
15
2
35 ∗
+ (100 + ) ∗ 10 = 1003,51
182
32
Exercise 6
a)
1000 = 87,5 ∗
1 − (1 + 𝑖)−20
1000
+
(1 + 𝑖)20
𝑖
So 𝑖 = 8,75%
The first is priced at 580, the other at 1000 which is close to 1050.
1050
= 81% very good if it’s called.
580
Either
1050
= 5%, so quite good
1000
b)
We would prefer the first one, and it’s the reason why the yield is lower.
c)
Exercise 7
(1 − (1 + 𝑖)−10 )
1000
900 = 140 ∗
+
=> 𝑖 = 16,075%
(1 + 𝑖)10
𝑖
(1 − (1 + 𝑖)−10 )
1000
900 = 70 ∗
+
=> 𝑖 = 8,52%
(1
𝑖
+ 𝑖)10
Exercise 8
a)
110
= 100,9174
1,09
1
110
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝐵𝑜𝑛𝑑 1 =
∗(
)=1
100,9174 1,09
100
𝑃𝑟𝑖𝑐𝑒 𝑏𝑜𝑛𝑑 2 =
= 77,2183
1,093
1
110
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝐵𝑜𝑛𝑑 2 =
∗ (3 ∗
)=3
77,2183
1,093
1 − 1,09−4
𝑃𝑟𝑖𝑐𝑒 𝐵𝑜𝑛𝑑 3 = 20 ∗
+ 100 ∗ 1,09−4 = 135,6369
0,09
1
20
20
20
120
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝐵𝑜𝑛𝑑 3 =
∗(
+2∗
+3∗
+4∗
) = 3,232
2
3
135,6369 1,09
1,09
1,09
1,094
𝑃𝑟𝑖𝑐𝑒 𝑏𝑜𝑛𝑑 1 =
b)
Bond1+Bond4
130
20
20
120
𝐵𝑜𝑛𝑑 5 =
+
+
+
= 236,55
2
3
1,09 1,09
1,09
1,094
1
130
20
20
120
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑏𝑜𝑛𝑑 5 =
∗(
+2∗
+
3
∗
+
4
∗
) = 2,2799
236,5543 1,09
1,092
1,093
1,094
c)
29
Maths
Exercises
𝑥 + (1 − 𝑥) ∗ 3,232 = 2
𝑥 = 0,552, (1 − 𝑥) = 0,448
Exercise 9
a)
Bond with lower price, higher yield, so less duration
b)
A : non callable,
Lower coupon, so high duration
c)
By going to annual to semi annual, reduce duration
d)
Lower coupon higher duration
So it’s B
e)
Lower yield higher duration so it’s Baa who has the higher duration
Exercise 10
1
10
4
21,5093
∗(
+5∗
)=
= 1,8583
5
10
4
1,1
1,1
11,5746
+
1,1 1,15
𝐹𝑉
11,5746 =
=> 𝐹𝑉 = 13,8174
1,11,8583
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 =
Exercise 11
a)
1,16
= 7,25𝑦
0,16
7,25 = 4 ∗ 𝑥 + 11 ∗ (1 − 𝑥)
11 − 7,25
𝑥=
= 0,5357
7
1 − 𝑥 = 0,4643
𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛𝑝𝑒𝑟𝑝 =
b)
2000000
= 12,5𝑀
0,16
𝑃𝑉5𝑦 = 0,5357 ∗ 12,5𝑀 = 6,696𝑀
𝑃𝑉20𝑦 = 0,4642 ∗ 12,5𝑀 = 5,804𝑀
𝑃𝑉 =
1 − 1,16−20 1000
+
= 407,12
0,16
1,1620
5,804𝑀
So 407,12 = 14256,24
𝑃𝑉20𝑦 = 60 ∗
Exercise 12
a)
1000
𝑃𝑉 =
= 374,84
1,0812,75
1000
𝑃𝑉 =
= 333,28
1,0912,75
30
Maths
Exercises
1
𝑉𝑎𝑟 = −1% ∗ 11,81 + ∗ 1%2 ∗ 150,3 = 11,06%
2
1 − 1,08−30 1000
𝑃𝑉 = 60 ∗
+
= 774,84
0,08
1,0830
−30
1 − 1,09
1000
𝑃𝑉 = 60 ∗
+
= 691,79
0,09
1,0930
1
𝑉𝑎𝑟 = −1% ∗ 11,79 + ∗ 1%2 ∗ 231,2 = 10,63%
2
333,28 − 374,84
= 11,09% 𝑙𝑜𝑠𝑠
374,84
691,79 − 774,89
𝑉𝑎𝑟 =
= 10,72%
774,89
𝑉𝑎𝑟 =
b)
if decrease of 1%
𝑉𝑎𝑟 = 12,59% 𝑔𝑎𝑖𝑛
𝑉𝑎𝑟 = 12,56% 𝑔𝑎𝑖𝑛
𝑉𝑎𝑟 = 13,04% 𝑔𝑎𝑖𝑛
𝑉𝑎𝑟 = 12,95% 𝑔𝑎𝑖𝑛
c)
formula gives a good approximation of the change.
d)
convexity higher, yield lower.
Problem +
31
Maths
Exercises
Problem Set 8
Exercise 1
(1 + 𝑖2 )2 = (1 + 𝑖1 )(1 + 1𝑓2)
Exercise 2
a)
100 = 107 ∗ (1 + 𝑖1 )−1
=> 𝑖1 = 7%
102 = 8 ∗ (1 + 𝑖1 )−1 + 108 ∗ (1 + 𝑖2 )−2 = 8 ∗ (1,07)−1 + 108 ∗ (1 + 𝑖2 )−2
1
2
108
=> 𝑖2 = (
)
− 1 = 6,8913%
102 − 8 ∗ 1,07−1
95 = 6,7 ∗ (1 + 𝑖1 )−1 + 6,7 ∗ (1 + 𝑖2 )−2 + 106,7 ∗ (1 + 𝑖)−3
= 6,7 ∗ 1,07−1 + 6,7 ∗ 1,068913−2 + 106,7 ∗ (1 + 𝑖)−3
1
3
106,7
=> 𝑖3 = (
) − 1 = 8,7881%
−1
−2
95 − 6,7 ∗ 1,07 + 6,7 ∗ 1,068913
93 = 7 ∗ (1 + 𝑖1 )−1 + 7 ∗ (1 + 𝑖2 )−2 + 7 ∗ (1 + 𝑖)−3 + 107 ∗ (1 + 𝑖)−4
= 7 ∗ 1,07−1 + 7 ∗ 1,068913−2 + 7 ∗ 1,087881−3 + 107 ∗ (1 + 𝑖)−4
1
4
107
=> 𝑖4 = (
) − 1 = 9,3285%
−1
−2
−3
93 − 7 ∗ 1,07 + 7 ∗ 1,068913 + 7 ∗ 1,087881
109 = 12 ∗ (1 + 𝑖1 )−1 + 12 ∗ (1 + 𝑖2 )−2 + 12 ∗ (1 + 𝑖)−3 + 12 ∗ (1 + 𝑖)−4 + 112 ∗ (1 + 𝑖)−5
= 12 ∗ 1,07−1 + 12 ∗ 1,068913−2 + 12 ∗ 1,087881−3 + 12 ∗ 1,093285−4 + 112
∗ (1 + 𝑖)−5
1
5
112
=> 𝑖5 = (
) − 1 = 10%
−1
−2
−3
−4
109 − 12 ∗ 1,07 + 12 ∗ 1,068913 + 12 ∗ 1,087881 + 12 ∗ 1,093285
12%
10%
8%
6%
Spot rate
4%
2%
0%
0
2
4
6
c)
1,0689132
− 1 = 6,7827%
1,07
In fact we have : (1 + 𝑖) ∗ 1,07 = 1,068913 ∗ 1,068913 !
