Exercises
Financial Economics Maths Exercises Problem Set 1 Problem 1 a) 8000 ∗ 1,053 = 9261€ 8000 ∗ 1,0513 = 15085,15€ 32000 = 1,05𝑛𝑛 8000 32000 𝑛𝑛𝑛𝑛𝑛𝑛1,05 = ln � � 8000 𝑛𝑛 = 28,4𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦
b) 𝑥𝑥 ∗ 1,055 = 50000 => 𝑥𝑥 = 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 = 10,47% 𝐸𝐸𝐸𝐸𝐸𝐸 �1 + 12 10% 0,1 365 �1 + � − 1 = 10,52% 𝐸𝐸𝐸𝐸𝐸𝐸 365 e)
𝑥𝑥 2 5% = �1 + � − 1 => 𝑥𝑥 = 4,939% 2 𝑥𝑥 12 5% = �1 + � − 1 => 𝑥𝑥 = 4,889% 12 f) 1000 ∗ 1,0510 = 1628,89€
0,05 12 � − 1 = 5,116% 12 1000 ∗ 1,0511610 = 1646,98€
�1 +
0,05 𝑛𝑛 � − 1 = 𝑒𝑒 0,05 − 1 = 5,127% lim �1 + 𝑛𝑛→∞ 𝑛𝑛 1000 ∗ 1,0512710 = 1648,7€ Problem 2 a)
𝑁𝑁𝑁𝑁𝑁𝑁 = −10000 +
500 1500 10000 + + = −2609,36 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 1,06 1,062 1,0610
Maths
Exercises
𝑁𝑁𝑁𝑁𝑁𝑁 = −10000 + b)
𝐴𝐴: 𝑁𝑁𝑁𝑁𝑁𝑁 = −10 +
500 1500 10000 + + = 135,43 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 1,02 1,022 1,0210
20 = 8,18 1,1
5 = 9,54 1,1 10 𝐶𝐶: 𝑁𝑁𝑁𝑁𝑁𝑁 = 20 − = 10,9 1,1 =>C =>B,C 𝐵𝐵: 𝑁𝑁𝑁𝑁𝑁𝑁 = 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 = 12 993,1242€ 1200 ∗ 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 ∗ b)
1,06 −12 −1,06 −30 0,06
∗ 1,0612 = 12 993,1242€
1,0517 − 1,117 ∗ 1,1−17 1,05 − 1,1 𝑥𝑥 = 21 861 455,8€
𝑥𝑥 = 2 000 000 ∗ Problem 5 a)
350 000 − 50 000 = 𝑎𝑎 ∗ 𝑎𝑎 =
1 − 1,07−36 0,07
300 000 ∗ 0,07 => 𝑎𝑎 = 24 175,92€ 1 − 1,07−30
300 000 = 23 500 ∗ 𝑥𝑥 = 63 848,0342€
𝑥𝑥 1 − 1,07−30 + 0,07 1,0730
2
Maths b)
Exercises
2 000 000 = 𝑎𝑎 ∗
1,0536 − 1 0,05
2 000 000 = 𝑎𝑎 ∗
1,0736 − 1,0536 => 𝑎𝑎 = 7102,1138€ 1,07 − 1,05
𝑎𝑎 = 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.
c)
1 − 1,05−𝑛𝑛 > 200000 0,05 200000 ∗ 0,05 − 1� 1,05−𝑛𝑛 < − � 25000 200000 ∗ 0,05 − 1�) ln(− � 25000 −𝑛𝑛 < ln 1,05 𝑛𝑛 > 10,47 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 (11) 25000 ∗
d)
(1,02𝑛𝑛 − 1,08𝑛𝑛 ) 1,35 − 1,085 ∗ 1,08−𝑛𝑛 ∗ 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 = 1 294 767,23€ 100 000 ∗ 0,07 1,0735 − 1 1 294 767,23 = 𝑎𝑎 ∗ 0,07 𝑎𝑎 = 9366,29€
𝑥𝑥 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 1,0235 − 1,0735 𝑥𝑥 ∗ 75000 ∗ = 1 294 767,23€ 1,02 − 1,07 𝑥𝑥 = 9,948% Must find n 1000000 − 50000 ∗ (1,05𝑛𝑛 ) > 0 𝑛𝑛 = 62
1 − 1,06−𝑛𝑛 1,05𝑛𝑛 − 1,06𝑛𝑛 −𝑛𝑛 ∗ 1,06 + 1 000 000 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁 = −10 000 000 − 50 000 ∗ 1,05 − 1,06 0,06 With 𝑛𝑛 = 62 𝑁𝑁𝑁𝑁𝑁𝑁 = 3 995 073,97
3
Maths
Exercises
Problem Set 2 Problem 1 a) First find 𝑖𝑖1 100 = 94 1 + 𝑖𝑖1 𝑖𝑖1 = 6,3830% 𝑖𝑖2
100 = 85 (1 + 𝑖𝑖2 )2 𝑖𝑖2 = 8,4652%
𝑃𝑃 =
100 1,063830
𝑃𝑃 =
1,063830 50 1,063830
+
100 1,084652 2
= 179
+
1,084652 100 1,084652 2
= 132
We can also do 94+85… 100 500 𝑃𝑃 = + 2 = 519 There an arbitrage opportunity (>130) you buy.
b) Case 1 Asset 1 𝑃𝑃1 = 0,5 = 1 − 0,5 𝑃𝑃2 = 3 > 5 − 2,5 Asset 2 is too expensive We sell it / buy Asset 1 Case 2 Asset 1 𝑃𝑃1 = 0,5 =
𝑖𝑖 = 224% ? 𝑃𝑃2 = 2,5 =
𝑖𝑖 = 169% ?
1 2 + 1 + 𝑖𝑖 (1 + 𝑖𝑖)2 3 10 + 1 + 𝑖𝑖 (1 + 𝑖𝑖)2
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 + 𝑖𝑖 𝑖𝑖 = 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 𝜎𝜎𝑝𝑝 = �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: 𝑥𝑥 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝐴𝐴, 𝑦𝑦 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝐵𝐵 min 𝜎𝜎𝑝𝑝 = 𝑥𝑥² ∗ 0,2² + 𝑦𝑦² ∗ 0,25² − 2 ∗ 0,4 ∗ 𝑥𝑥 ∗ 𝑦𝑦 ∗ 0,2 ∗ 0,25 𝑥𝑥 + 𝑦𝑦 = 1 So min 𝜎𝜎𝑝𝑝 = 𝑥𝑥² ∗ 0,2² + (1 − 𝑥𝑥)2 ∗ 0,252 − 2 ∗ 0,4 ∗ 𝑥𝑥 ∗ (1 − 𝑥𝑥) ∗ 0,2 ∗ 0,25 = 0,1425𝑥𝑥² − 0,165𝑥𝑥 + 0,0625 min′ 𝜎𝜎𝑝𝑝 = 0,285𝑥𝑥 − 0,165 Minimum for 𝑥𝑥 = 0,5789 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ∶ 𝑥𝑥 = 0,5789, 𝑦𝑦 = 0,4211, 𝐸𝐸 = 17,9%, 𝜎𝜎 = 12,14% d)
⎡ ⎢ 𝐴𝐴 � 4 × 4 ⎢ 𝐵𝐵 ⎢ 𝐶𝐶 ⎣ 𝐷𝐷
𝐴𝐴 𝐵𝐵 𝐶𝐶 𝐷𝐷 ⎤ 10 −10 5 12 ⎥ −10 15 10 −5⎥ 5 10 20 0 ⎥ 12 −5 0 12 ⎦
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 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 𝜎𝜎 = 2,12% 1,3 𝛽𝛽𝐴𝐴 = = 0,2892 4,495 4,75 𝛽𝛽𝐵𝐵 = = 1,0567 4,495 9,25 𝛽𝛽𝐶𝐶 = = 2,0578 4,495 2,8 = 0,6229 𝛽𝛽𝐷𝐷 = 4,495 The proportion of 𝐴𝐴 is 0,2892 ∗ 0,25 = 7,23%, 𝑒𝑒𝑒𝑒𝑒𝑒 … 5
Maths
Exercises
Problem 4 a) Maximize expected returns : security 2 Minimize risk : security 1 b) 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 1 So 100% security 1 c) 𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑥𝑥² ∗ 0,05² + 𝑦𝑦² ∗ 0,08² − 2 ∗ 𝑥𝑥𝑥𝑥 ∗ 0,05 ∗ 0,08 = 𝑥𝑥² ∗ 0,0025 + 𝑦𝑦² ∗ 0,0064 − 0,0080 ∗ 𝑥𝑥𝑥𝑥 𝑥𝑥 + 𝑦𝑦 = 1 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 25𝑥𝑥² + 64(𝑥𝑥 2 − 2𝑥𝑥 + 1) − 80 ∗ (𝑥𝑥 − 𝑥𝑥 2 ) = 169𝑥𝑥² − 208𝑥𝑥 + 64 = 0 2082 − 4 ∗ 169 ∗ 64 = 0 𝑏𝑏 208 𝑥𝑥 = − = = 0,6154 2𝑎𝑎 2 ∗ 169 𝑥𝑥 = 61,54% 𝑦𝑦 = 38,46% d) 𝐸𝐸 = 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, 𝑆𝑆𝑆𝑆 = 1 + 2 + 3 … + 99 𝑆𝑆𝑆𝑆 = 99 + 98 + ⋯ + 1 2 ∗ 𝑆𝑆𝑆𝑆 = 100 ∗ 99 9900 = 4950 𝑆𝑆𝑆𝑆 = 2 b)
1 2 … 𝑛𝑛 ⎡ ⎤ 1 0,3² = 0,09 0,036 … 0,036 ⎢ ⎥ … 0,036⎥ ⎢ 2 0,3² ∗ 0,4 = 0,036 0,09 … … … 0,036⎥ ⎢… ⎣ 𝑛𝑛 0,036 0,036 0,036 0,09 ⎦ 𝑥𝑥1 = 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ𝑡𝑡 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 1 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑥𝑥2 = ⋯ 𝑥𝑥100 = 0,01 𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃 = 𝑥𝑥1 ∗ (𝑥𝑥1 ∗ 0,09 + 𝑥𝑥2 ∗ 0,036 … + 𝑥𝑥100 ∗ 0,036) + 𝑥𝑥2 ∗ (𝑥𝑥1 ∗ 0,036 + 𝑥𝑥2 ∗ 0,09 … + 𝑥𝑥100 ∗ 0,036) + ⋯ + 𝑥𝑥100 ∗ (𝑥𝑥1 ∗ 0,036 + 𝑥𝑥2 ∗ 0,036 … + 𝑥𝑥100 ∗ 0,09) 𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃 = 0,01 ∗ 0,09 + 0,01 ∗ 0,036 … + 0,01 ∗ 0,036 𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃 = 0,01 ∗ 0,09 + 0,99 ∗ 0,036 = 0,03654 𝜎𝜎𝜎𝜎 = 19,1154% c)
𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃 =
1 1 ∗ 0,09 + �1 − � ∗ 0,036 𝑛𝑛 𝑛𝑛
1 ∗ 0,09 + �1 − � ∗ 0,036 = 0,036 𝑛𝑛 𝜎𝜎𝜎𝜎 = 18,9737% lim
1
𝑛𝑛→∞ 𝑛𝑛
6
Maths
Exercises
Problem 6 a) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0,5² ∗ 0,627² + 0,5² ∗ 0,507² + 2 ∗ 0,5 ∗ 0,5 ∗ 0,66 ∗ 0,627 ∗ 0,507 = 0,267448 𝜎𝜎𝑃𝑃 = 51,7154%
b) Correlation TBill with asset =0 1 1 1 1 1 𝜎𝜎𝑃𝑃 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 𝜎𝜎𝑃𝑃 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 𝜎𝜎𝑃𝑃 = 34,4769% c) 0,517 ∗ 2 = 1,034 103,4%
d) 𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽 = 2,21 𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽 = 1,81 𝜎𝜎𝑃𝑃 ² = 15% 𝜎𝜎𝑝𝑝𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 0,15 ∗ 2,21 = 33,15% 𝜎𝜎𝑝𝑝𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 0,15 ∗ 1,81 = 27,15%
Problem 7 a) 𝐸𝐸(𝐴𝐴1) = 0,5 ∗ 0,08 + 0,3 ∗ −0,02 + 0,2 ∗ 0,12 = 5,8% 𝐸𝐸(𝐴𝐴2) = 0,5 ∗ −0,05 + 0,3 ∗ 0,14 + 0,2 ∗ 0,09 = 3,5%
b) 𝑉𝑉𝑉𝑉𝑉𝑉(𝐴𝐴1) = (8 − 5,8)2 ∗ 0,5 + (−2 − 5,8)2 ∗ 0,3 + (12 − 5,8)2 ∗ 0,2 = 28,36 𝑉𝑉𝑉𝑉𝑉𝑉(𝐴𝐴2) = (−5 − 3,5)² ∗ 0,5 + (14 − 3,5)2 ∗ 0,3 + (9 − 3,5)2 ∗ 0,2 = 75,25
c) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐1𝐴𝐴2 = (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 = −0,586628 𝜌𝜌 = − √28,36 ∗ �75,25 d) 5,8 + 3,5 = 4,65% 2 2 2 𝜎𝜎𝑃𝑃 = 0,5 ∗ 28,36 + 0,52 ∗ 75,25 + 2 ∗ 0,52 ∗ (−27,1) = 12,3525% 𝜎𝜎𝑝𝑝 = 3,515%
7
Maths
Exercises
Problem Set 3 Problem 1 a) b) Find 𝐸𝐸(3) 𝑤𝑤𝑤𝑤 ℎ𝑎𝑎𝑎𝑎𝑎𝑎: 𝛽𝛽 = 1,8 ∶ 𝑥𝑥 = 12% 𝛽𝛽 = 0,8 ∶ 𝑥𝑥 = 7% 12−7 So the slope is : = 0,05 1,8−0,8
Assume 𝛽𝛽 = 0,8 𝑖𝑖𝑖𝑖 𝛽𝛽 ∗ = 0 Then 𝛽𝛽 = 1,2 𝑖𝑖𝑖𝑖 𝛽𝛽 ∗ = 0,4 So 𝐸𝐸 = 0,05𝛽𝛽 ∗ + 0,07 𝐸𝐸(3) = 9%
OR 𝑟𝑟𝑖𝑖 = 𝑟𝑟𝑓𝑓 + 𝛽𝛽𝑖𝑖 ∗ (𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 = 𝑟𝑟𝑓𝑓 + 1.8 ∗ �𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 � = 12 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 = 𝑟𝑟𝑓𝑓 + 0.8 ∗ �𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 � = 8 𝑟𝑟𝑚𝑚 = 8%, 𝑟𝑟𝑓𝑓 = 3% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 3 = 3% + 1,2(8% − 3%) = 9%
c) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 4 = 3% + 2(8% − 3%) = 13% Real is 16%, price is underpriced (too low = too much return) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 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 𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑟𝑟𝑖𝑖 = 𝑟𝑟𝑓𝑓 + 𝛽𝛽𝑖𝑖 ∗ (𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ) 𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑟𝑟𝑖𝑖 = 𝑟𝑟𝑓𝑓 + 𝜎𝜎𝑖𝑖 ∗ 𝜎𝜎𝑚𝑚
a) 𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑟𝑟𝑖𝑖 = 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 𝜎𝜎𝑖𝑖 = 𝜎𝜎𝑀𝑀 0,15 − 0,06 𝐶𝐶𝐶𝐶𝐶𝐶 = 𝑟𝑟𝑖𝑖 = 0,06 + 0,15 ∗ = 0,15 0,15 c)
𝛽𝛽 =
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 𝐵𝐵𝐵𝐵 = 𝐶𝐶𝐶𝐶 − 𝐶𝐶𝐶𝐶 = 𝜎𝜎𝐴𝐴 − 𝜎𝜎𝐵𝐵 CML 𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ∗ 𝜎𝜎𝐵𝐵 𝑟𝑟𝐵𝐵 = 𝑟𝑟𝑓𝑓 + 𝜎𝜎𝑀𝑀 SML 𝑟𝑟𝐴𝐴 = 𝑟𝑟𝑓𝑓 + (𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ) ∗ 𝛽𝛽𝐴𝐴 𝑟𝑟𝐴𝐴 = 𝑟𝑟𝐵𝐵 => Lecture 4 slide 23 𝐴𝐴𝐴𝐴 = 𝜎𝜎𝐴𝐴 − 𝜎𝜎𝐵𝐵 = 𝜎𝜎𝐴𝐴 − 𝛽𝛽𝐴𝐴 𝜎𝜎𝑀𝑀 = 50 − 2 ∗ 15 = 20%
Problem 3 a)
𝐷𝐷 𝐸𝐸 + 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 0 because it’s risk free. So we have 𝐸𝐸 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 𝐷𝐷 + 𝐸𝐸 Competitors Estimated 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∗
Rimi foods Sony Electronics Dow chemicals b)
0,8 1,6 1,2
𝐷𝐷 𝐷𝐷 + 𝐸𝐸 0,3 0,2 0,4
𝐷𝐷 𝐸𝐸 + 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 0,6 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,56 ∗ 0,5 + 1,28 ∗ 0,3 + 0,72 ∗ 0,2 = 0,808 0,808 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = = 1,35 0,6
𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∗
c) 𝑟𝑟𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 0,07 + 0,56 ∗ (0,15 − 0,07) = 11,48% 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,07 + 1,28 ∗ (0,15 − 0,07) = 17,24% 𝑟𝑟𝑐𝑐ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,07 + 0,72 ∗ (0,15 − 0,07) = 12,76% 𝑟𝑟𝐿𝐿 = 0,07 + 0,808 ∗ (0,15 − 0,07) = 13,464% d) 𝛽𝛽𝐴𝐴𝐴𝐴 = 0,2 ∗ 0,3 + 0,8 ∗ 0,7 = 0,62 𝛽𝛽𝐴𝐴𝐴𝐴 = 0,2 ∗ 0,2 + 1,6 ∗ 0,8 = 1,32 𝛽𝛽𝐴𝐴𝐴𝐴 = 0,2 ∗ 0,4 + 1,2 ∗ 0,6 = 0,8
𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,62 ∗ 0,5 + 1,32 ∗ 0,3 + 0,8 ∗ 0,2 = 0,866 𝑟𝑟𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 0,07 + 0,62 ∗ (0,15 − 0,07) = 11,96%
9
𝐸𝐸 𝐷𝐷 + 𝐸𝐸 0,7 0,8 0,6
𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
0,8 ∗ 0,7 = 0,56 1,6 ∗ 0,8 = 1,28 1,2 ∗ 0,6 = 0,72
Maths
Exercises
𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,07 + 1,32 ∗ (0,15 − 0,07) = 17,56% 𝑟𝑟𝑐𝑐ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,07 + 0,8 ∗ (0,15 − 0,07) = 13,4% 𝑟𝑟𝐿𝐿 = 0,07 + 0,866 ∗ (0,15 − 0,07) = 13,298%
Problem 4 𝑟𝑟𝑓𝑓 = 2% 𝑟𝑟𝑚𝑚 = 8% + 2% = 10% 𝜎𝜎𝑚𝑚 = 𝜎𝜎𝑆𝑆𝑆𝑆 = 20%
a) �𝑟𝑟𝑡𝑡 − 𝑟𝑟𝑓𝑓 � = 𝛼𝛼 + 𝛽𝛽�𝑟𝑟𝑚𝑚𝑚𝑚 − 𝑟𝑟𝑓𝑓 � + 𝜀𝜀 𝑟𝑟𝑡𝑡 = 0 + 1 ∗ 0,08 + 0,02 + 0 = 0,10 = 10% b)
0,10 − 0,02 = 0,4 0,2 10 − 2 = = 0,2828 √800
𝑆𝑆𝑆𝑆&𝑃𝑃 = 𝑆𝑆𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
𝜎𝜎𝑖𝑖 ² =
2 2 𝛽𝛽𝑖𝑖𝑖𝑖 202 𝜎𝜎𝑚𝑚2 𝛽𝛽𝑖𝑖𝑖𝑖 𝜎𝜎𝑚𝑚2 = = 1 ∗ = 800 2 0,5 𝑅𝑅² 𝜌𝜌𝑖𝑖𝑖𝑖
c) The IUEF is inefficient so we don’t take it. 𝐸𝐸(𝑃𝑃) = 8% = 𝑥𝑥 ∗ 0,02 + 𝑦𝑦 ∗ 0,10 𝑥𝑥 + 𝑦𝑦 = 1 𝑥𝑥 = 0,25 𝑦𝑦 = 0,75 d) Now we take IUEF
𝑟𝑟𝑡𝑡 = 0,02 + 1 ∗ 0,08 + 0,02 = 12% 𝐸𝐸(𝑃𝑃) = 𝑥𝑥 ∗ 0,02 + 𝑦𝑦 ∗ 0,10 + 𝑧𝑧 ∗ 0,12 = 0,08 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 = 1 We’ve 𝑥𝑥 ∗ 2 + 𝑦𝑦 ∗ 10 + (1 − 𝑥𝑥 − 𝑦𝑦) ∗ 12 = 8 � 𝑧𝑧 = 1 − 𝑥𝑥 − 𝑦𝑦 −𝑥𝑥 ∗ 10 − 𝑦𝑦 ∗ 2 + 12 = 8 � 𝑧𝑧 = 1 − 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 = 0,4 − 0,2𝑦𝑦 � 𝑧𝑧 = 0,6 − 0,8𝑦𝑦
20 202 + 𝑦𝑦𝑦𝑦 ∗ �0,5 ∗ 20 ∗ = 400𝑦𝑦² + 800𝑧𝑧² + 800𝑦𝑦𝑦𝑦 0,5 �0,5 𝑉𝑉𝑉𝑉𝑉𝑉 = 400𝑦𝑦² + 800 ∗ (0,6 − 0,8𝑦𝑦)2 + 800 ∗ 𝑦𝑦 ∗ (0,6 − 0,8𝑦𝑦) = 400𝑦𝑦² + 800 ∗ (0,82 𝑦𝑦 2 − 0,8 ∗ 2 ∗ 0,6𝑦𝑦 + 0,62 ) + 800 ∗ 0,6𝑦𝑦 − 800 ∗ 0,8𝑦𝑦² = 272𝑦𝑦² − 288𝑦𝑦 + 288 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 544𝑦𝑦 − 288 𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑦𝑦² ∗ 20² + 𝑧𝑧² ∗
𝑥𝑥 = 29,40% 𝑦𝑦 = 52,95% 𝑧𝑧 = 17,65%
10
Maths
Exercises
𝑟𝑟𝑚𝑚∗ = 10,4998%
Other method 2 2 𝜎𝜎𝑝𝑝2 = 𝑦𝑦² ∗ 𝜎𝜎𝑀𝑀 + 𝑧𝑧² ∗ 𝜎𝜎𝐼𝐼𝐼𝐼 + 2 ∗ 𝑦𝑦 ∗ 𝑧𝑧 ∗ 𝜎𝜎𝐼𝐼𝐼𝐼,𝑀𝑀 = 𝑦𝑦² ∗ 400 + 𝑧𝑧² ∗ 800 + 2 ∗ 𝑦𝑦 ∗ 𝑧𝑧 ∗ 400 𝜎𝜎 𝑖𝑖𝑖𝑖 (𝛽𝛽𝑖𝑖𝑖𝑖 = 2 => 𝜎𝜎𝐼𝐼𝐼𝐼,𝑚𝑚 = 𝛽𝛽𝐼𝐼𝐼𝐼𝐼𝐼 𝜎𝜎𝑚𝑚2 = 400) 𝜎𝜎𝑚𝑚
𝐿𝐿(𝑦𝑦, 𝑧𝑧, 𝛾𝛾) = 𝑦𝑦² ∗ 400 + 𝑧𝑧² ∗ 800 + 𝑦𝑦 ∗ 𝑧𝑧 ∗ 400 + 𝛾𝛾(8 ∗ 𝑦𝑦 + 10𝑧𝑧 − 6) 𝜕𝜕𝜕𝜕(𝑦𝑦, 𝑧𝑧, 𝛾𝛾) = 800𝑦𝑦 + 800𝑧𝑧 + 8𝛾𝛾 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕(𝑦𝑦, 𝑧𝑧, 𝛾𝛾) = 1600𝑧𝑧 + 800𝑦𝑦 + 10𝛾𝛾 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕(𝑦𝑦, 𝑧𝑧, 𝛾𝛾) = 8𝑦𝑦 + 10𝑧𝑧 − 6 𝜕𝜕𝜕𝜕 Then 𝑦𝑦 = 0,5294 𝑧𝑧 = 0,1765 𝑠𝑠𝑠𝑠 𝑥𝑥 = 0,2941 Problem 5 a) 1,3 ∗ 𝑥𝑥 + 0,9 ∗ 𝑦𝑦 = 1 𝑥𝑥 + 𝑦𝑦 = 1 1,3𝑥𝑥 + 0,9 − 0,9𝑥𝑥 = 1 𝑥𝑥 = 0,25 𝑦𝑦 = 0,75
b) on a 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀 = 𝑥𝑥1 (𝑥𝑥1 ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀1 + 𝑥𝑥2 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀1 𝜀𝜀2 … + 𝑥𝑥𝑛𝑛 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀1 𝜀𝜀𝑛𝑛 ) + 𝑥𝑥2 (𝑥𝑥1 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀2 𝜀𝜀1 + 𝑥𝑥2 ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀2 … + 𝑥𝑥𝑛𝑛 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀2 𝜀𝜀𝑛𝑛 ) + ⋯ + 𝑥𝑥𝑛𝑛 (𝑥𝑥1 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀𝑛𝑛 𝜀𝜀1 + 𝑥𝑥2 ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀𝑛𝑛 𝜀𝜀2 … + 𝑥𝑥𝑛𝑛 ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀𝑛𝑛 ) 1 𝑥𝑥1 = 𝑥𝑥2 = ⋯ = 𝑥𝑥𝑛𝑛 = 𝑥𝑥 = 𝑛𝑛 𝜌𝜌 = 0 So 1 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀 = ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀𝑖𝑖 𝑛𝑛 1 lim ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀𝑖𝑖 = 0 𝑛𝑛→∞ 𝑛𝑛 c) ∝≠ 0 so the CAPM doesn’t hold d)
1 1 ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀𝑖𝑖 + �1 − � ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀𝑖𝑖 𝜀𝜀𝑗𝑗 𝑛𝑛 𝑛𝑛 1 1 lim ∗ 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀𝑖𝑖 + �1 − � ∗ 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀𝑖𝑖 𝜀𝜀𝑗𝑗 = 𝐶𝐶𝐶𝐶𝐶𝐶 𝜀𝜀𝑖𝑖 𝜀𝜀𝑗𝑗 𝑛𝑛→∞ 𝑛𝑛 𝑛𝑛 𝑉𝑉𝑉𝑉𝑉𝑉 𝜀𝜀 =
Problem 6 a) 𝑋𝑋 = 1,75 ∗ 0,04 + 0,25 ∗ 0,08 = 0,09 = 9% 𝑌𝑌 = −1 ∗ 0,04 + 2 ∗ 0,08 = 12% 𝑍𝑍 = 2 ∗ 0,04 + 1 ∗ 0,08 = 16%
(= 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 𝜀𝜀𝑖𝑖 )
11
Maths
Exercises
b) 80 + 60 − 40 = 100 60 40 80 ∗ 1,75 − −2∗ =0 100 100 100 60 40 80 ∗ 0,25 + ∗2−1∗ =1 100 100 100 𝑟𝑟𝑝𝑝 = 0 ∗ 0,04 + 1 ∗ 0,08 = 0,08
c) 1600 + 20 − 80 = 1540 20 80 1600 ∗1− −2∗ =1 1540 1540 1540 20 80 1600 ∗ 0,25 + ∗2−1∗ =0 1540 1540 1540 𝑟𝑟𝑝𝑝 = 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 60
Buy Put return
40 20
Portfolia Return
0 0 20 40 60 80 100120140
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
0
-50
20
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 -50
Portfolio return 0
25
50
75
100
125
-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
-50
0
25
50
75
100
125
Short Call E4 Portfolio Return
-100 -150
Problem 2 ASSUME always take 𝑆𝑆 + 𝑃𝑃 = 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶, 𝑖𝑖𝑖𝑖 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 a) 𝐶𝐶 = 10, 𝐸𝐸 = 110, 𝑆𝑆 = 120, 𝑡𝑡 = 1, 𝑟𝑟 = 10% (𝑆𝑆 + 𝑃𝑃 = 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶 𝑠𝑠𝑠𝑠 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 𝑆𝑆 ≤ 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶)
Real price 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶 = 110 ∗ 𝑒𝑒 −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 ∗ 𝑒𝑒 −0,10∗1 = 99,53 +10.4679
Risk free profit today
b) (𝑆𝑆 + 𝑃𝑃 ≥ 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 )
CF at the date of maturity S1E -S1 S1-110 110
110-S1 P=max[0,110-S1]
0
1
𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 = 165 ∗ 𝑒𝑒 −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 ∗ 𝑒𝑒 − 6 = 164,4509 +3,4509
