Exercises

Financial Economics Maths Exam Exam 2002 Problem 1 a) Bank 50 000 000 ∗ 0,1% = 50 000€ per year

b) we have : 𝑃𝑃(𝐵𝐵1) = 220 ∗ 1,08−1 𝑃𝑃(𝐵𝐵2) = 5 ∗ 1,085−1 + 105 ∗ 1,085−2 𝑃𝑃(𝐵𝐵3) = 10 ∗ 1,09−1 + 10 ∗ 1,09−2 + 110 ∗ 1,09−3

We seek 𝑖𝑖1, 𝑖𝑖2, 𝑖𝑖3 𝑤𝑤𝑤𝑤𝑤𝑤ℎ

𝑃𝑃(𝐵𝐵1) = 220 ∗ (1 + 𝑖𝑖1)−1 � 𝑃𝑃(𝐵𝐵2) = 5 ∗ (1 + 𝑖𝑖1)−1 + 105 ∗ (1 + 𝑖𝑖2)−2 𝑃𝑃(𝐵𝐵3) = 10 ∗ (1 + 𝑖𝑖1)−1 + 10 ∗ (1 + 𝑖𝑖2)−2 + 110 ∗ (1 + 𝑖𝑖3)−3 So that 𝑖𝑖1 = 8% Then

𝑃𝑃(𝐵𝐵1) = 220 ∗ 1,08−1 𝑃𝑃(𝐵𝐵2) = 5 ∗ 1,08−1 + 105 ∗ (1 + 𝑖𝑖2)−2 � 𝑃𝑃(𝐵𝐵3) = 10 ∗ 1,08−1 + 10 ∗ (1 + 𝑖𝑖2)−2 + 110 ∗ (1 + 𝑖𝑖3)−3 105 5∗1,085 −1 +105∗1,085 −2 −5∗1,08 −1

We have 𝑖𝑖2 = �

Then on the same way, We have 𝑖𝑖3 =

− 1 = 8,513

c) The duration is the actuarial life mean of an asset. The dollar duration is the actuarial life mean of the dollar ? 𝑃𝑃(𝐵𝐵2) = 5 ∗ 1,085−1 + 105 ∗ 1,085−2 = 93,8 1 ∗ (5 ∗ 1,085−1 + 2 ∗ 105 ∗ 1,085−2 ) = 1,95 𝐷𝐷(𝐵𝐵2) = 93,8 d) 𝑃𝑃(𝐵𝐵4) = 8 ∗ (1,08−1 ) + 108 ∗ (1,08513−2 ) = 99,1266 At the 102, the price is too high. The profit is 102 − 99,1266 = 2,873 e)

1

110 3 � � 10∗1,09−1 +10∗1,09−2 +110∗1,09−3 −10∗1,08 −1 −10∗1,08513 −2

− 1 = 9,069%

Maths

Exercises

Problem 2 (Ex02) a) 1 1 1 1 1 𝜎𝜎² = ∗ � ∗ 0,52 + ∗ 0,5 ∗ 0,4 ∗ 0,1 + ∗ 0,5 ∗ 0,3 ∗ 0,1� + 3 3 3 3 3 1 1 1 1 ∗ � ∗ 0,4 ∗ 0,5 ∗ 0,1 + ∗ 0,4² + ∗ 0,4 ∗ 0,3 ∗ 0,1� + 3 3 3 3 1 1 1 ∗ � ∗ 0,3 ∗ 0,5 ∗ 0,1 + ∗ 0,3 ∗ 0,4 ∗ 0,1 + ∗ 0,3²� = 0,066 3 3 3 𝜎𝜎 = 25,69% 1 1 1 2 1 3 ∗ 0,5 + 3 ∗ 0,5 ∗ 0,4 ∗ 0,1 + 3 ∗ 0,5 ∗ 0,3 ∗ 0,1 𝑥𝑥𝐴𝐴 = ∗ = 47,98% 0,066 3 1 1 1 ∗ 0,4 ∗ 0,5 ∗ 0,1 + 3 ∗ 0,4² + 3 ∗ 0,4 ∗ 0,3 ∗ 0,1 3 𝑥𝑥𝐵𝐵 = = 32,32% 0,066 1 1 1 ∗ 0,3 ∗ 0,5 ∗ 0,1 + 3 ∗ 0,3 ∗ 0,4 ∗ 0,1 + 3 ∗ 0,3² 3 𝑥𝑥𝐶𝐶 = = 19,69% 0,066

b) 22 = 5 + 𝛽𝛽𝑃𝑃 ∗ 6 𝛽𝛽𝑝𝑝 = 2,833 𝛽𝛽𝑚𝑚 = 1 𝛽𝛽𝑓𝑓 = −1,833 𝜎𝜎𝑝𝑝 = 2,833 ∗ 𝜎𝜎𝑀𝑀 = 2,833 ∗ 25,69 = 72,78% c)

1 1 1 ∗ 0,52 + 3 ∗ 0,5 ∗ 0,4 ∗ 0,1 + 3 ∗ 0,5 ∗ 0,3 ∗ 0,1 3 𝛽𝛽𝐴𝐴 = = 1,439 0,066 𝑟𝑟𝐴𝐴 = 𝑟𝑟𝑓𝑓 + 𝛽𝛽𝐴𝐴 �𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 � = 4 + 1,439 ∗ 6 = 13,634% 𝑟𝑟𝑝𝑝 = 𝑥𝑥𝑎𝑎 ∗ 𝑟𝑟𝑎𝑎 + 𝑥𝑥𝑓𝑓 ∗ 𝑟𝑟𝑓𝑓 = 22% 𝑥𝑥𝑎𝑎 + 𝑥𝑥𝑓𝑓 = 1 22 = 𝑥𝑥𝑎𝑎 ∗ 13,634 + (1 − 𝑥𝑥𝑎𝑎 ) ∗ 5 𝑥𝑥𝑎𝑎 = 1,969 𝑥𝑥𝑓𝑓 = −0,969 𝜎𝜎𝑃𝑃 = 𝑥𝑥𝑎𝑎 ∗ 𝜎𝜎𝑎𝑎 = 1,969 ∗ 0,5

d) Firm A Asset 200 liquid => 3% 400 equity 800 600 debt at 6% Total 1000 Total 1000 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∗ 0,6 + 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 0,4 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 13,634 (firm A => share A !!, see question c) 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 6% 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 9,0536% 2 8 ∗ 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 + ∗ 𝑟𝑟 = 9,0536% 10 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 10 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 10,567% Firm E Asset 150 : 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

1

=3

100 equity 2

Maths

Exercises

150 : 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 Total 300

2

= 0 because uncorrelated

200 debt at 6,5% Total 300

Firm A+E Asset 100 liquid (because -100 to buy firm E) at 3% 300 at 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 1,5 800 at 10,567% Total 1200 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 1,5 𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 5 + 1,5 ∗ 6 = 14% 100

800

400 equity 200 debt at 6,5% 600 debt at 6% Total 1200

300

So 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = ∗3+ ∗ 10,567 + ∗ 14 = 10,795% 1200 1200 1200 10,795 = 5 + 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 6 => 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,966 200 600 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ∗ 6,5 + ∗ 6 = 6,125% 800 800 400 800 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 10,795 = ∗ 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + ∗ 6,125 1200 1200 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 20,135% 𝑟𝑟𝑒𝑒𝑒𝑒𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 20,135 = 5 + 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 6 => 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 2,523

3

Maths

Exercises

RE-Exam 2002 Problem 1 a) 0,5 ∗ 12,9 + 0,35 ∗ 15,4 + 0,15 ∗ 14,5 = 14,015

b) 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0,5 ∗ (0,5 ∗ 0,32 + 0,35 ∗ 0,3 ∗ 0,4 ∗ 0,3 + 0,15 ∗ 0,3 ∗ 0,6 ∗ 0,1) + 0,35 ∗ (0,5 ∗ 0,3 ∗ 0,4 ∗ 0,3 + 0,35 ∗ 0,42 + 0,15 ∗ 0,4 ∗ 0,6 ∗ 0,1) + 0,15 ∗ (0,5 ∗ 0,1 ∗ 0,3 ∗ 0,6 + 0,35 ∗ 0,1 ∗ 0,4 ∗ 0,6 + 0,15 ∗ 0,62 ) = 0,06802 𝜎𝜎𝜎𝜎 = 26,8319% 0,5 ∗ 0,32 + 0,35 ∗ 0,3 ∗ 0,4 ∗ 0,3 + 0,15 ∗ 0,3 ∗ 0,6 ∗ 0,1 = 0,8865 𝛽𝛽𝐴𝐴 = 0,06802 0,5 ∗ 0,3 ∗ 0,4 ∗ 0,3 + 0,35 ∗ 0,42 + 0,15 ∗ 0,4 ∗ 0,6 ∗ 0,1 = 1,1408 𝛽𝛽𝐵𝐵 = 0,06802 0,5 ∗ 0,1 ∗ 0,3 ∗ 0,6 + 0,35 ∗ 0,1 ∗ 0,4 ∗ 0,6 + 0,15 ∗ 0,62 = 1,0497 𝛽𝛽𝐶𝐶 = 0,06802

c) Yes because there are two value >14%... Example with stock A and C 14 = 14,5 ∗ 𝑥𝑥𝑐𝑐 + 12,9 ∗ 𝑥𝑥𝑎𝑎 = 14,5 ∗ 𝑥𝑥𝑐𝑐 + (1 − 𝑥𝑥𝑐𝑐 ) ∗ 12,9 = 1,6 ∗ 𝑥𝑥𝑐𝑐 + 12,9 14 − 12,9 = 1,6 ∗ 𝑥𝑥𝑐𝑐 => 𝑥𝑥𝑐𝑐 = 68,75% Example with stock C and Tbill 14 = 14,5 ∗ 𝑥𝑥𝑐𝑐 + 4 ∗ 𝑥𝑥𝑓𝑓 = 10,5𝑥𝑥𝑐𝑐 + 4 => 𝑥𝑥𝑐𝑐 = 0,9524

d) 𝐷𝐷 𝐸𝐸 1 𝐷𝐷 = 1 => 𝐷𝐷 = 𝐸𝐸 => = = 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 2 𝐸𝐸 1 1 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = ∗ 0,8865 + ∗ 0,1 = 0,493 2 2 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 4 + 0,493 ∗ (14,015 − 4) = 8,94

e) With 30 different stocks there is diversification, the volatility of the portfolio will be lower than 60% With only 1 stock the volatility of portfolio will be 60%, it’s very dangerous, more than with 30 stocks.

4

Maths

Exercises

Exam 2003 Problem 2 a)

100 = 83,4 1,0952 10 110 𝑃𝑃𝑃𝑃2 = + = 100,918 1,09 1,0952 ⎨𝑃𝑃𝑃𝑃3 = 10 + 10 + 110 = 100,159 1,09 1,0952 1,13 ⎪ ⎪ 100 ⎪ 𝑃𝑃𝑃𝑃4 = = 67,07 ⎩ 1,1054 1 2 ∗ 100 𝐷𝐷𝐷𝐷1 = ∗ =2 ⎧ 83,4 1,0952 ⎪ 1 10 2 ∗ 110 ⎪ ⎪ 𝐷𝐷𝐷𝐷2 = ∗� + � = 1,9 100,918 1,09 1,0952 1 10 2 ∗ 10 3 ∗ 110 ⎨𝐷𝐷𝐷𝐷3 = ∗� + + � = 2,734 100,159 1,09 1,0952 ⎪ 1,13 ⎪ 1 4 ∗ 100 ⎪ 𝐷𝐷𝐷𝐷4 = ∗� �=4 ⎩ 67,07 1,1054 ⎧ ⎪ ⎪ ⎪

𝑃𝑃𝑃𝑃1 =

b)

20 = 10𝑏𝑏 + 10𝑐𝑐 20 = 100𝑎𝑎 + 110𝑏𝑏 + 10𝑐𝑐 � 20 = 110𝑐𝑐 220 = 100𝑑𝑑 20 − 110 ∗ 1,818 − 10 ∗ 0,182 = −1,818 ⎧𝑎𝑎 = 100 ⎪ ⎪ 𝑏𝑏 = 2 − 0,182 = 1,818 20 𝑐𝑐 = = 0,182 ⎨ 110 ⎪ 220 ⎪ = 2,2 𝑑𝑑 = ⎩ 110

5

Maths

Exercises

Exam 2004 Problem 1 (Ex04) a) Paalzowoil 𝑃𝑃(10%) = 0,4 𝑃𝑃(5%) = 0,6 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,4 ∗ 10 + 0,6 ∗ 5 = 7%

Hillercomputer 𝐸𝐸 1 𝐷𝐷 = = 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 2 𝛽𝛽ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 2 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 0,2 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = (2 − 0,5 ∗ 0,2) ∗ 2 = 3,8 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑟𝑟𝑓𝑓 + 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ �𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 � => 𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 = 5% (𝑟𝑟𝑚𝑚 = 9%) 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 4 + 3,8 ∗ 5 = 23%

b) 𝑟𝑟𝑃𝑃 = 30% 𝑥𝑥𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,1 𝑥𝑥ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑥𝑥𝑓𝑓 = 0,9 30 = 0,1 ∗ 7 + 𝑥𝑥ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ 23 + �0,9 − 𝑥𝑥ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 � ∗ 4 29,3 − 3,6 𝑥𝑥ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = = 1,353 19 𝑥𝑥𝑓𝑓 = −0,453 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑖𝑖𝑖𝑖 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 = 1,353 ∗ 5 000 000 = 6 765 000 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,1 ∗ 5 000 000 = 500 000 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = −0,453 ∗ 5 000 000 = −2 265 000 c) 𝜌𝜌𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

= 0,5

𝜎𝜎𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = �0,4 ∗ 3² + 0,6 ∗ (−2)² = 2,449%

𝜎𝜎ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 0,2 𝑥𝑥𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,1 𝑥𝑥ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 1,353

𝑉𝑉𝑉𝑉𝑉𝑉𝐾𝐾𝐾𝐾 = 0,1 ∗ (0,1 ∗ 0,02449² + 1,353 ∗ 0,02449 ∗ 0,2 ∗ 0,5) + 1,353 ∗ (0,1 ∗ 0,02449 ∗ 0,2 ∗ 0,5 + 1,353 ∗ 0,22 ) = 0,07389 𝑟𝑟𝐾𝐾𝐾𝐾 = 27,18%

d) 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 8% 𝜎𝜎𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 0,8 𝜎𝜎𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 15% 𝑟𝑟𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 10% + 4% = 14% 𝑟𝑟𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 4 + 0,8 ∗ 5 = 8% 𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

14 + 7 + 23 = 14,67% 3 = 0,6

𝑤𝑤𝑤𝑤𝑤𝑤 ℎ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

=

6

Maths 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

Exercises 𝑤𝑤𝑤𝑤𝑤𝑤 ℎ 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 =

=

3,8 + 0,6 + 3,8 = 2,733 3

14,67 − 4 = 3,9 (𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛 5,009 ? ) 2,73

8 + 7 + 23 = 12,667% 3 0,8 + 0,6 + 3,8 𝛽𝛽𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑤𝑤𝑤𝑤𝑤𝑤 ℎ 𝑚𝑚𝑚𝑚𝑚𝑚 = = 1,733 3 12,667 − 4 = 5,001 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 1,733

𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝

𝑤𝑤𝑤𝑤𝑤𝑤 ℎ 𝑚𝑚𝑚𝑚𝑚𝑚

=

7

Maths

Exercises

Problem 3 (Ex04) Pizzabarn FA : 4000 CA : 3000 7000 𝐸𝐸𝐸𝐸𝐸𝐸: 0,2 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,4 𝑔𝑔 = 10% 𝑟𝑟𝑒𝑒 = 15% 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 200$ 𝑟𝑟𝑑𝑑 = 9% 𝑡𝑡𝑐𝑐 = 20% 𝑟𝑟𝑓𝑓 = 5% 𝑟𝑟𝑚𝑚𝑚𝑚 = 6% 𝐸𝐸 𝐷𝐷 𝑟𝑟𝑎𝑎 = 𝑟𝑟𝑒𝑒 ∗ + 𝑟𝑟𝑑𝑑 ∗ ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝑉𝑉 𝑉𝑉

E : 5000 D : 2000 7000

So we must find E, D, V 𝑀𝑀𝑀𝑀(𝐸𝐸) = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 5000 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 =

𝐷𝐷𝐷𝐷𝐷𝐷 𝑟𝑟𝑒𝑒 − 𝑔𝑔

𝐷𝐷𝐷𝐷𝐷𝐷 => 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐸𝐸𝐸𝐸𝐸𝐸 ∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐸𝐸𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝐷𝐷 = 0,2 ∗ 0,4 = 0,08 0,08 = 1,6$ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 0,15 − 0,1

𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =

𝑀𝑀𝑀𝑀(𝐸𝐸) = 5000 ∗ 1,6 = 8000$ 𝑀𝑀𝑀𝑀(𝐷𝐷) = 200 ∗

1 − 1,09−3 2000 + = 2050,6259$ 0,09 1,093

𝑀𝑀𝑀𝑀(𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴) = 8000 + 2050,6259 = 10050,6259$

So we can now find 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 8000 2050,6259 𝑟𝑟𝐴𝐴 = 0,15 ∗ + 0,09 ∗ ∗ (1 − 0,20) = 13,4086% 10050,6259 10050,6259 If the leverage is 0,5, D=constant

1 𝐷𝐷 = 𝐷𝐷 + 𝐸𝐸 3 𝐷𝐷 𝑟𝑟𝑒𝑒 + ∗ 𝑟𝑟𝑑𝑑 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 𝑟𝑟𝑢𝑢 = 𝐷𝐷 1 + 𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 2050,6259 (1 0,15 + 8000 ∗ 0,09 ∗ − 0,2) = 13,9790% 𝑟𝑟𝑢𝑢 = 2050,6259 (1 1+ 8000 ∗ − 0,2) 𝐷𝐷 𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑢𝑢 + (𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑑𝑑 ) ∗ ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 𝑟𝑟𝑒𝑒 = 0,13979 + (0,13979 − 0,09) ∗ 0,5 ∗ 0,8 = 15,9706% So we can now find 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊

8

Maths

Exercises

2 1 𝑟𝑟𝐴𝐴 = 0,159706 ∗ + 0,09 ∗ ∗ 0,8 = 13,0471% 3 3

b) The problem : You buy a machine, you sell an old one, you save money in salaries with the new machine. And to achieve this project, you buy with same amount of equity and debt. You save money from tax, you lose money with the cost of raising debt and equity So at the end, you should have the following 𝐴𝐴𝐴𝐴𝐴𝐴 = −𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜𝑜 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

You can sell the old machine at its book value. It’s midlife, so you can sell it 320 Then you need : 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = −1000 + 320 = −680 You save the salaries : 200 per year 1 − 1,13979−10 = 835,2629$ 200 ∗ (1 − 0,2) ∗ 0,13979

The depreciation of the machine is on 10 years, 100 per year ! 1 − 1,13979−10 = 104,4079$ 100 ∗ 0,2 ∗ 0,13979 The depreciation of the old machine was on 8 year, 80 per year ! 4 years left !! 1 − 1,13979−4 = 46,6395$ 80 ∗ 0,2 ∗ 0,13979

1 − 1,09−10 2050,6259 � ∗ 0,2 ∗ 0,09 ∗ = 16,0270$ 0,09 10050,6259 (=>Need to raise 680 with the proportion of debt : 2050,6259/10050,6259 ; times cost of debt= interest saving, times tax shield (0,2)= interest tax saving ; times 10 year (annuity)) 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇? ) = �680 ∗

Finally, there is also a cost 7% for the equity and 3% for the debt 2050,6259 �680 ∗ � ∗ 0,8 ∗ 0,03 10050,6259 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = = 3,4327$ (1 − 0,03) 8000 �680 ∗ � ∗ 0,8 ∗ 0,07 10050,6259 = 32,5920$ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = (1 − 0,07)

So APV : 𝐴𝐴𝐴𝐴𝐴𝐴 = −680 + 835,2629 + 16,0270 + 104,4079 − 46,6395 − 3,4327 − 32,5920 𝐴𝐴𝐴𝐴𝐴𝐴 = 193,0336 c)

Cost Buy U Sell L Borrow at 10% Total 50 𝑅𝑅𝑅𝑅𝑅𝑅 𝑈𝑈 = = 12,5% 400 30 𝑅𝑅𝑅𝑅𝑅𝑅 𝐿𝐿 = = 10% 300

Receive −400 300 100 0

9

50 −30 −10 10

Maths

Exercises

Problem 4 (Ex04) a) 𝐸𝐸 = 80 𝑆𝑆 = 100 𝑟𝑟𝑟𝑟 = 3,75% 𝑢𝑢 = 1,15 𝑑𝑑 = 0,85 0,0375

𝑒𝑒 12 − 0,85 = 0,5104 𝑝𝑝 = 1,15 − 0,85 (1 − 𝑝𝑝) = 0,4896 𝑆𝑆𝑢𝑢 = 115 − 2% = 112,7

𝑆𝑆 = 100 𝐶𝐶𝑢𝑢 = 𝐶𝐶𝑑𝑑 =

0,5104 ∗ 49,6 + 0,4896 ∗ 15,8 0,0375

𝑒𝑒 12 0,5104 ∗ 15,8 + 0,4896 ∗ 0

𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸 =

0,0375 𝑒𝑒 12

= 32,948

= 8,039

0,5104 ∗ 32,948 + 0,4896 ∗ 8,039 0,0375 𝑒𝑒 12

49,6 − 15,8 =1 112,7 ∗ 0,3 15,8 𝛼𝛼𝑑𝑑∗ = = 0,632 83,3 ∗ 0,3 49,6 − 1 ∗ 129,6 = −79,75 𝐵𝐵𝑢𝑢 = 0,0375 𝑒𝑒 12 15,8 − 0,632 ∗ 95,8 = −44,606 𝐵𝐵𝑑𝑑 = 0,0375 𝑒𝑒 12 𝐶𝐶𝑢𝑢 = 1 ∗ 112,7 − 79,75 = 32,95 𝐶𝐶𝑑𝑑 = 0,632 ∗ 83,3 − 44,606 = 8,04 32,95 − 8,04 𝛼𝛼 = = 0,83 100 ∗ 0,3 32,95 − 0,83 ∗ 115 𝐵𝐵 = = −62,3 0,0375 𝑒𝑒 12 𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸 = 0,83 ∗ 100 − 62,3 = 20,7 𝛼𝛼𝑢𝑢∗ =

𝑆𝑆𝑑𝑑 = 85 − 2% = 83,3

= 20,69

35 − 8,04 = 0,899 100 ∗ 0,3 35 − 0,83 ∗ 115 𝐵𝐵1 = = −68,17 0,0375 12 𝑒𝑒 𝐶𝐶𝑈𝑈𝑈𝑈𝑈𝑈 = 21,73

𝛼𝛼1 =

b) 𝜎𝜎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 => 𝑙𝑙𝑙𝑙1,15 = 0,4841 𝜎𝜎1 = 1 � 12

10

𝑆𝑆𝑢𝑢𝑢𝑢 = 129,6 𝐶𝐶𝑢𝑢𝑢𝑢 = 49,6 𝑆𝑆𝑢𝑢𝑢𝑢 = 95,8 𝐶𝐶𝑢𝑢𝑢𝑢 = 15,8 𝑆𝑆𝑑𝑑𝑑𝑑 = 70,8

Maths 𝜎𝜎1 = −

Exercises 𝑙𝑙𝑙𝑙0,85

= 0,563 1 � 12 0,563 + 0,4841 = 0,5236 𝜎𝜎 = 2

1

𝑆𝑆 ∗ = 𝑆𝑆 − 𝐷𝐷 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 = 100 − 2% ∗ 100 ∗ 𝑒𝑒 −0,0375∗12 = 98,01

𝑑𝑑1 = 𝑑𝑑2 =

0,52362 2 98,01 ln � 80 � + �0,0375 + 2 � ∗ 12

2 0,5236 ∗ �12 0,52362 2 98,01 ln � 80 � + �0,0375 − 2 � ∗ 12

𝑁𝑁(𝑑𝑑1 ) = 0,8612 𝑁𝑁(𝑑𝑑2 ) = 0,8084

2 0,5236 ∗ �12

= 1,0860 = 0,8722

2

𝐶𝐶 = 98,01 ∗ 0,8612 − 80 ∗ 𝑒𝑒 −0,0375 ∗12 ∗ 0,8084 𝐶𝐶 = 20,14

c) 𝐶𝐶𝑚𝑚 = 19 Buy call and short sell stock.

d) A convertible bond is a bond which can be converted into share.