1𝑓2 =
1,0878813
− 1 = 12,6833%
1,0689132
1,0932854
3𝑓4 =
− 1 = 10,97%
1,0878813
2𝑓3 =
32
Maths
4𝑓5 =
Exercises
1,15
− 1 = 12,7275%
1,0932854
e)
1,0878813 = 1,07 ∗ (1 + 𝑖)2
1,0878813
𝑖=√
− 1 = 9,6933%
1,07
1,0932854 = 1,0689132 ∗ (1 + 𝑖)2
1,0932854
𝑖=√
− 1 = 11,8213%
1,0689132
Exercise 3
5
5
5
105
𝑃1 =
+
+
+
= 102,5968
2
3
1,05 1,0475
1,045
1,04254
10
10
10
110
𝑃2 =
+
+
+
= 120,5302
2
3
1,05 1,0475
1,045
1,04254
1 − (1 + 𝑖)−4
100
+
=> 𝑖 = 4,2799%
(1 + 𝑖)4
𝑖
1 − (1 + 𝑖)−4
100
𝑃2 = 10 ∗
+
=> 𝑖 = 4,3036%
(1 + 𝑖)4
𝑖
𝑃1 = 5 ∗
1,05 ∗ (1 + 𝑖) = 1,04752
1,04752
𝑖=
−1
1,05
𝑖 = 4,5%
1,04752 ∗ (1 + 𝑖) = 1,0453
1,0453
𝑖=
− 1 = 4%
1,04752
1,04254
− 1 = 3,504%
1,0453
Exercise 4
𝐹𝑉 = 100
𝑡=6
𝐶 = 6 => 𝑖 = 12%
𝐶 = 10 => 𝑖 = 8%
1 − 1,12−6
100
+
= 75,3316
0,12
1,126
1 − 1,08−6
100
𝑃2 = 10 ∗
+
= 109,2458
0,08
1,086
𝑃1 = 6 ∗
If we do 𝑃1 − 0,6 ∗ 𝑃2 we have a zero coupon bond of 6 years.
𝑃1 − 0,6 ∗ 𝑃2 = 9,7841
At the end we have 106 − 0,6 ∗ 110 = 40
40
= (1 + 𝑖)6
9,7841
𝑖 = 26,4513%
33
Maths
Exercises
Exercise 5
a)
(1 + 𝑖2 )2
1𝑓2 =
− 1 = 6,4%
1 + 𝑖1
2𝑓3 = 5,5
3𝑓4 = 4,31%
4𝑓5 = −0,065%
The last one is wrong
Lending for 4 y gives a better return than for 5 y
b)
so strategy
Borrow for 5 y
Lend for 4y
Net
Gain
Now
+100$
-100$
0
Y=4
Y=5
−100 ∗ 1,0465 = 125,216
+100 ∗ 1,0584 = 125,298
125,298
Exercise 6
a)
HPR=6%
1𝑓2 = 6,01%
2𝑓3 = 7,014%
b)
mkt expects higher 2f3 than you
=> 𝑚𝑘𝑡 𝑒𝑥𝑝𝑒𝑐𝑡𝑠 𝑙𝑜𝑤𝑒𝑟 𝑝𝑟𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 𝑎𝑡 𝑡 = 1
=> 𝑚𝑘𝑡 𝑒𝑥𝑝𝑒𝑐𝑡𝑠 𝑙𝑜𝑤𝑒𝑟 𝐻𝑃𝑅
Exercise 7
1y=5%
2y=6%
12
112
+
= 111,108
1,05 1,062
12
112
+
= 111,399
1,058 1,0582
There is a little difference for arbitrage, 0,280…
111,399 is too expensive, you could sell it
Exercise 8
a)
(1 + 𝑖2 )2
1,064 =
1,06
𝑖2 = 6,1998%
(1 + 𝑖3 )3
1,071 =
1,0619982
𝑖3 = 6,4990%
(1 + 𝑖4 )4
1,073 =
1,0649903
𝑖4 = 6,6987%
(1 + 𝑖5 )5
1,082 =
1,0669873
34
125,216
125,298 − 125,216 = 0,082
Maths
Exercises
𝑖5 = 6,9973%
b)
future interest rate up
c)
borrow PV(100M) for 4y
100𝑀
borrow
4 = 77,151𝑀
1,067
invest 77,151M for 5y => net cash flow at t=0 = 0
at t=4
repay the loan : −77,151 ∗ 1,0674 = −100𝑀
at t=5
receive the 5y investment : 77,151 ∗ 1,075 = 108,208𝑀
(with the future, but in the exercise it’s not the purpose : 100 ∗ 1,082 = 108,2𝑀 > 107)
d)
6,93%
Exercise 9
a)
The price of cement is not volatile,
no demand
if people were interested, there will be standardized contracts.
b)
to reduce cost of transaction, future : on big quantities
and future, no exchange of the underlying !
no need to have all the cash
Exercise 10
a)
1477,2 ∗ 250 ∗ 0,1 = 36930
b)
(1500 − 1477,2) ∗ 250 = 5700
=> 15,43%
c)
1477,2 ∗ 0,99 = 1462,428
(1462,428 − 1477,2) ∗ 250 = −3693 => −10%
(by logic, we lose 10 times the market, so 1%=>10%)
Exercise 11
? Interest swap fixe vs floating, receive fixe, give floating ?
Buy a long T-bond contract.
Exercise 12
a)
short because we think it’s going to be down
b)
35
Maths
Exercises
13,5𝑀
= 40
1350 ∗ 250
c)
beta 0,6
40 ∗ 0,6 = 24
Exercise 13
3
671,5
= (1 + 𝑖3𝑚 )12
664,3
𝑖3𝑚 = 4,406%
9
690
= (1 + 𝑖6𝑚 )12
664,3
𝑖6𝑚 = 5,19%
𝑖15𝑚 = 4,85%
𝑖21𝑚 = 5,06
Exercise 14
𝑆 = 1300
𝑟𝑓 = 4%, 𝑑𝑖𝑣 = 1%
𝐹1𝑦 = 1330
𝐹1𝑦 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 1300 ∗ 1,03 = 1339
𝐹𝑢𝑡𝑢𝑟𝑒 𝑖𝑠 𝑢𝑛𝑑𝑒𝑟𝑝𝑟𝑖𝑐𝑒𝑑
Buy future
Short index
Lend
T=0
0
1300
-1300
0
T=12
𝑆12 − 1330
−𝑆12 − 0,01 ∗ 1300
1300 ∗ 1,04
9
Exercise 15
a)
£
= 2.00
$
so
2∗
1,04
= 1,9627
1,06
(foreign/domestic in the point of view of £/$=2, the foreign is US)
b)
2,03 is too high, you sell it.