CF at the date of maturity S1E +S1 0 -165
0
S1-165 C=Max[0,S1-165]
15
Maths
Exercises
c) Buy a call 100 ∗ 𝑒𝑒 −0,5∗0,10 = 95,1229$ 95,1229 − 90 = 5,1229$ +8$ = 13,123$ Problem 3 a) 300 250
235
200
200 Firm Asset
150
Loan AA Loan BB
100
100
100
Equity 35
50
35
0 0
50
100
150
200
250
300
350
𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 (𝐹𝐹𝐹𝐹 100) − 𝑃𝑃𝑃𝑃𝑃𝑃 (𝐸𝐸 = 100) 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵𝐵𝐵 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝐸𝐸 = 100) − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝐸𝐸 = 200) 𝑆𝑆𝑆𝑆𝑆𝑆 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 (𝐸𝐸 = 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 -50
0
50
100
150
200
250
-100 -150
16
300
350
Maths
Exercises
The seller is the company, the buyer is the bank (who sell the credit) 𝑃𝑃𝑃𝑃(𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙) = 𝑃𝑃𝑃𝑃(𝐹𝐹𝐹𝐹 = 100) − 𝑃𝑃(𝐸𝐸 = 100) 100 100 𝑃𝑃(𝐸𝐸 = 100) = 𝑃𝑃𝑃𝑃(𝐹𝐹𝐹𝐹 = 100) − 𝑃𝑃𝑃𝑃(𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴𝐴𝐴) = − = 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 -50
0
50
100
150
200
250
300
350
-100 -150
𝐶𝐶(𝐸𝐸 = 200) = 35 𝐸𝐸 + 𝐶𝐶 𝑆𝑆 + 𝑃𝑃 = 1 + 𝑖𝑖 𝑆𝑆 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆 + 𝑃𝑃(𝐸𝐸 = 100) = 𝑃𝑃𝑃𝑃(𝐹𝐹𝐹𝐹 = 100) + 𝐶𝐶(𝐸𝐸 = 100)
𝐶𝐶(𝐸𝐸 = 100) = −𝑃𝑃𝑃𝑃(𝐹𝐹𝐹𝐹 = 100) + 𝑆𝑆 + 𝑃𝑃(𝐸𝐸 = 100) = 200 + 0,4267 − 𝑃𝑃𝑃𝑃(𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿) = 107,8341 − 35 = 72,8341 Problem 4 a) 𝑢𝑢 = 1,5 𝑑𝑑 = 0,75
𝑆𝑆0 = 100 𝑆𝑆𝑢𝑢 = 150 𝑆𝑆𝑑𝑑 = 75 𝐸𝐸 = 105
𝐶𝐶𝑢𝑢 = 150 − 105 = 45 𝐶𝐶𝑑𝑑 = 0
∝∗ =
𝐶𝐶𝑢𝑢 − 𝐶𝐶𝑑𝑑 45 = = 0,6 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 100 ∗ 0,75
17
100 = 107,8341 1,08
Maths b) 𝛽𝛽 ∗ =
Exercises
𝑢𝑢𝐶𝐶𝑑𝑑 − 𝑑𝑑𝐶𝐶𝑢𝑢 1,5 ∗ 0 − 0,75 ∗ 45 = = −42,857 𝑟𝑟(𝑢𝑢 − 𝑑𝑑) 1,05 ∗ 0,75
c) 𝐶𝐶 = ∝∗ ∗ 𝑆𝑆 + 𝛽𝛽 ∗ = 0,6 ∗ 100 − 42,857 = 17,143 Or 1,05 − 0,75 𝑝𝑝 = = 0,4 1,5 − 0,75 0,4 ∗ 45 + 0,6 ∗ 0 𝐶𝐶 = = 17,143 1,05
Problem Set 5 Problem 1 𝐶𝐶𝑢𝑢 − 𝐶𝐶𝑑𝑑 ∝∗ = 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 𝑢𝑢𝐶𝐶𝑑𝑑 − 𝑑𝑑𝐶𝐶𝑢𝑢 𝛽𝛽 ∗ = 𝑟𝑟(𝑢𝑢 − 𝑑𝑑) 𝑡𝑡 = 0
50
𝑒𝑒 𝑟𝑟𝑟𝑟 −𝑑𝑑
𝑡𝑡 = 1
𝑡𝑡 = 2 72
50 ∗ 1,2 = 60
𝑒𝑒 𝑟𝑟𝑟𝑟 −𝑑𝑑 𝑢𝑢−𝑑𝑑
64,8 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 24,8
54
50 ∗ 0,9 = 45
𝑝𝑝 = in case t=1 => 𝑝𝑝 = 𝑢𝑢−𝑑𝑑 So that 1,05 − 0,9 𝑝𝑝 = = 0,5 1,2 − 0,9 1 − 𝑝𝑝 = 0,5
𝑡𝑡 = 3 86,4 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 46,4
48,6 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 8,6
40,5 =
36,45 𝐶𝐶𝑑𝑑𝑑𝑑𝑑𝑑 = 0
𝑟𝑟−𝑑𝑑 𝑢𝑢−𝑑𝑑
𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 0,5 ∗ (46,4 + 24,8) = = 33,9048 1,05 𝑟𝑟 𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 0,5 ∗ (24,8 + 8,6) = = 15,9048 𝐶𝐶𝑢𝑢𝑢𝑢 = 1,05 𝑟𝑟 𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑑𝑑𝑑𝑑𝑑𝑑 0,5 ∗ (8,6) = = 4,0952 𝐶𝐶𝑑𝑑𝑑𝑑 = 1,05 𝑟𝑟 𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑢𝑢𝑢𝑢 0,5 ∗ (33,9048 + 15,9048) = = 23,7188 𝐶𝐶𝑢𝑢 = 1,05 𝑟𝑟 𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑑𝑑𝑑𝑑 0,5 ∗ (15,9048 + 4,0952) = = 9,5238 𝐶𝐶𝑑𝑑 = 1,05 𝑟𝑟 𝑝𝑝 ∗ 𝐶𝐶𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝐶𝐶𝑑𝑑 0,5 ∗ (23,7188 + 9,5238) = = 15,8298 𝐶𝐶 = 1,05 𝑟𝑟
𝐶𝐶𝑢𝑢𝑢𝑢 =
OR
𝐶𝐶 =
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. 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 46,4 − 24,8 ∝∗𝑢𝑢𝑢𝑢 = = =1 𝑆𝑆𝑢𝑢𝑢𝑢 (𝑢𝑢 − 𝑑𝑑) 72 ∗ (1,2 − 0,9) 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 −∝∗𝑢𝑢𝑢𝑢 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 46,4 − 1 ∗ 86,4 𝛽𝛽𝑢𝑢𝑢𝑢 = = = −38,0952 𝑟𝑟 1,05 𝐶𝐶𝑢𝑢𝑢𝑢 =∝∗𝑢𝑢𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 + 𝛽𝛽𝑢𝑢𝑢𝑢 = 1 ∗ 72 − 38,0952 = 33,9048 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 =1 𝑆𝑆𝑢𝑢𝑢𝑢 (𝑢𝑢 − 𝑑𝑑) 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 −∝∗𝑢𝑢𝑢𝑢 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 24,8 − 1 ∗ 64,8 = = = −38,0952 𝑟𝑟 1,05 =∝∗𝑢𝑢𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 + 𝛽𝛽𝑢𝑢𝑢𝑢 = 1 ∗ 54 − 38,0952 = 15,9048
∝∗𝑢𝑢𝑢𝑢 =
𝛽𝛽𝑢𝑢𝑢𝑢 𝐶𝐶𝑢𝑢𝑢𝑢
𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑑𝑑𝑑𝑑𝑑𝑑 8,6 = = 0,7078 𝑆𝑆𝑑𝑑𝑑𝑑 (𝑢𝑢 − 𝑑𝑑) 40,5 ∗ (1,2 − 0,9) 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 −∝∗𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 8,6 − 0,7078 ∗ 40,5 = = = −24,5706 𝑟𝑟 1,05 = ∝∗𝑑𝑑𝑑𝑑 ∗ 𝑆𝑆𝑑𝑑𝑑𝑑 + 𝛽𝛽𝑑𝑑𝑑𝑑 = 0,7078 ∗ 40,5 − 24,5706 = 4,0953
∝∗𝑑𝑑𝑑𝑑 = 𝛽𝛽𝑑𝑑𝑑𝑑 𝐶𝐶𝑑𝑑𝑑𝑑
𝐶𝐶𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢 =1 𝑆𝑆𝑢𝑢 (𝑢𝑢 − 𝑑𝑑) 𝐶𝐶𝑢𝑢𝑢𝑢 −∝∗𝑢𝑢 𝑆𝑆𝑢𝑢𝑢𝑢 33,9048 − 1 ∗ 72 𝛽𝛽𝑢𝑢 = = = −36,2811 𝑟𝑟 1,05 𝐶𝐶𝑢𝑢 =∝∗𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢 + 𝛽𝛽𝑢𝑢 = 1 ∗ 60 − 36,2811 = 23,7189
∝∗𝑢𝑢 =
𝐶𝐶𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑑𝑑𝑑𝑑 15,9048 − 4,0953 = = 0,8748 𝑆𝑆𝑢𝑢 (𝑢𝑢 − 𝑑𝑑) 45 ∗ (1,2 − 0,9) 𝐶𝐶𝑢𝑢𝑢𝑢 −∝∗𝑢𝑢 𝑆𝑆𝑢𝑢𝑢𝑢 15,9048 − 0,8748 ∗ 54 𝛽𝛽𝑑𝑑 = = = −29,8423 𝑟𝑟 1,05 ∗ 𝐶𝐶𝑑𝑑 =∝𝑑𝑑 ∗ 𝑆𝑆𝑑𝑑 + 𝛽𝛽𝑑𝑑 = 0,8748 ∗ 45 − 29,8423 = 9,5237
∝∗𝑑𝑑 =
𝐶𝐶𝑢𝑢 − 𝐶𝐶𝑑𝑑 23,7189 − 9,5237 = = 0,9463 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 50 ∗ (1,2 − 0,9) ∗ 𝐶𝐶𝑢𝑢 −∝ ∗ 𝑆𝑆𝑢𝑢 = −31,4848 𝛽𝛽 = 𝑟𝑟 ∗ 𝐶𝐶 =∝ ∗ 𝑆𝑆 + 13 = 0,9463 − 31,4848 = 15,8302
∝∗ =
𝑡𝑡 = 0
𝑆𝑆 = 50
𝑡𝑡 = 1
𝑆𝑆𝑢𝑢 = 50 ∗ 1,2 = 60 ∝∗𝑢𝑢 = 1 𝛽𝛽𝑢𝑢 = −33,2811 𝐶𝐶𝑢𝑢 = 23,7189
50 ∗ 0,9 = 45
𝑡𝑡 = 2
𝑆𝑆𝑢𝑢𝑢𝑢 = 72 ∝∗𝑢𝑢𝑢𝑢 = 1 𝛽𝛽𝑢𝑢𝑢𝑢 = −38,0952 𝐶𝐶𝑢𝑢𝑢𝑢 = 33,9048
𝑆𝑆𝑢𝑢𝑢𝑢 = 54 ∝∗𝑢𝑢𝑢𝑢 = 1 𝛽𝛽𝑢𝑢𝑢𝑢 = −38,0952 𝐶𝐶𝑢𝑢𝑢𝑢 = 15,9048 19
𝑡𝑡 = 3 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 86,4 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 46,4
𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 64,8 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 24,8
𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 48,6
Maths
Exercises ∝∗𝑑𝑑 = 0,8748 𝛽𝛽𝑑𝑑 = −29,8423 𝐶𝐶𝑑𝑑 = 9,5237
𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 = 8,6 40,5 ∝∗𝑑𝑑𝑑𝑑 = 0,7078 𝛽𝛽𝑑𝑑𝑑𝑑 = −24,5706 𝐶𝐶𝑑𝑑𝑑𝑑 = 4,0953
𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑 = 36,45 𝐶𝐶𝑑𝑑𝑑𝑑𝑑𝑑 = 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) 𝐶𝐶𝑀𝑀 = 20, 𝐶𝐶𝑇𝑇 = 15,8302 𝑃𝑃𝑃𝑃𝑃𝑃ℎ 1 ∶ 𝑆𝑆 => 𝑑𝑑𝑑𝑑 => 𝑢𝑢𝑢𝑢𝑢𝑢 => 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑃𝑃𝑃𝑃𝑃𝑃ℎ 2 ∶ 𝑆𝑆 => 𝑑𝑑𝑑𝑑 => 𝑑𝑑𝑑𝑑𝑑𝑑 => 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Period/node 𝑡𝑡 = 0
CF in period t +20 +31,4848 −0,9463 ∗ 50 = −47,315 (−20 − 15,8302) = −4,1698 (total 0) 𝑡𝑡 = 1 Sell (0,9463 − 0,8748) 𝑎𝑎𝑎𝑎 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) 𝑎𝑎𝑎𝑎 𝑆𝑆𝑢𝑢𝑢𝑢 = 54 𝑡𝑡 = 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 𝑡𝑡 = 3 exercised +64,8 Sell a full stock at 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 64,8 −38,0944 ∗ 1,05 = −39,99912 Repay the debt off 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0! At the end we have (𝐶𝐶𝑀𝑀 − 𝐶𝐶𝑇𝑇 ) ∗ (1 + 𝑟𝑟)𝑛𝑛 = (20 − 15,8302) ∗ 1,053 = 4,8271 Period/node 𝑡𝑡 = 0 𝑡𝑡 = 1 𝑡𝑡 = 2
Strategy Write 1 call Borrow 31,4848 Buy ∝∗ of a stock at 𝑆𝑆 = 50 Deposit
Strategy Write 1 call Borrow 31,4848 Buy ∝∗ of a stock at 𝑆𝑆 = 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) 𝑎𝑎𝑎𝑎 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) 𝑎𝑎𝑎𝑎 𝑆𝑆𝑑𝑑𝑑𝑑 = 40,5 Repaid 6,6,7635 = 6,7635 −6,7635 Debt outstanding : 29,84154 ∗ 1,05 − 6,7635 = (total 0) 20
Maths
Exercises
𝑡𝑡 = 3
24,570117 The call is out of the money, won’t be exercised Sell 0,7077 at 𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑 = 36,45 Repay the debt
Problem 2 a) 𝑡𝑡 = 0
𝑡𝑡 = 1
𝑡𝑡 = 2 𝑆𝑆𝑢𝑢𝑢𝑢 = 138,11 𝐶𝐶𝑢𝑢𝑢𝑢 = 48,11
𝑆𝑆𝑢𝑢 = 117,52
𝑆𝑆 = 100
𝑆𝑆𝑢𝑢𝑢𝑢 = 104,09 𝐶𝐶𝑢𝑢𝑢𝑢 = 14,09
𝑆𝑆𝑑𝑑 = 88,57
𝑆𝑆𝑑𝑑𝑑𝑑 = 78,45 𝐶𝐶𝑑𝑑𝑑𝑑 = 0
𝑢𝑢 = 1,1752 𝑑𝑑 = 0,8857 Risk neutral valuation 𝑡𝑡
0,7073 ∗ 36,45 = 25,79931 −24,570117 ∗ 1,05 = −25,79862 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0!