11

Maths

Exercises

RE-Exam 2004 Problem 1 a) 𝐸𝐸(𝑟𝑟𝑎𝑎 ) = 20 ∗ 0,5 + 7 ∗ 0,2 = 11,4% 𝐸𝐸(𝑟𝑟𝑏𝑏 ) = 15 ∗ 0,5 − 5 ∗ 0,3 + 5 ∗ 0,2 = 7% 𝑉𝑉𝑉𝑉𝑉𝑉(𝐴𝐴) = (20 − 11,4)2 ∗ 0,5 + 11,42 ∗ 0,3 + 4,42 ∗ 0,2 = 79,84 𝑉𝑉𝑉𝑉𝑉𝑉(𝐵𝐵) = 82 ∗ 0,5 + 122 ∗ 0,3 + 22 ∗ 0,2 = 76 𝜎𝜎𝐴𝐴 = 8,935% 𝜎𝜎𝐵𝐵 = 8,718% 𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴, 𝐵𝐵) = (8,6 − 8) ∗ 0,5 + 11,4 ∗ 12 ∗ 0,3 + 4,4 ∗ 2 ∗ 0,2 = 77,2 77,2 𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴, 𝐵𝐵) = 𝜌𝜌 = 𝜎𝜎𝐴𝐴 𝜎𝜎𝐵𝐵 √76 ∗ �79,54 𝜌𝜌 = 0,991 b) Janis : 0,5 ∗ 𝑟𝑟𝑓𝑓 + 0,5 ∗ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐵𝐵 𝑉𝑉𝑉𝑉𝑉𝑉 𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 = 0,52 ∗ 76 = 19 𝜎𝜎𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 = 4,3589% 1 3

2 3

Andris : ∗ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴 + ∗ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐶𝐶

1 2 2 2 𝑉𝑉𝑉𝑉𝑉𝑉 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = � � ∗ 79,84 + � � ∗ 62 = 24,8711 3 3 𝜎𝜎𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 4,9871% 𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 =

𝜎𝜎𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 6,2%

1 1 1 1 1 1 1 1 ∗ � ∗ 79,84 + ∗ 77,2 + 0� + ∗ � ∗ 77,2 + ∗ 76 + 0� + ∗ � ∗ 62 � = 38,4711 3 3 3 3 3 3 3 3

1 1 �3 ∗ 79,84 + 0� = 0,3567 𝑥𝑥𝐴𝐴 ∗ 𝛽𝛽𝐴𝐴 = ∗ 24,8711 3 2 2 �3 ∗ 6²� = 0,6433 𝑥𝑥𝐶𝐶 ∗ 𝛽𝛽𝐶𝐶 = ∗ 3 24,8711

c) Beta measures the sensibility with the market 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽 = 0,5 ∗ 4 + 0,5 ∗ 7 = 5,5% 1 1 1 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∗ 11,4 + ∗ 7 + ∗ 12 = 10,1333 3 3 3 5,5 = 4 + 𝛽𝛽𝐽𝐽 ∗ (10,1333 − 4) (5,5 − 4) 𝛽𝛽𝐽𝐽 = = 0,2446 6,1333 2 1 ∗ 11,4 + ∗ 12 = 11,8 3 3 5,5 = 4 + 𝛽𝛽𝐽𝐽𝐽𝐽 ∗ (11,8 − 4) 𝛽𝛽𝐽𝐽𝐽𝐽 = 0,1923

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 =

d) 𝜎𝜎1 = 𝜎𝜎2 = 𝜎𝜎 𝑥𝑥 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦 ∶ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 1 & 2 𝑦𝑦 = 1 − 𝑥𝑥 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑥𝑥 2 ∗ 𝜎𝜎 2 + 𝑦𝑦 2 ∗ 𝜎𝜎 2 + 2 ∗ 𝜌𝜌 ∗ 𝜎𝜎 2 ∗ 𝑥𝑥 ∗ 𝑦𝑦

12

Maths

Exercises

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑥𝑥 2 ∗ 𝜎𝜎 2 + (𝑥𝑥 2 − 2𝑥𝑥 + 1) ∗ 𝜎𝜎 2 + 2 ∗ 𝜌𝜌 ∗ 𝜎𝜎 2 ∗ (𝑥𝑥 − 𝑥𝑥 2 ) 𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 2𝑥𝑥 ∗ 𝜎𝜎 2 + 2𝑥𝑥 ∗ 𝜎𝜎 2 − 2𝜎𝜎 2 + 2 ∗ 𝜌𝜌 ∗ 𝜎𝜎 2 − 4𝑥𝑥 ∗ 𝜌𝜌 ∗ 𝜎𝜎 2 = 0 4𝑥𝑥 − 2 + 2𝜌𝜌 − 4𝑥𝑥 ∗ 𝜌𝜌 = 0 2𝑥𝑥(1 − 𝜌𝜌) = (1 − 𝜌𝜌) 2𝑥𝑥 = 1 𝑥𝑥 = 0,5 𝑦𝑦 = 0,5

e) Market portfolio is a portfolio with all the asset in the market… The market portfolio doesn’t have the minimum variance portfolio because we can’t eliminate all the risks, there is still systemic risk.

13

Maths

Exercises

Problem 2 (Reex04) a) 𝑡𝑡𝑐𝑐 = 0,25 𝑟𝑟𝑑𝑑 = 6% 𝑟𝑟𝑒𝑒 = 14% 𝑅𝑅𝑅𝑅𝑅𝑅 = 20% 𝐵𝐵𝐵𝐵(𝐷𝐷) ∗ �𝑅𝑅𝑅𝑅𝑅𝑅 − 𝑟𝑟𝑑𝑑 ∗ (1 − 𝑡𝑡𝑐𝑐 )� 𝑅𝑅𝑅𝑅𝑅𝑅 = 𝑅𝑅𝑅𝑅𝑅𝑅 + 𝐵𝐵𝐵𝐵(𝐸𝐸) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝑅𝑅𝑅𝑅𝑅𝑅 = 𝐵𝐵𝐵𝐵(𝐷𝐷) + 𝐵𝐵𝐵𝐵(𝐸𝐸)

1000 ∗ �𝑅𝑅𝑅𝑅𝑅𝑅 − 0,06 ∗ (1 − 0,25)� 4000 4000 1000 𝑅𝑅𝑅𝑅𝑅𝑅 = ∗ �0,2 + ∗ 0,06 ∗ (1 − 0,25)� = 16,90% 5000 4000 𝑅𝑅𝑅𝑅𝑅𝑅 ∗ �𝐵𝐵𝐵𝐵(𝐷𝐷) + 𝐵𝐵𝐵𝐵(𝐸𝐸)� 0,1690 ∗ 5000 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = = = 1126,6667$ (1 − 𝑡𝑡𝑐𝑐 ) 1 − 0,25

𝑅𝑅𝑅𝑅𝑅𝑅 = 0,2 = 𝑅𝑅𝑅𝑅𝑅𝑅 +

𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = (1126,6667 − 1000 ∗ 0,06) ∗ (1 − 0,25) = 800$ b)

800 = 5714,2857$ 0,14 𝑀𝑀𝑀𝑀(𝐷𝐷) = 1000$ 5714,2857 1000 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = ∗ 0,14 + ∗ 0,06 ∗ (1 − 0,25) 6714,2857 6714,2857 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 12,5851%

𝑀𝑀𝑀𝑀(𝐸𝐸) =

c) (see ex04 p3)

1000 ∗ 0,06 ∗ (1 − 0,25) 5714,2857 = 13,0718% 𝑟𝑟𝑢𝑢 = 1000 (1 1+ ∗ − 0,25) 5714,2857 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = −600$ 1 − 1,130718−3 400 ∗ 0,75 + ∗ 1,130718−3 (! 𝑃𝑃𝑃𝑃 3 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦!) = 1941,2752$ 𝑃𝑃𝑃𝑃 = 200 ∗ 0,75 ∗ 0,130718 0,130718 600 ∗ 0,5 ∗ 0,75 ∗ 0,02 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = = 4,5918$ 0,98 0,14 +

Debt 600, for 5y, dep straight line method, 1 − 1,130718−5 = 105,3323$ 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 120 ∗ 0,25 ∗ 0,130718 1 − 1,06−5 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 300 ∗ 0,06 ∗ 0,25 ∗ = 18,9556$ 0,06

𝐴𝐴𝐴𝐴𝐴𝐴 = −600 + 1941,2752 − 4,5918 + 105,3323 + 18,9556 𝐴𝐴𝐴𝐴𝐴𝐴 = 1460,9713$

14

Maths

Exercises

Problem 3 (Reex04) a)

1000 => 𝑖𝑖1 = 10% … 1,12 1100 100 + => 𝑖𝑖2 = 10% … 𝐵𝐵2 = 1,12 1,1

𝐵𝐵1 =

b) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵1 = 2 1100 100 +2∗ 1,1 1,12 = 1909,0909 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵1 = 100 1100 + 1,1 1,12 1 = 𝑥𝑥 ∗ 2 + (1 − 𝑥𝑥) ∗ 1,90909 1 − 1,909090 = 0,090909 𝑥𝑥 = −10 𝑦𝑦 = 11 c)

𝑡𝑡 = 1 0

𝐵𝐵1

𝑡𝑡 = 2 1000 1,12 1100 1,12 0

100 1,1 0

𝐵𝐵2 𝐵𝐵3

50 1,1

𝐵𝐵4

𝑡𝑡 = 3

50 1,12

𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐵𝐵1 = 𝑥𝑥; 𝐵𝐵2 = 𝑦𝑦; 𝐵𝐵3 = 𝑧𝑧 We have then : 100 50 ⎧ 𝑦𝑦 ∗ = 1,1 1,1 ⎪ ⎪ 1000 1100 50 𝑥𝑥 ∗ + 𝑦𝑦 ∗ = 1,12 1,12 1,12 ⎨ 1000 1050 ⎪ ⎪ 𝑧𝑧 ∗ = 3 ⎩ 1,105 1,1053 We can see that there is in fact no need to write the rate, it’s easier to write : 𝑦𝑦 ∗ 100 = 50 �𝑥𝑥 ∗ 1000 + 𝑦𝑦 ∗ 1100 = 50 𝑧𝑧 ∗ 1000 = 1050 𝑠𝑠𝑠𝑠 𝑧𝑧 = 1,05, 𝑦𝑦 = 0,5 50−0,5∗1100 Then 𝑥𝑥 = 1000 𝑥𝑥 = −0,5 d)

Year1

Year2 100

Year3 100

S1&S2 For S2, we pay 170 For S1, we need to calculate the price of the bond, but it’s in 1year So we need the one year rate in one year, and the 2 years rate in one year too. 1,102 − 1 = 10% 1𝑓𝑓1 = 1,1 15

1000 1,1053 1050 1,1053

Maths

Exercises

1,1053 2𝑓𝑓1 = � − 1 = 10,7509% 1,1 (=> (1 + 𝑖𝑖)2 ∗ 1,1 = 1,1053 )

Then we have 100 100 + = 172,437 1,1 1,1075092 So the S2 is better

16

Maths

Exercises

Problem 4 (Reex04) 𝑆𝑆𝑢𝑢 = 109,54

𝑆𝑆 = 100

𝑆𝑆𝑑𝑑 = 89,44

a) b) 𝑢𝑢 = 1,0954 𝑑𝑑 = 0,8944 𝑒𝑒

0,05 2

= 1,025315 0,05

𝑒𝑒 2 − 0,8944 𝑝𝑝 = = 0,6513 1,0954 − 0,8944 20 ∗ 0,65132 = 8,0701 𝐶𝐶 = 1,0253152 c) 100 − 8,0701 = 91,9299

𝑖𝑖𝑖𝑖 𝑤𝑤𝑤𝑤 𝑢𝑢𝑢𝑢𝑢𝑢 𝑝𝑝𝑝𝑝𝑝𝑝: (1 − 0,6513) ∗ 2,0274 𝑃𝑃𝑢𝑢 = = 0,6895 1,025315 20 ∗ (1 − 0,6513) + 2,0274 ∗ 0,6513 = 8,0897 𝑃𝑃𝑑𝑑 = 1,025315 8,0897 ∗ (1 − 0,6513) + 0,6895 ∗ 0,6513 = 3,1892 𝑃𝑃 = 1,025315 100 ∗ 𝑒𝑒 −0,05 = 95,1229 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 − 𝑝𝑝𝑝𝑝𝑝𝑝 = 95,1229 − 3,1892 = 91,9337

d) 𝑢𝑢6𝑚𝑚 = 𝑒𝑒 𝜎𝜎 √𝑡𝑡 𝑑𝑑6𝑚𝑚 = 𝑒𝑒 −𝜎𝜎 √𝑡𝑡 ln(1,0954) = 12,8863% 𝜎𝜎𝑢𝑢 = �0,5 ln(0,8944) 𝜎𝜎𝑑𝑑 = − = 15,7829% �0,5 12,8863 + 15,7829 = 14,3346% 𝜎𝜎 = 2 𝑑𝑑1 = 𝑑𝑑2 =

0,14332 100 � ln �100� + �0,05 + 2 0,1433

0,14332 100 ln �100� + �0,05 − 2 �

= 0,4206

= 0,2773 0,1433 𝑁𝑁(𝑑𝑑1 ) = 0,6628 + 0,06 ∗ (0,6664 − 0,6628) = 0,6630 𝑁𝑁(𝑑𝑑2 ) = 0,6064 + 0,73 ∗ (0,6103 − 0,6064) = 0,6092 𝐶𝐶 = 100 ∗ 0,6630 − 100 ∗ 𝑒𝑒 −0,05 ∗ 0,6092 𝐶𝐶 = 8,3511 17

𝑆𝑆𝑢𝑢𝑢𝑢 = 120

𝑆𝑆𝑢𝑢𝑢𝑢 = 97,9726 𝑃𝑃𝑢𝑢𝑢𝑢 = 2,0274 𝑆𝑆𝑑𝑑𝑑𝑑 = 80

Maths

Exercises

Also good but on one period, it’s less close to the result 𝑢𝑢 = 𝑒𝑒 𝜎𝜎 𝑑𝑑 = 𝑒𝑒 −𝜎𝜎 𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙1,2 − 𝑙𝑙𝑙𝑙0,8 𝜎𝜎 = = = 10,1361% 2 2 𝑑𝑑1 =

0,1013612 100 � ln �100� + �0,05 + 2

= 0,5440 0,101361 0,1013612 100 � ln �100� + �0,05 − 2 = 0,4426 𝑑𝑑2 = 0,101361 𝑁𝑁(𝑑𝑑1 ) = 0,7054 + 0,4 ∗ (0,7088 − 0,7054) = 0,7068 𝑁𝑁(𝑑𝑑2 ) = 0,6700 + 0,26 ∗ (0,6736 − 0,6700) = 0,6709 𝐶𝐶 = 100 ∗ 0,7068 − 100 ∗ 𝑒𝑒 −0,05 ∗ 0,6709 𝐶𝐶 = 6,862

e) binomial ? 60*1/2/1,05*0,4 ??

18

Maths

Exercises

Exam 2005 Problem 1 (Ex05) Stockmadame 800 100 => 𝛽𝛽 = 0 Total 900 𝜌𝜌𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 & 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ℎ𝑢𝑢𝑢𝑢𝑢𝑢 = 0,2 𝜎𝜎𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0,5 𝜎𝜎𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ℎ𝑢𝑢𝑢𝑢𝑢𝑢 = 0,4 𝑟𝑟𝑓𝑓 = 5% 𝑟𝑟𝑚𝑚 = 9%

600 debt at 6% 300 equity Total 900

6−5 = 0,25 4 𝜎𝜎𝑀𝑀 ² = 0,1² ∗ 0,5² + 0,9² ∗ 0,4² + 2 ∗ 0,2 ∗ 0,1 ∗ 0,9 ∗ 0,5 ∗ 0,4 = 0,1393 𝑐𝑐𝑐𝑐𝑣𝑣𝐴𝐴𝐴𝐴 = 𝑥𝑥𝐴𝐴 ∗ 𝑐𝑐𝑐𝑐𝑣𝑣𝐴𝐴𝐴𝐴 + 𝑥𝑥𝐵𝐵 ∗ 𝑐𝑐𝑐𝑐𝑣𝑣𝐴𝐴𝐴𝐴 … 𝑐𝑐𝑐𝑐𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 & 𝑀𝑀 = 0,1 ∗ 0,5² + 0,9 ∗ 0,2 ∗ 0,5 ∗ 0,4 = 0,061 0,061 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = = 0,4379 0,1393 1 2 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,4379 ∗ + 0,25 ∗ = 0,3126 3 3 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 =

b) Baltman

Total 175 (6,5 − 5) 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = = 0,375 4 75 100 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = ∗ 0,5 + ∗ 0,375 = 0,4286 175 175

c) Stockmadame after acquisition 800 25 => 𝛽𝛽 = 0 175 Total 1000 600 100 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 6% ∗ + 6,5% ∗ = 6,0714% 700 700 6,0714 − 5 = 0,2679 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 4

75 equity => 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 0,5 100 => 6,5% Total 175

600 debt at 6% 300 equity 100 debt => 6,5% Total 1000

𝑟𝑟175 = 5 + 0,4286 ∗ 4 = 6,7144 𝑟𝑟25 = 5 𝑟𝑟800 =>

800 100 ∗ 𝑟𝑟800 + ∗ 5 => 𝑟𝑟800 = 6,4% 900 900 800 25 175 ∗ 6,4 + ∗5+ ∗ 6,7144 = 6,42 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 1000 1000 1000 6,42 − 5 = 0,355 𝛽𝛽𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 4 700 300 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,355 = ∗ 0,2679 + ∗ 𝛽𝛽 1000 1000 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,558

𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 5 + 0,3126 ∗ 4 = 6,2504 =

19

Maths 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

Exercises 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

= 5 + 0,558 ∗ 4 = 7,232%

d) All information is already included in prices Inefficient : why high volatility, why changes without news, why calendar effect, etc.

e) CAPM : 𝑟𝑟𝑖𝑖 = 𝑟𝑟𝐴𝐴 + 𝛽𝛽(𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ) Three factor model : 𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑓𝑓 = 𝑏𝑏𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ 𝑟𝑟𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∗ 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑏𝑏𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ 𝑟𝑟𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

20

𝑘𝑘𝑘𝑘𝑘𝑘

Maths

Exercises

Problem 3 (Ex05) a) 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 5𝑀𝑀 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 4€ 𝑀𝑀𝑀𝑀(𝐷𝐷) = 75𝑀𝑀€ 𝐸𝐸𝐸𝐸𝐸𝐸 = 0,6€ 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 0,4 𝑟𝑟𝑑𝑑 = 5% 𝑡𝑡𝑐𝑐 = 35% 𝑟𝑟𝑚𝑚𝑚𝑚 = 7%

80 𝑐𝑐𝑐𝑐𝑐𝑐 = = 0,8 𝑣𝑣𝑣𝑣𝑣𝑣 100 𝑟𝑟𝑒𝑒 = 5% + 0,8 ∗ 7% = 10,6% 0,6 𝐸𝐸𝐸𝐸𝐸𝐸 = = 15% 𝑅𝑅𝑅𝑅𝑅𝑅 = 4 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑔𝑔 = 𝑅𝑅𝑅𝑅𝑅𝑅 ∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 15% ∗ (1 − 0,4) = 9% 𝐷𝐷𝐷𝐷𝑣𝑣0 = 0,6 ∗ 0,4 = 0,24 0,24 ∗ 1,09 ∗ 5𝑀𝑀 = 81,750𝑀𝑀$ 𝑀𝑀𝑀𝑀(𝐸𝐸) = 0,106 − 0,09

𝛽𝛽 =

81,750 75 ∗ 0,106 + ∗ 0,05 ∗ (1 − 0,35) 81,750 + 75 81,750 + 75 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 7,0833%

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 =

b) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑉𝑉(𝐸𝐸)𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 − 𝑉𝑉(𝐸𝐸)𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛

Non growth : payout = 1 0,6 ∗ 5 = 81,75 − 28,301 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 81,750 − 0,106 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 53,448𝑀𝑀

𝐸𝐸𝐸𝐸𝐸𝐸 0,55 = = 13,75% 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 4 𝑔𝑔 = 13,75% ∗ 0,6 = 8,25% 0,55 ∗ 0,4 ∗ 1,0825 0,55 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = ∗ 5𝑀𝑀 − ∗ 5𝑀𝑀 = 24,7268𝑀𝑀 0,106 − 0,0825 0,106

𝑅𝑅𝑅𝑅𝑅𝑅 =

c)

75 ∗ 0,05 ∗ (1 − 0,35) 81,75 𝑟𝑟𝑢𝑢 = = 8,508% 75 1+ ∗ (1 − 0,35) 81,75 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = −6𝑀𝑀 1 − 1,08508−6 = 5,9182𝑀𝑀€ 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 2𝑀𝑀 ∗ 0,65 ∗ 0,08508 −5 1 − 1,05 = 0,2273𝑀𝑀€ 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 3𝑀𝑀 ∗ 0,35 ∗ 0,05 ∗ 0,05 −6 1 − 1,08508 = 1,5934𝑀𝑀€ 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 1𝑀𝑀 ∗ 0,35 ∗ 0,08508 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 0,2 ∗ 0,65 = 0,13𝑀𝑀€ 𝐴𝐴𝐴𝐴𝐴𝐴 = −6 + 5,9182 + 0,2273 + 1,5934 − 0,07 𝐴𝐴𝐴𝐴𝐴𝐴 = 1,6089𝑀𝑀€ 0,106 +

21

Maths

Exercises

Problem 4 (Ex05) a)

𝑆𝑆 = 90

𝑆𝑆𝑢𝑢𝑢𝑢 = 152,1 𝑃𝑃𝑢𝑢𝑢𝑢 = 0

𝑆𝑆𝑢𝑢 = 117

𝑆𝑆𝑢𝑢𝑢𝑢 = 99,45 𝑃𝑃𝑑𝑑𝑑𝑑 = 0,55

𝑆𝑆𝑑𝑑 = 76,5

𝑆𝑆𝑑𝑑𝑑𝑑 = 65,025 𝑃𝑃𝑑𝑑𝑑𝑑 = 34,975

𝑟𝑟𝑟𝑟 = 3,75% 𝑢𝑢 = 1,3 𝑑𝑑 = 0,85 1,035 − 0,85 𝑝𝑝 = = 0,4167 1,3 − 0,85 (1 − 𝑝𝑝) = 0,5833

0,55 ∗ 0,5833 = 0,3092 1,0375 0,55 ∗ 0,4167 + 34,975 𝑃𝑃𝑑𝑑𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = = 19,8844 1,0375 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 ∶ 𝑃𝑃𝑑𝑑𝑒𝑒𝑒𝑒 = 23,5

𝑃𝑃𝑢𝑢 =

0,3092 ∗ 0,4167 + 19,8844 ∗ 0,5833 1,0375 = 11,3 0,3092 ∗ 0,4167 + 23,5 ∗ 0,5833 = 1,0375 = 13,336

𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸 =

𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸 𝑃𝑃𝑈𝑈𝑈𝑈𝑈𝑈 𝑃𝑃𝑈𝑈𝑈𝑈𝑈𝑈

b) Put-call parity 𝐸𝐸 𝑆𝑆 + 𝑃𝑃 = + 𝐶𝐶 𝑟𝑟 𝐸𝐸 100 𝐶𝐶 = 𝑆𝑆 + 𝑃𝑃 − = 90 + 11,3 − = 8,398 𝑟𝑟 1,0375 Volatility 𝑢𝑢 = 𝑒𝑒 𝜎𝜎∗√𝑡𝑡 𝜎𝜎1 = 𝑙𝑙𝑙𝑙1,3 = 26,2364% 𝜎𝜎2 = −𝑙𝑙𝑙𝑙0,85 = 16,25% 𝜎𝜎 = 21,25%

0,41672 ∗ (90 ∗ 1,32 − 90) + 2 ∗ 0,4167 ∗ 0,5833 ∗ (90 ∗ 1,3 ∗ 0,85 − 90) = 14,28 𝐶𝐶 = 1,03752 c) Repurchase 20% of shares 100 ∗ 1 = 100 80 ∗ 𝑥𝑥 = 100 𝑥𝑥 = 25% 𝑆𝑆 = 90

𝑆𝑆𝑢𝑢 = 117 ∗ 1,25 = 146,25

𝑆𝑆𝑑𝑑 = 76,5 ∗ 1,25 = 95,625 22

𝑆𝑆𝑢𝑢𝑢𝑢 = 190,125 𝑆𝑆𝑢𝑢𝑢𝑢 = 124,313

Maths

Exercises 𝑆𝑆𝑑𝑑𝑑𝑑 = 81,281

𝑃𝑃𝑢𝑢 = 0 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑃𝑃𝑑𝑑𝑒𝑒𝑒𝑒 = 23,5 18,719 ∗ 0,5833 = 10,5241 𝑃𝑃𝑑𝑑𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = 1,0375 10,5241 ∗ 0,5833 𝑃𝑃𝐸𝐸𝐸𝐸 = = 5,917 1,0375 23,5 ∗ 0,5833 𝑃𝑃𝑈𝑈𝑈𝑈 = = 13,212 1,0375

𝑃𝑃𝐸𝐸𝐸𝐸 = 5,917 𝑃𝑃𝑈𝑈𝑈𝑈 = 13,212 d) ???