T=0
Sell 1£ forward
1£
Buy 1,06 in spot market
Borrow in US
−
2$
= −1,887
1,06
+1,887
36
T=1
Pay 1£, collect 2,03$:
2,03$ − E1$
Collect 1£ from UK loan +
collect to $
E1$
Repay −1,887 ∗ 1,04 = 1,962
Gain = 0,068$
Maths
Exercises
Exercise 16
𝐷𝑀 = 8𝑦
𝐹0 = 100 ; 𝐷𝑀 = 6𝑦
𝑙𝑒𝑡 ′ 𝑠 𝑡𝑎𝑘𝑒 1𝑏𝑝
𝑉𝑎𝑟𝑃𝑏𝑜𝑛𝑑𝑠 = 8 ∗ 0,0001 ∗ 10𝑀 = 8000$
𝐶𝐹 𝑓𝑟𝑜𝑚 1 𝑓𝑢𝑡𝑢𝑟𝑒𝑠 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡 = 6 ∗ 0,0001 ∗ 100000 = 60$
8000
𝑠𝑜 𝑤𝑒 𝑛𝑒𝑒𝑑
= 133 𝑐𝑜𝑛𝑡𝑟𝑎𝑐𝑡𝑠
60
Exercise 17
𝑈𝑆 = 4%; 𝐸𝑈𝑅 = 3%
𝐸𝑈𝑅
= 1,5
𝐷𝑜𝑙𝑙𝑎𝑟
The swap will call for an exchange of 1 million € for a given number of dollars each year
So we have first the forward for each period.
1,04
𝐹1𝑦 = 1,5 ∗
= 1,5146
1,03
1,04²
𝐹2𝑦 = 1,5 ∗
= 1,5293
1,03²
1,043
𝐹3𝑦 = 1,5 ∗
= 1,5441
1,033
So the “mean”
𝐹∗
𝐹∗
𝐹∗
1 − 1,04−3 1,5146 1,5293 1,5441
∗
+
+
=
𝐹
∗
=
+
+
1,04 1,042 1,043
0,04
1,04
1,042
1,043
∗
𝐹 = 1,5289
Exercise 18
The last payment was 2 months ago, so the next is in 4 months, and the last one 6 months later,
so in 10 months
Time
Rate
Fixed cash flow : 12%
Floating CF : 9,6%
6
0,1
1
T=4 months
100𝑀 + 100𝑀 ∗ (𝑒 0,096∗12
𝑒 3 = 1,033895
100𝑀 ∗ 12% ∗ = 6𝑀
= 1/3y
2
− 1)
6
𝑃𝑉 =
= 104,9171
1,033895
104,9171
𝑃𝑉 =
= 101,4775
1,033895
0,1∗10
1
T=10 months
𝑒 12 = 1,086904
100𝑀 + 100𝑀 ∗ 12% ∗
= 10/12y
2
= 106𝑀
106
𝑃𝑉 =
1,086904
6
106
Total
𝑃𝑉 = 101,4775
𝑃𝑉 =
+
1,033895 1,086904
= 103,328
𝐵𝑓𝑖𝑥𝑒𝑑 = 103,3280𝑀
𝐵𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔 = 101,4775𝑀
𝑉𝑠𝑤𝑎𝑝(𝑡𝑜 𝑝𝑎𝑟𝑡𝑦 𝑝𝑎𝑦 𝑓𝑙𝑜𝑎𝑡𝑖𝑛𝑔) = 1,85𝑀
Exercise 19
From £ payers perspective
Time
3 months
Cf $
Cf £
6% ∗ 30𝑀 = 1,8𝑀$
37
−10% ∗ 20𝑀£ = 2𝑀£
Maths
Exercises
1,8𝑀$
15 months
−2𝑀£
Current
£/$ = 1,85
(1£ = 1,85$)
Future
3
𝐹3𝑚
1,04 12
= 1,85 ∗ (
) = 1,8369
1,07
15
𝐹15𝑚
1,04 12
= 1,85 ∗ (
) = 1,7854
1,07
2𝑀 ∗ 1,8369 = 3,6738𝑀$
2𝑀 ∗ 1,7854 = 3,5708𝑀$
𝑃𝑉 =
𝑃𝑉 =
3,6738 − 1,8
3
1,0415
3,5708 − 1,8
15
1,0412
= 1,8555𝑀$
= 1,6861𝑀$
At the end the 2 parties exchange the currencies, so 30𝑀$, and 20𝑀£ = 20 ∗ 1,7854𝑀 = 35,708𝑀$
35,708 − 30
𝑃𝑉 =
= 5,4349𝑀$
15
1,0412
For the view of the party paying £, he loses : 5,4349 + 1,6861 + 1,8555 = 8,9765𝑀$
38
Maths
Exercises
Problem Set 9
Exercise 1
a)
𝑟𝑒 = 0,16
𝐷1 = 2$
𝐷1
𝑃=
𝑟𝑒 − 𝑔
𝐷1
𝑔 = 𝑟𝑒 −
𝑃
𝑔 = 0,16 −
𝑔 = 12%
2
50
b)
𝑃=
𝑃
𝐸
2
= 18,18
0,16 − 0,05
=>down; earning constant, price down. PVGO is falling.
Exercise 2
a)
𝐸𝑃𝑆 = 6
𝑔 = 8%
𝑃𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜 =
𝑟𝑒 = 12%
𝑃𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜 =
1
3
𝐷1
𝐸𝑃𝑆1
1
∗6=2
3
2
𝑃=
= 50$
0,12 − 0,08
𝐷1 =
b)
𝑔 = 𝑝𝑙𝑜𝑤𝑏𝑎𝑐𝑘𝑟𝑎𝑡𝑖𝑜 ∗ 𝑅𝑂𝐸 = (1 − 𝑝𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜) ∗ 𝑅𝑂𝐸
𝑔
𝑅𝑂𝐸 =
1 − 𝑝𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜
0,08
𝑅𝑂𝐸 =
= 0,12
1
1−3
Doesn’t need it
𝐸𝑃𝑆
+ 𝑃𝑉𝐺𝑂
𝑟𝑒
6
𝑃𝑉𝐺𝑂 = 50 −
=0
0,12
𝑃0 =
c)
𝑅𝑂𝐸 = 10%
1
𝑔 = 0,10 ∗ (1 − ) = 0,0667
3
2
𝑃0 =
= 37,5
0,12 − 0,0667
𝑃𝑉𝐺𝑂 = −12,5$
39
Maths
Exercises
d)
𝑅𝑂𝐸 = 15%
1
𝑔 = 0,15 ∗ (1 − ) = 0,1
3
2
𝑃=
= 100
0,12 − 0.10
𝑃𝑉𝐺𝑂 = 50$
Exercise 3
𝐸𝑃𝑆 = 1$
𝑟𝑒 = 15%
𝑃𝑎𝑅 = 0,5
𝑅𝑂𝐸 = 20%
a)
𝑔 = 0,5 ∗ 0,2 = 0,1
𝐷0 = 0,5$
0,5 ∗ 1,1
𝑃0 =
= 11$
0,15 − 0,1
b)
𝑔 = 0,4 ∗ 0,15
𝑔 = 6%
BE CAREFUL, payout ratio has changed, so div change too !