1
𝑒𝑒 𝑟𝑟 𝑛𝑛 − 𝑑𝑑 𝑒𝑒 0,06∗2 − 𝑑𝑑 𝑝𝑝 = = 𝑢𝑢 − 𝑑𝑑 𝑢𝑢 − 𝑑𝑑 1
𝑒𝑒 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 = 𝐶𝐶 = 1,03042 0,4998 ∗ 48,11 + 0,5002 ∗ 14,0875 𝐶𝐶𝑢𝑢 = = 30,1744 1,0304 0,4998 ∗ 14,0875 + 0,5002 ∗ 0 𝐶𝐶𝑑𝑑 = = 6,8332 1,0304 0,4998 ∗ 30,1744 + 0,5002 ∗ 6,8332 𝐶𝐶 = = 17,9533 1,0304
𝑝𝑝 =
b)
𝑡𝑡 = 0
100 𝛼𝛼 = 0,8130 𝛽𝛽 = −66,0168 𝐶𝐶 = 15,2832 ∗
𝑡𝑡 = 1 𝐷𝐷𝐷𝐷𝐷𝐷 = 5
𝑆𝑆𝑢𝑢 = 117,52 𝑆𝑆𝑢𝑢𝑑𝑑𝑑𝑑𝑑𝑑 = 117,52 − 5 = 112,52
𝑆𝑆𝑑𝑑 = 88,57 − 5 = 83,57 𝛼𝛼𝑑𝑑∗ = 0,3354 𝛽𝛽𝑑𝑑 = −24,3803 𝐶𝐶𝑑𝑑 = 3,9833 21
𝑡𝑡 = 2
𝑆𝑆𝑢𝑢𝑢𝑢 = 132,2452 𝐶𝐶𝑢𝑢𝑢𝑢 = 42,2452 𝑆𝑆𝑢𝑢𝑢𝑢 = 99,659 𝐶𝐶𝑢𝑢𝑢𝑢 = 9,6590 𝑆𝑆𝑑𝑑𝑑𝑑 = 98,2115 𝐶𝐶𝑑𝑑𝑑𝑑 = 8,2115
𝑆𝑆𝑑𝑑𝑑𝑑 = 74,0149 𝐶𝐶𝑑𝑑𝑑𝑑 = 0
Maths
Exercises
0,4998 ∗ 42,2452 + 0,5002 ∗ 9,6590 = 25,1801 1,0304 𝐶𝐶𝑢𝑢Exercise = 𝑚𝑚𝑚𝑚𝑚𝑚{0, 𝑆𝑆𝑢𝑢 − 𝐸𝐸 } = 117,52 − 90 = 27,52 𝐶𝐶𝑢𝑢 = 𝑚𝑚𝑚𝑚𝑚𝑚 = 27,52 0,4998 ∗ 8,2115 𝐶𝐶𝑑𝑑𝑁𝑁𝑁𝑁 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = = 3,9830 1,0304 𝐶𝐶𝑑𝑑Exercise = 0 𝐶𝐶𝑑𝑑 = 𝑚𝑚𝑚𝑚𝑚𝑚 = 3,9830 0,4998 ∗ 27,52 + 0,5002 ∗ 3,9830 𝐶𝐶 𝑁𝑁𝑁𝑁 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = = 15,2822 1,0304 𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 10 𝐶𝐶𝑢𝑢No exercise =
c)
𝛼𝛼𝑢𝑢∗ =
𝐶𝐶𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢 = 𝑑𝑑𝑑𝑑𝑑𝑑 (𝑢𝑢 𝑆𝑆𝑢𝑢 ∗ − 𝑑𝑑) 𝐶𝐶𝑑𝑑𝑑𝑑 − 𝐶𝐶𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑𝑑𝑑 (𝑢𝑢 𝑆𝑆𝑑𝑑 ∗ − 𝑑𝑑) 𝐶𝐶𝑑𝑑𝑑𝑑 − 𝛼𝛼𝑑𝑑∗ ∗ 𝑆𝑆𝑑𝑑𝑑𝑑
𝑛𝑛𝑛𝑛𝑛𝑛ℎ𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑤𝑤𝑒𝑒 ′ 𝑣𝑣𝑣𝑣 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ‼
8,2215 = 0,3394 83,57 ∗ (1,1152 − 0,8857) 8,2115 − 0,3394 ∗ 98,2215 = = −24,3803 𝛽𝛽𝑑𝑑 = 𝑟𝑟 1,0304 𝐶𝐶𝑑𝑑 = 𝛼𝛼𝑑𝑑∗ ∗ 𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝛽𝛽𝑑𝑑 = 0,3394 ∗ 83,57 − 24,3803 = 3,9833
𝛼𝛼𝑑𝑑∗ =
𝐶𝐶𝑢𝑢 − 𝐶𝐶𝑑𝑑 27,52 − 3,9833 = = 0,8130 𝑆𝑆 ∗ (𝑢𝑢 − 𝑑𝑑) 100 ∗ (1,1752 − 0,8857) 𝐶𝐶𝑢𝑢 − 𝛼𝛼 ∗ ∗ 𝑆𝑆𝑢𝑢 27,52 − 0,8130 ∗ 117,52 = = −66,0168 𝛽𝛽 = 𝑟𝑟 1,0304 𝐶𝐶 = 𝛼𝛼 ∗ ∗ 𝑆𝑆 + 𝛽𝛽 = 0,8150 ∗ 100 − 66,0168 = 15,2832
𝛼𝛼 ∗ =
𝐶𝐶𝑀𝑀 = 10 𝐶𝐶𝑇𝑇 = 15,2832 Period/node 𝑡𝑡 = 0 𝑡𝑡 = 1 𝑡𝑡 = 2
d) 𝐸𝐸 = 110 𝑃𝑃𝑢𝑢𝑢𝑢 = 0 𝑃𝑃𝑢𝑢𝑢𝑢 = 5,9125 𝑃𝑃𝑑𝑑𝑑𝑑 = 31,5536
Strategy Buy 1 call Short sell 𝛼𝛼 ∗ at 𝑆𝑆 = 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 (𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 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 (𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ∶ 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 𝑆𝑆𝑑𝑑𝑑𝑑 = −33,33298 (𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0)
22
Maths
Exercises
𝑝𝑝 ∗ 𝑃𝑃𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝑃𝑃𝑢𝑢𝑢𝑢 (0,4998 ∗ 0 + (1 − 0,4998) ∗ 5,1125) = = 2,8702 𝑟𝑟 1,0304 𝑃𝑃𝑢𝑢𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = max{0, 𝐸𝐸 − 𝑆𝑆𝑢𝑢 } = 0 𝑝𝑝 ∗ 𝑃𝑃𝑢𝑢𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝑃𝑃𝑑𝑑𝑑𝑑 (0,4998 ∗ 5,1125 + (1 − 0,4998) ∗ 31,5536) 𝑃𝑃𝑑𝑑𝑛𝑛𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = = = 18,1853 𝑟𝑟 1,0304 𝑃𝑃𝑑𝑑𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = max{0, 𝐸𝐸 − 𝑆𝑆𝑢𝑢 } = 21,43 𝑃𝑃𝑢𝑢𝑛𝑛𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑝𝑝 ∗ 𝑃𝑃𝑢𝑢 + (1 − 𝑝𝑝) ∗ 𝑃𝑃𝑑𝑑 = 11,7952 𝑟𝑟 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 10 𝑃𝑃 𝑃𝑃 = 𝑚𝑚𝑚𝑚𝑚𝑚 = 11,7952
𝑃𝑃𝑛𝑛𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
If E=90 It’s wrong to do that : 𝑆𝑆 + 𝑃𝑃 = 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶 𝑃𝑃 = 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶 − 𝑆𝑆 = 5,293398 wrong!! Problem 3
𝑆𝑆 = 100
𝑆𝑆𝑢𝑢 = 140 𝑆𝑆𝑑𝑑 = 75
1,04 − 0,75 𝑝𝑝 = = 0,4462 1,4 − 0,75 1 − 𝑝𝑝 = 0,5538 0,4462 ∗ 96 + 0,5538 ∗ 5 𝐶𝐶𝑢𝑢 = = 43,486 1,04 0,4462 ∗ 25 𝐶𝐶𝑑𝑑 = = 10,73 1,04 43,846 ∗ 0,4462 + 10,73 ∗ 0,5538 𝐶𝐶 = = 24,52 1,04
𝐶𝐶𝑑𝑑𝑑𝑑 − 𝐶𝐶𝑑𝑑𝑑𝑑 25 = = 0,513 𝑆𝑆𝑑𝑑 ∗ (𝑢𝑢 − 𝑑𝑑) 75 ∗ (1,4 − 0,75) 𝐶𝐶𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢 96 − 5 𝛼𝛼𝑢𝑢 = = =1 𝑆𝑆𝑢𝑢 ∗ (𝑢𝑢 − 𝑑𝑑) 140 ∗ (1,4 − 0,75) 𝐶𝐶𝑢𝑢𝑢𝑢 − 𝛼𝛼𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 96 − 1 ∗ 196 = = −96,1538 𝛽𝛽𝑢𝑢 = 𝑟𝑟 1,04 𝐶𝐶𝑢𝑢𝑢𝑢 − 𝛼𝛼𝑑𝑑 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 25 − 0,513 ∗ 105 𝛽𝛽𝑑𝑑 = = = −27,75 𝑟𝑟 1,04 𝐶𝐶𝑢𝑢 = 1 ∗ 140 − 96,1538 = 43,8462 𝐶𝐶𝑑𝑑 = 0,513 ∗ 75 − 27,75 = 10,73 43,8462 − 10,73 = 0,5096 𝛼𝛼 = 100 ∗ (1,4 − 0,75) 43,8462 − 0,5096 ∗ 140 𝛽𝛽 = = −26,4402 1,04 𝐶𝐶 = 0,5096 ∗ 100 − 26,4402 = 24,5192
𝛼𝛼𝑑𝑑 =
b)
23
𝑆𝑆𝑢𝑢𝑢𝑢 = 196 𝑆𝑆𝑢𝑢𝑢𝑢 = 105 𝐶𝐶𝑢𝑢𝑢𝑢 = 5 𝐶𝐶𝑑𝑑𝑑𝑑 = 25
𝑆𝑆𝑑𝑑𝑑𝑑 = 56,25
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) 𝐸𝐸 = 100 𝑆𝑆 = 120 𝜎𝜎 = 0,4 𝑟𝑟𝑟𝑟 = 0,0618 (𝑢𝑢 = 𝑒𝑒 0,4√0,25 = 1,2214 𝑑𝑑 = 𝑒𝑒 −0,4√0,25 = 0,81873) 𝑑𝑑1 = 𝑑𝑑2 =
𝜎𝜎 2 𝑆𝑆 ln �𝐸𝐸 � + �𝑟𝑟 + 2 � ∗ 𝑡𝑡 𝜎𝜎√𝑡𝑡
𝜎𝜎 2 𝑆𝑆 ln �𝐸𝐸 � + �𝑟𝑟 − 2 � ∗ 𝑡𝑡
=
0,42 120 ln �100� + �0,0618 + 2 � ∗ 0,25
0,4 ∗ �0,25 0,42 120 ln �100� + �0,0618 − 2 � ∗ 0,25
= 𝜎𝜎√𝑡𝑡 0,4 ∗ �0,25 ) (0,8621 𝑁𝑁(𝑑𝑑1 = 0,8599 + 0,89 ∗ − 0,8599) = 0,8619 𝑁𝑁(𝑑𝑑2 ) = 0,8106 + 0,89 ∗ (0,8133 − 0,8106) = 0,8130
= 1,0889 = 0,88886
𝐶𝐶 = 𝑆𝑆 ∗ 𝑁𝑁(𝑑𝑑1 ) − 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 ∗ 𝑁𝑁(𝑑𝑑2 ) = 120 ∗ 0,8619 − 100 ∗ 𝑒𝑒 −0,25∗0,0618 ∗ 0,8130 𝐶𝐶 = 23,3744 𝑃𝑃 = −𝑆𝑆 ∗ �1 − 𝑁𝑁(𝑑𝑑1 )� + 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 ∗ �1 − 𝑁𝑁(𝑑𝑑2 )� = −120 ∗ (1 − 0,8619) + 100 ∗ 𝑒𝑒 −0,25∗0,0618 ∗ (1 − 0,8130) 𝑃𝑃 = 1,8413 b) 𝐸𝐸 = 25 𝑆𝑆 = 25 𝜎𝜎 = 0,3 𝑟𝑟𝑟𝑟 = 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
𝜎𝜎 2 𝑆𝑆 ln �𝐸𝐸 � + �𝑟𝑟 + 2 � ∗ 𝑡𝑡
0,32 25 ln � � + �0,0618 + 2 � ∗ 1 25 𝑑𝑑1 = = = 0,356 𝜎𝜎√𝑡𝑡 0,3 ∗ √1 𝜎𝜎 2 0,32 𝑆𝑆 25 ln �𝐸𝐸 � + �𝑟𝑟 − 2 � ∗ 𝑡𝑡 ln � � + �0,0618 − 2 � ∗ 1 25 𝑑𝑑2 = = = 0,056 𝜎𝜎√𝑡𝑡 0,3 ∗ √1 𝑁𝑁(𝑑𝑑1 ) = 0,6368 + 0,6 ∗ (0,6406 − 0,6368) = 0,639 𝑁𝑁(𝑑𝑑2 ) = 0,5199 + 0,6 ∗ (0,5239 − 0,5199) = 0,522
𝑃𝑃 = −𝑆𝑆 ∗ �1 − 𝑁𝑁(𝑑𝑑1 )� + 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 ∗ �1 − 𝑁𝑁(𝑑𝑑2 )� = −25 ∗ (1 − 0,639) + 25 ∗ 𝑒𝑒 −0,0618 ∗ (1 − 0,522) 𝑃𝑃 = 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) 𝑆𝑆 = 200 𝐸𝐸 = 180 𝜎𝜎 = 22,3% 𝑖𝑖 = 21% 𝑑𝑑1 =
0,2232 200 ln �180� + �0,21 + 2 � ∗ 1
= 1,5257 0,223 ∗ √1 0,2232 200 ln �180� + �0,21 − 2 � ∗ 1 𝑑𝑑2 = = 1,3027 0,223 ∗ √1 𝑁𝑁(𝑑𝑑1 ) = 0,9357 + 0,57 ∗ (0,9370 − 0,9357) = 0,9364 𝑁𝑁(𝑑𝑑2 ) = 0,9032 + 0,27 ∗ (0,9049 − 0,9032) = 0,9037
𝐶𝐶 = 200 ∗ 0,9364 − 180 ∗ 𝑒𝑒 −0,21 ∗ 0,9037 𝐶𝐶 = 55,4255
b) We should use replicating portfolio 𝑢𝑢 = 𝑒𝑒 𝜎𝜎 √𝑡𝑡 = 𝑒𝑒 0,223 = 1,2498 𝑑𝑑 = 𝑒𝑒 −𝜎𝜎 √𝑡𝑡 = 0,8001 𝑆𝑆𝑢𝑢 = 1,2498 ∗ 200 = 249,96 𝑆𝑆𝑑𝑑 = 0,8001 ∗ 200 = 160,02 𝑒𝑒 0,21 − 0,8001 = 0,9641 𝑝𝑝 = 1,2498 − 0,8001 0,9641 ∗ (249,96 − 180) + (1 − 0,9641) ∗ 0 𝐶𝐶 = 𝑒𝑒 0,21 𝐶𝐶 = 54,67 c) We should use replicating portfolio 𝑢𝑢 = 𝑒𝑒 𝜎𝜎 √𝑡𝑡 = 𝑒𝑒 0,223 √0,5 = 1,1708 𝑑𝑑 = 𝑒𝑒 −𝜎𝜎 √𝑡𝑡 = 0,8541 𝑆𝑆 = 200
𝑆𝑆𝑢𝑢 = 1,1708 ∗ 200 = 234,16 𝑆𝑆𝑑𝑑 = 0,8541 ∗ 200 = 170,82 25
𝑆𝑆𝑢𝑢𝑢𝑢 = 1,1708 ∗ 234,16 = 274,15 𝑆𝑆𝑑𝑑𝑑𝑑 = 200
Maths
Exercises
0,21∗0,5
𝑒𝑒 − 0,8541 = 0,8103 1,1708 − 0,8541 𝑒𝑒 0,21∗0,5 = 1,1107 0,8103 ∗ (274,15 − 180) + (1 − 0,8103) ∗ 20 = 72,10 𝐶𝐶𝑢𝑢 = 1,1107 0,8103 ∗ 20 = 14,59 𝐶𝐶𝑑𝑑 = 1,1107 0,8103 ∗ 72,10 + 14,59 ∗ (1 − 0,8103) 𝐶𝐶 = 1,1107 𝐶𝐶 = 55,14 𝑝𝑝 =
𝑆𝑆𝑑𝑑𝑑𝑑 = 0,8541 ∗ 170,82 = 145,897
d)
𝐶𝐶𝑢𝑢 − 𝐶𝐶𝑑𝑑 72,10 − 14,59 = = 0,908 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 200 ∗ (1,1708 − 0,8541) (274,15 − 180) − 20 𝛼𝛼𝑢𝑢∗ = =1 234,16 ∗ (1,1708 − 0,8541) 20 𝛼𝛼𝑑𝑑∗ = = 0,3697 170,82 ∗ (1,1708 − 0,8541)
𝛼𝛼 ∗ =
Node
Stock price
BMS model 2 period binomial model 𝐶𝐶 ∆ 𝐵𝐵 𝐶𝐶 ∆ 𝐵𝐵 𝑆𝑆𝑢𝑢 200 55,4255 0,9364 131,8515 56,7273 0,9195 127,1727 𝑆𝑆𝑑𝑑 170,82 15,4166 0,6601 97,3436 14,8996 0,3696 48,2355 𝑆𝑆 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 𝛼𝛼 𝑎𝑎𝑎𝑎𝑎𝑎 𝛽𝛽 even if the prices of the call are similar. Problem 3 a) 300 250 200 150 100 50 0
Commercial Loan Tranche 1 Tranche 2 Tranche 3 0
100
200
300
400
b) Tranche 1 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑇𝑇1 = 𝑉𝑉 − 𝐶𝐶(𝐸𝐸 = 𝐷𝐷1) We also can do Put write+bond(D1) but not in this problem… Tranche 2 Call buy (D1)+Call write (D1+D2) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑇𝑇2 = 𝐶𝐶(𝐸𝐸 = 𝐷𝐷1) − 𝐶𝐶(𝐸𝐸 = 𝐷𝐷1 + 𝐷𝐷2) Tranche 3 Call buy (D2) Call write (D1+D2) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑇𝑇3 = 𝐶𝐶(𝐸𝐸 = 𝐷𝐷1 + 𝐷𝐷2)