Present 𝑢𝑢

*

𝑑𝑑 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

Stock 0 56,511 36,95 1,521 0,995 99,45 65,025

Bond 0 −56,82

−56,82 −1,521 −1,521 −100 −100

Stock 𝛼𝛼 ∗ ∗ 𝑆𝑆 = 0,48 ∗ 90 = 43,47 𝛼𝛼𝑢𝑢∗ ∗ 𝑆𝑆𝑢𝑢 = 1,17 76,5 0 0 0 0

𝑃𝑃𝑢𝑢 − 𝑃𝑃𝑑𝑑 0,3092 − 19,8844 = = −0,483 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 90 ∗ (1,3 − 0,85) 𝑃𝑃𝑢𝑢𝑢𝑢 − 𝑃𝑃𝑢𝑢𝑢𝑢 = −0,1 𝛼𝛼𝑢𝑢∗ = 𝑆𝑆𝑢𝑢 ∗ (𝑢𝑢 − 𝑑𝑑) 𝛼𝛼𝑑𝑑∗ = 1 𝛼𝛼 ∗ =

Bond −54,7660 −1,466

−36,386 0 0 0 0

23

S B −43,47 54,7660 55,341

−39,55 1,521 0,995 99,45 65,025

P E −5,917 +5,385

−55,354 39,55 −1,521 −1,521 −100 −100

0 0 0,55 0,55 34,975

0 0 0 0 0

Maths

Exercises

Problem 5 (Ex05) 𝑆𝑆 = 3,422𝑏𝑏 𝐸𝐸 = 2,875𝑏𝑏 𝑡𝑡 = 17𝑦𝑦 𝑟𝑟𝑟𝑟 = 6,7% 𝜎𝜎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ² = 22,4% a)

1200 1000 800 600 400 200 0 0

b)

𝑑𝑑1 = 𝑑𝑑2 =

1000

2000 3000 4000 Call E=2875 S=3422

5000

16 3,422 ∗ 17 0,224 ln � � + �𝑙𝑙𝑙𝑙1,067 + 2 � ∗ 17 2,875

√0,224 ∗ √17 16 3,422 ∗ 17 0,224 ln � � + �𝑙𝑙𝑙𝑙1,067 − 2 � ∗ 17 2,875

= 1,5989

= −0,3526 √0,224 ∗ √17 𝑁𝑁(𝑑𝑑1 ) = 0,9441 + 0,89 ∗ (0,9452 − 0,9441) = 0,9451 𝑁𝑁(𝑑𝑑2 ) = 0,3632 + 0,26 ∗ (0,3594 − 0,3632) = 0,3622

𝐶𝐶 = 3,422 ∗ c)

16 ∗ 0,9451 − 2,875 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,067∗17 ∗ 0,3622 = 2698,116𝑀𝑀 17

1 − 1,07−12 = 397,13𝑀𝑀 0,07 So the total of the asset is: 2698,12 + 397,13 = 3095,25𝑀𝑀 50 ∗

d) 𝐷𝐷 = 1 => 𝐷𝐷 = 0,5 => 𝐸𝐸 = 0,5 𝐸𝐸 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 5,13% 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 10𝑀𝑀

3095,25 = 1547,625𝑀𝑀 2 10𝑀𝑀 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 => 154,7625 $/𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐸𝐸 = 70 0,224 154,7625 ln � 70 � + �𝑙𝑙𝑙𝑙1,0513 + 2 � ∗ 5 = 1,5152 𝑑𝑑1 = √0,224 ∗ √5 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =

24

Maths

Exercises

0,224 154,7625 ln � 70 � + �𝑙𝑙𝑙𝑙1,0513 − 2 � ∗ 5 𝑑𝑑2 = = 0,4569 √0,224 ∗ √5 𝑁𝑁(𝑑𝑑1 ) = 0,9345 + 0,52 ∗ (0,9357 − 0,9345) = 0,9351 𝑁𝑁(𝑑𝑑2 ) = 0,6736 + 0,69 ∗ (0,6772 − 0,6736) = 0,6761 𝐶𝐶 = 154,7625 ∗ 0,9351 − 70 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,0513∗5 ∗ 0,6761 = 107,865

25

Maths

Exercises

RE-Exam 2005 Problem 1 (Reex05) a) 𝑟𝑟𝑓𝑓 = 7% 𝑟𝑟𝑚𝑚 = 15% 𝛽𝛽𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1,5 𝐷𝐷 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ∶ = 0,6 𝐸𝐸

𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 7 + 1,5 ∗ 8 = 19% 𝐷𝐷 𝐷𝐷 0,6 𝐷𝐷 = = = = 0,375 𝐷𝐷 + 𝐸𝐸 0,6𝐸𝐸 + 𝐸𝐸 1,6𝐸𝐸 1,6 𝐸𝐸 = 0,625 𝐷𝐷 + 𝐸𝐸 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,625 ∗ 19 + 0,375 ∗ 7 = 14,5% b)

𝑁𝑁𝑁𝑁𝑁𝑁 = −1 200 000 + 600 000 ∗ It’s ok

1 − 1,145−3 = 181 375,23 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 0,145

c) 𝐷𝐷 𝐸𝐸 𝐷𝐷 = 1 => = 0,5 = 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 𝐸𝐸

𝐷𝐷 ! 𝐸𝐸 There is more debt, so it’s riskier for the equity holders 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 14,5 = 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 0,5 + 7 ∗ 0,5 => 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 22% 𝑁𝑁𝑁𝑁𝑁𝑁 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎 𝑏𝑏 = 181 375,23 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (It’s ok) 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ‼! 𝑤𝑤𝑤𝑤 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

d) 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 0,3 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 22% 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 7 + 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ∗ 8 15 = 1,875 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 8 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 1,875 ∗ 0,5 + 0,3 ∗ 0,5 = 1,0875 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 + 1,0875 ∗ 8 = 15,7%

26

Maths

Exercises

Problem 2 (Reex05) a) 10 10 10 10 200 𝑃𝑃𝑃𝑃 = 0,95 ∗ + 0,9 ∗ +1∗ + 0,95 ∗ + = 199,003279 𝐿𝐿𝐿𝐿𝐿𝐿/𝑚𝑚² 2 3 4 1,045 1,047 1,048 1,049 1,0494 If we invest 1000m², the price is: 1000 ∗ 199,003279 = 199003,279 𝐿𝐿𝐿𝐿𝐿𝐿 The price 195000 is underpriced

b) To change, not with spot rate must find the yield !! 10 10 10 200 10 + 2 ∗ 0,9 ∗ +3∗1∗ + 4 ∗ 0,95 ∗ +4∗ 0,95 ∗ 1,045 1,0494 1,0494 743,632235 1,0472 1,0483 = 𝐷𝐷 = 199,003279 199,003279 = 3,7368𝑦𝑦 𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝐷𝐷1 = 4 & 𝐷𝐷2 = 0,5 3,7368 = 𝑥𝑥 ∗ 4 + (1 − 𝑥𝑥) ∗ 0,5 𝑥𝑥 = 0,9248 𝑦𝑦 = 0,0752 c) 0,9 𝐿𝐿𝐿𝐿𝐿𝐿/𝐺𝐺𝐺𝐺𝐺𝐺 1 1 𝐿𝐿𝐿𝐿𝐿𝐿 = 𝐺𝐺𝐺𝐺𝐺𝐺; 1𝐺𝐺𝐺𝐺𝐺𝐺 = 0,9𝐿𝐿𝐿𝐿𝐿𝐿 0,9

1 − 1,045−3 = 4,4898% 1,042−1 + 1,043−2 + 1,045−3 1 − 1,0465−3 = 4,6462% 𝐼𝐼𝐼𝐼𝐼𝐼 𝐺𝐺𝐺𝐺𝐺𝐺 = 1,045−1 + 1,046−2 + 1,0465−3 CF LVL CF GBP 4646,2 1 100000 ∗ 0,044898 = 4489,8 𝐿𝐿𝐿𝐿𝐿𝐿 = 5162,44 𝐺𝐺𝐺𝐺𝐺𝐺 0,9 2 4489,8 𝐿𝐿𝐿𝐿𝐿𝐿 5162,44 𝐺𝐺𝐺𝐺𝐺𝐺 104646,2 3 104489,8 𝐿𝐿𝐿𝐿𝐿𝐿 = 116273,56 𝐺𝐺𝐺𝐺𝐺𝐺 0,9 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐿𝐿𝐿𝐿𝐿𝐿 =

After one year 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐺𝐺𝐺𝐺𝐺𝐺 = 4,55%; 4,65%; 4,7% CF LVL 4489,8 2 = 4308,83 𝐿𝐿𝐿𝐿𝐿𝐿 1,042 3

Total 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

CF GBP

5162,44 = 4937,77 𝐺𝐺𝐺𝐺𝐺𝐺 1,0455 => 4937,77 ∗ 0,92 = 4542,75 𝐿𝐿𝐿𝐿𝐿𝐿 116273,56 = 106170,17 𝐺𝐺𝐺𝐺𝐺𝐺 1,04652 => 106170,17 ∗ 0,92 = 97676,56 𝐿𝐿𝐿𝐿𝐿𝐿 102219,31 𝐿𝐿𝐿𝐿𝐿𝐿

104489,8 = 96051,75 𝐿𝐿𝐿𝐿𝐿𝐿 1,0432 100360,58 𝐿𝐿𝐿𝐿𝐿𝐿 1858,73 𝐿𝐿𝐿𝐿𝐿𝐿

27

Maths

Exercises

Problem 4 (Reex05) a) 350 300 250 200 150 100 50 0 0

100

200

300

400

500

600

Put E=300 S=360

b) 𝐸𝐸 = 300 𝑆𝑆 = 360 𝜎𝜎 2 = 0,09 𝜎𝜎 = 0,3 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∶ 10% 𝑢𝑢 = 𝑒𝑒 0,3 = 1,3499 𝑑𝑑 = 𝑒𝑒 −0,3 = 0,7408 𝑝𝑝 = 0,5076 𝑟𝑟𝑐𝑐 = 𝑙𝑙𝑙𝑙1,05 = 0,04879 2 binomial model 𝑆𝑆 = 360

𝑆𝑆𝑢𝑢 = 485,964 ∗ 0,9 = 437,37 𝑆𝑆𝑑𝑑 = 266,69 ∗ 0,9 = 240,02

8,3983 ∗ (1 − 0,5076) = 3,9384 1,05 𝑃𝑃𝑑𝑑 = 69,7013 𝑃𝑃 = 34,5905

𝑃𝑃𝑢𝑢 =

𝑆𝑆𝑢𝑢𝑢𝑢 = 590,4025 ∗ 0,9 = 531,3623 𝑃𝑃𝑢𝑢𝑢𝑢 = 0

𝑆𝑆𝑢𝑢𝑢𝑢 = 324,0019 ∗ 0,9 = 291,6017 𝑃𝑃𝑢𝑢𝑢𝑢 = 8,3983 𝑆𝑆𝑑𝑑𝑑𝑑 = 177,8062 ∗ 0,9 = 160,0256 𝑃𝑃𝑑𝑑𝑑𝑑 = 139,9744

c) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐: −380 𝐶𝐶𝐶𝐶 = 0,4 ∗ 900 = 360 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 = +39905 𝑁𝑁𝑁𝑁𝑁𝑁 = 14990,5

d) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐: −570 𝐶𝐶𝐶𝐶 = 0,6 ∗ 900 = 540 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = −34990,5 −64990,5

𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 64990,5 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐ℎ 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑔𝑔𝑔𝑔 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑, 𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑒𝑒𝑒𝑒 ∶ 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 64990,5 ∗

e) When do budgeting, account for option

28

3 = 97486. 2

Maths

Exercises

Problem 5 (Reex05) a) 3000 2500 2000 1500 1000 500 0 0

500

1000 1500 2000 2500 Call E=2000, S=1021,63

3000

𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵 ∶ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2000, 𝐶𝐶𝐶𝐶 150 𝑓𝑓𝑓𝑓𝑓𝑓 15 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 1 − 1,12−15 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 150 ∗ = 1021,63 0,12 b) 𝜎𝜎 = 0,4 𝑟𝑟𝑐𝑐 = 6,11%

0,42 1021,63 ln � 2000 � + �0,0611 + 2 � ∗ 10 𝑑𝑑1 = = 0,5844 0,4 ∗ √10 𝑤𝑤𝑒𝑒 ′ 𝑣𝑣𝑣𝑣 10 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝑖𝑖 10 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛 𝑡𝑡ℎ𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ! 0,42 1021,63 ln � 2000 � + �0,0611 − 2 � ∗ 10 = −0,6805 𝑑𝑑2 = 0,4 ∗ √10 𝑁𝑁(𝑑𝑑1 ) = 0,7190 + 0,44 ∗ (0,7224 − 0,7190) = 0,7205 𝑁𝑁(𝑑𝑑2 ) = 0,2483 + 0,05 ∗ (0,2451 − 0,2483) = 0,2481 𝐶𝐶 = 1021,63 ∗ 0,7205 − 2000 ∗ 𝑒𝑒 −0,0611 ∗10 ∗ 0,2481 = 466,743 c) 𝑁𝑁𝑁𝑁𝑁𝑁 = −270 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 466,743 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 196743 > 0 d)

3000 2500 2000

Loan

1500

Bonds

1000

Equity

500

Asset

0 0

1000

2000

(Usually loan senior than bond) Fixed data Total asset 2500 + 196,743 = 2696,743 𝜎𝜎 = 0,15 𝑡𝑡 = 7

3000

29

Maths

Exercises

𝑟𝑟𝑟𝑟 = 0,0611

Calcul of the equity 𝐸𝐸 = 1800 0,152 2696,743 ln � 1800 � + �0,0611 + 2 � ∗ 7 𝑑𝑑1 = = 2,2948 0,15 ∗ √7 0,152 2696,743 ln � 1800 � + �0,0611 − 2 � ∗ 7 𝑑𝑑2 = = 1,8979 0,15 ∗ √7 𝑁𝑁(𝑑𝑑1 ) = 0,9890 + 0,48 ∗ (0,9893 − 0,9890) = 0,9891 𝑁𝑁(𝑑𝑑2 ) = 0,9706 + 0,79 ∗ (0,9713 − 0,9706) = 0,9712 𝐶𝐶1800 = 2696,743 ∗ 0,9891 − 1800 ∗ 𝑒𝑒 −0,0611∗7 ∗ 0,9712 = 1527,54 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1527,54

Calcul of the bond 𝐸𝐸 = 1500 0,152 2696,743 � + �0,0611 + 2 � ∗ 7 ln � 1500 𝑑𝑑1 = = 2,7542 0,15 ∗ √7 0,152 2696,743 � + �0,0611 − 2 � ∗ 7 ln � 1500 𝑑𝑑2 = = 2,3573 0,15 ∗ √7 𝑁𝑁(𝑑𝑑1 ) = 0,9970 + 0,42 ∗ (0,9971 − 0,9970) = 0,9970 𝑁𝑁(𝑑𝑑2 ) = 0,9906 + 0,73 ∗ (0,9909 − 0,9906) = 0,9908

𝐶𝐶1500 = 2696,743 ∗ 0,9970 − 1500 ∗ 𝑒𝑒 −0,0611∗7 ∗ 0,9908 = 1719,64 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑜𝑜𝑜𝑜 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 1719,64 − 1527,54 = 192,1

Calcul of the loan You can consider it as a put with E=1500, and PV of bond of 1500. You have then: 1500 ∗ 𝑒𝑒 −0,0611∗7 = 978,0105 𝐶𝐶1500 = −2696,743 ∗ (1 − 0,9970) + 1500 ∗ 𝑒𝑒 −0,0611 ∗7 ∗ (1 − 0,9908) = 0,9075 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 = 978,0105 − 0,9075 = 977,103

Or 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 + 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 − 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 2696,743 − 192,1 − 1527,54 = 977,103

30

Maths

Exercises

Exam 2006 Problem 1 (Ex06) a) 𝐸𝐸(𝑟𝑟𝑎𝑎 ) = 0,25 ∗ −20 + 0,25 ∗ 10 + 30 ∗ 0,25 + 50 ∗ 0,25 = 17,5% 𝐸𝐸(𝑟𝑟𝑏𝑏 ) = 0,25 ∗ 5 + 0,25 ∗ 20 + 0,25 ∗ −12 + 0,25 ∗ 9 = 5,5% 𝐸𝐸(𝑟𝑟𝑐𝑐 ) = 0,25 ∗ −5 + 0,25 ∗ 5 + 0,25 ∗ 5 + 0,25 ∗ −3 = 0,5%

𝜎𝜎𝐴𝐴 = �(−37,5)2 ∗ 0,25 + (−7,5)2 ∗ 0,25 + 12,52 ∗ 0,25 + 32,52 ∗ 0,25 = 25,86% 𝜎𝜎𝐵𝐵 = �(−0,5)2 ∗ 0,25 + 14,52 ∗ 0,25 + (−17,5)2 ∗ 0,25 + 3,52 ∗ 0,25 = 11,5% 𝜎𝜎𝐶𝐶 = �(−5,5)2 ∗ 0,25 + 4,52 ∗ 0,25 + 4,52 ∗ 0,25 + (−3,5)2 ∗ 0,25 = 4,5552%

𝜎𝜎𝐴𝐴𝐴𝐴 = −37,5 ∗ −0,5 ∗ 0,25 − 7,5 ∗ 14,5 ∗ 0,25 + 12,5 ∗ −17,5 ∗ 0,25 + 32,5 ∗ 3,5 ∗ 0,25 = −48,75 𝜎𝜎𝐴𝐴𝐴𝐴 = −37,5 ∗ −5,5 ∗ 0,25 − 7,5 ∗ 4,5 ∗ 0,25 + 12,5 ∗ 4,5 ∗ 0,25 + 32,5 ∗ −3,5 ∗ 0,25 = 28,75 𝜎𝜎𝐵𝐵𝐵𝐵 = −0,5 ∗ −5,5 ∗ 0,25 + 14,5 ∗ 4,5 ∗ 0,25 − 17,5 ∗ 4,5 ∗ 0,25 + 3,5 ∗ −3,5 ∗ 0,25 = −5,75 −48,75 = −0,164 25,86 ∗ 11,5 28,75 = = 0,244 25,86 ∗ 4,5552 −5,75 = = −0,11 11,5 ∗ 4,5552

𝜌𝜌𝐴𝐴𝐴𝐴 = 𝜌𝜌𝐴𝐴𝐴𝐴

𝜌𝜌𝐵𝐵𝐵𝐵

b) Portfolio D 𝑥𝑥𝐵𝐵 = 𝑥𝑥𝑐𝑐 𝜎𝜎𝐷𝐷 ² = 0,52 ∗ 11,52 + 0,52 ∗ 4,55522 + 2 ∗ −5,75 ∗ 0,52 = 35,375 𝜎𝜎𝐷𝐷 = 5,9477% D Mean Depression 0 Recession 12,5 Normal -3,5 Boom 3 Mean 0 ∗ 0,25 + 12,5 ∗ 0,25 − 3,5 ∗ 0,25 + 3 ∗ 0,25 = 3 𝜎𝜎𝐴𝐴𝐴𝐴 = −37,5 ∗ (0 − 3) ∗ 0,25 − 7,5 ∗ (12,5 − 3) ∗ 0,25 + 12,5 ∗ (−3,5 − 3) ∗ 0,25 + 32,5 ∗ (3 − 3) ∗ 0,25 = −10 𝑐𝑐𝑐𝑐𝑐𝑐(𝐴𝐴, 𝐷𝐷) 10 =− = −0,065 𝜌𝜌𝐴𝐴𝐴𝐴 = 𝜎𝜎𝐴𝐴 𝜎𝜎𝐷𝐷 25,86 ∗ 5,9477 c) 𝜎𝜎𝑃𝑃 ² = 𝑥𝑥𝐴𝐴 ² ∗ 𝜎𝜎𝐴𝐴 ² + 𝑥𝑥𝐷𝐷 ² ∗ 𝜎𝜎𝐷𝐷 ² + 2 ∗ 𝑥𝑥𝐴𝐴 ∗ 𝑥𝑥𝐷𝐷 ∗ 𝜎𝜎𝐴𝐴𝐴𝐴 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐷𝐷 = 1 𝜎𝜎𝑃𝑃 ² = 𝑥𝑥𝐴𝐴 ² ∗ 𝜎𝜎𝐴𝐴 ² + (𝑥𝑥𝐴𝐴2 − 2𝑥𝑥𝐴𝐴 + 1) ∗ 𝜎𝜎𝐷𝐷2 + 2 ∗ (𝑥𝑥𝐴𝐴 − 𝑥𝑥𝐴𝐴2 ) ∗ 𝜎𝜎𝐴𝐴𝐴𝐴

𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉 𝑃𝑃 = 2 ∗ 𝑥𝑥𝐴𝐴 ∗ 𝜎𝜎𝐴𝐴2 + 2 ∗ 𝑥𝑥𝐴𝐴 ∗ 𝜎𝜎𝐷𝐷2 − 2𝜎𝜎𝐷𝐷2 + 2 ∗ 𝜎𝜎𝐴𝐴𝐴𝐴 − 4𝑥𝑥𝐴𝐴 ∗ 𝜎𝜎𝐴𝐴𝐴𝐴 = 0 𝑥𝑥𝐴𝐴 ∗ (𝜎𝜎𝐴𝐴2 + 𝜎𝜎𝐷𝐷2 − 2 ∗ 𝜎𝜎𝐴𝐴𝐴𝐴 ) = 𝜎𝜎𝐷𝐷2 − 𝜎𝜎𝐴𝐴𝐴𝐴 𝜎𝜎𝐷𝐷2 − 𝜎𝜎𝐴𝐴𝐴𝐴 𝑥𝑥𝐴𝐴 = 2 𝜎𝜎𝐴𝐴 + 𝜎𝜎𝐷𝐷2 − 2 ∗ 𝜎𝜎𝐴𝐴𝐴𝐴 35,575 + 10 𝑥𝑥𝐴𝐴 = = 0,063 25,862 + 5,94772 + 20 𝑥𝑥𝐴𝐴 = 6,3% 𝑥𝑥𝐷𝐷 = 93,7% d) semi deviation : only take 𝑟𝑟𝑖𝑖 > 𝑟𝑟̅ 𝑜𝑜𝑜𝑜 𝑟𝑟𝑖𝑖 < 𝑟𝑟̅

31

Maths

Exercises

Problem 2 (Ex06) a) 103 155 144 206 + + + = 461,971 2 3 1,095 1,097 1,1 1,124 92,5 82 113,5 134 + + + = 323,05 2 3 1,095 1,097 1,1 1,124 1𝐹𝐹2 => (1 + 1𝐹𝐹2) ∗ (1,097)2 = 1,13 1𝐹𝐹2 = 10,6025% 2𝐹𝐹2 => (1 + 2𝐹𝐹2)2 ∗ 1,0972 = 1,124 2𝐹𝐹2 = 14,3482% 206 144 + = 287,742 1,106025 1,1434822 134 113,5 + = 205,101 2 1,143482 1,106028

b) Duration K1 K9 What is the yield : 103 155 144 206 103 155 144 206 + + + = 461,971 = + + + 1,095 1,0972 1,13 1,124 1, 𝑖𝑖1 1, 𝑖𝑖1 2 1, 𝑖𝑖1 3 1, 𝑖𝑖1 4 𝑖𝑖1 = 10,78% 92,5 82 113,5 134 92,5 82 113,5 134 + + + = 323,05 = + + + 2 2 3 4 1,095 1,097 1,1 1,12 1, 𝑖𝑖2 1, 𝑖𝑖2 1, 𝑖𝑖2 3 1, 𝑖𝑖2 4 𝑖𝑖2 = 10,74% 155 144 206 103 +2∗ +3∗ +4∗ 1,1078 1,10784 1210,46 1,10782 1,10783 = = 2,6202 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 1 = 461,971 461,971 92,5 82 113,5 134 824,4 1,1074 + 2 ∗ 1,10742 + 3 ∗ 1,10743 + 4 ∗ 1,10744 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 2 = = = 2,5519 323,05 323,05 461,971 323,05 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2,6202 ∗ + 2,5519 ∗ = 2,5921 461,971 + 323,05 461,971 + 323,05 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 2,5921 Bond : 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 3 & 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 7 3 ∗ 𝑥𝑥1 + 7 ∗ 𝑥𝑥2 = 2,5921 𝑥𝑥1 + 𝑥𝑥2 = 1 3 ∗ 𝑥𝑥1 + (1 − 𝑥𝑥1 ) ∗ 7 = 2,5921 7 − 2,5921 = 4𝑥𝑥1 𝑥𝑥1 = 1,1020 𝑥𝑥2 = −0,1020

c) 1 𝑈𝑈𝑈𝑈𝑈𝑈 𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑜𝑜𝑜𝑜 = 0,9 => 1$ = 0,9€ 𝑈𝑈𝑈𝑈𝑈𝑈 0,9 𝐸𝐸𝐸𝐸𝐸𝐸 1 − 1,055−3 = 5,4804% 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐸𝐸𝐸𝐸𝐸𝐸 = 1,05−1 + 1,052−2 + 1,055−3 1 − 1,051−3 = 5,0884% 𝐼𝐼𝐼𝐼𝐼𝐼 𝑈𝑈𝑈𝑈𝑈𝑈 = 1,048−1 + 1,049−2 + 1,051−3 1

CF EUR 200000 ∗ 0,054804 = 10960,8€

32

CF USD 10176,8€ => 11307,56$

Maths

Exercises 10960,8€

2

210960,8€

3

d) 𝑛𝑛𝑛𝑛𝑛𝑛 𝑈𝑈𝑈𝑈𝑈𝑈 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑢𝑢𝑢𝑢𝑢𝑢ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, 𝑛𝑛𝑛𝑛𝑛𝑛 𝐸𝐸𝐸𝐸𝐸𝐸 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ∶ 5,1%; 5,3%; 5,6% CF EUR 10960,8 2 = 10428,92€ 1,051 3