𝑑2 0,605
𝐸𝑃𝑆2 =
=
= 1,21$
0,5
0,5
𝐸𝑃𝑆3 = 𝐸𝑃𝑆2 ∗ 1,06 = 1,2826$
𝑑3 = 1,2826 ∗ 0,6 = 0,76956
0,5 ∗ 1,1 0,5 ∗ 1,1²
0,76956
𝑃0 =
+
+
2
1,15
1,15 ∗ (0,15 − 0,06)
1,15²
𝑃0 = 7,4$
0,76956
(𝑃2 = 0,15−0,06 = 8,5507$)
c)
𝑃0 = 11
𝑃1 = 11 ∗ 1,1 = 12,1
12,1 + 0,55 − 11
𝑟1 =
= 15%
11
𝑃2 = 8,5507
𝑟2 = −23,3%
𝑃3 = 𝑃2 ∗ 1,06 = 9,064$
𝑟3 = 15%
Exercise 4
𝐶𝐹 = 2𝑀$
𝑔 = 5%
𝑝𝑟𝑒𝑡𝑎𝑥 𝑖𝑛𝑣𝑒𝑠𝑡 = 20%
𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 = 35%
𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 = 200000$
𝑟𝑎𝑠𝑠𝑒𝑡 = 12%
𝑑𝑒𝑏𝑡 = 2𝑀$
40
Maths
Exercises
(2𝑀 ∗ (1 − 0,35) + 200000 − 400000) ∗ 1,05
= 16,5𝑀$
0,12 − 0,05
16,5 − 2𝑀
𝑃0 =
= 14,5$
1𝑀
𝑃𝑉(𝐹𝐶𝐹) =
Exercise 5
𝐸𝑃𝑆 = 10$
𝑅𝑂𝐸 = 20% => 5𝑦
6𝑡ℎ 𝑦𝑒𝑎𝑟 𝑅𝑂𝐸 = 15%, 𝑝𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜 = 40%
𝑟𝑒 = 15%
a)
𝑔 = 20% => 5𝑦
𝑔 = 0,6 ∗ 0,15 = 9%
𝐸𝑃𝑆5 = 10 ∗ 1,25 = 24,8832$
𝐸𝑃𝑆6 = 24,8832 ∗ 1,09 = 27,1227$
𝑑6 = 𝐸𝑃𝑆6 ∗ 𝑝𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑜𝑖 = 27,1227 ∗ 0,4 = 10,8491$
𝑑6
10,8491
𝑃5 =
=
= 180,8183$
0,15 − 0,09 0,15 − 0,09
𝑃5
180,8183
𝑃0 =
=
= 89,899$
(1 + 𝑖)5
1,155
b)
𝑟𝑒 = 𝑑𝑖𝑣 𝑦𝑖𝑒𝑙𝑑 + 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛
𝐷1
𝑟𝑒 =
+𝑔
𝑃0
first year: only capital gain, so return of 15%
from 5th, there is div yield so capital gain = 9%
c)
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑦𝑖𝑒𝑙𝑑 = 𝑟𝑒 − 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑔𝑎𝑖𝑛
So div yield = 6%
d)
𝜌 = 0,2
It doesn’t change, pvgo constant, roe=re=15%
Exercise 6 (no correction)
𝑅𝑂𝐸 = 9%
a)
𝑟𝑒 = 6 + 1,25 ∗ (14 − 6) = 16%
𝐸𝑃𝑆 = 3$
2
𝑔 = 0,09 ∗ = 6%
3
1
𝐷0 = 3$ ∗ = 1$
3
𝐷0 ∗ 1,06
𝑃0 =
= 10,6$
0,1
b)
𝐹𝑜𝑟𝑤𝑎𝑟𝑑𝑃𝐸 =
𝑃0
𝑝𝑎𝑦𝑜𝑢𝑡𝑟𝑎𝑡𝑖𝑜
=
𝐸𝑃𝑆1
𝑟𝑒 − 𝑔
41
Maths
Exercises
1
𝐹𝑃𝐸 = 3 = 3,33
0,1
c)
𝑃𝑉𝐺𝑂 = 𝑃0 −
𝐸𝑃𝑆
3
= 10,6 −
= −8,15
𝑟𝑒
0,16
d)
1
𝑃𝑙𝑜𝑤𝐵𝑎𝑘 =
3
1,03
𝑃0 = 2$ ∗
= 15,85
0,13
Exercise 7
𝐷𝑖𝑣 = 0,5$
𝑟𝑎𝑡𝑒 = 6%
𝑡 = 20𝑦
𝑟𝑒 = 9%
𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
𝑑20 = 𝑑0 ∗ (1 + 𝑔)20 = 1,6056
𝑑20
𝑑1
𝑟𝑒 − 𝑔
𝑃𝑉𝑑1 𝑑2 =
−
= 6,2705$
𝑟𝑒 − 𝑔 (1 + 𝑟𝑒 )19
𝑑20
𝑟𝑒
𝑃𝑉𝑑20 𝑑∞ =
= 3,4654$
(1 + 𝑟𝑒 )19
𝑃𝑉𝑡𝑜𝑡 = 9,7359$
𝑆𝐻𝑂𝑈𝐿𝐷 𝑊𝑂𝑅𝐾 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑖𝑡𝑖𝑒𝑠, 𝑤ℎ𝑒𝑟𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑠𝑡𝑎𝑘𝑒 …
1,0619 − 1,0919
𝑉20 = 0,5 ∗ (
) ∗ 1,09−19 = 6,85
1,06 − 1,09
0,5 ∗ 1,0620
∗ 1,09−20 = 3,18
0,09
𝑉𝑡𝑜𝑡 = 10 …
Exercise 8
a)b)
1𝑀 𝑠ℎ𝑎𝑟𝑒 = 20𝑀$
𝐷𝑖𝑣 = 1𝑀$
1
20 =
=> 𝑟 = 10%
𝑟 − 0,05
1,05
𝑉1 =
= 21𝑀$
0,05
We have :
21𝑀
1𝑀 + 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒
𝑎𝑛𝑑 𝑃1 ∗ 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒 = 1𝑀$
𝑠𝑜 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒 𝑖𝑠 50000
𝑎𝑛𝑑 𝑃1 = 20$
𝑃1 =
c)
new number of share = 1050000
42
Maths
Exercises
1,05𝑀
= 1$
1,05
So 𝑑3 = 1,05$
… and so on…
𝑛𝑒𝑤 𝑑𝑖𝑣2 =
d)
1
2
0,1 − 0,05
𝑃𝑉 =
+
= 20𝑀$
1,1
1,1
e)
old shareholder looses, new shareholder gains.
Exercise 9 no correction
Exercise 10
a)
The institutions make the price here
b)
15 + 5
= 12% => 𝑃𝑙𝑜𝑤 = 166,667
𝑃𝑙𝑜𝑤
5+5
100 ∗
= 12% => 𝑃𝑚𝑒𝑑 = 83,333
𝑃𝑚𝑒𝑑
30 + 0
100 ∗
= 12% => 𝑃ℎ𝑖𝑔ℎ = 250
𝑃ℎ𝑖𝑔ℎ
100 ∗
c)
𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠
=> 5 ∗ 0,5 + 15 ∗ 0,5 = 2,5 + 2,25 = 4,75
15,25
100 +
= 9,15%
166,667
=> 5 ∗ 0,5 + 5 ∗ 0,15 = 2,5 + 0,75 = 3,25
6,75
100 ∗
= 8,1%
83,33
=> 30 ∗ 0,5 = 15
15
100 ∗
= 6%
250
𝑐𝑜𝑟𝑝𝑜𝑟𝑎𝑡𝑒𝑠
=> 5 ∗ 0,05 + 15 ∗ 0,15 = 0,25 + 2,25 = 2,5
17,5
100 +
= 8,7%
166,667
=> 5 ∗ 0,05 + 5 ∗ 0,35 = 0,25 + 1,75 = 2
8
100 ∗
= 9,6%
83,33
=> 30 ∗ 0,5 = 15
27,5
100 ∗
= 11,4%
250
d)
we assume that the investors invest where return max
Low
Med
80b indiv
80b
43
High
Maths
10b corporate
Rest institutions
Total
Exercises
20b
100b
50b
50b
10b
110b
120b
Exercise 11
Payout
2009
2008
2007
2006
2005
2004
Average
No massive difference
Div yield
0,3
= 14,63%
2,05
0,3
= 19,35%
1,55
0,25
= 14,71%
1,7
0,25
= 27,72%
0,9
0,25
= 25,4%
0,85
0,25
= 24,51%
1,02
21,73%
0,3
= 1,48%
20,3
0,3
= 2,44%
12,3
0,25
= 1,76%
14,2
0,25
= 2,55%
9,80
0,25
= 2,94%
8,5
0,25
= 2,45%
10,2
2,27%
b)
(𝑃𝑐𝑢𝑚 − 𝑃𝑒𝑥 ) ∗ (1 − 𝑡𝑔 ) = 𝑑𝑖𝑣 ∗ (1 − 𝑡𝑑 )
𝑃𝑐𝑢𝑚 − 𝑃𝑒𝑥
1−(
∗ (1 − 𝑡𝑔 )) = 𝑡𝑑
𝑑𝑖𝑣
𝑃
−𝑃𝑒𝑥
We know that 0,6 < 𝑐𝑢𝑚
< 0,8
𝑑𝑖𝑣
And 𝑡𝑔 = 15%
so
32% ≤ 𝑡𝑑 ≤ 49%
c)
yes,
the difference before and after the dividend = 0,7 in average (0,8 and 0,6)
so hedge fund can buy just before the dividend, receive the dividend (1 div) and sell just after (-0,7
div)
the payoff is
𝑃𝑎𝑦𝑜𝑓𝑓 = 1 𝑑𝑖𝑣 − 0,7𝑑𝑖𝑣 = 0,3𝑑𝑖𝑣
With that you bear overnight risk.