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 + 100 ∗ 1,04−5 = 108,9 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 6 ∗ 0,04 b)
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 30 ∗ Exercise 2 a)
1 − 1,02−10 + 1000 ∗ 1,02−10 = 1089,8 0,02
1 − 1,0375−8 + 100000 ∗ (1,0375−8 ) = 108503,49 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑃𝑃 = 5000 ∗ 0,0375 1 − 1,0375−16 + 100000 ∗ (1,0375−16 ) = 114837,7 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑄𝑄 = 5000 ∗ 0,0375 b)
1 − 1,06−8 + 100000 ∗ (1,06−8 ) = 93790,2 0,06 1 − 1,06−16 + 100000 ∗ (1,06−16 ) = 89894,1 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑄𝑄 = 5000 ∗ 0,06
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑃𝑃 = 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 + 𝑖𝑖)4 => 𝑖𝑖 = 10% b) 10% semi annual 𝑥𝑥 2 �1 + � − 1 = 0,10 2 𝑥𝑥 = 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 + 100 ∗ 1,04−5 = 104,452 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 = 5 ∗ 0,04
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 5 ∗
b) 104,452 + 5 − 105,242 = 4% 105,242 c)
28
Maths
Exercises
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 = 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 ∗
1000 1 − (1 + 𝑖𝑖)−20 + (1 + 𝑖𝑖)20 𝑖𝑖
So 𝑖𝑖 = 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 + 𝑖𝑖)−10 ) 1000 + => 𝑖𝑖 = 16,075% 900 = 140 ∗ (1 + 𝑖𝑖)10 𝑖𝑖 (1 − (1 + 𝑖𝑖)−10 ) 1000 900 = 70 ∗ + => 𝑖𝑖 = 8,52% (1 𝑖𝑖 + 𝑖𝑖)10 Exercise 8 a)
110 = 100,9174 1,09 110 1 ∗� �=1 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 1 = 100,9174 1,09 100 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 2 = = 77,2183 1,093 110 1 ∗ �3 ∗ �=3 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 2 = 1,093 77,2183 1 − 1,09−4 + 100 ∗ 1,09−4 = 135,6369 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 3 = 20 ∗ 0,09 1 20 20 20 120 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 3 = ∗� +2∗ +3∗ +4∗ � = 3,232 2 3 135,6369 1,09 1,09 1,09 1,094 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 1 =
b) Bond1+Bond4 20 20 120 130 + + + = 236,55 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 5 = 2 3 1,09 1,094 1,09 1,09 1 130 20 20 120 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 5 = ∗� +2∗ + 3 ∗ + 4 ∗ � = 2,2799 236,5543 1,09 1,092 1,093 1,094 c)
29
Maths
Exercises
𝑥𝑥 + (1 − 𝑥𝑥) ∗ 3,232 = 2 𝑥𝑥 = 0,552, (1 − 𝑥𝑥) = 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 𝐹𝐹𝐹𝐹 11,5746 = => 𝐹𝐹𝐹𝐹 = 13,8174 1,11,8583
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 =
Exercise 11 a)
1,16 = 7,25𝑦𝑦 0,16 7,25 = 4 ∗ 𝑥𝑥 + 11 ∗ (1 − 𝑥𝑥) 11 − 7,25 𝑥𝑥 = = 0,5357 7 1 − 𝑥𝑥 = 0,4643
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑛𝑛𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =
b)
2000000 = 12,5𝑀𝑀 0,16 𝑃𝑃𝑉𝑉5𝑦𝑦 = 0,5357 ∗ 12,5𝑀𝑀 = 6,696𝑀𝑀 𝑃𝑃𝑉𝑉20𝑦𝑦 = 0,4642 ∗ 12,5𝑀𝑀 = 5,804𝑀𝑀
𝑃𝑃𝑃𝑃 =
1 − 1,16−20 1000 + = 407,12 0,16 1,1620 5,804𝑀𝑀 So = 14256,24 𝑃𝑃𝑉𝑉20𝑦𝑦 = 60 ∗ 407,12
Exercise 12 a) 1000 𝑃𝑃𝑃𝑃 = = 374,84 1,0812,75 1000 = 333,28 𝑃𝑃𝑃𝑃 = 1,0912,75
30
Maths
Exercises
1 𝑉𝑉𝑉𝑉𝑉𝑉 = −1% ∗ 11,81 + ∗ 1%2 ∗ 150,3 = 11,06% 2 1 − 1,08−30 1000 𝑃𝑃𝑃𝑃 = 60 ∗ + = 774,84 0,08 1,0830 −30 1000 1 − 1,09 + = 691,79 𝑃𝑃𝑃𝑃 = 60 ∗ 0,09 1,0930 1 𝑉𝑉𝑉𝑉𝑉𝑉 = −1% ∗ 11,79 + ∗ 1%2 ∗ 231,2 = 10,63% 2 333,28 − 374,84 = 11,09% 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 374,84 691,79 − 774,89 = 10,72% 𝑉𝑉𝑉𝑉𝑉𝑉 = 774,89
𝑉𝑉𝑉𝑉𝑉𝑉 =
b) if decrease of 1% 𝑉𝑉𝑉𝑉𝑉𝑉 = 12,59% 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑉𝑉𝑉𝑉𝑉𝑉 = 12,56% 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
𝑉𝑉𝑉𝑉𝑉𝑉 = 13,04% 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑉𝑉𝑉𝑉𝑉𝑉 = 12,95% 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
c) formula gives a good approximation of the change. d) convexity higher, yield lower. Problem +
31
Maths
Exercises
Problem Set 8 Exercise 1 (1 + 𝑖𝑖2 )2 = (1 + 𝑖𝑖1 )(1 + 1𝑓𝑓2)
Exercise 2 a) 100 = 107 ∗ (1 + 𝑖𝑖1 )−1 => 𝑖𝑖1 = 7%
102 = 8 ∗ (1 + 𝑖𝑖1 )−1 + 108 ∗ (1 + 𝑖𝑖2 )−2 = 8 ∗ (1,07)−1 + 108 ∗ (1 + 𝑖𝑖2 )−2 1
2 108 � − 1 = 6,8913% => 𝑖𝑖2 = � 102 − 8 ∗ 1,07−1
95 = 6,7 ∗ (1 + 𝑖𝑖1 )−1 + 6,7 ∗ (1 + 𝑖𝑖2 )−2 + 106,7 ∗ (1 + 𝑖𝑖)−3 = 6,7 ∗ 1,07−1 + 6,7 ∗ 1,068913−2 + 106,7 ∗ (1 + 𝑖𝑖)−3 1
3 106,7 � − 1 = 8,7881% => 𝑖𝑖3 = � −1 −2 95 − 6,7 ∗ 1,07 + 6,7 ∗ 1,068913
93 = 7 ∗ (1 + 𝑖𝑖1 )−1 + 7 ∗ (1 + 𝑖𝑖2 )−2 + 7 ∗ (1 + 𝑖𝑖)−3 + 107 ∗ (1 + 𝑖𝑖)−4 = 7 ∗ 1,07−1 + 7 ∗ 1,068913−2 + 7 ∗ 1,087881−3 + 107 ∗ (1 + 𝑖𝑖)−4 1
4 107 � − 1 = 9,3285% => 𝑖𝑖4 = � −1 −2 −3 93 − 7 ∗ 1,07 + 7 ∗ 1,068913 + 7 ∗ 1,087881
109 = 12 ∗ (1 + 𝑖𝑖1 )−1 + 12 ∗ (1 + 𝑖𝑖2 )−2 + 12 ∗ (1 + 𝑖𝑖)−3 + 12 ∗ (1 + 𝑖𝑖)−4 + 112 ∗ (1 + 𝑖𝑖)−5 = 12 ∗ 1,07−1 + 12 ∗ 1,068913−2 + 12 ∗ 1,087881−3 + 12 ∗ 1,093285−4 + 112 ∗ (1 + 𝑖𝑖)−5 1
5 112 � − 1 = 10% => 𝑖𝑖5 = � −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 + 𝑖𝑖) ∗ 1,07 = 1,068913 ∗ 1,068913 ! 1𝑓𝑓2 =
1,0878813 − 1 = 12,6833% 1,0689132 1,0932854 − 1 = 10,97% 3𝑓𝑓4 = 1,0878813
2𝑓𝑓3 =
32
Maths 4𝑓𝑓5 =
Exercises 1,15 − 1 = 12,7275% 1,0932854
e) 1,0878813 = 1,07 ∗ (1 + 𝑖𝑖)2
1,0878813 − 1 = 9,6933% 𝑖𝑖 = � 1,07
1,0932854 = 1,0689132 ∗ (1 + 𝑖𝑖)2 1,0932854 𝑖𝑖 = � − 1 = 11,8213% 1,0689132
Exercise 3 5 5 5 105 𝑃𝑃1 = + + + = 102,5968 2 3 1,05 1,0475 1,045 1,04254 10 10 10 110 + + + = 120,5302 𝑃𝑃2 = 2 3 1,05 1,0475 1,045 1,04254 1 − (1 + 𝑖𝑖)−4 100 + => 𝑖𝑖 = 4,2799% (1 + 𝑖𝑖)4 𝑖𝑖 1 − (1 + 𝑖𝑖)−4 100 𝑃𝑃2 = 10 ∗ + => 𝑖𝑖 = 4,3036% (1 + 𝑖𝑖)4 𝑖𝑖
𝑃𝑃1 = 5 ∗
1,05 ∗ (1 + 𝑖𝑖) = 1,04752 1,04752 −1 𝑖𝑖 = 1,05 𝑖𝑖 = 4,5%
1,04752 ∗ (1 + 𝑖𝑖) = 1,0453 1,0453 − 1 = 4% 𝑖𝑖 = 1,04752 1,04254 − 1 = 3,504% 1,0453 Exercise 4 𝐹𝐹𝐹𝐹 = 100 𝑡𝑡 = 6 𝐶𝐶 = 6 => 𝑖𝑖 = 12% 𝐶𝐶 = 10 => 𝑖𝑖 = 8%
1 − 1,12−6 100 + = 75,3316 0,12 1,126 1 − 1,08−6 100 + = 109,2458 𝑃𝑃2 = 10 ∗ 0,08 1,086
𝑃𝑃1 = 6 ∗
If we do 𝑃𝑃1 − 0,6 ∗ 𝑃𝑃2 we have a zero coupon bond of 6 years. 𝑃𝑃1 − 0,6 ∗ 𝑃𝑃2 = 9,7841 At the end we have 106 − 0,6 ∗ 110 = 40 40 = (1 + 𝑖𝑖)6 9,7841 𝑖𝑖 = 26,4513%
33
Maths
Exercises
Exercise 5 a) (1 + 𝑖𝑖2 )2 − 1 = 6,4% 1𝑓𝑓2 = 1 + 𝑖𝑖1 2𝑓𝑓3 = 5,5 3𝑓𝑓4 = 4,31% 4𝑓𝑓5 = −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 +100 ∗ 1,0584 = 125,298 125,298
Exercise 6 a) HPR=6% 1𝑓𝑓2 = 6,01% 2𝑓𝑓3 = 7,014%
b) mkt expects higher 2f3 than you => 𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎𝑎𝑎 𝑡𝑡 = 1 => 𝑚𝑚𝑚𝑚𝑚𝑚 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝐻𝐻𝐻𝐻𝐻𝐻
Exercise 7 1y=5% 2y=6%
112 12 + = 111,108 1,05 1,062 112 12 + = 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 + 𝑖𝑖2 )2 1,064 = 1,06 𝑖𝑖2 = 6,1998% (1 + 𝑖𝑖3 )3 1,071 = 1,0619982 𝑖𝑖3 = 6,4990% (1 + 𝑖𝑖4 )4 1,073 = 1,0649903 𝑖𝑖4 = 6,6987% (1 + 𝑖𝑖5 )5 1,082 = 1,0669873
34
Y=5 −100 ∗ 1,0465 = 125,216
125,216 125,298 − 125,216 = 0,082
Maths
Exercises
𝑖𝑖5 = 6,9973%
b) future interest rate up
c) borrow PV(100M) for 4y 100𝑀𝑀 borrow 4 = 77,151𝑀𝑀 1,067
invest 77,151M for 5y => net cash flow at t=0 = 0 at t=4 repay the loan : −77,151 ∗ 1,0674 = −100𝑀𝑀 at t=5 receive the 5y investment : 77,151 ∗ 1,075 = 108,208𝑀𝑀
(with the future, but in the exercise it’s not the purpose : 100 ∗ 1,082 = 108,2𝑀𝑀 > 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
13,5𝑀𝑀 = 40 1350 ∗ 250
c) beta 0,6 40 ∗ 0,6 = 24
Exercise 13 3 671,5 = (1 + 𝑖𝑖3𝑚𝑚 )12 664,3 𝑖𝑖3𝑚𝑚 = 4,406% 9 690 = (1 + 𝑖𝑖6𝑚𝑚 )12 664,3 𝑖𝑖6𝑚𝑚 = 5,19%
𝑖𝑖15𝑚𝑚 = 4,85% 𝑖𝑖21𝑚𝑚 = 5,06
Exercise 14 𝑆𝑆 = 1300 𝑟𝑟𝑓𝑓 = 4%, 𝑑𝑑𝑑𝑑𝑑𝑑 = 1% 𝐹𝐹1𝑦𝑦 = 1330 𝐹𝐹1𝑦𝑦 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 1300 ∗ 1,03 = 1339 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 Buy future Short index Lend
T=0 0 1300 -1300 0
T=12
9
𝑆𝑆12 − 1330 −𝑆𝑆12 − 0,01 ∗ 1300 1300 ∗ 1,04
Exercise 15 a) £ = 2.00 $ so 2∗
1,04 = 1,9627 1,06
(foreign/domestic in the point of view of £/$=2, the foreign is US) b) 2,03 is too high, you sell it.