210960,8 = 190258,92€ 1,0532

Total 200687,84€ Gain 44625,47€ You pay CF EUR, you receive CF USD, so you gain

33

10176,8€ => 11307,56$ 210176,8€ => 233529,78$ CF USD 11307,56 = 10789,66$ 1,048 => 11868,63€ 233529,78 = 212222,44$ 1,0492 => 233444,68€ 245313,31€

Maths

Exercises

Problem 3 (Ex06) a) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 1,2𝑀𝑀 𝑟𝑟𝑚𝑚𝑚𝑚 = 4% 𝑡𝑡𝑐𝑐 = 0,4 𝑟𝑟𝑑𝑑 = 5% = 𝑟𝑟𝑓𝑓 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 =

𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 2,5 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿

2,5 = 1𝑀𝑀 2,5 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝐶𝐶𝐶𝐶 + 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 1,2𝑀𝑀 𝑀𝑀𝑀𝑀(𝐷𝐷) = 𝐵𝐵𝐵𝐵(𝐷𝐷) = 1,2𝑀𝑀 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 1,2𝑀𝑀 ∗ 0,05 = 0,06𝑀𝑀 𝐶𝐶𝐶𝐶 =

𝐷𝐷 = 0,25 𝐸𝐸 𝑀𝑀𝑀𝑀(𝐸𝐸) = 4,8𝑀𝑀

𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 4,8𝑀𝑀 𝑟𝑟𝑒𝑒 𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 − 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼) ∗ (1 − 𝑡𝑡𝑐𝑐 ) = (2,5 − 0,06) ∗ (1 − 0,4) = 0,684 0,684 𝑟𝑟𝑒𝑒 = = 14,25% 4,8

𝑀𝑀𝑀𝑀(𝐸𝐸) =

𝛽𝛽𝑒𝑒 =

14,25 − 5 = 2,3125 4

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 14,25 ∗

1,2 4,8 +5∗ ∗ (1 − 0,4) = 12% 6 6

b) 𝐷𝐷 = 2, constant ! 𝐸𝐸 𝐷𝐷 𝑟𝑟𝑒𝑒 + ∗ 𝑟𝑟𝑑𝑑 14,25 + 0,25 ∗ 5 𝐸𝐸 = = 12,4% 𝑟𝑟𝑢𝑢 = 𝐷𝐷 1 + 0,25 1 + 𝐸𝐸 𝐷𝐷 𝑟𝑟𝑒𝑒 = 𝑟𝑟𝑢𝑢 + (𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑑𝑑 ) ∗ = 12,4 + (12,4 − 5) ∗ 2 = 27,2% 𝐸𝐸 2 1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 27,2 ∗ + 5 ∗ ∗ (1 − 0,4) = 11,0667% 3 3

c) Poodle: 𝛽𝛽𝑑𝑑 = 0,25 𝛽𝛽𝑒𝑒 = 1,2 𝐷𝐷 = 0,6 𝑉𝑉 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 0,48𝑀𝑀€ 𝑔𝑔 = 2%

𝑟𝑟𝑒𝑒 = 5 + 1,2 ∗ 4 = 9,8% 𝑟𝑟𝑑𝑑 = 5 + 0,25 ∗ 4 = 6% 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,8 ∗ 0,4 + 6 ∗ 0,6 ∗ (1 − 0,4) = 6,08%

34

Maths

Exercises

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) ∗ (1 + 𝑔𝑔) 0,48 ∗ 0,6 ∗ 1,02 = = 7,2𝑀𝑀 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 𝑔𝑔 0,0608 − 0,02 𝑀𝑀𝑀𝑀(𝐸𝐸) = 0,4 ∗ 7,2 = 2,88𝑀𝑀

𝑉𝑉𝐴𝐴 =

𝑁𝑁𝑁𝑁𝑁𝑁(𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃) = −3 + 2,88 = −0,12𝑀𝑀

Debt and equity in equal amount : /2. 1 − 1,05−5 3 = 0,1299𝑀𝑀 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = ∗ 0,4 ∗ 0,05 ∗ 0,05 2 3 ∗ (1 − 0,4) ∗ 0,05 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐸𝐸 = 2 = 0,0474 0,95 3 ∗ (1 − 0,4) ∗ 0,03 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐷𝐷 = 2 = 0,0278 0,97 𝐴𝐴𝐴𝐴𝐴𝐴 = −0,12 + 0,1299 − 0,0474 − 0,0278 = −0,0653𝑀𝑀€

35

Maths

Exercises

Problem 4 (Ex06) a) 40 30 Asset

20

Equity

10

Loan

0 0

10

20

30

40

b) 𝑡𝑡 = 10 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 Problem 10 = 𝑥𝑥 ∗ (1 + 𝑟𝑟)10 𝑥𝑥 = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝑝𝑝𝑝𝑝𝑝𝑝

What is the price of the put ? 0,362 20 ln � � + �𝑙𝑙𝑙𝑙1,04 + 2 � ∗ 10 10 𝑑𝑑1 = = 1,5226 0,36 ∗ √10 0,362 20 ln �10� + �𝑙𝑙𝑙𝑙1,04 − 2 � ∗ 10 𝑑𝑑2 = = 0,3842 0,36 ∗ √10 𝑁𝑁(𝑑𝑑1 ) = 0,9357 + 0,26 ∗ (0,9370 − 0,9357) = 0,936 𝑁𝑁(𝑑𝑑2 ) = 0,6480 + 0,42 ∗ (0,6517 − 0,6480) = 0,6496 𝑃𝑃 = −20 ∗ (1 − 0936) + 10 ∗ e− ln (1,04)∗10 ∗ (1 − 0,6496) = 1,0872 Of the bond ? 10 𝐹𝐹𝐹𝐹 = = 6,7556 1,0410

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 = 6,7556 − 1,0872 = 5,6684

10 = 5,6684 ∗ (1 + 𝑟𝑟)10 𝑟𝑟 = 5,84%

c) 𝐸𝐸 = 11 𝑆𝑆 = 20 − 5,6684 = 14,3316 14,3316 = 0,7166 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 20 5,6684 = 0,2834 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 20 𝑡𝑡 = 1 𝑟𝑟𝑚𝑚 = 7% 𝜌𝜌𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 &𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 0,3 𝜎𝜎𝑚𝑚 = 0,1 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 0,04 + 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ∗ 0,03 0,3 ∗ 𝜎𝜎𝐿𝐿 0,0584 = 0,04 + ∗ 0,03 0,1 0,1 = 20,44% 𝜎𝜎𝐿𝐿 = (0,0584 − 0,04) ∗ 0,3 ∗ 0,03

36

Maths

Exercises

𝐸𝐸 2 𝐷𝐷 2 2 = ∗ � � + 𝜎𝜎𝐷𝐷 ∗ � � + 2𝑐𝑐𝑐𝑐𝑣𝑣𝐸𝐸𝐸𝐸 𝐴𝐴 𝐴𝐴 2𝑐𝑐𝑐𝑐𝑣𝑣𝐸𝐸𝐸𝐸 => 𝑤𝑤𝑤𝑤 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑚𝑚 𝑡𝑡ℎ𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐 = 0 … 𝑜𝑜𝑜𝑜 … 36² = 𝜎𝜎𝐸𝐸2 ∗ 0,72² + 20,44² ∗ 0,28² 𝜎𝜎𝐸𝐸 = 49,36% 𝜎𝜎𝐴𝐴2

𝜎𝜎𝐸𝐸2

0,49362 14,3316 ln � 11 � + �𝑙𝑙𝑙𝑙1,04 + 2 �∗1 𝑑𝑑1 = = 0,5908 0,4936 ∗ √1 0,49362 14,3316 �∗1 ln � 11 � + �𝑙𝑙𝑙𝑙1,04 − 2 𝑑𝑑2 = = 0,0972 0,4936 ∗ √1 𝑁𝑁(𝑑𝑑1 ) = 0,7224 + 0,08 ∗ (0,7257 − 0,7224) = 0,7227 𝑁𝑁(𝑑𝑑2 ) = 0,5359 + 0,72 ∗ (0,5398 − 0,5359) = 0,5387 𝐶𝐶 = 14,3316 ∗ 0,7227 − 11 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙1,04 ∗ 0,5387 = 4,66

37

Maths

Exercises

Problem 5 (Ex06) 25𝑀𝑀 𝑎𝑎𝑎𝑎 20$ 𝐸𝐸 = 18$ 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1,4 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 18% 𝑟𝑟𝑚𝑚𝑚𝑚 = 10% 𝐷𝐷𝐷𝐷𝐷𝐷 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 = 3$ 𝜎𝜎 = 30% 𝜎𝜎𝐴𝐴 = 25%

18 = 𝑟𝑟𝑓𝑓 + 1,4 ∗ 10 => 𝑟𝑟𝑓𝑓 = 4% 2

𝑆𝑆 ∗ = 20 − 3 ∗ 𝑒𝑒 −0,04∗12 = 17,02

0,32 17,02 ln � 18 � + �0,04 + 2 � ∗ 1 𝑑𝑑1 = = 0,0967 0,3 0,32 17,02 ln � 18 � + �0,04 − 2 � ∗ 1 𝑑𝑑2 = = −0,203 0,3 𝑁𝑁(𝑑𝑑1 ) = 0,5359 + 0,67 ∗ (0,5398 − 0,5359) = 0,5385 𝑁𝑁(𝑑𝑑2 ) = 0,4207 + 0,3 ∗ (4168 − 0,4207) = 0,4195 𝐶𝐶 = 17,02 ∗ 0,5385 − 18 ∗ 𝑒𝑒 −0,04 ∗ 0,4195 𝐶𝐶 = 1,91 b) Put + 2$ =>Sell it Buy Call Short stock c) 𝑞𝑞 =

𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 5 = = 0,2 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 25

38

Maths

Exercises

RE-Exam 2006 Problem 1 (Reex06) a) 𝐷𝐷𝐷𝐷𝐷𝐷 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑆𝑆ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡 − 𝑔𝑔 1,45 + 0,045 = 𝑡𝑡 = 14,167 15

b) Must write CAPM holds 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,8 ∗ 14,167 + 0,2 ∗ 4 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 12,1336 12,1336 − 4 = 1,0167 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 8 14,167 − 4 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = = 1,2709 8 Snake before acquisition

Total Catorade before acquisition

Total Snake after acquisition

Equity : 0,8 : => 14,167% 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 1,2709 1500000 Debt : 0,2 =>4% 375000 Total 1875000 Equity : 0,6 200000 => 𝛽𝛽 = 1,3 Debt : 0,4 133333,33 => 𝑟𝑟𝑑𝑑 = 5% Total 333333,33 Equity 1500000 Debt => 375000 + 200000 = 575000 133333,33 => 𝑟𝑟𝑑𝑑 = 5% Total 2208333,33

Total 𝐸𝐸 = 0,679 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 = 0,3208 𝐷𝐷 + 𝐸𝐸 575000 133333,33 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 4 ∗ +5∗ = 4,188% 708333,33 708333,33 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 12,1336% 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 1,3 ∗ 8 + 4 = 14,4% 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = 14,4 ∗ 0,6 + 5 ∗ 0,4 = 10,64% 1875000 333333,33 + 10,64 ∗ = 11,9082% 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 12,1336 ∗ 2208333,33 2208333,33 708333,33 1500000 11,9082 = 4,188 ∗ + ∗ 𝑟𝑟 2208333,33 2208333,33 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 15,55% 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 4 + 8 ∗ 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 39

Maths 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 c)

Exercises 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

= 1,44

𝑁𝑁𝑁𝑁𝑁𝑁 = −200000 + (500000 − 325000 − 10000) ∗ d)

𝑁𝑁𝑁𝑁𝑁𝑁 = −200000 + (500000 − 325000 − 10000) ∗ e)

𝑁𝑁𝑁𝑁𝑁𝑁 = −200000 + (500000 − 325000 − 10000) ∗

1 − 1,119−3 = 196984,16 0,119

1 − 1,121336−3 = 195394,11 0,121336 1 − 1,04−3 = 257890,02 0,04

40

Maths

Exercises

Problem 2 (Reex06) a) 5 5 105 𝑃𝑃𝐵𝐵1 = + + = 87,6836 2 1,09 1,095 1,13 100 = 75,1315 𝑃𝑃𝐵𝐵2 = 1,13 200 𝑃𝑃𝐵𝐵3 = = 166,8022 1,0952 10 10 10 210 + + + = 163,3611 𝑃𝑃𝐵𝐵4 = 2 3 1,09 1,095 1,1 1,114 20 20 20 220 + + + = 194,976 𝑃𝑃𝐵𝐵5 = 1,09 1,0952 1,13 1,114

b) 𝑎𝑎 => 𝐵𝐵1 ; 𝑏𝑏 => 𝐵𝐵2 ; 𝑐𝑐 => 𝐵𝐵3 ; 𝑑𝑑 => 𝐵𝐵4 5 10 20 ⎧𝑎𝑎 ∗ + 𝑑𝑑 ∗ = = 𝑎𝑎 ∗ 5 + 𝑑𝑑 ∗ 10 = 20 => 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑡𝑡ℎ𝑒𝑒 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 1,09 1,09 ⎪ 1,09 𝑎𝑎 ∗ 5 + 𝑐𝑐 ∗ 200 + 𝑑𝑑 ∗ 10 = 20 ⎨ 𝑎𝑎 ∗ 105 + 𝑏𝑏 ∗ 100 + 𝑑𝑑 ∗ 10 = 20 ⎪ ⎩ 𝑑𝑑 ∗ 210 = 220 𝑑𝑑 = 1,0476 20 − 10 ∗ 1,0476 = 1,9048 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 ∶ 𝑎𝑎 = 5 20 − 1,9048 ∗ 5 − 1,0476 ∗ 10 =0 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 ∶ 𝑐𝑐 = 200 20 − 1,9048 ∗ 105 − 1,0476 ∗ 10 = −1,9048 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 ∶ 𝑏𝑏 = 100

c) need to know two years rate in two years and also the 1 year rate in two years. 1,103 − 1 = 11% 𝑖𝑖1 = 1,0952 1,114 = 1,0952 ∗ (1 + 𝑖𝑖2 )2 1,114 𝑖𝑖2 = � − 1 = 12,5209% 1,0952 15 165 200 + + = 301,81 2 1,11 1,12522 1,1252

d) Be careful, 0,18 𝐿𝐿𝐿𝐿𝐿𝐿/𝑃𝑃𝑃𝑃𝑃𝑃 equal 𝐿𝐿𝐿𝐿𝐿𝐿/𝑃𝑃𝑃𝑃𝑃𝑃 = 1𝑃𝑃𝑃𝑃𝑃𝑃 = 0,18 𝐿𝐿𝐿𝐿𝐿𝐿

1 ‼ 0,18

1 − 𝐷𝐷𝑛𝑛 𝐷𝐷1 + 𝐷𝐷2 + ⋯ 𝐷𝐷𝑛𝑛 1 − 1,068−3 = 6,769% 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐿𝐿𝐿𝐿𝐿𝐿 = 1,06−1 + 1,065−2 + 1,068−3 1 − 1,076−3 = 7,566% 𝐼𝐼𝐼𝐼𝐼𝐼 𝑃𝑃𝑃𝑃𝑃𝑃 = 1,07−1 + 1,072−2 + 1,076−3 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =

Year1 Year2 Year3

CF LVL (pay) 1000000 ∗ 0,06769 = 67690 67690 ∗ 0,18 = 12184,2 𝐿𝐿𝐿𝐿𝐿𝐿 12184,2 𝐿𝐿𝐿𝐿𝐿𝐿 1000000 ∗ 0,18 + 12184,2 = 192184,2 𝐿𝐿𝐿𝐿𝐿𝐿 41

CF PLN (receive) 75660 𝑃𝑃𝑃𝑃𝑃𝑃 75660 1075660

Maths

Exercises

At year 2 CF Year2 Year3 Total Difference loss

CF

12184,2 = 11472,88 1,062 192184,2 = 168806,37 1,0672 180279,25 𝐿𝐿𝐿𝐿𝐿𝐿 10999,65 𝐿𝐿𝐿𝐿𝐿𝐿

42

75660 = 70710,28 1,07 1075660 = 936020,69 1,0722 1006730,97 𝑃𝑃𝑃𝑃𝑃𝑃 ∗ 0,19 = 191278,88 𝐿𝐿𝐿𝐿𝐿𝐿

Maths

Exercises

Problem 3 (Reex06) CORRECTION : Assume 𝑀𝑀𝑀𝑀(𝐷𝐷) = 𝐵𝐵𝐵𝐵(𝐷𝐷) = 0,9 𝑛𝑛𝑛𝑛𝑛𝑛ℎ = 2𝑀𝑀 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 0,5𝑀𝑀 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑜𝑜𝑜𝑜 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 60000 𝑓𝑓𝑓𝑓𝑓𝑓 6𝑦𝑦 𝑟𝑟𝑑𝑑 = 5,56% 80 𝑐𝑐𝑐𝑐𝑐𝑐 = = 0,8 𝛽𝛽 = 𝑣𝑣𝑣𝑣𝑣𝑣 100 𝑟𝑟𝑚𝑚 = 10% 𝐼𝐼𝐼𝐼 = 4% 𝑡𝑡𝑐𝑐 = 35%

a) 𝑟𝑟𝑒𝑒 = 5,56 + 0,8 ∗ (10 − 5,56) = 9,112% 𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 (0,5 − 0,9 ∗ 0,0556) ∗ (1 − 0,35) = = 3,2098𝑀𝑀 𝑀𝑀𝑀𝑀(𝐸𝐸) = 𝑟𝑟𝑒𝑒 0,09112 𝑀𝑀𝑀𝑀(𝐷𝐷) = 0,9𝑀𝑀 𝑀𝑀𝑀𝑀(𝐴𝐴) = 3,2098 + 0,9 = 4,1098𝑀𝑀 0,5 ∗ (1 − 0,35) = 4,1098𝑀𝑀 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 7,9079%

𝑀𝑀𝑀𝑀(𝐴𝐴) =

b) D/A=0,3 D=constant ! 0,9 0,09112 + 3,2098 ∗ 0,0556 ∗ (1 − 0,35) = 8,5644% 𝑟𝑟𝑢𝑢 = 0,9 1 + 3,2098 ∗ (1 − 0,35)

Here we can both assume that D is constant or D/E constant 0,3 ∗ (1 − 0,35) = 9,4013% 𝑟𝑟𝑒𝑒 = 0,085644 + (0,085644 − 0,0556) ∗ 0,7 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,4013 ∗ 0,7 + 5,56 ∗ 0,3 ∗ (1 − 0,35) = 7,6651% Or 0,3 𝑟𝑟𝑒𝑒 = 0,085644 + (0,085644 − 0,0556) ∗ = 9,852% 0,7 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,852 ∗ 0,7 + 5,56 ∗ 0,3 ∗ (1 − 0,35) = 7,9806% c) Project 2 Invest with 100% debt 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∶ 𝑋𝑋

1 − 1,085644−5 = 0,56256 0,085644 1 − 1,0556−5 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 𝑋𝑋 ∗ 0,35 ∗ 0,0556 ∗ = 𝑋𝑋 ∗ 0,0829 0,0556 Depreciation 5y 1 − 1,085644−5 1 = 𝑋𝑋 ∗ 0,2754 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 𝑋𝑋 ∗ ∗ 0,35 ∗ 0,085644 5 𝑋𝑋 ∗ 0,04 ∗ 0,65 𝐼𝐼𝐼𝐼 𝐷𝐷 = = 𝑋𝑋 ∗ 0,02708 0,96 So 𝐴𝐴𝐴𝐴𝐴𝐴 = 0,5626 + 𝑋𝑋 ∗ (−1 + 0,0829 + 0,2754 − 0,02708) = 0,5626 − 𝑋𝑋 ∗ (0,6688) 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 0,22 ∗ (1 − 0,35) ∗

43

Maths

Exercises

𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑖𝑖𝑖𝑖 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤ℎ 𝑖𝑖𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴 > 0, 𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤, 𝑖𝑖𝑖𝑖 𝑋𝑋 >

0,5626 = 0,8412 0,6688

Project 3 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 2,1𝑀𝑀 𝛽𝛽𝐷𝐷 = 0,6 𝛽𝛽𝐸𝐸 = 1,2 𝐸𝐸𝐸𝐸𝐸𝐸𝑇𝑇(0) = 0,42 𝑔𝑔 = 3% 𝐷𝐷 =2 𝐸𝐸 𝑟𝑟𝑒𝑒 = 5,56 + 1,2 ∗ (10 − 5,56) = 10,888% 𝑟𝑟𝑑𝑑 = 5,56 + 0,6 ∗ (10 − 5,56) = 8,224% 2 1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 10,888 ∗ + 5,56 ∗ ∗ (1 − 0,35) = 7,1931% 3 3 (0,42 ∗ 1,03)(1 − 0,35) = 6,7060𝑀𝑀 𝑀𝑀𝑀𝑀(𝐴𝐴) = 0,071931 − 0,03 𝑀𝑀𝑀𝑀(𝐸𝐸) =

1 ∗ 𝑀𝑀𝑀𝑀(𝐴𝐴) = 2,2354𝑀𝑀 3

𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 ∶ 𝐸𝐸 = 𝐷𝐷 = 1,05

𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 1,05 ∗ 0,35 ∗ 0,0556 ∗

1 − 1,0556−5 = 0,08711𝑀𝑀 0,0556

2,1 ∗ 0,65 ∗ 0,04 = 0,056875 0,96 𝐴𝐴𝐴𝐴𝐴𝐴 = −2,1 + 2,2354 + 0,08711 − 0,056875 = 0,1656𝑀𝑀 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =

44

Maths

Exercises

Problem 4 (Reex06) a) ln(1,014) ∗ 3 = 4,17% 0,15 1 �0,0417 + 2 � ∗ 3 = 0,174 𝑑𝑑1 = 1 �0,15 ∗ �3 0,15 1 �0,0417 − 2 � ∗ 3 𝑑𝑑2 = = −0,0496 1 �0,15 ∗ �3 𝑁𝑁(𝑑𝑑1 ) = 0,5675 + 0,4 ∗ (0,5714 − 0,5675) = 0,569 𝑁𝑁(𝑑𝑑2 ) = 0,48 1

𝐶𝐶 = 15 ∗ 0,569 − 15 ∗ 𝑒𝑒 −0,04∗3 ∗ 0,48 = 1,433

b) Stock split 100 ∗ 3 = 300 300 ∗ 1 = 300 So 𝑥𝑥 = 0,33

𝑆𝑆𝑢𝑢 = 17,5695 ∗ 0,33 = 5,8565

𝑆𝑆 = 15 𝑢𝑢 =

1 √0,15∗�6 𝑒𝑒

𝑆𝑆𝑑𝑑 = 12,807 ∗ 0,33 = 4,269

= 1,1713

1 −√0,15∗� 6 = 0,8538 𝑑𝑑 = 𝑒𝑒 1 (𝑒𝑒 0,04∗3 = 1,014 … => 2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ

= 1,006976) 1,006976 − 0,8538 𝑝𝑝 = = 0,4824 1,1713 − 0,8538 𝐸𝐸 = 5, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑖𝑖𝑡𝑡 ′ 𝑠𝑠 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 1,8597 ∗ 0,48242 = 0,4268 𝐶𝐶𝐸𝐸𝐸𝐸 = 1,0069762 Now we have also 3 call, because of the split. ∗ 𝐶𝐶𝐸𝐸𝐸𝐸 = 1,28 The difference is only due to rounding

c) The firm does a split when the share price is too expensive for example. d) 1200 1150 1100 1050 1000 950 900

Partindex

E=I

45

𝑆𝑆𝑢𝑢𝑢𝑢 = 6,8597 𝑆𝑆𝑢𝑢𝑢𝑢 = 5

𝑆𝑆𝑑𝑑𝑑𝑑 = 3,6469

Maths

Exercises

e) 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐼𝐼𝐼𝐼 𝐼𝐼 ∗ > 𝐼𝐼 𝐼𝐼 ∗ ≤ 𝐼𝐼

𝑁𝑁𝑁𝑁𝑁𝑁ℎ𝑖𝑖𝑖𝑖𝑖𝑖 𝐼𝐼 ∗ − 𝐼𝐼 ∗ 1000 𝐼𝐼 𝑁𝑁𝑁𝑁𝑁𝑁ℎ𝑖𝑖𝑖𝑖𝑖𝑖 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

−1000 +1000 +1000 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵

46

−1000 𝐼𝐼 ∗ − 𝐼𝐼 1000 ∗ �1 + � 𝐼𝐼 +1000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇

Maths

Exercises

Problem 5 (Reex06)

𝑆𝑆 = 32

𝑆𝑆𝑢𝑢𝑑𝑑𝑑𝑑𝑑𝑑 = 36,8 − 3 = 33,8

𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 28,8 − 3 = 25,8

𝐸𝐸 = 28 a) 1,03 − 0,9 = 0,52 𝑝𝑝 = 1,15 − 0,9 1 − 𝑝𝑝 = 0,48 0,52 ∗ 10,87 + 0,48 ∗ 2,42 𝐶𝐶𝑢𝑢𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = = 6,6155 1,03 𝐶𝐶𝑢𝑢𝑒𝑒𝑒𝑒 = 36,8 − 28 = 8,8 => 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 0,52 ∗ 1,67 + 0 = 0,843 => 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 1,03 = 0,8

𝐶𝐶𝑑𝑑𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 = 𝐶𝐶𝑑𝑑𝑒𝑒𝑒𝑒

0,52 ∗ 8,8 + 0,48 ∗ 0,843 1,03 𝐶𝐶 = 4,8356

𝐶𝐶 = b)

1,67 − 0 25,8 ∗ (1,15 − 0,9) 𝛼𝛼𝑑𝑑∗ = 0,2589 1,67 − 0,2589 ∗ 29,67 = −5,8365 𝐵𝐵𝑑𝑑 = 1,03 𝐶𝐶𝑑𝑑 = 𝛼𝛼𝑑𝑑∗ ∗ 𝑆𝑆𝑑𝑑 + 𝛽𝛽𝑑𝑑 = 0,2589 ∗ 25,8 − 5,8365 = 0,843

𝛼𝛼𝑑𝑑∗ =

𝛼𝛼𝑢𝑢∗ => 𝑛𝑛𝑛𝑛‼ 𝑤𝑤𝑒𝑒 ′ 𝑣𝑣𝑣𝑣 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ! 𝐶𝐶𝑢𝑢 = 8,8 !