Exercise 12
𝐴1 = 50𝑀$
𝐴2 = 50𝑀$
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒 1000𝑀
Here we assume 𝑉 = 1000%$ ???
𝑉𝐴1 = 500𝑀$
𝑉𝐴2 = 500𝑀$
𝑉 = 1000𝑀$
𝑑𝑖𝑣 = 100𝑀$
𝑑 = 0,1
Want to increase div by 50 cent,
So we need 500𝑀$
44
Maths
Exercises
We have:
???
𝑁 ∗ 𝑃 ∗ 0,995 = 500
1000𝑀
𝑉=
1000𝑀 + 𝑁
𝑁 = 1010𝑀
𝑃 = 0,4975$
Old share wealth=0,4975 + 0,5 = 0,9975
Exercise 13
With CAPM
𝑟𝑒1 = 21%
𝑟𝑒2 = 18%
𝑟𝑒3 = 15%
𝑤𝑖𝑡ℎ 𝐼𝑅𝑅, 𝑏𝑒𝑠𝑡 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑖𝑠 𝐵 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 20 > 18%, 𝑡ℎ𝑒𝑛 𝐴 (22% > 21%), 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝐶 𝑖𝑠 𝑎𝑏𝑎𝑛𝑑𝑜𝑛𝑒𝑑
𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝐴 + 𝐵 = 1100$
𝐹𝐶𝐹 𝐸1 = (𝑛𝑒𝑡 𝑖𝑛𝑐𝑜𝑚𝑒1 + 𝑑𝑒𝑝𝑟𝑒𝑐 − 𝑐𝑎𝑝𝑒𝑥 − 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑤𝑜𝑟𝑘 𝑐𝑎𝑝1)
𝐹𝐶𝐹 = 1000 + 500 − 1100 − 5000 ∗ 0,25 ∗ 0,08 = 300$
Pay 300$ of dividend
45
Maths
Exercises
Problem Set 10
Exercise 1
A
B
D/A
0,3
0,1
E/A
0,7
0,9
a)
If you own 1% of the stock A, then you own 1% ∗ 0,7 of the value of A
So you own : 0,01 ∗ 0,7 𝑉𝐴 = 0,007 𝑉𝐴
Then you own 1% of the stock A, so you also gain 1% of the profit of the firm.
And you “loose” 1% ∗ 0,3 of the value of A, due to the debt at 𝑟𝑓
So entitlement = 0,01 ∗ (𝑃𝑟𝑜𝑓𝑖𝑡𝐴 − 0,3 ∗ 𝑉𝐴 ∗ 𝑟𝑓 ) = 0,01 ∗ 𝑃𝑟𝑜𝑓𝑖𝑡 𝐴 − 0,003 ∗ 𝑟𝑓 ∗ 𝑉𝐴
To have the same with B :
You buy 1% of B, so you own 0,01 ∗ 0,9 𝑉𝐵 = 0,009 𝑉𝐵
So you to have the same 0,007, you borrow 1% of (𝐷𝐴 − 𝐷𝐵 ) = 0,01 ∗ (0,3𝑉 − 0,1𝑉) = 0,002 𝑉𝐵
𝑁𝑒𝑡 𝑜𝑢𝑙𝑡𝑎𝑦 = 0,007 𝑉𝐵
𝑁𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛 = 0,01 ∗ (𝑃𝑟𝑜𝑓𝑖𝑡𝐵 − 0,1 ∗ 𝑉𝐵 ∗ 𝑟𝑓 ) − 0,002 ∗ 𝑉𝐵 ∗ 𝑟𝑓 = 0,01 ∗ 𝑃𝑟𝑜𝑓𝑖𝑡𝐵 − 0,003 ∗ 𝑉𝐵 ∗ 𝑟𝑓
b)
If you own 2% of the stock B, then you own 2% ∗ 0,9 𝑉𝐵 = 0,018 𝑉𝐵
And you are entitled to 0,02 ∗ 𝑃𝑟𝑜𝑓𝑖𝑡 𝐵 − 0,02 ∗ 0,1 ∗ 𝑉𝐵 ∗ 𝑟𝑓
If you buy 2% of stock A, you have 𝑉𝐴 = 0,014 𝑉𝐴
You lend 0,004 𝑉𝐴 such that 𝑛𝑒𝑡 𝑜𝑢𝑡𝑙𝑎𝑦 = 0,018 𝑉𝐴
𝑁𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛 = 0,02 ∗ (𝑃𝑟𝑜𝑓𝑖𝑡 𝐴 − 0,3 𝑉𝐴 ∗ 𝑟𝑓) + 0,004 ∗ 𝑉𝐴 ∗ 𝑟𝑓 = 0,02 𝑃𝑟𝑜𝑓𝑖𝑡 𝐴 − 0,002 𝑉𝐴 ∗ 𝑟𝑓
c)
If 𝑉𝐴 < 𝑉𝐵
𝑐𝑜𝑠𝑡 𝐴 < 𝑐𝑜𝑠𝑡 𝐵
𝑃𝑎𝑦𝑜𝑓𝑓 𝐴 > 𝑃𝑎𝑦𝑜𝑓𝑓 𝐵
So you choose A
It works if the capital structure is the same.