T=0
Sell 1£ forward Buy
1£ 1,06
in spot market
Borrow in US
−
2$ = −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 𝐷𝐷𝐷𝐷 = 8𝑦𝑦 𝐹𝐹0 = 100 ; 𝐷𝐷𝐷𝐷 = 6𝑦𝑦
𝑙𝑙𝑙𝑙𝑡𝑡 ′ 𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1𝑏𝑏𝑏𝑏 𝑉𝑉𝑉𝑉𝑉𝑉𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 8 ∗ 0,0001 ∗ 10𝑀𝑀 = 8000$ 𝐶𝐶𝐶𝐶 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 1 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 6 ∗ 0,0001 ∗ 100000 = 60$ 8000 = 133 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 60
Exercise 17 𝑈𝑈𝑈𝑈 = 4%; 𝐸𝐸𝐸𝐸𝐸𝐸 = 3% 𝐸𝐸𝐸𝐸𝐸𝐸 = 1,5 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 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 = 1,5146 𝐹𝐹1𝑦𝑦 = 1,5 ∗ 1,03 1,04² = 1,5293 𝐹𝐹2𝑦𝑦 = 1,5 ∗ 1,03² 1,043 𝐹𝐹3𝑦𝑦 = 1,5 ∗ = 1,5441 1,033
So the “mean” 𝐹𝐹 ∗ 𝐹𝐹 ∗ 1 − 1,04−3 1,5146 1,5293 1,5441 𝐹𝐹 ∗ ∗ + + = 𝐹𝐹 ∗ = + + 1,04 1,042 1,043 0,04 1,04 1,042 1,043 ∗ 𝐹𝐹 = 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 100𝑀𝑀 + 100𝑀𝑀 ∗ (𝑒𝑒 0,096∗12 𝑒𝑒 3 = 1,033895 100𝑀𝑀 ∗ 12% ∗ = 6𝑀𝑀 = 1/3y 2 − 1) 6 𝑃𝑃𝑃𝑃 = = 104,9171 1,033895 104,9171 𝑃𝑃𝑃𝑃 = = 101,4775 1,033895 0,1∗10 1 T=10 months 𝑒𝑒 12 = 1,086904 100𝑀𝑀 + 100𝑀𝑀 ∗ 12% ∗ = 10/12y 2 = 106𝑀𝑀 106 𝑃𝑃𝑃𝑃 = 1,086904 6 106 Total 𝑃𝑃𝑃𝑃 = 101,4775 𝑃𝑃𝑃𝑃 = + 1,033895 1,086904 = 103,328 𝐵𝐵𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 103,3280𝑀𝑀 𝐵𝐵𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 101,4775𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉(𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓) = 1,85𝑀𝑀 Exercise 19 From £ payers perspective Time 3 months
Cf $
Cf £ 6% ∗ 30𝑀𝑀 = 1,8𝑀𝑀$ 37
−10% ∗ 20𝑀𝑀£ = 2𝑀𝑀£
Maths
Exercises 1,8𝑀𝑀$
15 months Current £/$ = 1,85 (1£ = 1,85$) Future 𝐹𝐹3𝑚𝑚
−2𝑀𝑀£
3
1,04 12 = 1,85 ∗ � � = 1,8369 1,07
𝐹𝐹15𝑚𝑚
15
1,04 12 = 1,85 ∗ � � = 1,7854 1,07
2𝑀𝑀 ∗ 1,8369 = 3,6738𝑀𝑀$ 2𝑀𝑀 ∗ 1,7854 = 3,5708𝑀𝑀$ 𝑃𝑃𝑃𝑃 =
𝑃𝑃𝑃𝑃 =
3,6738 − 1,8 3 1,0415
3,5708 − 1,8 15 1,0412
= 1,8555𝑀𝑀$ = 1,6861𝑀𝑀$
At the end the 2 parties exchange the currencies, so 30𝑀𝑀$, and 20𝑀𝑀£ = 20 ∗ 1,7854𝑀𝑀 = 35,708𝑀𝑀$ 35,708 − 30 𝑃𝑃𝑃𝑃 = = 5,4349𝑀𝑀$ 15 1,0412 For the view of the party paying £, he loses : 5,4349 + 1,6861 + 1,8555 = 8,9765𝑀𝑀$
38
Maths
Exercises
Problem Set 9 Exercise 1 a) 𝑟𝑟𝑒𝑒 = 0,16 𝐷𝐷1 = 2$ 𝐷𝐷1 𝑃𝑃 = 𝑟𝑟𝑒𝑒 − 𝑔𝑔 𝐷𝐷1 𝑔𝑔 = 𝑟𝑟𝑒𝑒 − 𝑃𝑃
𝑔𝑔 = 0,16 − 𝑔𝑔 = 12%
b)
𝑃𝑃 = 𝑃𝑃 𝐸𝐸
2 50
2 = 18,18 0,16 − 0,05
=>down; earning constant, price down. PVGO is falling.
Exercise 2 a) 𝐸𝐸𝐸𝐸𝐸𝐸 = 6 𝑔𝑔 = 8%
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
𝑟𝑟𝑒𝑒 = 12%
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =
1 3
𝐷𝐷1 𝐸𝐸𝐸𝐸𝑆𝑆1
1 ∗6=2 3 2 = 50$ 𝑃𝑃 = 0,12 − 0,08
𝐷𝐷1 =
b) 𝑔𝑔 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ∗ 𝑅𝑅𝑅𝑅𝑅𝑅 = (1 − 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝) ∗ 𝑅𝑅𝑅𝑅𝑅𝑅 𝑔𝑔 𝑅𝑅𝑅𝑅𝑅𝑅 = 1 − 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 0,08 = 0,12 𝑅𝑅𝑅𝑅𝑅𝑅 = 1 1−3 Doesn’t need it 𝐸𝐸𝐸𝐸𝐸𝐸 + 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑟𝑟𝑒𝑒 6 =0 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 50 − 0,12
𝑃𝑃0 =
c) 𝑅𝑅𝑅𝑅𝑅𝑅 = 10%
1 𝑔𝑔 = 0,10 ∗ �1 − � = 0,0667 3 2 = 37,5 𝑃𝑃0 = 0,12 − 0,0667 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = −12,5$
39
Maths
Exercises
d) 𝑅𝑅𝑅𝑅𝑅𝑅 = 15%
1 𝑔𝑔 = 0,15 ∗ �1 − � = 0,1 3 2 = 100 𝑃𝑃 = 0,12 − 0.10 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 50$ Exercise 3 𝐸𝐸𝐸𝐸𝐸𝐸 = 1$ 𝑟𝑟𝑒𝑒 = 15% 𝑃𝑃𝑃𝑃𝑃𝑃 = 0,5 𝑅𝑅𝑅𝑅𝑅𝑅 = 20%
a) 𝑔𝑔 = 0,5 ∗ 0,2 = 0,1 𝐷𝐷0 = 0,5$ 0,5 ∗ 1,1 𝑃𝑃0 = = 11$ 0,15 − 0,1
b) 𝑔𝑔 = 0,4 ∗ 0,15 𝑔𝑔 = 6% BE CAREFUL, payout ratio has changed, so div change too ! 𝑑𝑑2 0,605 𝐸𝐸𝐸𝐸𝑆𝑆2 = = = 1,21$ 0,5 0,5 𝐸𝐸𝐸𝐸𝑆𝑆3 = 𝐸𝐸𝐸𝐸𝑆𝑆2 ∗ 1,06 = 1,2826$ 𝑑𝑑3 = 1,2826 ∗ 0,6 = 0,76956 0,5 ∗ 1,1 0,5 ∗ 1,1² 0,76956 + + 𝑃𝑃0 = 2 1,15 1,15 ∗ (0,15 − 0,06) 1,15² 𝑃𝑃0 = 7,4$ (𝑃𝑃2 =
0,76956 0,15−0,06
= 8,5507$)
c) 𝑃𝑃0 = 11 𝑃𝑃1 = 11 ∗ 1,1 = 12,1 12,1 + 0,55 − 11 𝑟𝑟1 = = 15% 11 𝑃𝑃2 = 8,5507 𝑟𝑟2 = −23,3% 𝑃𝑃3 = 𝑃𝑃2 ∗ 1,06 = 9,064$ 𝑟𝑟3 = 15%
Exercise 4 𝐶𝐶𝐶𝐶 = 2𝑀𝑀$ 𝑔𝑔 = 5% 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 20% 𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 35% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 200000$ 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 12% 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 2𝑀𝑀$
40
Maths
Exercises
(2𝑀𝑀 ∗ (1 − 0,35) + 200000 − 400000) ∗ 1,05 = 16,5𝑀𝑀$ 0,12 − 0,05 16,5 − 2𝑀𝑀 𝑃𝑃0 = = 14,5$ 1𝑀𝑀
𝑃𝑃𝑃𝑃(𝐹𝐹𝐹𝐹𝐹𝐹) =
Exercise 5 𝐸𝐸𝐸𝐸𝐸𝐸 = 10$ 𝑅𝑅𝑅𝑅𝑅𝑅 = 20% => 5𝑦𝑦 6𝑡𝑡ℎ 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑅𝑅𝑅𝑅𝑅𝑅 = 15%, 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 40% 𝑟𝑟𝑒𝑒 = 15% a) 𝑔𝑔 = 20% => 5𝑦𝑦 𝑔𝑔 = 0,6 ∗ 0,15 = 9%
𝐸𝐸𝐸𝐸𝑆𝑆5 = 10 ∗ 1,25 = 24,8832$ 𝐸𝐸𝐸𝐸𝑆𝑆6 = 24,8832 ∗ 1,09 = 27,1227$ 𝑑𝑑6 = 𝐸𝐸𝐸𝐸𝑆𝑆6 ∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 27,1227 ∗ 0,4 = 10,8491$ 𝑑𝑑6 10,8491 𝑃𝑃5 = = = 180,8183$ 0,15 − 0,09 0,15 − 0,09 𝑃𝑃5 180,8183 𝑃𝑃0 = = = 89,899$ (1 + 𝑖𝑖)5 1,155
b) 𝑟𝑟𝑒𝑒 = 𝑑𝑑𝑑𝑑𝑑𝑑 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝐷𝐷1 + 𝑔𝑔 𝑟𝑟𝑒𝑒 = 𝑃𝑃0 first year: only capital gain, so return of 15% from 5th, there is div yield so capital gain = 9% c) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 = 𝑟𝑟𝑒𝑒 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 So div yield = 6% d) 𝜌𝜌 = 0,2 It doesn’t change, pvgo constant, roe=re=15% Exercise 6 (no correction) 𝑅𝑅𝑅𝑅𝑅𝑅 = 9% a) 𝑟𝑟𝑒𝑒 = 6 + 1,25 ∗ (14 − 6) = 16% 𝐸𝐸𝐸𝐸𝐸𝐸 = 3$ 2 𝑔𝑔 = 0,09 ∗ = 6% 3 1 𝐷𝐷0 = 3$ ∗ = 1$ 3 𝐷𝐷0 ∗ 1,06 = 10,6$ 𝑃𝑃0 = 0,1 b) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 =
𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑃𝑃0 = 𝐸𝐸𝐸𝐸𝑆𝑆1 𝑟𝑟𝑒𝑒 − 𝑔𝑔 41
Maths
Exercises
1 𝐹𝐹𝐹𝐹𝐹𝐹 = 3 = 3,33 0,1 c)
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑃𝑃0 − d)
𝐸𝐸𝐸𝐸𝐸𝐸 3 = 10,6 − = −8,15 𝑟𝑟𝑒𝑒 0,16
1 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 3 1,03 = 15,85 𝑃𝑃0 = 2$ ∗ 0,13 Exercise 7 𝐷𝐷𝐷𝐷𝐷𝐷 = 0,5$ 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 6% 𝑡𝑡 = 20𝑦𝑦 𝑟𝑟𝑒𝑒 = 9%
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑20 = 𝑑𝑑0 ∗ (1 + 𝑔𝑔)20 = 1,6056 𝑑𝑑20 𝑑𝑑1 𝑟𝑟𝑒𝑒 − 𝑔𝑔 𝑃𝑃𝑃𝑃𝑑𝑑1 𝑑𝑑2 = − = 6,2705$ 𝑟𝑟𝑒𝑒 − 𝑔𝑔 (1 + 𝑟𝑟𝑒𝑒 )19 𝑑𝑑20 𝑟𝑟𝑒𝑒 𝑃𝑃𝑃𝑃𝑑𝑑20 𝑑𝑑∞ = = 3,4654$ (1 + 𝑟𝑟𝑒𝑒 )19 𝑃𝑃𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡 = 9,7359$ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 … 1,0619 − 1,0919 𝑉𝑉20 = 0,5 ∗ � � ∗ 1,09−19 = 6,85 1,06 − 1,09 0,5 ∗ 1,0620 ∗ 1,09−20 = 3,18 0,09 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 10 … Exercise 8 a)b) 1𝑀𝑀 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 20𝑀𝑀$ 𝐷𝐷𝐷𝐷𝐷𝐷 = 1𝑀𝑀$ 1 20 = => 𝑟𝑟 = 10% 𝑟𝑟 − 0,05 1,05 𝑉𝑉1 = = 21𝑀𝑀$ 0,05 We have :
21𝑀𝑀 1𝑀𝑀 + 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃1 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 1𝑀𝑀$ 𝑠𝑠𝑠𝑠 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 50000 𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃1 = 20$ 𝑃𝑃1 =
c) new number of share = 1050000
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1,05𝑀𝑀 = 1$ 1,05 So 𝑑𝑑3 = 1,05$ … and so on… 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑣𝑣2 = d)
1 2 0,1 − 0,05 + = 20𝑀𝑀$ 𝑃𝑃𝑃𝑃 = 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% => 𝑃𝑃𝑙𝑙𝑙𝑙𝑙𝑙 = 166,667 𝑃𝑃𝑙𝑙𝑙𝑙𝑙𝑙 5+5 100 ∗ = 12% => 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 = 83,333 𝑃𝑃𝑚𝑚𝑚𝑚𝑚𝑚 30 + 0 = 12% => 𝑃𝑃ℎ𝑖𝑖𝑖𝑖ℎ = 250 100 ∗ 𝑃𝑃ℎ𝑖𝑖𝑖𝑖ℎ
100 ∗
c) 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 => 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
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 => 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
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%
b) (𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑃𝑃𝑒𝑒𝑒𝑒 ) ∗ �1 − 𝑡𝑡𝑔𝑔 � = 𝑑𝑑𝑑𝑑𝑑𝑑 ∗ (1 − 𝑡𝑡𝑑𝑑 ) 𝑃𝑃𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑃𝑃𝑒𝑒𝑒𝑒 ∗ �1 − 𝑡𝑡𝑔𝑔 �� = 𝑡𝑡𝑑𝑑 1−� 𝑑𝑑𝑑𝑑𝑑𝑑 𝑃𝑃 −𝑃𝑃 We know that 0,6 < 𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑𝑑𝑑𝑑𝑑 𝑒𝑒𝑒𝑒 < 0,8 And 𝑡𝑡𝑔𝑔 = 15% so 32% ≤ 𝑡𝑡𝑑𝑑 ≤ 49%
Div yield
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%
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 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 1 𝑑𝑑𝑑𝑑𝑑𝑑 − 0,7𝑑𝑑𝑑𝑑𝑑𝑑 = 0,3𝑑𝑑𝑑𝑑𝑑𝑑 With that you bear overnight risk.