8,8 − 0,843 = 0,9946 32 ∗ (1,15 − 0,9) 8,8 − 0,9946 ∗ 36,8 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶‼ = −27 𝐵𝐵 = 1,03 𝐶𝐶 = 0,9946 ∗ 32 − 27 𝐶𝐶 = 4,8272

𝛼𝛼 =

At the end with two down step 𝛽𝛽𝑑𝑑𝑑𝑑 = 𝛽𝛽𝑑𝑑 ∗ 1,03 = −6,012 So we have : 𝐶𝐶𝐶𝐶(𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵)𝑑𝑑𝑑𝑑 = −6,012 Then 𝐶𝐶𝐶𝐶(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝑑𝑑𝑑𝑑 = 0,2589 ∗ 23,22 = 6,012 So 𝐶𝐶𝐶𝐶(𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜) = 0

c) With an American put it wouldn’t pay to exercise it early. 47

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 38,87 𝐶𝐶𝑢𝑢𝑢𝑢 = 10,87 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 30,42 𝐶𝐶𝑢𝑢𝑢𝑢 = 2,42 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑 = 29,67 𝐶𝐶𝑑𝑑𝑑𝑑 = 1,67 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 23,22

Maths

Exercises

Exam 2007 Problem 1 (Ex07) a) 𝜎𝜎² = 0,49 ∗ (0,49 ∗ 0,252 + 0,23 ∗ 0,2 ∗ 0,25 ∗ 0,35 + 0,28 ∗ 0,3 ∗ 0,25 ∗ 0,45) + 0,23 ∗ (0,49 ∗ 0,2 ∗ 0,25 ∗ 0,35 + 0,23 ∗ 0,35² + 0,28 ∗ 0,4 ∗ 0,35 ∗ 0,45) + 0,28 ∗ (0,49 ∗ 0,3 ∗ 0,25 ∗ 0,45 + 0,23 ∗ 0,4 ∗ 0,35 ∗ 0,45 + 0,28 ∗ 0,45²) = 0,5868 𝜎𝜎 = 24,2245% 0,49 ∗ 0,252 + 0,23 ∗ 0,2 ∗ 0,25 ∗ 0,35 + 0,28 ∗ 0,3 ∗ 0,25 ∗ 0,45 = 0,7515 5,868 0,49 ∗ 0,2 ∗ 0,25 ∗ 0,35 + 0,23 ∗ 0,35² + 0,28 ∗ 0,4 ∗ 0,35 ∗ 0,45 = 0,9269 𝛽𝛽𝐵𝐵 = 5,868 0,49 ∗ 0,3 ∗ 0,25 ∗ 0,45 + 0,23 ∗ 0,4 ∗ 0,35 ∗ 0,45 + 0,28 ∗ 0,45² = 1,495 𝛽𝛽𝐶𝐶 = 5,868

𝛽𝛽𝐴𝐴 =

With CAPM 𝐸𝐸(𝑟𝑟𝑚𝑚 ) = 0,49 ∗ 0,14 + 0,23 ∗ 0,15 + 0,18 ∗ 0,28 = 15,35 14 − 10 𝛽𝛽𝐴𝐴 = = 0,7477 15,35 − 10 15 − 10 𝛽𝛽𝐵𝐵 = = 0,9346 15,35 − 10 18 − 10 = 1,4953 𝛽𝛽𝐶𝐶 = 15,35 − 10 𝐴𝐴: 0,49 ∗ 0,7515 = 36,8235% 𝐵𝐵: 0,23 ∗ 0,9269 = 21,3187% 𝐶𝐶: 0,28 ∗ 1,495 = 41,86%

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ∶

𝜎𝜎𝑖𝑖 − 𝛽𝛽𝑖𝑖 ∗ 𝜎𝜎𝑚𝑚 𝜎𝜎𝑖𝑖

0,25 − 0,7515 ∗ 24,2245 = 27,18% 0,25 0,35 − 0,9269 ∗ 24,2245 𝐵𝐵 ∶ = 35,8466% 0,35 0,45 − 1,495 ∗ 24,2245 𝐶𝐶 ∶ = 19,5208% 0,45

𝐴𝐴 ∶

b) 45 = 𝑥𝑥𝑚𝑚 ∗ 15,35 + 𝑥𝑥𝑓𝑓 ∗ 10 = 𝑥𝑥𝑚𝑚 ∗ 15,35 + (1 − 𝑥𝑥𝑚𝑚 ) ∗ 10 = 5,35𝑥𝑥𝑚𝑚 + 10 => 𝑥𝑥𝑚𝑚 = 6,542 𝑥𝑥𝑓𝑓 = −5,542 𝜎𝜎𝑃𝑃2 = 6,5422 ∗ 0,5868 = 2,51137 𝜎𝜎𝑃𝑃 = 158,5% c)

𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑓𝑓 𝑀𝑀² = �� � ∗ 𝜎𝜎𝑚𝑚 � − (𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 ) 𝜎𝜎𝑖𝑖 14 − 10 ∗ 24,2245 − 5,35 = −1,474 𝑀𝑀𝐴𝐴2 = 25 15 − 10 ∗ 24,2245 − 5,35 = −1,889 𝑀𝑀𝐵𝐵2 = 35 18 − 10 𝑀𝑀𝐶𝐶2 = ∗ 24,2245 − 5,35 = −1,0434 45 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 =

𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑓𝑓 𝛽𝛽𝑖𝑖

48

Maths

Exercises

14 − 10 = 5,3497 0,7477 15 − 10 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐵𝐵 = = 5,3499 0,9346 18 − 10 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐶𝐶 = = 5,35 1,4953

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝐴𝐴 =

d) 𝐸𝐸�𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑓𝑓 � = 𝛽𝛽�𝑟𝑟𝑚𝑚 − 𝑟𝑟𝑓𝑓 � + 𝛽𝛽2 ∗ 𝑆𝑆𝑆𝑆𝑆𝑆 (=> 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) + 𝛽𝛽3 ∗ 𝐻𝐻𝐻𝐻𝐻𝐻 (=> 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚) + 𝛽𝛽4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

49

Maths

Exercises

Problem 2 (Ex07) a) 5000 10000 10000 10000 25000 𝑃𝑃1 = + + + + = 49299,03 1,045 1,0472 1,053 1,0554 1,065 5000 5000 5000 5000 45000 𝑃𝑃2 = + + + + = 51327,75 2 3 4 1,045 1,047 1,05 1,055 1,065 25000 30000 𝑃𝑃3 = + = 49338,57 1,045 1,053 10000 10000 15000 25000 𝑃𝑃4 = + + + = 48550,43 1,0472 1,053 1,0554 1,065

b) 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 = 5,81% 5000 5000 5000 45000 5000 +2∗ +3∗ +4∗ +5∗ 4 2 3 1,0581 1,0581 1,0581 1,0581 1,05815 211923,7102 𝐷𝐷2 = = = 4,128589 5000 5000 5000 45000 5000 51330,78912 + + + + 5 ∗ 1,0581 1,05812 1,05813 1,05814 1,05815 30000 25000 +3∗ 1,0488 1,04883 101849,3157 = = 2,043487 𝐷𝐷3 = 30000 25000 49840,94913 + 1,0488 1,04883 4,128589 = 3,901889 𝑀𝑀𝐷𝐷2 = 1,0581 2,043487 = 1,948405 𝑀𝑀𝐷𝐷3 = 1,0488 So when the yield raises of 1%, price fell of 3,901889% and 1,948405% respectively 𝑥𝑥 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 %

3,9019 1,9484 ) ≤ 49840,9491 ∗ (1 − 𝑥𝑥 ∗ ) 100 100 51330,7891 − 49840,9491 𝑥𝑥 ≥ 3,9019 1,9484 51330,7891 ∗ 100 − 49840,9491 ∗ 100 𝑥𝑥 ≥ 1,444%

51330,7891 ∗ (1 − 𝑥𝑥 ∗

c) 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 ∶ 21 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 2006, 100𝑀𝑀 1 𝐸𝐸𝐸𝐸𝐸𝐸/𝑈𝑈𝑈𝑈𝑈𝑈 = 0,70

1 − 1,0465−3 = 4,6462% 1,045−1 + 1,046−2 + 1,0465−3 1 − 1,042−3 = 4,1945% 𝐼𝐼𝐼𝐼𝐼𝐼 𝑈𝑈𝑈𝑈𝑈𝑈 = 1,04−1 + 1,041−2 + 1,042−3 Time CF EUR 1 100𝑀𝑀 ∗ 0,046462 = 4,6462𝑀𝑀 𝑈𝑈𝑈𝑈𝑈𝑈 3,25234€ 2 4,6462𝑀𝑀 3,25234€ 3 104,6462𝑀𝑀 73,25234𝑀𝑀€ 𝐼𝐼𝐼𝐼𝐼𝐼 𝐸𝐸𝐸𝐸𝐸𝐸 =

d) Suppose 1 year later !!! PFF Don’t compute the new swap rate, it’s USELESS 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐸𝐸𝐸𝐸𝐸𝐸 => 4,65, 4,75, 4,8% 50

CF USD 4,1945𝑀𝑀$ 4,1945𝑀𝑀$

104,1945𝑀𝑀$

Maths

Exercises CF E 3,25234 = 3,1078𝑀𝑀€ 1,0465 73,25234 = 66,7596𝑀𝑀€ 1,04752 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 69,8674𝑀𝑀€

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷

69,8674 − 75,1364 = −5,269𝑀𝑀€

51

CF USD 4,1945 = 4,0332𝑀𝑀$ 1,04 104,1945 = 96,1487𝑀𝑀$ 1,0412 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 100,1819𝑀𝑀$ 100,1819𝑀𝑀$ ∗ 0,75 = 75,1364𝑀𝑀€

Maths

Exercises

Problem 3 (Ex07) CORRECTION on dividend 𝐷𝐷𝐷𝐷𝐷𝐷 = 14$ 𝑛𝑛𝑛𝑛𝑛𝑛ℎ = 50𝑀𝑀 𝐸𝐸𝐸𝐸𝐸𝐸 = 16$ a)

𝐷𝐷𝐷𝐷𝐷𝐷 14 = = 0,875 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝐸𝐸𝐸𝐸𝐸𝐸 16 𝐸𝐸𝐸𝐸𝐸𝐸 𝑅𝑅𝑅𝑅𝑅𝑅 = = 16% 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑔𝑔 = 𝑅𝑅𝑅𝑅𝑅𝑅 ∗ (1 − 𝑝𝑝) = 16 ∗ 0,125 = 2%

200 𝐼𝐼𝐼𝐼𝐼𝐼 = = 5000 𝑟𝑟𝑑𝑑 − 𝑔𝑔 0,06 − 0,02 50 ∗ 14 ∗ 1,02 𝑀𝑀𝑀𝑀(𝐸𝐸) = = 5492,3077 0,15 − 0,02

𝑀𝑀𝑀𝑀(𝐷𝐷) =

We should also take the dividend at t=0 𝑀𝑀𝑀𝑀(𝐸𝐸 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) = 5492,3077 + 14 ∗ 50 = 6192,3077 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = b)

6192,3077 5000 ∗ 0,15 + ∗ 0,06 ∗ (1 − 0,3) = 10,1753% 11192,3077 11192,3077

𝐷𝐷 0,15 + 0,06 ∗ 5000 𝑟𝑟𝑒𝑒 + 𝑟𝑟𝑑𝑑 ∗ 6192,3077 𝐸𝐸 = 𝑟𝑟𝑢𝑢 = = 10,9794% 𝐷𝐷 5000 1 + 𝐸𝐸 1 + 6192,3077 𝑟𝑟𝑙𝑙𝑙𝑙 = 0,109794 + (0,109794 − 0,06) ∗ 1 = 15,9588% 1 1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 0,159588 ∗ + 0,06 ∗ ∗ (1 − 0,3) = 10,0794% 2 2 c) 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 250𝑀𝑀 20% ∗ 250 = 50𝑀𝑀 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 50 = 12,5𝑀𝑀 𝑝𝑝𝑝𝑝𝑝𝑝 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 4 1 − 1,109794−4 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 12,5 ∗ 0,3 ∗ = 11,6393𝑀𝑀 0,109794 30 ∗ 0,7 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = = 191,2673𝑀𝑀 0,109794 Equal amount of debt and equity : 250/2=125 1 − 1,06−5 = 9,4778𝑀𝑀 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 125 ∗ 0,06 ∗ 0,3 ∗ 0,06 125 ∗ 0,05 ∗ 0,7 = 4,6053𝑀𝑀 0,95 125 ∗ 0,03 ∗ 0,7 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐷𝐷 = = 2,7062𝑀𝑀 0,97

𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐸𝐸 =

𝐴𝐴𝐴𝐴𝐴𝐴 = −250 + 11,6393 + 191,2673 + 9,4778 − 4,6053 − 2,7062 = −44,9271𝑀𝑀 52

Maths

Exercises

Problem 4 (Ex07) a) 𝑆𝑆 = 54$ 𝐸𝐸 = 55$ 𝑡𝑡 = 9 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 = 0,75 𝐷𝐷𝐷𝐷𝐷𝐷 𝑖𝑖𝑖𝑖 6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 = 4$ 𝜎𝜎 = 0,35 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟 => 100 = 93,47 ∗ 𝑒𝑒 𝑟𝑟∗0,75 => 𝑟𝑟 = 9% 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

Good method, short ! We actualize the dividend as follow. 4 ∗ 𝑒𝑒 −0,09∗0,5 = 3,824 So that 𝑆𝑆 ∗ = 𝑆𝑆 − 3,824 = 50,176 We then use B&S formula 0,352 50,176 � + �0,09 + 2 � ∗ 0,75 ln � 55 𝑑𝑑1 = = 0,0714 0,35 ∗ �0,75 0,352 50,176 � + �0,09 − 2 � ∗ 0,75 ln � 55 𝑑𝑑2 = = −0,2317 0,35 ∗ �0,75 𝑁𝑁(𝑑𝑑1 ) = 0,5279 + 0,14 ∗ (0,5319 − 0,5279) = 0,5285 𝑁𝑁(𝑑𝑑2 ) = 0,4090 + 0,17 ∗ (0,4052 − 0,4090) = 0,4084 𝐶𝐶 = 50,176 ∗ 0,5285 − 55 ∗ 𝑒𝑒 −0,09∗0,75 ∗ 0,4084 𝐶𝐶 = 5,522

𝑃𝑃 = −50,176 ∗ (1 − 0,5285) + 55 ∗ 𝑒𝑒 −0,09∗0,75 ∗ (1 − 0,4084) 𝑃𝑃 = 6,756

This following method is very long, but should be ok, to use only if mentioned in the question ! 𝑢𝑢 = 𝑒𝑒 0,35∗√0,25 = 1,1912 𝑑𝑑 = 𝑒𝑒 −0,35∗√0,25 = 0,8395 𝑒𝑒 0,09∗0,25 − 0,8395 = 0,5211 𝑝𝑝 = 1,1912 − 0,8395 (1 − 𝑝𝑝) = 0,4789 𝑡𝑡 = 0

3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚

𝑆𝑆𝑢𝑢 = 64,32

𝑆𝑆 = 54

𝐶𝐶𝑢𝑢𝑢𝑢 =

𝑆𝑆𝑑𝑑 = 45,333

6 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐷𝐷𝐷𝐷𝐷𝐷 = 4$

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 76,62 − 4 = 72,62

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢

= 54 − 4 = 50

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑 = 38,057 − 4 = 34,057

31,505 ∗ 0,5211 + 5,965 ∗ 0,4789 = 18,845 𝑒𝑒 0,09∗0,25

53

𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢 𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢

9 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 86,505 = 86,505 − 55 = 31,505

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 60,965 = 60,965 − 55 = 5,965

𝐶𝐶𝑢𝑢𝑢𝑢𝑢𝑢

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢𝑢𝑢 = 59,56 = 59,56 − 55 = 4,56 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑 = 41,975 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑 = 40,569 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑 = 28,59

Maths

Exercises

4,56 ∗ 0,5211 = 2,323 𝑒𝑒 0,09∗0,25 18,845 ∗ 0,5211 + 2,323 ∗ 0,4789 = 10,689 𝐶𝐶𝑢𝑢 = 𝑒𝑒 0,09∗0,25 2,323 ∗ 0,5211 = 1,1836 𝐶𝐶𝑑𝑑 = 𝑒𝑒 0,09∗0,25 10,689 ∗ 0,5211 + 1,1836 ∗ 0,4789 𝐶𝐶 = 𝑒𝑒 0,09∗0,25 𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸 = 6 𝐶𝐶𝑢𝑢𝑢𝑢 =

14,431 ∗ 0,5211 + 26,41 ∗ 0,4789 = 19,719 𝑒𝑒 0,09∗0,25 13,025 ∗ 0,4789 = 6,099 𝑃𝑃𝑑𝑑𝑑𝑑 = 𝑒𝑒 0,09∗0,25 6,099 ∗ 0,5211 + 19,719 ∗ 0,4789 = 12,3408 𝑃𝑃𝑑𝑑 = 𝑒𝑒 0,09∗0,25 6,099 ∗ 0,4789 = 2,8558 𝑃𝑃𝑢𝑢 = 𝑒𝑒 0,09∗0,25 2,8558 ∗ 0,5211 + 12,3408 ∗ 0,4789 𝑃𝑃 = 𝑒𝑒 0,09∗0,25 𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸 = 7,234 The difference should be due to the round 𝑃𝑃𝑑𝑑𝑑𝑑 =

b) For the put, the 𝛼𝛼 ∗ = −�1 − 𝑁𝑁(𝑑𝑑1 )� = −0,4715 And the 𝐵𝐵 = 55 ∗ 𝑒𝑒 −0,09∗0,75 ∗ (1 − 0,4084) = 30,414 (In fact we have 𝑃𝑃 = 𝛼𝛼 ∗ ∗ 𝑆𝑆 ∗ + 𝐵𝐵) 𝑆𝑆 = −0,472 ∗ 50,176 = −23,683 If you do S+B you have P

c) 𝐸𝐸 = 55

𝑡𝑡 = 9 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 =

𝑆𝑆 = 50 𝜎𝜎 = 0,35 𝑟𝑟 = 9% 𝑢𝑢 = 𝑒𝑒

𝑑𝑑 =

3 0,35∗� 4

3 4

= 1,354

3 −0,35∗� 4 = 0,7385 𝑒𝑒 3 𝑒𝑒 0,09∗4 − 0,7385

(𝑝𝑝 =

= 0,5383) 1,354 − 0,7385 𝑅𝑅𝑅𝑅𝐶𝐶𝑢𝑢 − 𝑅𝑅𝑅𝑅𝐶𝐶𝑑𝑑 (67,7 − 55) − (55 − 36,925) = = −0,1812 𝛼𝛼 ∗ = 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 50 ∗ (1,354 − 0,7385) 12,2 − (−0,1812 ∗ 67,7) 𝐵𝐵∗ = = 22,87 3 𝑒𝑒 0,09∗4 𝑆𝑆 = −0,1812 ∗ 50 + 22,87 = 13,81

54

Maths

Exercises

Problem 5 (Ex07)

Return of the Oilbond 1200 1150 1100 1050 1000 950 0

20

40

60

80

b) We have: 𝜎𝜎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,36 ∗ �0,5 = 25,4558% 𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑙𝑙𝑙𝑙1,03 ∗ 3 = 0,0886764 𝑢𝑢 = 𝑒𝑒 0,254558 ∗√0,5 = 1,1972 𝑑𝑑 = 𝑒𝑒 −0,254558 ∗√0,5 = 0,8353 2 period 𝑆𝑆 = 50

𝑆𝑆𝑢𝑢 = 59,86 𝑆𝑆𝑑𝑑 = 41,765

𝑆𝑆𝑢𝑢𝑢𝑢 = 71,66 𝐶𝐶𝑢𝑢𝑢𝑢 = 21,66 𝑆𝑆𝑢𝑢𝑢𝑢 = 50 𝐶𝐶𝑢𝑢𝑢𝑢 = 0

𝑆𝑆𝑑𝑑𝑑𝑑 = 34,886 THEN, we should use the dynamic portfolio to calculate the price of the call, because of the next question, we will have all the information we need in the question c). On the period, the rate is : 𝑒𝑒 0,0886764 ∗0,5 = 1,04534 21,66 ) 𝛼𝛼𝑢𝑢 = 1 (= 59,86 ∗ (1,1972 − 0,8353) 𝛼𝛼𝑑𝑑 = 0 21,66 − 1 ∗ 71,66 = −47,8313 𝐵𝐵𝑢𝑢 = 1,04534 𝐶𝐶𝑢𝑢 = 1 ∗ 59,86 − 47,8313 = 12,0287 12,0287 𝛼𝛼 = = 0,6648 50 ∗ (1,1972 − 0,8353) 12,0287 − 0,6648 ∗ 59,86 = −26,5619 𝐵𝐵 = 1,04534 𝐶𝐶 = 0,6648 ∗ 50 − 26,5619 = 6,678 We could have the same result in this way, but the question c) needs alpha, so it’s a waste of time 1,04534 − 0,8353 𝑝𝑝 = = 0,5803 1,1972 − 0,8353 0,58032 ∗ 21,66 = 6,675 𝐶𝐶 = 1,045342 So we have then 10 call per oilbond, and we must calculate the present value of this oilbond We have : 1000 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 = + 6,675 ∗ 10 = 915,13 + 66,75 = 981,88 1,045342

55

Maths

Exercises

c) We have all the information, so At t=0, the oilbond (O) is too expensive we sell 𝑆𝑆 => −𝛼𝛼 ∗ 𝑆𝑆 ∗ 10 = 0,6648 ∗ 50 ∗ 10 = −332,4 it (so we receive money so +), we buy S (stock) 𝑂𝑂 => +1000 1000 (so -), and lend (B) (-). 𝐵𝐵 => −26,5616 ∗ 10 + = 649,52 And there are 10 call 1,045342 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 18,08 (𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔) At t=1, there are 2 possibilities up or down If up, then we must buy stock, so we lend. 𝑆𝑆 => −(𝛼𝛼𝑢𝑢 − 𝛼𝛼) ∗ 𝑆𝑆𝑢𝑢 ∗ 10 = −(1 − 0,6648) ∗ 59,86 ∗ 10 = −200,65 𝐵𝐵 => 200,65 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 If down, then we must sell stock (so +), so we 𝑆𝑆 => (𝛼𝛼 − 𝛼𝛼𝑑𝑑 ) ∗ 𝑆𝑆𝑑𝑑 ∗ 10 = 0,6648 ∗ 41,765 ∗ 10 borrow… = 277,65 𝐵𝐵 => −277,65 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 At t=2, it’s the end If upup, we exercise, so we give back the 𝑂𝑂 => −1000 − 10 ∗ 21,66 = −1216,66 oilbond, 𝑆𝑆 => 𝛼𝛼𝑢𝑢𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 ∗ 10 = 1 ∗ 71,66 ∗ 10 = 716,66 We sell all the stock we have. 𝐵𝐵 => +500 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 If updown, we must give back the stock 𝑂𝑂 => −1000 (𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 0!) The oilbond, 𝑆𝑆 => 𝛼𝛼𝑢𝑢𝑢𝑢 ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 ∗ 10 = 1 ∗ 50 ∗ 10 = 500 And lend to compensate 𝐵𝐵 => +500 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 If downdown, or downup, we don’t have any 𝑆𝑆 => 0 more stock, so we just give back the oilbond, 𝑂𝑂 => −1000 and lend to equilibrate. 𝐵𝐵 => +1000 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 Correction, but too long !! Enter port Exit port S B S Start 0 0 332,3 𝑢𝑢 397,863 679,1015 598,65 𝑑𝑑 277,537 679,1015 0 𝑢𝑢𝑢𝑢 716,764 500 0 𝑢𝑢𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

500 500 0 0 1000 0 0 1000 0 𝐶𝐶 − 𝐶𝐶 𝑢𝑢 𝑑𝑑 𝛼𝛼 ∗ = = 0,6646 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) ∗ 𝛼𝛼 ∗ 𝑆𝑆 ∗ 10 = 332,3 𝐶𝐶𝑢𝑢 − 𝛼𝛼 ∗ 𝑆𝑆𝑢𝑢 ∗ 𝐵𝐵 = 𝑟𝑟 𝐵𝐵 = 10 ∗ 𝐵𝐵∗ + 𝑃𝑃𝑃𝑃(1000) = 649,6490

Cash flow B S B Oilbond 649,649 −332,3 −649,649 1000 478,3158 −200,787 200,787 − 956,6306 277,537 −277,537 − 0 716,764 500 −1000 − 10 ∗ 21,6764 0 500 500 −1000 0 0 1000 −1000 0 0 1000 −1000

𝑆𝑆 ∗ = 0,6646 ∗ 𝑆𝑆𝑢𝑢 = 0,6646 ∗ 59,865 ∗ 10 = 397,863 𝐵𝐵𝑢𝑢∗ = 𝐵𝐵∗ ∗ (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) = 679,1015 𝐶𝐶𝑢𝑢𝑢𝑢 − 𝐶𝐶𝑢𝑢𝑢𝑢 =1 𝛼𝛼𝑢𝑢∗ = 𝑆𝑆𝑢𝑢 ∗ (𝑢𝑢 − 𝑑𝑑) 𝑆𝑆𝑢𝑢∗ = 59,865 ∗ 10 ∗ 1 = 598,65

56

Total 18,051 0 0 0 0 0 0

Maths 𝐵𝐵∗ = End

Exercises

𝐶𝐶𝑢𝑢𝑢𝑢 − 𝛼𝛼𝑢𝑢∗ ∗ 𝑆𝑆𝑢𝑢𝑢𝑢 ∗ 10 + 𝑃𝑃𝑃𝑃(1000) = 478,3158 𝑟𝑟

d) It could be interesting because there is less cost transaction.