Exercise 2
a)
𝐷
After the refinancing, 𝐸 = 1
Before there is no debt, 𝐵𝐴 = 𝐵𝐸 = 0,8
After : 𝐵𝐴 = 0,8 = 0,5 ∗ 𝐵𝐸 + 0,5 ∗ 0
So 𝐵𝐸 = 1,6
b)
𝑟𝑢 = 8% = 5% + 0,8 ∗ (𝑟𝑚 − 𝑟𝑓) => 𝑟𝑚 − 𝑟𝑓 = 3,75%
𝑟𝑒 = 5% + 1,6 ∗ 3,75% = 11%
Or
𝑟𝑒 = 𝑟𝑢 + (𝑟𝑢 − 𝑟𝑓 ) ∗
𝐷
= 8% + (8% − 5%) ∗ 1 = 11%
𝐸
c)
𝑊𝐴𝐶𝐶 = 𝑟𝑎 = 11% ∗ 0,5 + 5% ∗ 0,5 = 8% = 𝑟𝑢
d)
𝐸𝑃𝑆1
=> 𝐸𝑃𝑆 = 𝑃0 ∗ 𝑟𝑒
𝑟𝑒
𝑃0 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑟𝑒 => 8% 𝑡𝑜 11% => +37,5%, 𝑠𝑜 𝐸𝑃𝑆: + 37,5%
𝑃0 =
46
Maths
Exercises
e)
𝑃0
1
1
= =
= 9,0909
𝐸𝑃𝑆1 𝑟𝑒 0,11
Exercise 3
a)
Save from the debt :
40𝑀$ ∗ 0,09 ∗ 0,35 = 1,26𝑀$
b)
Debt change permanent
𝐷 ∗ 𝑟𝑑 ∗ 𝑡𝑐
𝑃𝑉(𝑇𝑆) =
= 𝐷 ∗ 𝑡𝑐 = 40 ∗ 0,35 = 14𝑀$
𝑟𝑑
c)
𝑃𝑉(𝑇𝑆) = 1,26 ∗
1 − 1,09−10
= 8,09𝑀$
0,09
d)
Interest rate drop to 7%, but debt fixed rate
1,26
𝑃𝑉(𝑇𝑆)𝑝𝑒𝑟𝑝 =
= 18𝑀$
0,07
1 − 1,07−10
𝑃𝑉(𝑇𝑆)10𝑦 = 1,26 ∗
= 8,85𝑀$
0,07
Exercise 4
a)
𝑉𝐿 = 𝑉𝑈 + 𝑃𝑉(𝑇𝑆) = 𝑉𝑈 + 𝐷 ∗ 𝑡𝑐 = 40 ∗ 0,4 = 16
b)
𝑚𝑜𝑟𝑒 ∶ 20 ∗ 0,4 = 8
c)
𝑟𝑑 = 8%
Save from debt (annual) : 8% ∗ 40𝑀 ∗ 0,4 = 3,2𝑀$ ∗ 0,4 = 1,28𝑀$
1,28
𝐼𝑁 𝑓𝑎𝑐𝑡 𝑤𝑒 ℎ𝑎𝑣𝑒
= 16 => 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 !
0,08
𝑃𝑉(𝑇𝑆)(5𝑦!) = 1,28 ∗
1 − 1,08−5
= 5,1107𝑀$
0,08
So if there is no deductibility after 5 years, you lose : 16𝑀 − 5,1107𝑀 = 10,8893𝑀$
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 = 160 − 10,8893 = 149,1107𝑀$
Exercise 5
𝐸𝐵𝐼𝑇 = 2𝑀$ 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
𝑉 = 𝐸 = 12𝑀$
𝑡𝑐 = 0,4
𝛽=1
𝑟𝑚 = 15%
𝑟𝑓 = 9%
𝑟𝑑 = 12%
𝑃𝑉(𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑐𝑜𝑠𝑡) = 8𝑀$
47
Maths
Exercises
a)
𝑟𝑒 = 0,09 + 1 ∗ 0,06 = 15% = 𝑊𝐴𝐶𝐶
b)
Value Debt
2,5
5
Var Value
debt
2,5
2,5
Marginal Tax
Benefit
1𝑀
1𝑀
Probability
Default
0%
8%
Var Prob
Default
0%
8%
7,5
2,5
1𝑀
20,5%
12,5%
8
0,5
0,2𝑀
30%
9,5%
Marg cost
debt
0
0,08 ∗ 8
= 0,64𝑀
0,115 ∗ 8
= 1𝑀
0,095 ∗ 8
= 0,76𝑀
1,2𝑀
0,6𝑀
1,4𝑀
9
1
0,4𝑀
45%
15%
10
1
0,4𝑀
52,5%
7,5%
12,5
2,5
1𝑀
70%
17,5%
We see at Value Debt=7,5 it’s not any more interesting
So
𝑉𝐿 = 𝑉𝑈 + 𝑃𝑉(𝑇𝑆) − 𝐸(𝑃𝑉(𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑐𝑜𝑠𝑡)) = 12𝑀 + 1𝑀 + 1𝑀 − 0,64𝑀 = 13,36𝑀
For example is you have more debt, you have 13,36 + 1 − 1 = 13,36𝑀, it doesn’t increase the
value of the firm, and after it decreases the value : 13,36𝑀 + 0,2 − 0,76 = 12,8𝑀 …
Exercise 6
Exercise 7
𝑉0 = 1,7𝑀$
𝐷 = 0,5𝑀$
𝑟𝑑 = 10%
𝑡𝑐 = 34%
𝑟𝑢 = 20%
a)
𝑉𝐿 = 𝑉𝑈 + 𝑃𝑉(𝑇𝑆)
𝑉𝑈 = 𝑉𝐿 − 𝑃𝑉(𝑇𝑆) = 1 700 000 − 500 000 ∗ 0,34 = 1 530 000$
b)
𝐷
∗ (1 − 𝑡𝑐 )
𝐸
0,5
𝑟𝑒 = 0,2 + (0,2 − 0,1) ∗
∗ (1 − 0,34)
1,2
𝑟𝑒 = 22,75%
𝑟𝑒 = 𝑟𝑢 + (𝑟𝑢 − 𝑟𝑑 ) ∗
𝑉𝐸 =
𝑑𝑖𝑣
=> 𝑑𝑖𝑣 = 𝑟𝑒 ∗ 𝑉𝐸 = 0,2275 ∗ 1,2𝑀 = 273 000$
𝑟𝑒
Exercise 8
𝑟𝑓 = 4% ; 𝑟𝑚𝑝 = 5% ; 𝑡𝑐 = 29%
A
𝐷
= 0,35
𝐷+𝐸
𝑟𝑑 = 4,5%
B
𝐷
= 0,5
𝐷+𝐸
𝑟𝑑 = 5%
C
No debt
𝐵𝑒𝑞𝑢𝑖𝑡𝑦 = 1,4 = 𝐵𝑎𝑠𝑠𝑒𝑡
48
Maths
Exercises
𝐷
𝑟𝑒 = 𝑟𝑢 + (𝑟𝑢 − 𝑟𝑑 ) ∗ ∗ (1 − 𝑡𝑐 )
𝐸
0,35
𝑟𝑒 = 11,5 + (11,5 − 4,5) ∗
0,65
∗ (1 − 0,29)
= 14,1762%
𝑟𝑎 = 14,1762 ∗ 0,65
+ (1 − 0,29)
∗ 4,5 ∗ 0,35
= 10,3328%
𝑟𝑒 = 4,5 + 1,4 ∗ 5 = 11,5%
𝑟𝑒 = 11,5 + (11,5 − 5)
∗ (1 − 0,29)
= 16,115%
𝑟𝑎 = 16,115 ∗ 0,5 + (1 − 0,29)
∗ 5 ∗ 0,5
= 9,8325%
Exercise 9
a)
Dividend => gain for shareholders
b)
good for bondholder
c)
NPV=0
There is no added value and there are more bonds, the old bondholder lose
d)
NPV=2$
Less senior security
Bondholder ==
e)
it’s good for shareholders.