Exercise 12 𝐴𝐴1 = 50𝑀𝑀$ 𝐴𝐴2 = 50𝑀𝑀$ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 1000𝑀𝑀 Here we assume 𝑉𝑉 = 1000%$ ??? 𝑉𝑉𝐴𝐴1 = 500𝑀𝑀$ 𝑉𝑉𝐴𝐴2 = 500𝑀𝑀$ 𝑉𝑉 = 1000𝑀𝑀$ 𝑑𝑑𝑑𝑑𝑑𝑑 = 100𝑀𝑀$ 𝑑𝑑 = 0,1
Want to increase div by 50 cent, So we need 500𝑀𝑀$
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We have: ??? 𝑁𝑁 ∗ 𝑃𝑃 ∗ 0,995 = 500 1000𝑀𝑀 𝑉𝑉 = 1000𝑀𝑀 + 𝑁𝑁 𝑁𝑁 = 1010𝑀𝑀 𝑃𝑃 = 0,4975$
Old share wealth=0,4975 + 0,5 = 0,9975 Exercise 13 With CAPM 𝑟𝑟𝑒𝑒1 = 21% 𝑟𝑟𝑒𝑒2 = 18% 𝑟𝑟𝑒𝑒3 = 15%
𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝐼𝐼𝐼𝐼𝐼𝐼, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑖𝑖𝑖𝑖 𝐵𝐵 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 20 > 18%, 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝐴𝐴 (22% > 21%), 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐶𝐶 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐴𝐴 + 𝐵𝐵 = 1100$
𝐹𝐹𝐹𝐹𝐹𝐹 𝐸𝐸1 = (𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖1 + 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑐𝑐𝑐𝑐𝑐𝑐1) 𝐹𝐹𝐹𝐹𝐹𝐹 = 1000 + 500 − 1100 − 5000 ∗ 0,25 ∗ 0,08 = 300$ Pay 300$ of dividend
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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 𝑉𝑉𝐴𝐴 = 0,007 𝑉𝑉𝐴𝐴 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 𝑟𝑟𝑟𝑟 So entitlement = 0,01 ∗ �𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 0,3 ∗ 𝑉𝑉𝐴𝐴 ∗ 𝑟𝑟𝑓𝑓 � = 0,01 ∗ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐴𝐴 − 0,003 ∗ 𝑟𝑟𝑓𝑓 ∗ 𝑉𝑉𝐴𝐴
To have the same with B : You buy 1% of B, so you own 0,01 ∗ 0,9 𝑉𝑉𝐵𝐵 = 0,009 𝑉𝑉𝐵𝐵 So you to have the same 0,007, you borrow 1% of (𝐷𝐷𝐴𝐴 − 𝐷𝐷𝐵𝐵 ) = 0,01 ∗ (0,3𝑉𝑉 − 0,1𝑉𝑉) = 0,002 𝑉𝑉𝐵𝐵 𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 = 0,007 𝑉𝑉𝐵𝐵 𝑁𝑁𝑁𝑁𝑁𝑁 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 0,01 ∗ �𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 0,1 ∗ 𝑉𝑉𝐵𝐵 ∗ 𝑟𝑟𝑓𝑓 � − 0,002 ∗ 𝑉𝑉𝐵𝐵 ∗ 𝑟𝑟𝑟𝑟 = 0,01 ∗ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 − 0,003 ∗ 𝑉𝑉𝐵𝐵 ∗ 𝑟𝑟𝑟𝑟 b) If you own 2% of the stock B, then you own 2% ∗ 0,9 𝑉𝑉𝐵𝐵 = 0,018 𝑉𝑉𝐵𝐵 And you are entitled to 0,02 ∗ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵 − 0,02 ∗ 0,1 ∗ 𝑉𝑉𝐵𝐵 ∗ 𝑟𝑟𝑟𝑟 If you buy 2% of stock A, you have 𝑉𝑉𝐴𝐴 = 0,014 𝑉𝑉𝐴𝐴 You lend 0,004 𝑉𝑉𝐴𝐴 such that 𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 = 0,018 𝑉𝑉𝐴𝐴 𝑁𝑁𝑁𝑁𝑁𝑁 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,02 ∗ (𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐴𝐴 − 0,3 𝑉𝑉𝐴𝐴 ∗ 𝑟𝑟𝑟𝑟) + 0,004 ∗ 𝑉𝑉𝐴𝐴 ∗ 𝑟𝑟𝑟𝑟 = 0,02 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐴𝐴 − 0,002 𝑉𝑉𝐴𝐴 ∗ 𝑟𝑟𝑟𝑟 c) If 𝑉𝑉𝐴𝐴 < 𝑉𝑉𝐵𝐵 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐴𝐴 < 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐵𝐵 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐴𝐴 > 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵 So you choose A It works if the capital structure is the same.
Exercise 2 a) 𝐷𝐷 After the refinancing, = 1 𝐸𝐸 Before there is no debt, 𝐵𝐵𝐴𝐴 = 𝐵𝐵𝐸𝐸 = 0,8 After : 𝐵𝐵𝐴𝐴 = 0,8 = 0,5 ∗ 𝐵𝐵𝐸𝐸 + 0,5 ∗ 0 So 𝐵𝐵𝐸𝐸 = 1,6
b) 𝑟𝑟𝑢𝑢 = 8% = 5% + 0,8 ∗ (𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟) => 𝑟𝑟𝑟𝑟 − 𝑟𝑟𝑟𝑟 = 3,75% 𝑟𝑟𝑒𝑒 = 5% + 1,6 ∗ 3,75% = 11% Or
𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑢𝑢 + �𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑓𝑓 � ∗
𝐷𝐷 = 8% + (8% − 5%) ∗ 1 = 11% 𝐸𝐸
c) 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑟𝑟𝑎𝑎 = 11% ∗ 0,5 + 5% ∗ 0,5 = 8% = 𝑟𝑟𝑢𝑢 d)
𝐸𝐸𝐸𝐸𝑆𝑆1 => 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑃𝑃0 ∗ 𝑟𝑟𝑒𝑒 𝑟𝑟𝑒𝑒 𝑃𝑃0 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐, 𝑟𝑟𝑒𝑒 => 8% 𝑡𝑡𝑡𝑡 11% => +37,5%, 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸𝐸𝐸: + 37,5%
𝑃𝑃0 =
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e) 1 1 𝑃𝑃0 = = = 9,0909 𝐸𝐸𝐸𝐸𝑆𝑆1 𝑟𝑟𝑒𝑒 0,11
Exercise 3 a) Save from the debt : 40𝑀𝑀$ ∗ 0,09 ∗ 0,35 = 1,26𝑀𝑀$
b) Debt change permanent 𝐷𝐷 ∗ 𝑟𝑟𝑑𝑑 ∗ 𝑡𝑡𝑐𝑐 = 𝐷𝐷 ∗ 𝑡𝑡𝑐𝑐 = 40 ∗ 0,35 = 14𝑀𝑀$ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 𝑟𝑟𝑑𝑑 c)
𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 1,26 ∗
1 − 1,09−10 = 8,09𝑀𝑀$ 0,09
d) Interest rate drop to 7%, but debt fixed rate 1,26 = 18𝑀𝑀$ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇)𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,07 1 − 1,07−10 = 8,85𝑀𝑀$ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇)10𝑦𝑦 = 1,26 ∗ 0,07
Exercise 4 a) 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 𝑉𝑉𝑈𝑈 + 𝐷𝐷 ∗ 𝑡𝑡𝑐𝑐 = 40 ∗ 0,4 = 16 b) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∶ 20 ∗ 0,4 = 8
c) 𝑟𝑟𝑑𝑑 = 8% Save from debt (annual) : 8% ∗ 40𝑀𝑀 ∗ 0,4 = 3,2𝑀𝑀$ ∗ 0,4 = 1,28𝑀𝑀$ 1,28 = 16 => 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ! 𝐼𝐼𝐼𝐼 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑤𝑤𝑤𝑤 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 0,08 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇)(5𝑦𝑦!) = 1,28 ∗
1 − 1,08−5 = 5,1107𝑀𝑀$ 0,08
So if there is no deductibility after 5 years, you lose : 16𝑀𝑀 − 5,1107𝑀𝑀 = 10,8893𝑀𝑀$
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 160 − 10,8893 = 149,1107𝑀𝑀$
Exercise 5 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 2𝑀𝑀$ 𝑝𝑝𝑝𝑝𝑝𝑝 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑉𝑉 = 𝐸𝐸 = 12𝑀𝑀$ 𝑡𝑡𝑐𝑐 = 0,4 𝛽𝛽 = 1 𝑟𝑟𝑚𝑚 = 15% 𝑟𝑟𝑓𝑓 = 9% 𝑟𝑟𝑑𝑑 = 12% 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) = 8𝑀𝑀$
47
Maths
Exercises
a) 𝑟𝑟𝑒𝑒 = 0,09 + 1 ∗ 0,06 = 15% = 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 b) Value Debt 2,5 5
Var Value debt 2,5 2,5
Marginal Tax Benefit 1𝑀𝑀 1𝑀𝑀
Probability Default 0% 8%
Var Prob Default 0% 8%
8
0,5
0,2𝑀𝑀
30%
9,5%
7,5
2,5
1𝑀𝑀
20,5%
Marg cost debt 0 0,08 ∗ 8 = 0,64𝑀𝑀 0,115 ∗ 8 = 1𝑀𝑀 0,095 ∗ 8 = 0,76𝑀𝑀 1,2𝑀𝑀 0,6𝑀𝑀 1,4𝑀𝑀
12,5%
9 1 0,4𝑀𝑀 45% 15% 10 1 0,4𝑀𝑀 52,5% 7,5% 12,5 2,5 1𝑀𝑀 70% 17,5% We see at Value Debt=7,5 it’s not any more interesting So 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) − 𝐸𝐸�𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)� = 12𝑀𝑀 + 1𝑀𝑀 + 1𝑀𝑀 − 0,64𝑀𝑀 = 13,36𝑀𝑀
For example is you have more debt, you have 13,36 + 1 − 1 = 13,36𝑀𝑀, it doesn’t increase the value of the firm, and after it decreases the value : 13,36𝑀𝑀 + 0,2 − 0,76 = 12,8𝑀𝑀 … Exercise 6
Exercise 7 𝑉𝑉0 = 1,7𝑀𝑀$ 𝐷𝐷 = 0,5𝑀𝑀$ 𝑟𝑟𝑑𝑑 = 10% 𝑡𝑡𝑐𝑐 = 34% 𝑟𝑟𝑢𝑢 = 20%
a) 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) 𝑉𝑉𝑈𝑈 = 𝑉𝑉𝐿𝐿 − 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 1 700 000 − 500 000 ∗ 0,34 = 1 530 000$ b)
𝐷𝐷 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 0,5 ∗ (1 − 0,34) 𝑟𝑟𝑒𝑒 = 0,2 + (0,2 − 0,1) ∗ 1,2 𝑟𝑟𝑒𝑒 = 22,75% 𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑢𝑢 + (𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑑𝑑 ) ∗
𝑉𝑉𝐸𝐸 =
𝑑𝑑𝑑𝑑𝑑𝑑 => 𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑟𝑟𝑒𝑒 ∗ 𝑉𝑉𝐸𝐸 = 0,2275 ∗ 1,2𝑀𝑀 = 273 000$ 𝑟𝑟𝑒𝑒
Exercise 8 𝑟𝑟𝑓𝑓 = 4% ; 𝑟𝑟𝑚𝑚𝑚𝑚 = 5% ; 𝑡𝑡𝑐𝑐 = 29% A 𝐷𝐷 = 0,35 𝐷𝐷 + 𝐸𝐸 𝑟𝑟𝑑𝑑 = 4,5%
B 𝐷𝐷 = 0,5 𝐷𝐷 + 𝐸𝐸 𝑟𝑟𝑑𝑑 = 5% 48
C No debt 𝐵𝐵𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1,4 = 𝐵𝐵𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
Maths
Exercises
𝐷𝐷 𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑢𝑢 + (𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑑𝑑 ) ∗ ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 0,35 𝑟𝑟𝑒𝑒 = 11,5 + (11,5 − 4,5) ∗ 0,65 ∗ (1 − 0,29) = 14,1762% 𝑟𝑟𝑎𝑎 = 14,1762 ∗ 0,65 + (1 − 0,29) ∗ 4,5 ∗ 0,35 = 10,3328%
𝑟𝑟𝑒𝑒 = 11,5 + (11,5 − 5) ∗ (1 − 0,29) = 16,115%
𝑟𝑟𝑒𝑒 = 4,5 + 1,4 ∗ 5 = 11,5%
𝑟𝑟𝑎𝑎 = 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 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 120𝑀𝑀$ 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 10,5𝑀𝑀$ 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 15𝑀𝑀$ 𝑔𝑔 = 6%
a) 𝑟𝑟𝑒𝑒 = 8 + 1,05 ∗ 5,5 = 13,775% 𝑟𝑟𝑎𝑎 = 0,90909 ∗ 13,775% + 10,3% ∗ 0,090909 ∗ (1 − 0,34) = 13,14%
b) 𝐹𝐹𝐹𝐹𝐹𝐹0 = 𝐸𝐸𝐸𝐸𝐸𝐸𝑇𝑇0 ∗ (1 − 𝑡𝑡𝑐𝑐 ) + 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐹𝐹𝐹𝐹𝐹𝐹0 = 120 ∗ (1 − 0,34) + 10,5 − 15 = 74,7 74,7 = 1108,85 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0,13775 − 0,06 c)
𝐷𝐷 𝛽𝛽𝐸𝐸 = 𝛽𝛽𝑈𝑈 + (𝛽𝛽𝑈𝑈 − 𝛽𝛽𝐷𝐷 ) ∗ ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 1,05 = 0,9850 𝛽𝛽𝐴𝐴 = 𝛽𝛽𝑈𝑈 = 1 + 0,1 ∗ (1 − 0,34) d)
𝛽𝛽𝐸𝐸 = 0,9850 + (0,9850) ∗
𝐷𝐷 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸
49
𝑟𝑟𝑎𝑎 = 11,5%
Maths
Exercises
So that you can calculate 𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑓𝑓 + 𝛽𝛽𝐸𝐸 ∗ (𝐸𝐸(𝑟𝑟𝑚𝑚 ) − 𝑟𝑟𝑟𝑟) So that you can calculate 𝐸𝐸 𝐷𝐷 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑟𝑟𝑎𝑎 = 𝑟𝑟𝑒𝑒 ∗ + 𝑟𝑟𝑑𝑑 ∗ ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐴𝐴 𝐴𝐴 𝐹𝐹𝐹𝐹𝐹𝐹0 ∗ (1,06) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 0,06 So that you have now the value !!