57

Maths

Exercises

RE-Exam 2007 Problem 1 (Reex07) 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 60% : 𝛽𝛽 = 0,8 40% : 𝛽𝛽 = 1,2 1000

700M 300M at 5% 1000

𝐷𝐷 𝐸𝐸 𝐷𝐷 = 1 => = = 0,5 𝐷𝐷 + 𝐸𝐸 𝐷𝐷 + 𝐸𝐸 𝐸𝐸 Batslady pays 120M, so equity must be 120 and debt 120 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 240 Equity = 120 Debt = 120 𝛽𝛽 = 0,8 240 240 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 + 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 1000M (batslady) -30M of cash 𝛽𝛽 = 0,9897 240M (monika) 𝛽𝛽 = 0,8

700M 300M at 5% 90M at 5,5% 120M at 5% 1210

1210 a) 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

6 4 + 1,2 ∗ = 0,96 10 10 = 4,5 + 0,96 ∗ 6 = 10,26% = 0,8 ∗

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

7 ∗ 𝑟𝑟 10 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 12,5143%

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

= 10,26% =

𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

= 0,96 =

𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

b)

𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 & 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

Debt : 510, 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

+

3 ∗ 5% 10

ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 & 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

+

30 ∗0 1000

970 240 + 0,8 ∗ = 0,952 1210 1210 = 4,5 + 0,9534 ∗ 6 = 10,212% = 0,9897 ∗

300

90

120

= ∗5+ ∗ 5,5 + ∗ 5 = 5,088235% 510 510 510 1210 300 90 120 = ∗ �10,212 − � ∗5+ ∗ 5,5 + ∗ 5�� = 13,945% 700 1210 1210 1210

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

970 ∗ 𝛽𝛽 1000 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 0,9897

𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

We can also have the result with only the betas… 5 − 4,5 5,5 − 4,5 𝑛𝑛𝑛𝑛𝑛𝑛 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = = 0,083, 𝛽𝛽𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = = 0,167 6 6 We have 𝛽𝛽𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 so we can have 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

In theory market value doesn’t change… moreover you don’t expect to have the same CF after the acquisition. d) Efficient => arbitrage, all information already in news etc. 58

Maths

Exercises

Problem 3 (Reex07) a) 𝑅𝑅𝑅𝑅𝑅𝑅 = 15% 𝑟𝑟𝑑𝑑 = 8% 𝑡𝑡𝑐𝑐 = 30%

𝑉𝑉 = 1000𝑀𝑀 𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 700 ∗ 0,15 = 105𝑀𝑀 𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 105 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = + 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = + 300 ∗ 0,08 = 174𝑀𝑀 (1 − 𝑡𝑡𝑐𝑐 ) 0,7 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸(1 − 𝑡𝑡𝑐𝑐 ) => 𝑟𝑟𝑢𝑢 = 12,18% 𝑀𝑀𝑀𝑀(𝑈𝑈) = 𝑟𝑟𝑢𝑢 0,3 ∗ 300 ∗ 0,08 = 1090 𝑉𝑉𝑈𝑈 + 𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡 = 1000 + 0,08 𝑀𝑀𝑀𝑀(𝐸𝐸) = 1090 − 300 𝑀𝑀𝑀𝑀(𝐸𝐸) = 790

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 1090 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 11,1743%

b) 𝑏𝑏𝑏𝑏𝑏𝑏 5% 𝑜𝑜𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 ∶ 790 ∗ 0,05 = −39,5 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 5% 𝑜𝑜𝑜𝑜 105 = 5,25 ??*0,4 ?? why not 𝑏𝑏𝑏𝑏𝑏𝑏 5% 𝑜𝑜𝑜𝑜 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ∶ 1000 ∗ 0,05 = −50 Borrow at 8% 10,5 so that you have −39,5 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 5% 𝑜𝑜𝑜𝑜 174 ∗ 0,7 = 6,09 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∶ 8% ∗ 10,5 = 0,84 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 5,25 c) 𝑟𝑟𝑢𝑢 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = −1100 𝑀𝑀𝑀𝑀 = 1000

𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 550 ∗ 0,3 ∗ 0,08 ∗

1 − 1,08−10 = 88,5731 0,08

When the issue cost is not from gross proceed, we don’t divide. 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐸𝐸 = 550 ∗ 0,7 ∗ 0,03 = 11,55 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐷𝐷 = 550 ∗ 0,7 ∗ 0,02 = 7,7 𝐴𝐴𝐴𝐴𝐴𝐴 = −1100 + 1000 + 88,5731 − 11,55 − 7,7 = −30,6769 𝐴𝐴𝐴𝐴𝐴𝐴 = −30,6769

59

Maths

Exercises

Problem 4 (Reex07) a) 𝑟𝑟𝑓𝑓 = 𝑙𝑙𝑙𝑙1,08 𝜎𝜎 = 0,1 ∗ √12 𝐷𝐷𝐷𝐷𝐷𝐷 𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒, 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑢𝑢𝑢𝑢𝑢𝑢 𝑆𝑆 … 0,12 ∗ 12 3 80 �∗4 ln � � + �𝑙𝑙𝑙𝑙1,08 + 2 85 = 0,1403 𝑑𝑑1 = 3 0,1 ∗ √12 ∗ �4 𝑑𝑑2 = −0,1597 𝑁𝑁(𝑑𝑑1 ) = 0,558 𝑁𝑁(𝑑𝑑2 ) = 0,7366 3

𝐶𝐶 = 80 ∗ 0,558 − 85 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,08∗4 ∗ 0,7366 𝐶𝐶 = 9,4344

b) Three way 𝐶𝐶 = 76 ∗ 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐸𝐸) ∗ 𝑁𝑁(𝑑𝑑2 ) = 7,2112 ∆𝑐𝑐 = 0,5558 �𝑁𝑁(𝑑𝑑1 )� ∆𝐶𝐶 = ∆𝑐𝑐 ∗ ∆𝑆𝑆 = −2,2232 S ΩC = ∗ Δ = 4,713 C 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑆𝑆𝑛𝑛𝑛𝑛𝑛𝑛 = 5% 𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜 − 𝐶𝐶𝑛𝑛𝑛𝑛𝑛𝑛 = 5% ∗ Ω𝑐𝑐 = 0,2335 𝐶𝐶𝑜𝑜𝑜𝑜𝑜𝑜

c) Delta-neutral : portfolio which is perfectly hedge, the expected return is 0, because there are no cash flow involved. 𝛼𝛼 ∗ = 𝑁𝑁(𝑑𝑑1 ) = 0,5558 𝑆𝑆 ∗ = 44,464 𝐵𝐵∗ = −𝑁𝑁(𝑑𝑑2 ) ∗ 𝑃𝑃𝑃𝑃(𝐸𝐸) = −35,0296 d) Elasticity is omega. Option beta value = omega * stock beta value

60

Maths

Exercises

Problem 5 (Reex07) 50 40 30 20 10 0 0

10

20

30

40

50

b) 1000 With the simple stock we gain 5$/share, so finally we gain ∗ 5 = 200 25 At the end we have 1200 12 = 𝑟𝑟𝑟𝑟 + 1 ∗ 7 𝑟𝑟𝑟𝑟 = 5% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ! 𝐶𝐶25 => 0,2 2 25 ln � � + �0,04875 + 2 � ∗ 3 25 = 0,2717 𝑑𝑑1 = 2 √0,2 ∗ �3 𝑑𝑑2 = −0,0935 𝑁𝑁(𝑑𝑑1 ) = 0,6064 + 0,17 ∗ (0,6103 − 0,6064) = 0,6071 𝑁𝑁(𝑑𝑑2 ) = 0,4641 + 0,35 ∗ (0,4602 − 0,4641) = 0,4627 2

𝐶𝐶 = 25 ∗ 0,6071 − 25 ∗ 𝑒𝑒 −0,05∗3 ∗ 0,4627 𝐶𝐶25 = 3,989

We can buy

1000 3,989

= 250 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

1000 − 250 ∗ 3,989 = 2,75 We save 2,75 At the end we receive 5$/call, so we have 250 ∗ 5 = 1250

2

And we have the 2,75 with the interest, so we have 2,75 ∗ 𝑒𝑒 0,05∗3 = 2,84 Finally we have 1252,84 𝐶𝐶28 => 𝑑𝑑1 = −0,0387 𝑑𝑑2 = −0,4039 𝑁𝑁(𝑑𝑑1 ) = 0,4880 + 0,87 ∗ (0,4840 − 0,4880) = 0,4845 𝑁𝑁(𝑑𝑑2 ) = 0,3446 + 0,39 ∗ (0,3409 − 0,3446) = 0,3432 2

𝐶𝐶 = 25 ∗ 0,4845 − 28 ∗ 𝑒𝑒 −0,05∗3 ∗ 0,3432 𝐶𝐶28 = 2,8179

We can buy

1000 2,8179

= 354 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

We save 2,46 At the end we receive 2$/call, so we have 354 ∗ 2 = 708

2

And we have the 2,46 with the interest, so we have 2,46 ∗ 𝑒𝑒 0,05∗3 = 2,54 Finally we have 710,54 61

Maths

Exercises

c) 𝐸𝐸 = 25 𝑆𝑆 = 25

1 √0,2∗�

3 = 1,2946 𝑢𝑢 = 𝑒𝑒 𝑑𝑑 = 0,7724 𝑑𝑑𝑑𝑑𝑑𝑑 = 2% 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0,0162634

𝑆𝑆 = 25

𝑆𝑆𝑢𝑢 = 32,365 ∗ 0,98 = 31,7177 𝑆𝑆𝑑𝑑 = 19,31 ∗ 0,98 = 18,9238

−0,5 = −0,0302 31,7177 ∗ (1,2946 − 0,7724) 0,5 + 0,0302 ∗ 24,5 = 1,22 𝐵𝐵𝑢𝑢 = 1,0162634 𝑃𝑃𝑢𝑢 = −0,0302 ∗ 31,7177 + 1,22 = 0,2621 0,5 − 10,38 𝛼𝛼𝑑𝑑∗ = = −1 18,9238 ∗ (1,2946 − 0,7724) 10,38 + 14,6167 = 24,5967 𝐵𝐵𝑑𝑑 = 1,0162634 𝑃𝑃𝑑𝑑 = −18,9238 + 24,5967 = 5,6729 0,2621 − 5,6729 𝛼𝛼 ∗ = = −0,4145 25 ∗ (1,2946 − 0,7724) 5,6729 + 0,4145 ∗ 19,31 = 13,458 𝐵𝐵 = 1,0162634 𝑃𝑃 = −0,4145 ∗ 25 + 13,458 = 3,0955

𝛼𝛼𝑢𝑢∗ =

1,0162634 − 0,7724 = 0,467 1,2946 − 0,7724 0,5 ∗ (1 − 0,467) = 0,2622 𝑃𝑃𝑢𝑢 = 1,0162634 0,5 ∗ 0,467 + 10,38 ∗ (1 − 0,467) 𝑃𝑃𝑑𝑑 = = 5,6738 1,0162634 0,2622 ∗ 0,467 + 5,6738 ∗ (1 − 0,467) = 3,0962 𝑃𝑃 = 1,0162634

𝑝𝑝 =

62

𝑆𝑆𝑢𝑢𝑢𝑢 = 41,06 𝑆𝑆𝑢𝑢𝑢𝑢 = 24,5 𝑃𝑃𝑢𝑢𝑢𝑢 = 0,5

𝑆𝑆𝑑𝑑𝑑𝑑 = 14,6167 𝑃𝑃𝑑𝑑𝑑𝑑 = 10,38

Maths

Exercises

Exam 2008 Problem 2 (Ex08) 107,5 𝐵𝐵1 = 1,04 𝑖𝑖1 = 4% 10 110 10 110 + = + 𝐵𝐵3 = 2 1,0448 1,0448 1,04 (1 + 𝑖𝑖2 )2 1 2

110 � − 1 = 4,5029% 10 110 10 1,0448 + 1,04482 − 1,04 100 𝐵𝐵2 = 1,053 𝑖𝑖3 = 5% 100 𝐵𝐵4 = 1,075 𝑖𝑖5 = 7% 5 5 5 105 5 5 5 105 𝐵𝐵5 = + + + = + + + 1,059 1,0592 1,0593 1,0594 1,04 1,0450292 1,053 (1 + 𝑖𝑖4 )4 𝑖𝑖2 = �

1 4

105 � − 1 = 6% 5 5 5 105 5 5 5 + + + − − − 1,059 1,0592 1,0593 1,0594 1,04 1,0450292 1,053

𝑖𝑖4 = �

1year in 3y :

1,06 4 1,05 3

− 1 = 9,0575%

b) 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵: 𝐵𝐵5 − 𝐵𝐵3 5 5 105 5 +2∗ +3∗ +4∗ 2 3 1,059 1,0594 360,2059 1,059 1,059 𝐷𝐷5 = = = 3,7183 5 5 105 5 96,8743 + + + 1,059 1,0592 1,0593 1,0594 3,7183 = 3,5111 𝑀𝑀𝐷𝐷5 = 1,059 110 10 +2∗ 1,0448 1,04482 211,1089 = = 1,9133 𝐷𝐷3 = 110 10 110,3401 + 1,0448 1,04482 𝑀𝑀𝐷𝐷5 = 1,8313 (𝑃𝑃5 ) ∗ (−𝑥𝑥 ∗ 𝑀𝑀𝑀𝑀5) − (𝑃𝑃3 ) ∗ (−𝑥𝑥 ∗ 𝑀𝑀𝑀𝑀3) ≥ 3 1,8313 3,5111 � − 110,3401 ∗ �−𝑥𝑥 ∗ � ≥ −3 96,8743 ∗ �−𝑥𝑥 ∗ 100 100 𝑥𝑥 = 2,173%

c) Let’s compute the price today (after the 3 months) 5000 5000 5000 100000 𝑃𝑃𝑃𝑃𝑃𝑃 = + + + = 93606,074 0,75 1,75 2,75 1,07 1,07 1,07 1,073,25 Be careful on the last one there is no coupon ! 3 = 1250 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 100000 ∗ 5% ∗ 12 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 93606,074 − 1250 = 92356,074 𝑇𝑇ℎ𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 100, 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎 0,5%

63

Maths

Exercises

0,5 0,5 𝑎𝑎𝑎𝑎𝑎𝑎 92,356 − 𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡ℎ𝑒𝑒 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏𝑏𝑏 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 2 2 𝑠𝑠𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎: 92,606 ; 𝑏𝑏𝑏𝑏𝑏𝑏 ∶ 92,106 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ∶ 92,356 +

So what is the return ? We have bought the bond, so it’s the ask priced (high), we sell the bond now, so at the bid price (low). 92,106 + 1,25 − 95,64 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = = −2,388% 95,64 (don’t forget to add the 3 months coupon !, there is no spread on it) 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦, 𝑡𝑡ℎ𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑖𝑖𝑖𝑖 − 2,388 ∗ 4 = −9,552%

d) 𝑓𝑓 = 100,5 𝑆𝑆0 = 100 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 1𝑚𝑚: 1𝑃𝑃𝑃𝑃; 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 4𝑚𝑚 ∶ 1𝑃𝑃𝑃𝑃 𝑟𝑟1 = 3,6% 𝑟𝑟3 = 3,8% 𝑟𝑟6 = 4% 4𝑟𝑟6 = 4,1% 𝑃𝑃𝑃𝑃𝐷𝐷1 =

1 = 0,997 3,6% 1+ 12

𝐹𝐹𝐹𝐹𝐷𝐷1 = 0,997 ∗ �1 + 4% ∗ 𝐹𝐹𝐹𝐹𝐷𝐷2 = 1 ∗ �1 + 4,2% ∗

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 2,02377

6 � = 1,01694 12

2 � = 1,00683 12

6 � = 102 12 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 102 − 2,02377 = 99,9762

𝐹𝐹𝐹𝐹𝐹𝐹 = 100 ∗ �1 + 4% ∗ Sell the future Borrow Buy a share

T=0 0 +100 -100 0

T=6 100,5 -102 2,02377 0,52375

64

Maths

Exercises

Problem 3 (Ex08) 𝑀𝑀𝑀𝑀(𝐷𝐷) = 300𝑀𝑀 𝑔𝑔 = 3% 𝐸𝐸𝐸𝐸𝑆𝑆1 = 10 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 0,2 𝛽𝛽𝑈𝑈 = 1,7 𝑟𝑟𝑑𝑑 = 9,5% 𝑟𝑟𝑓𝑓 = 4,5% 𝑟𝑟𝑚𝑚𝑚𝑚 = 4% 𝑡𝑡𝑐𝑐 = 0,4 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 20𝑀𝑀

a) 𝑟𝑟𝑢𝑢 = 4,5 + 1,7 ∗ 4 = 11,3%

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) ∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑟𝑟𝑒𝑒 − 𝑔𝑔 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 𝐸𝐸𝐸𝐸𝐸𝐸 ∗ 𝑛𝑛𝑛𝑛𝑛𝑛ℎ + 𝑖𝑖𝑖𝑖𝑖𝑖 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 10 ∗ 20𝑀𝑀 + 300𝑀𝑀 ∗ 0,095 ∗ (1 − 0,4) = 217,1𝑀𝑀 𝑀𝑀𝑀𝑀(𝑈𝑈) =

217,1 ∗ 0,2 = 523,1325€ 0,113 − 0,03 300𝑀𝑀 ∗ 0,085 ∗ 0,4 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = = 137,3494€ 0,113 − 0,03 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 523,1325 + 137,3494 = 660,4819€

𝑀𝑀𝑀𝑀(𝑈𝑈) =

𝑀𝑀𝑀𝑀(𝐸𝐸) = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 − 𝑀𝑀𝑀𝑀(𝐷𝐷) = 660,4819 − 300 𝑀𝑀𝑀𝑀(𝐸𝐸) = 360,4819€ 𝑟𝑟𝑒𝑒 = 11,3 + (11,3 − 9,5) ∗

𝑟𝑟𝑒𝑒 = 12,7980%

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 12,7980 ∗ 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,5740%

300 360,4819

300 360,4819 + 9,5 ∗ ∗ (1 − 0,4) 660,4819 660,4819

b) 𝑟𝑟𝑒𝑒 = 12,798% 𝑁𝑁𝑁𝑁𝑁𝑁 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 10 ∗ 20𝑀𝑀 = = 1562,7442€ 𝑀𝑀𝑀𝑀(𝐸𝐸) = 0,12798 𝑟𝑟𝑒𝑒 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = 360,4819 − 1562,7442 = −1202,2623€ c) 𝑟𝑟𝑢𝑢 = 11,3% Target D/E 𝑟𝑟𝑙𝑙𝑙𝑙 = 11,3 + (11,3 − 6) ∗ 0,25 = 12,625% 1 4 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 12,625 ∗ + 6 ∗ ∗ 0,6 = 10,82% 5 5

Project 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 110𝑀𝑀 4 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 4 18 ∗ 0,6 = ∗ = 79,8521𝑀𝑀 𝑀𝑀𝑀𝑀(𝐸𝐸) = ∗ 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 5 0,1082 5 65

Maths

Exercises

1 − 1,095−10 = 9,5438𝑀𝑀 0,095 1 − 1,095−10 = 8,1624𝑀𝑀 𝑃𝑃𝑃𝑃(𝑆𝑆) = 20 ∗ (0,095 − 0,03) ∗ 0,095 1 − 1,095−10 = −3,2650𝑀𝑀 𝑃𝑃𝑃𝑃(𝑡𝑡𝑡𝑡𝑡𝑡 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = 20 ∗ (0,095 − 0,03) ∗ 0,4 ∗ 0,095 𝑃𝑃𝑃𝑃(𝑡𝑡𝑡𝑡𝑡𝑡) = 40 ∗ 0,095 ∗ 0,4 ∗

Here the cost of issuing is not from gross proceed, so we don’t divide 𝐼𝐼𝐼𝐼 𝐸𝐸 = 70 ∗ 0,05 ∗ 0,6 = 2,1 𝐼𝐼𝐼𝐼 𝐷𝐷 = 20 ∗ 0,045 ∗ 0,6 = 0,18 𝐴𝐴𝐴𝐴𝐴𝐴 = −110 + 79,8521 + 9,5438 + 8,1624 − 3,2650 − 2,1 − 0,18 𝐴𝐴𝐴𝐴𝐴𝐴 = −17,9867𝑀𝑀€

d) Increase D/E, target D/E=1, constant 𝑟𝑟𝑒𝑒 = 11,3 + (11,3 − 6) ∗ 1 = 16,6 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 16,6 ∗ 0,5 + 6 ∗ 0,5 ∗ 0,6 = 10,1% 𝑀𝑀𝑀𝑀(𝐸𝐸) =

1 18 ∗ 0,6 ∗ = 99,81𝑀𝑀€ 2 0,101

66

Maths

Exercises

Problem 4 (Ex08) a) Payman Island 𝑆𝑆 = 30$ 𝐸𝐸 = 30$ 𝑎𝑎𝑎𝑎 𝑡𝑡ℎ𝑒𝑒 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0,252 1 𝑙𝑙𝑙𝑙1 + �0,06 + 2 � ∗ 3 𝑑𝑑1 = = 0,2107 1 0,25 ∗ �3 0,252 1 𝑙𝑙𝑙𝑙1 + �0,06 − 2 � ∗ 3 = 0,0664 𝑑𝑑2 = 1 0,25 ∗ �3

𝑁𝑁(𝑑𝑑1 ) = 𝑁𝑁(0,21) + 0,07 ∗ �𝑁𝑁(0,22) − 𝑁𝑁(0,21)� = 0,5832 + 0,07 ∗ (0,5871 − 0,5832) = 0,5835 𝑁𝑁(𝑑𝑑2 ) = 𝑁𝑁(0,06) + 0,64 ∗ �𝑁𝑁(0,07) − 𝑁𝑁(0,06)� = 0,5239 + 0,64 ∗ (0,5279 − 0,5239) = 0,5265

𝐶𝐶 = 30 ∗ 𝑁𝑁(𝑑𝑑1 ) − 30 ∗ 𝑒𝑒 𝐶𝐶 = 2,023

−0,06 3

∗ 𝑁𝑁(𝑑𝑑2 ) = 30 ∗ 0,5835 − 30 ∗ 𝑒𝑒

−0,06 3

∗ 0,5265

0,06 3

𝑃𝑃 = −𝑆𝑆 ∗ �1 − 𝑁𝑁(𝑑𝑑1 )� + 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 ∗ �1 − 𝑁𝑁(𝑑𝑑2 )� = −30 ∗ (1 − 0,5835) + 30 ∗ 𝑒𝑒 − 𝑃𝑃 = 1,4287 With 2$ dividend

∗ (1 − 0,5265)

2

𝑆𝑆 ∗ = 𝑆𝑆 − 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 30 − 2 ∗ 𝑒𝑒 −0,06∗12 = 28,02 𝑑𝑑1 = −0,262 𝑑𝑑2 = −0,407 𝑁𝑁(𝑑𝑑1 ) = 0,397 𝑁𝑁(𝑑𝑑2 ) = 0,342 𝐶𝐶 = 1,067 𝑃𝑃 = 2,453

This following method is very long, but should be ok, like in a previous exercise, to use only if mentioned in the question ! 𝑢𝑢 = 𝑒𝑒 𝑑𝑑 =