Exercise 10
𝐸𝐵𝐼𝑇 = 120𝑀$
𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 = 10,5𝑀$
𝐶𝑎𝑝𝑒𝑥 = 15𝑀$
𝑔 = 6%
a)
𝑟𝑒 = 8 + 1,05 ∗ 5,5 = 13,775%
𝑟𝑎 = 0,90909 ∗ 13,775% + 10,3% ∗ 0,090909 ∗ (1 − 0,34) = 13,14%
b)
𝐹𝐶𝐹0 = 𝐸𝐵𝐼𝑇0 ∗ (1 − 𝑡𝑐 ) + 𝑑𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 − 𝐶𝑎𝑝𝑒𝑥
𝐹𝐶𝐹0 = 120 ∗ (1 − 0,34) + 10,5 − 15 = 74,7
74,7
𝑉𝑎𝑙𝑢𝑒 =
= 1108,85
0,13775 − 0,06
c)
𝐷
∗ (1 − 𝑡𝑐 )
𝐸
1,05
𝛽𝐴 = 𝛽𝑈 =
= 0,9850
1 + 0,1 ∗ (1 − 0,34)
𝛽𝐸 = 𝛽𝑈 + (𝛽𝑈 − 𝛽𝐷 ) ∗
d)
𝛽𝐸 = 0,9850 + (0,9850) ∗
𝐷
∗ (1 − 𝑡𝑐 )
𝐸
49
𝑟𝑎 = 11,5%
Maths
Exercises
So that you can calculate
𝑟𝑒 = 𝑟𝑓 + 𝛽𝐸 ∗ (𝐸(𝑟𝑚 ) − 𝑟𝑓)
So that you can calculate
𝐸
𝐷
𝑊𝐴𝐶𝐶 = 𝑟𝑎 = 𝑟𝑒 ∗ + 𝑟𝑑 ∗ ∗ (1 − 𝑡𝑐 )
𝐴
𝐴
𝐹𝐶𝐹0 ∗ (1,06)
𝑉𝑎𝑙𝑢𝑒 =
𝑊𝐴𝐶𝐶 − 0,06
So that you have now the value !!
50
Maths
Exercises
Problem Set 11
Exercise 1
𝑟𝑢 = 12%
600000 700000
+
= 93750$
1,12
1,122
300000 ∗ 0,08 ∗ 0,3 = 7200
300000
∗ 0,08 ∗ 0,3 = 3600
2
7200 3600
𝑃𝑉(𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑡𝑎𝑥 𝑠ℎ𝑖𝑒𝑙𝑑) =
+
= 9753,0864
1,08 1,082
𝐴𝑃𝑉 = 𝑁𝑃𝑉𝑏𝑎𝑠𝑒 = 93750 + 9753,0864 = 103503,0864$
𝑁𝑃𝑉 = −1000000 +
Exercise 2
𝑡𝑐 = 40%
𝐸𝑎𝑟𝑛𝑖𝑛𝑔 = 2000$
𝑔 = 3%
𝑟𝑓 = 5%
𝑟𝑚 = 11%
𝛽𝐴 = 1,11
a)
𝑟𝑈 = 𝑟𝐴 = 5 + 1,11 ∗ (11 − 5) = 11,66
2000 ∗ (1 − 0,4)
𝑉𝑈 =
= 13856,8129$
0,1166 − 0,03
b)
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠 = 5000 ∗ 0,05 = 250$
c)
𝑃𝑉(𝑇𝑆) =
250 ∗ 0,4
= 1154,7344
0,1166 − 0,03
d)
𝑉𝐿 = 𝑉𝑈 + 𝑃𝑉(𝑇𝑆) = 13856,8129 + 1154,7344 = 15011,5473$
So 𝐸 = 𝐴 − 𝐷 = 15011,5473 − 5000 = 10011,5473$
e)
𝐹𝐶𝐹1
𝑊𝐴𝐶𝐶 − 𝑔
𝐹𝐶𝐹1
2000 ∗ (1 − 0,4)
𝑊𝐴𝐶𝐶 =
−𝑔 =
+ 0,03 = 10,9938%
𝑉𝐿
15011,5473
𝑉𝐿 =
f)
𝐸
𝐷
𝐷
𝑉
∗ 𝑟𝑒 + ∗ 𝑟𝑑 ∗ (1 − 𝑡𝑐 ) => 𝑟𝑒 = (𝑟𝐴 − ∗ 𝑟𝑑 ∗ (1 − 𝑡𝑐 )) ∗
𝑉
𝑉
𝑉
𝐸
5000
15011,5473
𝑟𝑒 = (0,109938 −
∗ 0,05 ∗ (1 − 0,4)) ∗
= 14,9861%
15011,5473
10011,5473
𝑟𝐴 =
g)
??
𝐹𝐶𝐹𝐸1 = (𝐸𝐵𝐼𝑇1 − 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡1) ∗ (1 − 𝑡𝑐 ) + 𝑛𝑒𝑡 𝑏𝑜𝑟𝑟𝑜𝑤𝑖𝑛𝑔
𝐹𝐶𝐸𝐹1 = (2000 − 250) ∗ (1 − 0,4) + 150 = 1200
//3% ∗ 5000 = 150.
51
Maths
Exercises
𝐹𝐶𝐹𝐸 grow by 3%
𝐹𝐶𝐹𝐸1
1200
𝐸=
=
= 10 011,5473$
𝑟𝑒 − 𝑔
0,149862 − 0,03
Exercise 3
𝐷 = 400000$
𝑡𝑐 = 0,35
𝑟𝑒 = 0,1
𝑟𝑑 = 0,07
95000 ∗ (1 − 0,35)
𝑁𝑃𝑉 =
− 1000000 = −382 500$
0,1
a)
𝐷 ∗ 𝑟𝑑 ∗ 𝑡𝑐 400000 ∗ 0,07 ∗ 0,35
=
= 400000 ∗ 0,35 = 140 000$
𝑟𝑑
0,07
𝐴𝑃𝑉 = −242 500$
𝑃𝑉(𝑇𝑆) =
b)
Why 𝑟𝐴 ??
𝐷 ∗ 𝑟𝑑 ∗ 𝑡𝑐 400000 ∗ 0,07 ∗ 0,35
=
= 98 000$
𝑟𝐴
0,1
𝐴𝑃𝑉 = −284 000$
𝑃𝑉(𝑇𝑆) =
Exercise 4
No link between 𝑟𝑒 common stock and 𝑟𝑒
He needs to find the business risk of this project
Exercise 5
𝐸 = 24,27 𝑏$
𝐷 = 2,8 𝑏$
𝐴𝑠𝑠𝑒𝑡 = 24,27 + 2,8 = 27,07
𝛽𝐸 = 1,47
𝑡𝑐 = 0,4
𝑟𝑓 = 6,5%
𝑟𝑚𝑝 = 5,5%
a)
𝑟𝑑 = 6,5% + 0,3% = 6,8%
𝑟𝑒 = 6,5 + 1,47 ∗ 5,5 = 14,585%
𝐸
𝐷
𝑊𝐴𝐶𝐶 = ∗ 𝑟𝑒 + ∗ 𝑟𝑑 ∗ (1 − 𝑡𝑐 )
𝑉
𝑉
24,27
2,8
𝑊𝐴𝐶𝐶 =
∗ 0,14585 +
∗ 0,068 ∗ (1 − 0,4) = 13,4984%
27,07
27,07
b)
𝐷
= 0,3
𝐸
𝐷
𝐷
0,3
=
=
= 23,08%
𝐷 + 𝐸 1,3𝐸 1,3
𝐸
= 76,92%
𝑉
𝑟𝑑 = 6,5𝑀 + 2% = 8,5%
24,27
2,8
∗ 0,14585 +
∗ 0,068 = 13,7798%
27,07
27,07
𝑟𝑒 = 0,137798 + (0,137798 − 0,085) ∗ 0,3 = 15,3637%
𝑟𝑢 = 𝑟𝐴 =
52
Maths
Exercises
𝑊𝐴𝐶𝐶 = 0,152637 ∗ 0,7692 + 0,085 ∗ 0,2308 ∗ 0,6 = 12,9948%
Note: without tax shield we have : 𝑟𝐴 = 0,152637 ∗ 0,7692 + 0,085 ∗ 0,2308 = 13,7798% = 𝑟𝑢 !