50
Maths
Exercises
Problem Set 11 Exercise 1 𝑟𝑟𝑢𝑢 = 12%
600000 700000 + = 93750$ 1,12 1,122 300000 ∗ 0,08 ∗ 0,3 = 7200 300000 ∗ 0,08 ∗ 0,3 = 3600 2 7200 3600 𝑃𝑃𝑃𝑃(𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖) = + = 9753,0864 1,08 1,082 𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑁𝑁𝑁𝑁𝑉𝑉𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 93750 + 9753,0864 = 103503,0864$
𝑁𝑁𝑁𝑁𝑁𝑁 = −1000000 +
Exercise 2 𝑡𝑡𝑐𝑐 = 40% 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 2000$ 𝑔𝑔 = 3% 𝑟𝑟𝑓𝑓 = 5% 𝑟𝑟𝑚𝑚 = 11% 𝛽𝛽𝐴𝐴 = 1,11
a) 𝑟𝑟𝑈𝑈 = 𝑟𝑟𝐴𝐴 = 5 + 1,11 ∗ (11 − 5) = 11,66 2000 ∗ (1 − 0,4) = 13856,8129$ 𝑉𝑉𝑈𝑈 = 0,1166 − 0,03
b) 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 5000 ∗ 0,05 = 250$ c)
𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) =
250 ∗ 0,4 = 1154,7344 0,1166 − 0,03
d) 𝑉𝑉𝐿𝐿 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 13856,8129 + 1154,7344 = 15011,5473$ So 𝐸𝐸 = 𝐴𝐴 − 𝐷𝐷 = 15011,5473 − 5000 = 10011,5473$ e)
𝐹𝐹𝐹𝐹𝐹𝐹1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 𝑔𝑔 2000 ∗ (1 − 0,4) 𝐹𝐹𝐹𝐹𝐹𝐹1 − 𝑔𝑔 = + 0,03 = 10,9938% 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 15011,5473 𝑉𝑉𝐿𝐿 𝑉𝑉𝐿𝐿 = f)
𝑉𝑉 𝐸𝐸 𝐷𝐷 𝐷𝐷 ∗ 𝑟𝑟 + ∗ 𝑟𝑟 ∗ (1 − 𝑡𝑡𝑐𝑐 ) => 𝑟𝑟𝑒𝑒 = �𝑟𝑟𝐴𝐴 − ∗ 𝑟𝑟𝑑𝑑 ∗ (1 − 𝑡𝑡𝑐𝑐 )� ∗ 𝐸𝐸 𝑉𝑉 𝑒𝑒 𝑉𝑉 𝑑𝑑 𝑉𝑉 5000 15011,5473 𝑟𝑟𝑒𝑒 = �0,109938 − ∗ 0,05 ∗ (1 − 0,4)� ∗ = 14,9861% 15011,5473 10011,5473 𝑟𝑟𝐴𝐴 =
g) ?? 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹1 = (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸1 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖1) ∗ (1 − 𝑡𝑡𝑐𝑐 ) + 𝑛𝑛𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹1 = (2000 − 250) ∗ (1 − 0,4) + 150 = 1200 //3% ∗ 5000 = 150. 51
Maths
Exercises
𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 grow by 3% 1200 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹1 = = 10 011,5473$ 𝐸𝐸 = 0,149862 − 0,03 𝑟𝑟𝑒𝑒 − 𝑔𝑔
Exercise 3 𝐷𝐷 = 400000$ 𝑡𝑡𝑐𝑐 = 0,35 𝑟𝑟𝑒𝑒 = 0,1 𝑟𝑟𝑑𝑑 = 0,07 95000 ∗ (1 − 0,35) 𝑁𝑁𝑁𝑁𝑁𝑁 = − 1000000 = −382 500$ 0,1 a)
𝐷𝐷 ∗ 𝑟𝑟𝑑𝑑 ∗ 𝑡𝑡𝑐𝑐 400000 ∗ 0,07 ∗ 0,35 = = 400000 ∗ 0,35 = 140 000$ 𝑟𝑟𝑑𝑑 0,07 𝐴𝐴𝐴𝐴𝐴𝐴 = −242 500$ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) =
b) Why 𝑟𝑟𝐴𝐴 ??
𝐷𝐷 ∗ 𝑟𝑟𝑑𝑑 ∗ 𝑡𝑡𝑐𝑐 400000 ∗ 0,07 ∗ 0,35 = = 98 000$ 𝑟𝑟𝐴𝐴 0,1 𝐴𝐴𝐴𝐴𝐴𝐴 = −284 000$ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) =
Exercise 4 No link between 𝑟𝑟𝑒𝑒 common stock and 𝑟𝑟𝑒𝑒 He needs to find the business risk of this project
Exercise 5 𝐸𝐸 = 24,27 𝑏𝑏$ 𝐷𝐷 = 2,8 𝑏𝑏$ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 24,27 + 2,8 = 27,07 𝛽𝛽𝐸𝐸 = 1,47 𝑡𝑡𝑐𝑐 = 0,4 𝑟𝑟𝑓𝑓 = 6,5% 𝑟𝑟𝑚𝑚𝑚𝑚 = 5,5%
a) 𝑟𝑟𝑑𝑑 = 6,5% + 0,3% = 6,8% 𝑟𝑟𝑒𝑒 = 6,5 + 1,47 ∗ 5,5 = 14,585% 𝐷𝐷 𝐸𝐸 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = ∗ 𝑟𝑟𝑒𝑒 + ∗ 𝑟𝑟𝑑𝑑 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝑉𝑉 𝑉𝑉 2,8 24,27 ∗ 0,14585 + ∗ 0,068 ∗ (1 − 0,4) = 13,4984% 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 27,07 27,07 b) 𝐷𝐷 = 0,3 𝐸𝐸 𝐷𝐷 𝐷𝐷 0,3 = = = 23,08% 𝐷𝐷 + 𝐸𝐸 1,3𝐸𝐸 1,3 𝐸𝐸 = 76,92% 𝑉𝑉 𝑟𝑟𝑑𝑑 = 6,5𝑀𝑀 + 2% = 8,5% 24,27 2,8 ∗ 0,14585 + ∗ 0,068 = 13,7798% 27,07 27,07 𝑟𝑟𝑒𝑒 = 0,137798 + (0,137798 − 0,085) ∗ 0,3 = 15,3637%
𝑟𝑟𝑢𝑢 = 𝑟𝑟𝐴𝐴 =
52
Maths
Exercises
𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 0,152637 ∗ 0,7692 + 0,085 ∗ 0,2308 ∗ 0,6 = 12,9948%
Note: without tax shield we have : 𝑟𝑟𝐴𝐴 = 0,152637 ∗ 0,7692 + 0,085 ∗ 0,2308 = 13,7798% = 𝑟𝑟𝑢𝑢 !
c)
𝐹𝐹𝐹𝐹𝐹𝐹1 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐴𝐴 − 𝑔𝑔 𝐹𝐹𝐹𝐹𝐹𝐹1 𝑉𝑉𝐿𝐿𝐵𝐵 = 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐵𝐵 − 𝑔𝑔 𝑉𝑉𝐿𝐿𝐴𝐴 ∗ (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐴𝐴 − 𝑔𝑔) = 𝑉𝑉𝐿𝐿𝐵𝐵 ∗ (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐵𝐵 − 𝑔𝑔) (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐴𝐴 − 𝑔𝑔) (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐵𝐵 − 𝑔𝑔) − 𝑉𝑉𝐿𝐿𝐵𝐵 ∗ 𝑉𝑉𝐿𝐿𝐵𝐵 − 𝑉𝑉𝐿𝐿𝐴𝐴 = 𝑉𝑉𝐿𝐿𝐴𝐴 ∗ (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐵𝐵 − 𝑔𝑔) (𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐴𝐴 − 𝑔𝑔) 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶 13,4984 −6 − 𝑔𝑔 𝐴𝐴 − 1 = 24,27 ∗ −1= =? 𝑉𝑉𝐿𝐿𝐴𝐴 ∗ 12,9948 − 6 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶𝐵𝐵 − 𝑔𝑔 = 1,9485 𝑏𝑏$ 𝑉𝑉𝐿𝐿𝐴𝐴 =
Stock price increases =
Exercise 6 𝐷𝐷 = 527 𝑀𝑀$ 𝐸𝐸 = 1,76 𝑏𝑏$ 𝑉𝑉 = 2287 𝑀𝑀$ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 131 𝑀𝑀$ 𝑡𝑡𝑐𝑐 = 36% 𝑟𝑟𝑑𝑑 = 8% 𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) = 2,3% 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) = 30% 𝑟𝑟𝑓𝑓 = 6%
1,9485 24,27
= 8,0285%
a) 𝑉𝑉𝑢𝑢 = 𝑉𝑉𝐿𝐿 − 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) + 𝑃𝑃𝑃𝑃�𝐸𝐸(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐)� 𝑉𝑉𝑈𝑈 = 2287 − 527 ∗ 0,36 + 0,3 ∗ 0,023 ∗ (2287 − 527 ∗ 0,36) 𝑉𝑉𝑈𝑈 = 2111,7512 𝑀𝑀$
b) 𝐷𝐷 = 0,5 𝐸𝐸 0,5 1 𝐷𝐷 = = 𝐷𝐷 + 𝐸𝐸 1,5 3 𝑉𝑉𝐿𝐿2 = 𝑉𝑉𝑈𝑈 + 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) − 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) 1 1 𝑉𝑉𝐿𝐿2 = 2111,7512 + 0,36 ∗ ∗ 𝑉𝑉𝐿𝐿2 − 0,3 ∗ 0,4661 ∗ �𝑉𝑉𝐿𝐿2 − ∗ 𝑉𝑉𝐿𝐿2 ∗ 0,36� 3 3 𝑉𝑉𝐿𝐿2 = 2105,3 𝑀𝑀$ 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 = 9𝑀𝑀$
500000$
Total market value 4M$ 5M$ No economic gain, so total value= 4 + 5 = 9𝑀𝑀$ 700000 = 262172,2846 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 2,67 9𝑀𝑀 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = = 34,3286$ 262172,2846 34,3286 𝑃𝑃 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = = 12,8572 2,67 𝐸𝐸 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ∶ 262172,2846 − 1000000 = 162172,2846 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 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) 𝑡𝑡𝑐𝑐 = 0,5 𝐹𝐹𝐹𝐹𝐹𝐹𝐽𝐽𝐽𝐽𝐽𝐽 = 2000 ∗ (1 − 0,5) = 1000$ 𝐹𝐹𝐹𝐹𝐹𝐹𝑆𝑆𝑆𝑆𝑆𝑆 = 1600 ∗ (1 − 0,5) = 800$ 𝐹𝐹𝐹𝐹𝐹𝐹0 ∗ (1 + 𝑔𝑔) 1000 ∗ 1,04 = = 20800$ 𝑉𝑉𝐽𝐽𝐽𝐽𝐽𝐽 = 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 𝑔𝑔 0,09 − 0,04 800 ∗ 1,06 𝑉𝑉𝑆𝑆𝑆𝑆𝑆𝑆 = = 21200$ 0,1 − 0,06 𝑉𝑉 = 20800 + 21200 = 42000$ Rev -cogs EBIT g
No synergy 12000 −8400 3600 20,8 21,2 𝑔𝑔 = ∗ 0,04 + ∗ 0,06 42 42 = 5,0095% 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,5048%
Synergy 1 12000 −0,65 ∗ 12000 4200 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆
WACC 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 8400 = 0,7 12000 𝑛𝑛𝑛𝑛𝑛𝑛 = 0,65 12000 − 0,65 ∗ 12000 = 4200 4200 ∗ (1 − 0,5) ∗ (1,050095) − 42000 = 7055,6693$ 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠 1 = 0,095048 − 0,050095 3600 ∗ (1 − 0,5) ∗ 1,06 − 42000 = 12439,6257$ 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠 2 = 0,095048 − 0,06 Exercise 9 ?? 𝑟𝑟𝑟𝑟𝑣𝑣0 = 300 𝑀𝑀$ 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,5% 𝑡𝑡𝑐𝑐 = 0,3 𝑟𝑟𝑓𝑓 = 5,5% 𝑔𝑔 = 5% 𝑟𝑟𝑑𝑑 = 8% 𝑟𝑟𝑒𝑒 = 13% Date Rev
0
1 300 ∗ 1,05 ∗ 0,06 54
2 300 ∗ 1,052 ∗ 0,06
Synergy 2 12000 −8400 3600 6% 9,5048%
3 …
Maths
Exercises −40𝑀𝑀 −40 18 ∗ 1,05 −40 ∗ 0,7 18 ∗ 1,05 ∗ 0,7 18 ∗ 0,7 ∗ 1,05 = −40 ∗ 0,7 + = 137,3750 𝑀𝑀$ 0,13 − 0,05
Cost Ebit With tax 𝑁𝑁𝑁𝑁𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠
Exercise 10 𝑃𝑃𝐴𝐴 0,01 10 15 20 30 40 50
𝐸𝐸 𝑅𝑅 28 :1 0,01 28 :1 10 28 :1 15 1,4 ∶ 1 1,4 ∶ 1 1,4 ∶ 1 56 :1 50
18 ∗ 1,05 18 ∗ 1,052 ∗ 0,7
… … …
𝑛𝑛𝑛𝑛𝑛𝑛ℎ
𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑏𝑏𝑏𝑏𝑏𝑏 𝑝𝑝𝑝𝑝𝑝𝑝 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎
2,8
28
2800 1,87 1,4 1,4 1,4 1,12
55
2
28 28 28 42 56 56