𝑝𝑝 =

1 0,25∗� 6

= 1,1075

1 −0,25∗� 6 = 0,903 𝑒𝑒 0,06 𝑒𝑒 6 − 0,903

1,1075 − 0,903 (1 − 𝑝𝑝) = 0,4765

= 0,5235

𝑆𝑆 = 30

𝐶𝐶𝑢𝑢 =

4,28 ∗ 0,5235 0,06 𝑒𝑒 6

= 2,2183

𝐶𝐶𝑑𝑑 = 0 2,2183 ∗ 0,5235 𝐶𝐶 = = 1,1497 0,06 6 𝑒𝑒 ∆𝐶𝐶 = −0,87

𝑆𝑆𝑢𝑢𝑑𝑑𝑑𝑑𝑑𝑑

= 33,225 − 2 = 31,225

𝑆𝑆𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 27,09 − 2 = 25,09

67

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 34,58

𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑢𝑢𝑢𝑢 = 28,196 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑 = 27,787 𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑑𝑑𝑑𝑑 = 22,656

Maths

𝑃𝑃𝑢𝑢 =

𝑃𝑃𝑑𝑑 =

𝑃𝑃 =

Exercises

1,804 ∗ 0,4765 0,06 𝑒𝑒 6

= 0,8511

2,213 ∗ 0,5235 + 7,344 ∗ 0,4765 0,06 𝑒𝑒 6

0,8511 ∗ 0,5235 + 4,612 ∗ 0,4765

∆𝑃𝑃 = 1,188

𝑒𝑒

0,06 6

b) 𝐶𝐶𝑚𝑚 = 𝐶𝐶𝑇𝑇 + 2 𝐶𝐶𝑇𝑇 = 2,023 𝐶𝐶𝑚𝑚 = 4,023 𝐶𝐶𝑚𝑚 is too expensive you sell it Strategy 1 T=0 Sell call Buy Stock Buy Put Borrow rf Total Strategy 2 Sell call Buy forward Buy Put Total

= 4,612

= 2,6169

E>S 4,023 −30 −1,4287 +29,406 2

T=0

4

E>S

4,023 − −1,4287 2,594

E 𝐹𝐹𝐹𝐹(𝑆𝑆) = 30 ∗ 𝑒𝑒 0,06∗12 = 30,606 c) 𝑆𝑆 𝑆𝑆 ∗ ∆= ∗ 𝑁𝑁(𝑑𝑑1 ) = 8,538 𝐶𝐶 𝐶𝐶 𝑆𝑆 𝑆𝑆 ∗ ∆= − ∗ �1 − 𝑁𝑁(𝑑𝑑1 )� = −8,56 𝑃𝑃 𝑃𝑃 𝜎𝜎𝑐𝑐 = Ω ∗ 𝜎𝜎𝑠𝑠 = 2,135 𝜎𝜎𝑝𝑝 = −Ω ∗ 𝜎𝜎𝑠𝑠 = 2,14

68

E 𝐶𝐶𝑑𝑑𝑑𝑑 = 5,2 𝑆𝑆𝑑𝑑𝑑𝑑 = 10,952

16,54 − 0 = =1 27 ∗ (1,35 − 0,74) 5,2 = = 0,576 14,8 ∗ (1,35 − 0,74) 16,45 − 1 ∗ 36,45 = = −19,1439 1 0,0875∗ 2 𝑒𝑒 5,2 − 0,576 ∗ 20 = = −6,049 1 0,0875∗ 2 𝑒𝑒 = 27 − 19,1439 = 7,856 = 0,576 ∗ 14,8 − 6,049 = 2,476

7,856 − 2,476 = 0,441 20 ∗ (1,35 − 0,74) 7,856 − 0,441 ∗ 27 = −3,878 𝐵𝐵 = 1 0,0875∗ 2 𝑒𝑒 𝐶𝐶 = 0,441 ∗ 20 − 3,878 𝐶𝐶 = 4,942

𝛼𝛼 =

It won’t pay because 7 𝑖𝑖 = 1,085

b) If we find separate duration for B1 and B4 (after we found the yield for each at least and not use spot rate !!), we must have the average duration then, with bond which have not the same yield, so not possible in theory !… So one solution could be to add B1 and B4, find the yield and compute a duration with such a yield 𝑃𝑃𝐵𝐵1 + 𝑃𝑃𝐵𝐵4 = 35 ∗ 1, 𝑖𝑖1 −1 + 35 ∗ 1, 𝑖𝑖1 −2 + 35 ∗ 1, 𝑖𝑖1 −3 + 135 ∗ 1, 𝑖𝑖1 −4 + 230 ∗ 1, 𝑖𝑖1 −5 = 84,3414 + 232,734 = 317,08 𝑖𝑖1 = 10,44% 𝐷𝐷(𝑃𝑃) 35 ∗ 1,1044 −1 + 2 ∗ 35 ∗ 1,1044−2 + 3 ∗ 35 ∗ 1,1044−3 + 4 ∗ 135 ∗ 1,1044−4 + 5 ∗ 230 ∗ 1,1044 −5 = 317,08 1229,964 = = 3,879 317,08 3,879 = 3,5123 𝐷𝐷𝐷𝐷𝐷𝐷 = 1,1044 𝐶𝐶(𝑃𝑃) 2 ∗ 35 ∗ 1,1044 −3 + 2 ∗ 3 ∗ 35 ∗ 1,1044−4 + 3 ∗ 4 ∗ 35 ∗ 1,1044−5 + 4 ∗ 5 ∗ 135 ∗ 1,1044−6 + 5 ∗ 6 ∗ 230 ∗ 1,1044 = 317,08 = 16,5348 1 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = −3,5123 ∗ 0,01 + ∗ 16,5348 ∗ 0,012 = −3,4296% 2 1 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 3,5123 ∗ 0,01 + ∗ 16,5348 ∗ 0,012 = 3,595% 2 WE SHOULD NOT USE : In theory this following method is not possible, no spot rate !! We must calculate with yield (flat) ! 5 ∗ 1,085−1 + 2 ∗ 5 ∗ 1,09−2 + 3 ∗ 5 ∗ 1,095−3 + 4 ∗ 105 ∗ 1,1−4 311,3156 = = 3,6912 𝐷𝐷𝐵𝐵1 = 84,3414 84,3414 30 30 30 230 30 +2∗ +3∗ +4∗ +5∗ 4 2 3 1,085 1,1 1,09 1,095 1,115 911,1301 = = 3,9149 𝐷𝐷𝐵𝐵4 = 232,734 232,734 Completely wrong And so on… c) With a forward 2,3 𝐿𝐿𝐿𝐿𝐿𝐿/𝐽𝐽𝐽𝐽 => 𝐽𝐽𝐽𝐽/𝐿𝐿𝐿𝐿𝐿𝐿 = 2,3 1,04 = 2,204608 𝐹𝐹1 = 2,3 ∗ 1,085

71

Maths

Exercises

1,0452 = 2,114012 1,092 1,053 = 2,027932 𝐹𝐹3 = 2,3 ∗ 1,0953 𝐹𝐹2 = 2,3 ∗

CF LTL 1000000 𝐿𝐿𝐿𝐿𝐿𝐿

1

CF JD 1000000 = 453595,38 𝐽𝐽𝐽𝐽 2,204608 1000000 = 473034,21 𝐽𝐽𝐽𝐽 2,114012 1000000 = 493113,18 𝐽𝐽𝐽𝐽 2,027932

1000000 𝐿𝐿𝐿𝐿𝐿𝐿

2

1000000 𝐿𝐿𝐿𝐿𝐿𝐿

3

This is for a swap, not asked. 1𝐽𝐽𝐽𝐽 = 2,3𝐿𝐿𝐿𝐿𝐿𝐿

1 − 1,05−3 = 4,9674% 1,04−1 + 1,045−2 + 1,05−3 1 − 1,095−3 = 9,4395% 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑅𝑅𝑅𝑅 𝐽𝐽𝐽𝐽 = 1,085−1 + 1,09−2 + 1,095−3 CF LTL 1 1000000 ∗ 0,049674 = 49674 𝐿𝐿𝐿𝐿𝐿𝐿 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐿𝐿𝐿𝐿𝐿𝐿 =

2 3

49674 𝐿𝐿𝐿𝐿𝐿𝐿

1049674 𝐿𝐿𝐿𝐿𝐿𝐿

d) Forward 2,3 𝐿𝐿𝐿𝐿𝐿𝐿/𝐽𝐽𝐽𝐽 => 𝐽𝐽𝐽𝐽/𝐿𝐿𝐿𝐿𝐿𝐿 = 2,3 1,04 𝐹𝐹1 = 2,2 ∗ = 2,1185 1,08 1,0452 = 2,0408 𝐹𝐹2 = 2,2 ∗ 1,0852 CF LTL 2 1000000 𝐿𝐿𝐿𝐿𝐿𝐿 /2,1185 = 472032,1 𝐽𝐽𝐽𝐽 3 1000000 𝐿𝐿𝐿𝐿𝐿𝐿 /2,0408 = 490003,9 𝐽𝐽𝐽𝐽 Total 962036 TO CHANGE SOME THING HERE

CF JD 473034,21 𝐽𝐽𝐽𝐽 (∗ 2,1185 = 1002122,974) 493113,18 𝐽𝐽𝐽𝐽 (∗ 2,0408 = 1006345,38) 966147,39

Swap 𝑛𝑛𝑛𝑛𝑛𝑛 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐿𝐿𝐿𝐿𝐿𝐿 𝑢𝑢𝑢𝑢𝑢𝑢ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎, 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝐽𝐽𝐽𝐽 ∶ 8%; 8,5%; 9% CF LTL 49674 1 = 47763,46 𝐿𝐿𝐿𝐿𝐿𝐿 1,04 2

Total Loss

CF JD 94395 𝐿𝐿𝐿𝐿𝐿𝐿 41041,3 𝐽𝐽𝐽𝐽 94395 𝐿𝐿𝐿𝐿𝐿𝐿 41041,3 𝐽𝐽𝐽𝐽 1094395 𝐿𝐿𝐿𝐿𝐿𝐿 475823,91 𝐽𝐽𝐽𝐽

1049674 = 961217,92 𝐿𝐿𝐿𝐿𝐿𝐿 1,0452 1008981,38

72

CF JD

41041,3 = 38001,2 𝐽𝐽𝐽𝐽 1,08 => 83602,64 𝐿𝐿𝐿𝐿𝐿𝐿 475823,91 = 404191,14 𝐽𝐽𝐽𝐽 1,0852 => 889220,51 𝐿𝐿𝐿𝐿𝐿𝐿 972823,15 −36158,23

Maths

Exercises

Problem 3 (Reex08) a) 𝑟𝑟𝑑𝑑 = 6% 𝑔𝑔 = 1% 0,6 𝑀𝑀𝑀𝑀(𝐷𝐷) = = 12 0,06 − 0,01 1 𝑝𝑝 = 3 𝛽𝛽𝑙𝑙𝑙𝑙 = 1,2 𝑟𝑟𝑓𝑓 = 5% 𝑟𝑟𝑚𝑚 = 3% 𝑟𝑟𝑙𝑙𝑙𝑙 = 5 + 1,2 ∗ 3 = 8,6% 𝑡𝑡𝑐𝑐 = 25% 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 1,5𝑀𝑀

1 3 𝑀𝑀𝑀𝑀(𝐸𝐸) = = 6,5789 0,086 − 0,01 1,5 ∗

𝑀𝑀𝑀𝑀(𝐴𝐴) = 12 + 6,5789 = 18,5789 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 8,6 ∗ b)

6,5789 12 +6∗ ∗ (1 − 0,25) = 5,9518% 18,5789 18,5789

1,5 𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑡𝑡 = 0 1,01 = = 24,75% 𝑅𝑅𝑅𝑅𝑅𝑅 = 6 𝐵𝐵𝐵𝐵 𝑔𝑔 = 24,75 ∗ (1 − 𝑝𝑝) = 16,5% Actual g in inferior, destroy value, Or

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔 1% = = 1,5% 2 (1 − 𝑝𝑝) 3 1,5% < 𝑟𝑟𝑒𝑒

𝑅𝑅𝑅𝑅𝑅𝑅 = c)

12 ∗ 0,06 6,5789 = 6,9207% 𝑟𝑟𝑢𝑢 = 12 1+ 6,5789 𝑟𝑟𝑙𝑙𝑙𝑙 = 6,9207 + (6,9207 − 6) ∗ 2 = 8,7621% 2 1 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 8,7621 ∗ + 6 ∗ ∗ (1 − 0,25) = 5,9207% 3 3 0,086 +

𝑀𝑀𝑀𝑀(𝐴𝐴′ ) = 𝑀𝑀𝑀𝑀(𝐴𝐴) ∗

5,9519 − 1 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶0 − 𝑔𝑔 = 18,5789 ∗ = 18,6967 5,9207 − 1 𝑊𝑊𝑊𝑊𝑊𝑊𝐶𝐶1 − 𝑔𝑔

d) 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 2 + 1,5 − 1 = 2,5 200000 𝑝𝑝𝑝𝑝𝑝𝑝 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ, 6𝑀𝑀 𝑝𝑝𝑝𝑝𝑝𝑝 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 20𝑀𝑀, 10% 𝑢𝑢𝑢𝑢 𝑡𝑡𝑐𝑐 = 25% 𝑟𝑟𝑒𝑒 = 9% 𝑔𝑔 = 3% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 10% ∗ 20𝑀𝑀 = 2𝑀𝑀

73

Maths

Exercises

𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 12,5𝑀𝑀

5 1,035 − 1,095 ∗ 1,09−5 + = 15,5772 1,03 − 1,09 1,095 7,5 1 − 1,09−5 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = � ∗ 0,25� ∗ = 1,4586 5 0,09

𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = (6𝑀𝑀 − 2𝑀𝑀) ∗ 0,75 ∗

𝑁𝑁𝑁𝑁𝑁𝑁 = −12,5 + 15,5772 + 1,4586 = 4,5358𝑀𝑀

74

Maths

Exercises

Exam 2009 Problem 1 (Ex09) 𝐷𝐷 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 ∶ = 0,2 𝐸𝐸 𝐷𝐷 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ∶ = 1,6 𝐸𝐸

a) 𝜎𝜎𝑚𝑚 ² = 𝜎𝜎𝐾𝐾 ² ∗ 𝑥𝑥𝑘𝑘 ² + 𝜎𝜎𝑀𝑀 ² ∗ 𝑥𝑥𝑀𝑀 ² + 2 ∗ 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀 𝜎𝜎𝑚𝑚 ² = 10 𝜎𝜎𝐾𝐾 ² = 15 = 𝜎𝜎𝑀𝑀 ² 𝑥𝑥𝑀𝑀 + 𝑥𝑥𝐾𝐾 = 1 So 𝑥𝑥𝑀𝑀 = 1 − 𝑥𝑥𝐾𝐾

𝜎𝜎𝑚𝑚2 = 𝑥𝑥𝐾𝐾 [𝑥𝑥𝐾𝐾 𝜎𝜎𝐾𝐾2 + 𝑥𝑥𝑀𝑀 𝜎𝜎𝐾𝐾&𝑀𝑀 ] + 𝑥𝑥𝑀𝑀 [𝑥𝑥𝐾𝐾 𝜎𝜎𝐾𝐾&𝑀𝑀 + 𝑥𝑥𝑀𝑀 𝜎𝜎𝑀𝑀2 ] = 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀 =? We know :

𝑐𝑐𝑐𝑐𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀, 𝑚𝑚) 𝑉𝑉𝑉𝑉𝑉𝑉(𝑚𝑚) 𝑐𝑐𝑐𝑐𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀, 𝑚𝑚) = 0,21 ∗ 10 = 2,1 2 𝑐𝑐𝑐𝑐𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀, 𝑚𝑚) = 𝑥𝑥𝑘𝑘 ∗ 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 + (1 − 𝑥𝑥𝑘𝑘 ) ∗ 𝜎𝜎𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 2,1 − (1 − 𝑥𝑥𝑘𝑘 ) ∗ 15 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑥𝑥𝑘𝑘 So 𝜎𝜎𝑚𝑚2 = 10 = 𝑥𝑥𝐾𝐾 [𝑥𝑥𝐾𝐾 𝜎𝜎𝐾𝐾2 + 𝑥𝑥𝑀𝑀 𝜎𝜎𝐾𝐾&𝑀𝑀 ] + 𝑥𝑥𝑀𝑀 [𝑥𝑥𝐾𝐾 𝜎𝜎𝐾𝐾&𝑀𝑀 + 𝑥𝑥𝑀𝑀 𝜎𝜎𝑀𝑀2 ] 2,1 − (1 − 𝑥𝑥𝑘𝑘 ) ∗ 15 = 𝑥𝑥𝐾𝐾 ∗ �𝑥𝑥𝐾𝐾 ∗ 15 + (1 − 𝑥𝑥𝐾𝐾 ) ∗ � + (1 − 𝑥𝑥𝑘𝑘 ) ∗ 2,1 𝑥𝑥𝑘𝑘 So 10 = 𝑥𝑥𝐾𝐾2 ∗ 15 + (1 − 𝑥𝑥𝐾𝐾 ) ∗ (2,1 − (1 − 𝑥𝑥𝑘𝑘 ) ∗ 15) + (1 − 𝑥𝑥𝑘𝑘 ) ∗ 2,1 15𝑥𝑥 2 + 2,1 − 15 + 15𝑥𝑥 − 2,1𝑥𝑥 + 15𝑥𝑥 − 15𝑥𝑥 2 + 2,1 − 2,1𝑥𝑥 − 10 = 0 25,8𝑥𝑥 − 20,8 = 0 20,8 = 80,6% 𝑥𝑥𝑘𝑘 = 25,8 𝛽𝛽𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 0,21 =

𝑥𝑥𝑀𝑀 = 19,4%

b) 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = 18% ∗ 𝑥𝑥𝐾𝐾 + 7% ∗ 𝑥𝑥𝑀𝑀 = 18% ∗ 0,194 + 7% ∗ 0,806 = 15,866%

𝑟𝑟𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 7 = 𝑟𝑟𝑓𝑓 + 0,21 ∗ (15,866 − 7) 𝑟𝑟𝑓𝑓 = 4,643

𝑟𝑟𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 = 18 = 4,643 + 𝛽𝛽𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 (15,866 − 7) => 𝛽𝛽𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 =

18 − 4,643 = 1,190 15,866 − 4,643

2,1 − (1 − 𝑥𝑥𝑘𝑘 ) ∗ 15 2,1 − (1 − 0,806) ∗ 15 = 0,806 𝑥𝑥𝑘𝑘 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 2,1 − (1 − 0,806) ∗ 15 = = −0,067 𝜌𝜌 = 𝜎𝜎𝐾𝐾 ∗ 𝜎𝜎𝑀𝑀 0,806 ∗ 15 𝑐𝑐𝑐𝑐𝑣𝑣𝐾𝐾&𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = c)

75

Maths 𝛽𝛽𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

Exercises

= 0,21 𝐷𝐷 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 ∶ = 1,6 𝐸𝐸 𝐷𝐷 16 = 𝐷𝐷 + 𝐸𝐸 26 10 𝐸𝐸 = 𝐷𝐷 + 𝐸𝐸 26 10 16 +5∗ = 5,7692% 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 ∗ 26 26 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

180 20 = 5,7692 ∗ +3∗ = 5,492% 200 200 25 175 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑆𝑆𝑆𝑆 + ∗ 𝑟𝑟 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆𝑆𝑆 = 200 200 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑆𝑆𝑆𝑆 175 25 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑆𝑆𝑆𝑆 + ∗5 5,492 = 200 200 175 200 ∗ 5� ∗ = 8,936% 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑆𝑆𝑆𝑆 = �5,492 − 200 25 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

𝑆𝑆𝑆𝑆

d)

𝐷𝐷 = 0,2 𝐸𝐸 𝐸𝐸 = 75, 𝐷𝐷 = 15 Second breath FA : 180 Equity : 25+175=200 Bank deposit : 20 at 3% Debt : 175 Kaia asset : 75+15 Debt kaia : 15 Tbill : 100 Total 390 390 𝐹𝐹𝐹𝐹 = 5,7692 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 = 3% 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 4,643% 75 15 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 18 ∗ +5∗ = 15,833 90 90 180 20 90 100 𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝐾𝐾𝐾𝐾𝐾𝐾 = 5,7692 ∗ + ∗3+ ∗ 15,833 + ∗ 4,643 = 7,66% 390 390 390 390 200 190 7,66 = ∗ 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐾𝐾𝐾𝐾𝐾𝐾 + ∗5 390 390 => 𝑟𝑟𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐾𝐾𝐾𝐾𝐾𝐾 = 10,187% 𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 ∶

76

Maths

Exercises

Problem 2 (Ex09) a) 279 9579 279 +2∗ +3∗ 2 1,078 1,078 1,0783 23678,5477 𝐷𝐷3 = = = 2,907 279 9579 279 8145,4196 + + 1,078 1,0782 1,0783 𝐷𝐷𝑀𝑀3 = 2,6966 2,6966 � = 9024,017 8145,4196 ∗ �1 + (7,8 − 3,8) ∗ 100 279 279 9579 𝑁𝑁𝑁𝑁𝑁𝑁𝑃𝑃3 = + + = 9092,747 2 1,038 1,038 1,0383

With convexity 279 9579 279 +2∗3∗ +3∗4∗ 2∗ 1,0784 1,0783 1,0785 80645,1115 𝐶𝐶3 = = = 9,9007 279 9579 279 8145,4196 + + 1,078 1,0782 1,0783 1 𝑁𝑁𝑁𝑁𝑁𝑁𝑃𝑃3 = 8145,4196 �1 + (−2,6966) ∗ (−0,04) + ∗ 9,9007 ∗ (−0,04)2 � = 9088,53 2 b) compare the two bond bond 4 with 10000=> 10000 = 9433,962 1,06

The other better yield, riskier. 𝐸𝐸 =

10000 ∗ 9000 = 9540? ? ? 9433,962

c) 𝐹𝐹𝐹𝐹 = 10000 𝑡𝑡 = 3𝑦𝑦 7,5% ⎧ ⎪ ⎪ ⎪

𝑖𝑖1 = 6%

1 2

8882,5 𝑖𝑖2 = � � − 1 = 7,023% 382,5 8882,5 382,5 + − 1,07 1,06 1,072

⎨ 1 3 ⎪ 9579 ⎪ 𝑖𝑖 = � � − 1 = 7,8371% ⎪3 279 279 9579 279 279 1,078 + 1,0782 + 1,0783 − 1,06 − 1,070232 ⎩

1 4

10500 � − 1 = 9,37% 500 500 500 10500 500 500 500 1,092 + 1,0922 + 1,0923 + 1,0924 − 1,06 − 1,070232 − 1,0783713 1𝑓𝑓1 => 1,06 ∗ (1 + 1𝑓𝑓1) = 1,070232 => 1𝑓𝑓1 = 8,0559% 𝑖𝑖4 = �

1,0783713 2𝑓𝑓1 => 1,06 ∗ (1 + 2𝑓𝑓1)2 = 1,0783713 => 2𝑓𝑓1 = � − 1 = 8,7676% 1,06 1

1,09374 3 3𝑓𝑓1 = � � − 1 = 10,517% 1,06

77

Maths

𝑃𝑃 =

Exercises

750 750 10750 + + = 9291,88 2 1,080559 1,08766 1,105173

d) 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵3 = 2,907𝑦𝑦 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝐵𝐵4 = 1𝑦𝑦 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑃𝑃 = 0 = 𝑥𝑥 ∗ 2,907 + (1 − 𝑥𝑥) ∗ 1 1 = −0,5244 𝑥𝑥 => − 1,907 𝑦𝑦 = 1 − 𝑥𝑥 = 1,5244 change

279 7450 + 1,5244 ∗ = 1,078 1,06 279 𝐶𝐶𝐶𝐶2 = −0,5244 ∗ 1,0782 9579 𝐶𝐶𝐶𝐶3 = −0,5244 ∗ 1,0783

𝐶𝐶𝐶𝐶1 = −0,5244 ∗

78

Maths

Exercises

Problem 3 (Ex09) a) 𝑟𝑟𝑢𝑢 = 4 + 0,99 ∗ 6 = 9,96% 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 𝑁𝑁𝑁𝑁𝑁𝑁 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 𝐸𝐸𝐸𝐸𝐸𝐸 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁ℎ + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 5,975 ∗ 4𝑀𝑀 + 25𝑀𝑀 ∗ 0,06 ∗ (1 − 0,4) + 3,61 ∗ (1 − 0,4) = 26,9660𝑀𝑀 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 26,9660 = = 270,743𝑀𝑀 𝑀𝑀𝑀𝑀(𝑈𝑈) = 𝑟𝑟𝑢𝑢 − 𝑔𝑔 0,0996 1 − 1,062−5 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = (25𝑀𝑀 ∗ 0,06 ∗ 0,4 + 3,61 ∗ 0,4) ∗ 0,062 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 8,5634𝑀𝑀 𝑀𝑀𝑀𝑀(𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶) = 279,3064𝑀𝑀