c)
𝐹𝐶𝐹1
𝑊𝐴𝐶𝐶𝐴 − 𝑔
𝐹𝐶𝐹1
𝑉𝐿𝐵 =
𝑊𝐴𝐶𝐶𝐵 − 𝑔
𝑉𝐿𝐴 ∗ (𝑊𝐴𝐶𝐶𝐴 − 𝑔) = 𝑉𝐿𝐵 ∗ (𝑊𝐴𝐶𝐶𝐵 − 𝑔)
(𝑊𝐴𝐶𝐶𝐴 − 𝑔)
(𝑊𝐴𝐶𝐶𝐵 − 𝑔)
𝑉𝐿𝐵 − 𝑉𝐿𝐴 = 𝑉𝐿𝐴 ∗
− 𝑉𝐿𝐵 ∗
(𝑊𝐴𝐶𝐶𝐵 − 𝑔)
(𝑊𝐴𝐶𝐶𝐴 − 𝑔)
𝑊𝐴𝐶𝐶
−
𝑔
13,4984
−6
𝐴
=? 𝑉𝐿𝐴 ∗
− 1 = 24,27 ∗
−1=
𝑊𝐴𝐶𝐶𝐵 − 𝑔
12,9948 − 6
= 1,9485 𝑏$
𝑉𝐿𝐴 =
Stock price increases =
1,9485
24,27
= 8,0285%
Exercise 6
𝐷 = 527 𝑀$
𝐸 = 1,76 𝑏$
𝑉 = 2287 𝑀$
𝐸𝐵𝐼𝑇 = 131 𝑀$
𝑡𝑐 = 36%
𝑟𝑑 = 8%
𝑃(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) = 2,3%
𝑃𝑉(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) = 30%
𝑟𝑓 = 6%
a)
𝑉𝑢 = 𝑉𝐿 − 𝑃𝑉(𝑇𝑆) + 𝑃𝑉(𝐸(𝑑𝑒𝑓𝑎𝑢𝑙𝑡 𝑐𝑜𝑠𝑡𝑠))
𝑉𝑈 = 2287 − 527 ∗ 0,36 + 0,3 ∗ 0,023 ∗ (2287 − 527 ∗ 0,36)
𝑉𝑈 = 2111,7512 𝑀$
b)
𝐷
= 0,5
𝐸
𝐷
0,5 1
=
=
𝐷 + 𝐸 1,5 3
𝑉𝐿2 = 𝑉𝑈 + 𝑃𝑉(𝑇𝑆) − 𝑃𝑉(𝑑𝑒𝑓𝑎 𝑐𝑜𝑠𝑡)
1
1
𝑉𝐿2 = 2111,7512 + 0,36 ∗ ∗ 𝑉𝐿2 − 0,3 ∗ 0,4661 ∗ (𝑉𝐿2 − ∗ 𝑉𝐿2 ∗ 0,36)
3
3
𝑉𝐿2 = 2105,3 𝑀$
c)
No, you lose value
Exercise 7
EPS
Price per share
P/E ratio
Number of shares
Aldaris Corp
2$
40$
20
100000
Cesu Corp
2,5$
25$
10
200000
53
Merged firm
2,67$
34,3286
12,8572
262172,2846
Maths
Exercises
Total earnings
200000$
200000 + 500000
= 700000$
4 + 5 = 9𝑀$
500000$
Total market value
4M$
5M$
No economic gain, so total value= 4 + 5 = 9𝑀$
700000
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒 =
= 262172,2846
2,67
9𝑀
𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 =
= 34,3286$
262172,2846
𝑃
34,3286
𝑟𝑎𝑡𝑖𝑜 =
= 12,8572
𝐸
2,67
𝑖𝑠𝑠𝑢𝑒 𝑜𝑓 𝑠ℎ𝑎𝑟𝑒 ∶ 262172,2846 − 1000000 = 162172,2846
𝑐𝑜𝑠𝑡 = 162172,2846 ∗ 34,3286 = 5 567 147,49$
It’s not interesting, you pay 5,5M instead of 5M, you lose 567 147,49$
Exercise 8
a)
𝑡𝑐 = 0,5
𝐹𝐶𝐹𝐽𝑢𝑏 = 2000 ∗ (1 − 0,5) = 1000$
𝐹𝐶𝐹𝑆𝑒𝑛 = 1600 ∗ (1 − 0,5) = 800$
𝐹𝐶𝐹0 ∗ (1 + 𝑔) 1000 ∗ 1,04
𝑉𝐽𝑢𝑏 =
=
= 20800$
𝑊𝐴𝐶𝐶 − 𝑔
0,09 − 0,04
800 ∗ 1,06
𝑉𝑆𝑒𝑛 =
= 21200$
0,1 − 0,06
𝑉 = 20800 + 21200 = 42000$
Rev
-cogs
EBIT
g
No synergy
12000
−8400
3600
20,8
21,2
𝑔=
∗ 0,04 +
∗ 0,06
42
42
= 5,0095%
𝑊𝐴𝐶𝐶 = 9,5048%
Synergy 1
12000
−0,65 ∗ 12000
4200
𝑆𝑎𝑚𝑒
WACC
𝑆𝑎𝑚𝑒
8400
= 0,7
12000
𝑛𝑒𝑤 = 0,65
12000 − 0,65 ∗ 12000 = 4200
4200 ∗ (1 − 0,5) ∗ (1,050095)
𝑉𝑠𝑦𝑛1 =
− 42000 = 7055,6693$
0,095048 − 0,050095
3600 ∗ (1 − 0,5) ∗ 1,06
𝑉𝑠𝑦𝑛2 =
− 42000 = 12439,6257$
0,095048 − 0,06
Synergy 2
12000
−8400
3600
6%
9,5048%
Exercise 9
??
𝑟𝑒𝑣0 = 300 𝑀$
𝑊𝐴𝐶𝐶 = 9,5%
𝑡𝑐 = 0,3
𝑟𝑓 = 5,5%
𝑔 = 5%
𝑟𝑑 = 8%
𝑟𝑒 = 13%
Date
Rev
0
1
300 ∗ 1,05 ∗ 0,06
54
2
300 ∗ 1,052 ∗ 0,06
3
…
Maths
Exercises
−40𝑀
−40
18 ∗ 1,05
−40 ∗ 0,7
18 ∗ 1,05 ∗ 0,7
18 ∗ 0,7 ∗ 1,05
= −40 ∗ 0,7 +
= 137,3750 𝑀$
0,13 − 0,05
Cost
Ebit
With tax
𝑁𝑃𝑉𝑠𝑦𝑛
Exercise 10
𝑃𝐴
0,01
10
15
20
30
40
50
𝐸
𝑅
28
:1
0,01
28
:1
10
28
:1
15
1,4 ∶ 1
1,4 ∶ 1
1,4 ∶ 1
56
:1
50
55
2
18 ∗ 1,05
18 ∗ 1,052 ∗ 0,7
…
…
…
𝑛𝑜𝑠ℎ
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑖𝑑 𝑝𝑒𝑟 𝑡𝑎𝑟𝑔𝑒𝑡 𝑠ℎ𝑎𝑟𝑒
2800
28
2,8
28
1,87
28
1,4
1,4
1,4
1,12
28
42
56
56