𝑀𝑀𝑀𝑀(𝐷𝐷) = (25𝑀𝑀 ∗ 0,06 + 3,61) ∗

25 1 − 1,062−5 + = 39,9149𝑀𝑀 0,062 1,0625

𝑀𝑀𝑀𝑀(𝐸𝐸) = 𝑀𝑀𝑀𝑀(𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶) − 𝑀𝑀𝑀𝑀(𝐷𝐷) = 239,3915 𝑟𝑟𝑒𝑒 = 0,0996 + (0,0996 − 0,062) ∗

𝑟𝑟𝑒𝑒 = 10,3362%

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 10,3362 ∗

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 9,3907%

39,9149 ∗ 0,6 239,392

239,392 39,9149 + 6,2 ∗ ∗ 0,6 279,3064 279,3064

b) 𝑟𝑟𝑢𝑢 = 0,0996 𝑀𝑀𝑉𝑉𝑢𝑢 = 270,743𝑀𝑀 𝑟𝑟𝑑𝑑 = 7,5% 𝑀𝑀𝑀𝑀("𝑂𝑂𝑂𝑂𝑂𝑂" 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷) = 39,9149𝑀𝑀 𝑀𝑀𝑀𝑀(𝐸𝐸) = 239,3915𝑀𝑀 𝑋𝑋 = 𝑁𝑁𝑁𝑁𝑁𝑁 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) =

𝑋𝑋 ∗ 0,075 ∗ 0,4 = 0,3012 ∗ 𝑋𝑋 0,0996

𝐷𝐷 𝐷𝐷 = 1 ; = 0,5 𝑉𝑉 𝐸𝐸 ?? 1 𝑋𝑋 = (270,743 + 0,3012 ∗ 𝑋𝑋) 2 𝑋𝑋 = 159,3731 159,3731 − 39,9149 = 119,4582 159,3731 − 239,392 = −80,0189 119,4582 − 80,0189 = 39,4393 ?? c) Project 1 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 5𝑀𝑀

1 − 1,0996−10 = 2,8806 0,0996 1 − 1,0996−10 = 1,2310 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡𝑡𝑡) = 0,5 ∗ 0,4 ∗ 0,0996

𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 0,78 ∗ 0,6 ∗

79

Maths 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 2,5 ∗ 0,4 ∗ 0,075 ∗ Subsidized debt of 2M

Exercises 1 − 1,075−10 = 0,5148 0,075

1 − 1,075−10 = 0,3432 0,075 (1 − 1,075−10 ) = −0,1373 −𝑠𝑠𝑠𝑠𝑠𝑠(𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖) = 2 ∗ (0,075 − 0,05) ∗ 0,4 ∗ 0,075 Quicker to do 1 − 1,075−10 = 0,2059 𝑆𝑆 = 2 ∗ (0,075 − 0,05) ∗ 0,6 ∗ 0,075 𝑆𝑆 = 2 ∗ (0,075 − 0,05) ∗

5M half equity and debt : 2,5M 2,5 ∗ 0,05 ∗ 0,6 𝐼𝐼 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐸𝐸 = = 0,0789 0,95 No issue cost on 2M 0,5 ∗ 0,02 ∗ 0,6 𝐼𝐼 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐷𝐷 = = 0,0061 0,98

𝐴𝐴𝐴𝐴𝐴𝐴 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 1 = −5 + 2,8806 + 1,2310 + 0,5148 + 0,3432 − 0,1373 − 0,0789 − 0,0061 = −0,2527

Project 2 𝑟𝑟𝑒𝑒 = 6 + 4 ∗ 1,4 = 11,6% 1 2 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 11,6 ∗ + 7 ∗ ∗ 0,6 = 9,1333% 3 3 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 10,6 𝑀𝑀𝑀𝑀 = = = 9,7826 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 − 𝑔𝑔 0,091333 − 0,03 2 𝑀𝑀𝑀𝑀(𝐸𝐸) = ∗ 𝑀𝑀𝑀𝑀(𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶) = 6,5217 3 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 6,6𝑀𝑀 𝑀𝑀𝑀𝑀(𝐸𝐸) = 6,5217

𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 3,3 ∗ 0,075 ∗ 0,4 ∗

1 − 1,075−10 = 0,6795 0,075

3,3 ∗ 0,05 ∗ 0,6 = 0,1042 0,95 3,3 ∗ 0,02 ∗ 0,6 𝐼𝐼 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐷𝐷 = = 0,0404 0,98

𝐼𝐼 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐸𝐸 =

𝐴𝐴𝐴𝐴𝐴𝐴 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 2 = −6,6 + 6,5217 + 0,6795 − 0,1042 − 0,0404 = 0,4566

80

Maths

Exercises

Problem 4 (Ex09) a) 45 40 35 30 25

Price

20 15

Call

10 5 0 0

10

20

30

40

45 40 35 30 25 20 15 10 5 0

Price Debt

0

10

20

30

40

b) 𝜎𝜎 = 38,7% 𝑟𝑟𝑟𝑟 = 2% 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠, 𝑠𝑠𝑠𝑠 ln(1,02) ∗ 4 = 7,9211% 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 The equity is like a call, so we have : 0,3872 30 ln �10� + �0,079211 + 2 � ∗ 4 𝑑𝑑1 = = 2,2158 0,387 ∗ √4 0,3872 30 ln �10� + �0,079211 − 2 � ∗ 4 𝑑𝑑2 = = 1,4418 0,387 ∗ √4 The bond was issued 1 year ago with 5y of maturity ‼ so it remains 4y… 𝑁𝑁(𝑑𝑑1 ) = 0,9864 + 0,58 ∗ (0,9868 − 0,9864) = 0,9866 𝑁𝑁(𝑑𝑑2 ) = 0,9251 + 0,18 ∗ (0,9265 − 0,9251) = 0,9254 𝐶𝐶 = 30 ∗ 0,9866 − 10 ∗ 𝑒𝑒 −0,079211 ∗4 ∗ 0,9254 = 22,857 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 30𝑀𝑀 − 22,857 = 7,1423

We have also 𝑃𝑃 = −30 ∗ (1 − 0,9866) + 10 ∗ 𝑒𝑒 −0,079211 ∗4 ∗ (1 − 0,9254) = 0,1414 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 + 𝑃𝑃 = 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 => 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 − 𝑃𝑃 => 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 = 10𝑀𝑀 ∗ 𝑒𝑒 −0,079211 ∗4 − 0,1414 = 7,1423

Project of 1 year !

𝑆𝑆 = 22,857

𝑆𝑆𝑢𝑢 = �1,3 ∗ 𝑆𝑆 = 26,061

𝑆𝑆𝑑𝑑 = �0,8 ∗ 𝑆𝑆 = 20,4439 81

𝑆𝑆𝑢𝑢𝑢𝑢 = 1,3 ∗ 𝑆𝑆 = 29,7141 𝑆𝑆𝑢𝑢𝑢𝑢 = 23,31

Maths

Exercises 2

𝑟𝑟𝑟𝑟 = 1,02 = 1,0404 1,0404 − √0,8 𝑝𝑝 = = 0,594 √1,3 − √0,8 0,594 ∗ (29,7141 − 20) + (1 − 0,594) ∗ (21,31 − 20) 𝐶𝐶𝑢𝑢 = = 6,8378 1,0404 3,31 ∗ 0,594 𝐶𝐶𝑑𝑑 = = 1,8898 1,0404 0,594 ∗ 6,8378 + (1 − 0,594) ∗ 1,8898 = 4,6414 𝐶𝐶 = 1,0404 4,6414 = 2,3207 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 = 2 (because Vanno holds the half !) c) If the exercise price is 18, then the 𝐶𝐶𝑑𝑑 changes, also 𝐶𝐶𝑢𝑢 and then 𝐶𝐶

𝑆𝑆𝑑𝑑𝑑𝑑 = 18,286

d) 2 examples : - 2 assets quoted in different countries, prices can be different if investors buy at a higher price to diversify their portfolio - 2 assets can have different prices, if one is on an index, such the S&P500, investors have to buy asset who are on index, so higher price…

82

Maths

Exercises

Problem 5 (Ex09) a) 200 150 100 50 0 0

100 200 300 400 Call E=230k S1=100k & S2=400k

500

b) 20

77000 ∗ (1 + 𝑖𝑖𝑠𝑠 )20 = 100000 => 𝑖𝑖𝑠𝑠 = �

𝑖𝑖𝑐𝑐 = ln(1,013154) = 1,306824% 𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 ∶ 77000 ∗ 𝑒𝑒 𝑖𝑖𝑐𝑐 ∗20

𝑆𝑆 = 400 𝑑𝑑1 =

𝑑𝑑2 =

100000 − 1 = 1,3154% 77000

100000 ln⁡ ( 77000 ) = 100000 => 𝑖𝑖𝑐𝑐 = = 1,306824% 20

0,3662 400 ln �230� + �0,01306824 + 2 � ∗ 5 0,366 ∗ √5

0,3662 400 ln �230� + �0,01306824 − 2 � ∗ 5

= 1,1652

= 0,368 0,366 ∗ √5 𝑁𝑁(𝑑𝑑1 ) = 0,8770 + 0,52 ∗ (0,8790 − 0,8770) = 0,8780 𝑁𝑁(𝑑𝑑2 ) = 0,6331 + 0,68 ∗ (0,6368 − 0,6331) = 0,6356 𝐶𝐶 = 400 ∗ 0,8780 − 230 ∗ 𝑒𝑒 −0,01306824 ∗5 ∗ 0,6356 𝐶𝐶400 = 214258,714

𝑆𝑆 = 100 𝑑𝑑1 = +∞ => 𝑁𝑁(𝑑𝑑1 ) = 1 𝑑𝑑2 = −∞ => 𝑁𝑁(𝑑𝑑2 ) = 0 𝐶𝐶100 = 100000

𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0,7 ∗ 214258,714 + 0,3 ∗ 100 = 179981,0998 c)

60000 50000 40000 30000 20000 10000 0 0

500000 1000000 1500000 Binary Call E=600000 Vmax=50000

83

Maths

Exercises

(abscise market share) d) 𝑑𝑑2 =

0,3662 400 ln �600� + �0,01306824 − 2 � ∗ 3

= −0,8947 0,366 ∗ √3 𝑁𝑁(𝑑𝑑2 ) = 0,1867 + 0,47 ∗ (0,1841 − 0,1867) = 0,1855

No need to find 𝑑𝑑1 (= −0,2608 ) and 𝑁𝑁(𝑑𝑑1 ) (= 0,3971) because it’s a binary call 𝑁𝑁(𝑑𝑑2 ) is the probability to end up in the money, or in this case, that the company is purchased. 𝑁𝑁(𝑑𝑑2 ) ∗ 𝑉𝑉 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝐵𝐵𝐵𝐵 = (1 + 𝑖𝑖𝑠𝑠 )𝑛𝑛 Here we’ve 0,1855 ∗ 50000 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝐵𝐵𝐵𝐵 = = 8918,412 1,031543 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 214258,714 + 8918,412 = 223177,126

84

Maths

Exercises

RE-Exam 2009 Problem 2 (Reex09) a) 262,5 𝑃𝑃𝐵𝐵5 = => 𝑖𝑖1 = 5% 1,05 300 => 𝑖𝑖2 = 6% 𝑃𝑃𝐵𝐵4 = 1,06 11,25 11,25 161,25 11,25 11,25 161,25 𝑃𝑃𝐵𝐵3 = + + = + + 1,07 1,072 1,073 1,05 1,062 1, 𝑖𝑖3 3 1

161,25 3 𝑖𝑖3 = � � − 1 = 7,105% 131,2415 𝑃𝑃𝐵𝐵2 => 𝑖𝑖4 = 8% 1 − 1,09−5 100 10 10 10 10 110 + = + + + + 𝑃𝑃𝐵𝐵1 = 10 ∗ 5 2 3 4 (1, 0,09 1,09 1,05 1,06 1,07105 1,08 𝑖𝑖5 )5 1

110 5 � − 1 = 9,4682% 𝑖𝑖5 = � 69,9766

2 year rate in 3 year 1,071053 ∗ (1 + 2𝑓𝑓3)2 = 1,0946825 1

1,0946825 2 2𝑓𝑓3 = � � − 1 = 13,1111% 1,071053

b) Long B1 and short B3 1 − 1,09−5 100 + = 103,8897 𝑃𝑃𝐵𝐵1 = 10 ∗ 0,09 1,095 11,25 11,25 161,25 𝑃𝑃𝐵𝐵3 = + + = 151,9682 1,07 1,072 1,073

1 10 20 30 40 550 434,9727 ∗� + + + + �= = 4,1868 2 3 4 5 103,8897 1,09 1,09 1,09 1,09 1,09 103,8897 425,0505 𝐷𝐷𝐵𝐵3 = = 2,7970 151,9682

𝐷𝐷𝐵𝐵1 =

4,1868 = 3,8411 1,09 2,7970 = 2,6140 𝑀𝑀𝑀𝑀𝐵𝐵3 = 1,07

𝑀𝑀𝑀𝑀𝐵𝐵1 =

2,6140 3,8411 � − 151,9682 ∗ �−𝑥𝑥 ∗ � ≥ −2 100 100 𝑥𝑥 ∗ (151,9682 ∗ 0,02614 − 103,8897 ∗ 0,038411) ≥ −2 𝑥𝑥 = 110,75%

103,8897 ∗ �−𝑥𝑥 ∗

Quick Check If rate up 110,75 you have: −3,8411 ∗ 110,75% ∗ 103,8897 = −441,95 +(𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝑠𝑠ℎ𝑜𝑜𝑜𝑜𝑜𝑜) 2,6140 ∗ 110,75% ∗ 151,9682 = 439,95 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = −2 ! Question here, how can we lose -441,95 on a bond with a price of 103,8897… c) 𝐵𝐵6 = 3𝑦𝑦 ; 𝐹𝐹𝐹𝐹 = 250𝑀𝑀𝑀𝑀 ; 𝐶𝐶 = 10% = 25𝑀𝑀𝑀𝑀

85

Maths

Exercises

𝑃𝑃𝑃𝑃 = 252 𝑃𝑃𝐵𝐵6 = 25 ∗ 1,05−1 + 25 ∗ 1,06−2 + 275 ∗ 1,07105−3 = 269,818𝑀𝑀𝑀𝑀

To replicate 𝐵𝐵6 we can take 𝐵𝐵3 => 𝑧𝑧, 𝐵𝐵4 => 𝑦𝑦, 𝐵𝐵5 => 𝑥𝑥 We can write : 𝑥𝑥 ∗ 262,5 + 𝑧𝑧 ∗ 11,25 = 25 𝑦𝑦 ∗ 300 + 𝑧𝑧 ∗ 11,25 = 25 𝑧𝑧 ∗ 161,25 = 275

𝑧𝑧 = 1,7054 => 𝑦𝑦 = 0,01938 => 𝑥𝑥 = 0,02215 So you gain 269,818 − 252 = 17,818

86

Maths

Exercises

Problem 3 (Reex09) 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 11% 𝐷𝐷 = 1,5 𝐸𝐸 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 36$ 𝑟𝑟𝑑𝑑 = 6% 𝑟𝑟𝑓𝑓 = 5% 𝑡𝑡𝑐𝑐 = 0,4 𝑟𝑟𝑚𝑚 = 13% 𝑛𝑛𝑛𝑛𝑛𝑛ℎ = 5 a)

𝑀𝑀𝑀𝑀(𝐷𝐷) =

36 = 600 0,06

𝐷𝐷 = 1,5 => 𝑀𝑀𝑀𝑀(𝐸𝐸) = 400 𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸(1 − 𝑡𝑡𝑐𝑐 ) 𝑀𝑀𝑀𝑀(𝐴𝐴) = 1000 = => 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ∗ (1 − 𝑡𝑡𝑐𝑐 ) = 110 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸(1 − 𝑡𝑡𝑐𝑐 ) = 𝐸𝐸𝐸𝐸𝐸𝐸 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁ℎ + 𝐼𝐼𝐼𝐼𝐼𝐼(1 − 𝑡𝑡𝑐𝑐 ) => 110 − 36 ∗ (1 − 0,4) 84,4 = = 17,68$ 𝐸𝐸𝐸𝐸𝐸𝐸 = 5 5 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 𝑟𝑟𝑙𝑙𝑙𝑙 ∗ 0,4 + 𝑟𝑟𝑑𝑑 ∗ 0,6 ∗ 0,6 0,11 − 0,06 ∗ 0,6 ∗ 0,6 = 22,1% 𝑟𝑟𝑙𝑙𝑙𝑙 = 0,4 𝛽𝛽𝑙𝑙𝑙𝑙 = b)

22,1 − 5 = 2,1375 8

0,221 + 1,5 ∗ 0,06 ∗ (1 − 0,4) = 14,4737% 1 + 1,5 ∗ (1 − 0,4) 𝑔𝑔 = 𝑅𝑅𝑅𝑅𝑅𝑅 ∗ (1 − 𝑝𝑝) = 0,2 ∗ (1 − 0,5) = 0,1% 110 ∗ 0,5 = 1229,4074$ 𝑉𝑉(𝐴𝐴) = 𝑉𝑉(𝐸𝐸) = 0,144737 − 0,1

𝑟𝑟𝑢𝑢 =

𝐷𝐷 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 ∗ 𝑡𝑡𝑐𝑐 = 𝑡𝑡𝑐𝑐 ∗ ∗ 𝑉𝑉𝐴𝐴′ 𝐴𝐴 𝐷𝐷 ′ ′ 𝑉𝑉𝐴𝐴 = 𝑉𝑉𝑉𝑉 + 𝑡𝑡𝑐𝑐 ∗ ∗ 𝑉𝑉𝐴𝐴 𝐴𝐴 𝑉𝑉𝑉𝑉 ′ 𝑉𝑉(𝐴𝐴 ) = = 1617,647$ 𝐷𝐷 1 − 𝑡𝑡𝑐𝑐 ∗ 𝐴𝐴

c) Project A 𝑟𝑟𝑢𝑢𝑢𝑢 = 14,4737% 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 100 𝐷𝐷 =5 𝐸𝐸 𝑟𝑟𝑑𝑑 = 10%

𝐷𝐷 ∗ (1 − 𝑡𝑡𝑐𝑐 ) 𝐸𝐸 𝑟𝑟𝑙𝑙𝑙𝑙 = 0,144737 + (0,144737 − 0,1) ∗ 5 ∗ (1 − 0,4) = 27,8948%

𝑟𝑟𝑙𝑙𝑙𝑙 = 𝑟𝑟𝑢𝑢 + (𝑟𝑟𝑢𝑢 − 𝑟𝑟𝑑𝑑 ) ∗

87

Maths

Exercises

1 5 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 = 27,8948 ∗ + 10 ∗ ∗ (1 − 0,4) = 9,6491% 6 6 100 ∗ (1 − 0,4) = 621,8197 𝑀𝑀𝑀𝑀(𝐴𝐴) = 0,096491 1 𝑀𝑀𝑀𝑀(𝐸𝐸) = ∗ 𝑀𝑀𝑀𝑀(𝐴𝐴) = 103,6366 6 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑋𝑋

Project financed with same D/E=1,5 so D/V=0,6, E/V=0,4, and cost debt = 6% 1 − 1,06−10 = 0,1060 ∗ 𝑋𝑋 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇) = 𝑋𝑋 ∗ 0,6 ∗ 0,06 ∗ 0,4 ∗ 0,06 𝑋𝑋 ∗ 0,4 ∗ (1 − 0,4) ∗ 0,05 = 0,0126 ∗ 𝑋𝑋 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐸𝐸 = 0,95 𝑋𝑋 ∗ 0,6 ∗ (1 − 0,4) ∗ 0,02 = 0,0073 ∗ 𝑋𝑋 𝐼𝐼 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐷𝐷 = 0,98

𝐴𝐴𝐴𝐴𝐴𝐴 = −𝑋𝑋 + 103,6366 + 0,1060 ∗ 𝑋𝑋 − 0,0126 ∗ 𝑋𝑋 − 0,0073 ∗ 𝑋𝑋 = 103,6366 − 𝑋𝑋 ∗ 0,9139 The project is worth if 𝑋𝑋 ≤ 113,4004 Project A, with the subsidy

𝑃𝑃𝑃𝑃(𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆) = 𝑋𝑋 ∗ 0,6 ∗ (0,06 − 0,02 ∗ (1 + 0,4) ∗ 𝐴𝐴𝐴𝐴𝑉𝑉𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 103,6366 − 𝑋𝑋 ∗ 0,7726 The project is worth if 𝑋𝑋 ≤ 134,14

1 − 1,06−10 = 0,1413 0,06

Project B

1 − 1,144737−10 = 307,2661 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 100 ∗ 0,6 ∗ 0,144737 −10 1 − 1,144737 𝑋𝑋 ∗ 0,4 ∗ = 0,2048 ∗ 𝑋𝑋 𝑃𝑃𝑃𝑃(𝑑𝑑𝑑𝑑𝑑𝑑) = 0,144737 10

𝐴𝐴𝐴𝐴𝐴𝐴 = −𝑋𝑋 + 307,2661 + 0,2048 ∗ 𝑋𝑋 + 0,1060 ∗ 𝑋𝑋 − 0,0126 ∗ 𝑋𝑋 − 0,0073 ∗ 𝑋𝑋 = 307,2661 − 0,7091 ∗ 𝑋𝑋 The project is worth if 𝑋𝑋 ≤ 433,3184

Project A, with the subsidy 𝑃𝑃𝑃𝑃(𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆) = 0,1413 𝐴𝐴𝐴𝐴𝑉𝑉𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 307,2661 − 𝑋𝑋 ∗ 0,5678 The project is worth if 𝑋𝑋 ≤ 541,152

88

Maths

Exercises

Problem 4 (Reex09) a) 𝑆𝑆 = 55 𝑟𝑟𝑟𝑟 1𝑦𝑦 = 4% 𝜎𝜎 = 25% 𝑢𝑢 = 𝑒𝑒 0,25 √0,5 = 1,1934 𝑑𝑑 = 0,838 1

𝑒𝑒 0,04∗3 − 0,838 𝑝𝑝 = = 0,4936 1,1934 − 0,838

2

𝐸𝐸 = 𝐹𝐹 = 𝑆𝑆 ∗ 𝑒𝑒 𝑟𝑟𝑟𝑟 = 55 ∗ 𝑒𝑒 0,04∗3 𝐸𝐸 = 56,486

𝑆𝑆𝑢𝑢𝑢𝑢 = 78,331 𝑃𝑃𝑢𝑢𝑢𝑢 = 0

8 months model

𝑆𝑆𝑢𝑢 = 65,637

𝑆𝑆 = 55

𝑃𝑃𝑢𝑢 =

1,486 ∗ 0,5064

𝑃𝑃𝑑𝑑𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 =

1 𝑒𝑒 0,04∗3

𝑆𝑆𝑢𝑢𝑢𝑢 = 55 𝑃𝑃𝑢𝑢𝑢𝑢 = 1,486

𝑆𝑆𝑑𝑑 = 46,09 = 0,7425

1,486 ∗ 0,4936 + 17,863 ∗ 0,5064 1 𝑒𝑒 0,04∗3

𝑃𝑃𝑑𝑑𝑒𝑒𝑒𝑒 = 10,396 0,7425 ∗ 0,4936 + 0,65 ∗ 0,5064 𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸 = 1 𝑒𝑒 0,04∗3 𝑃𝑃𝐸𝐸𝐸𝐸𝐸𝐸 = 5,1837 𝑃𝑃𝑈𝑈𝑈𝑈𝑈𝑈 = 5,5565

𝑆𝑆𝑑𝑑𝑑𝑑 = 38,623 𝑃𝑃𝑑𝑑𝑑𝑑 = 17,863

= 9,65

b) Replicating portfolio : you replicate the option with stock and bond : opposite cash flow. Hedge portfolio : with different option.

𝑃𝑃𝑢𝑢 − 𝑃𝑃𝑑𝑑 0,7425 − 9,65 = 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 55 ∗ (1,1934 − 0,838) 𝛼𝛼 ∗ = −0,456 𝑃𝑃𝑑𝑑 − 𝛼𝛼 ∗ ∗ 𝑆𝑆𝑑𝑑 9,65 − 0,456 ∗ 46,09 𝐵𝐵 = = 1 𝑟𝑟 𝑒𝑒 0,04∗3 𝐵𝐵 = 30,261 𝑃𝑃𝑃𝑃𝑃𝑃: 5,1837 −𝛼𝛼 ∗ ∗ 𝑆𝑆 = 0,456 ∗ 55 = 25,08 𝐵𝐵 = −30,261 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 = 0 𝛼𝛼 ∗ =

𝑆𝑆 + 𝑃𝑃 = 𝐸𝐸 ∗ 𝑒𝑒 −𝑟𝑟𝑟𝑟 + 𝐶𝐶 𝑆𝑆 + 𝑃𝑃 = 𝑆𝑆 + 𝐶𝐶 𝑃𝑃 = 𝐶𝐶 c)

Total value (P)

Intrisic value (E-S) 89

Time value

Maths

Exercises

𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑢𝑢 𝑑𝑑 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

5,184 0,749 9,651 0 1,501 1,501 17,863

1,486 0 10,396 0 1,501 1,501 17,863

3,698 0,749 −0,745 0 0 0 0

d) 8 months model 𝑆𝑆𝑢𝑢 = 65,637 − 4 = 61,637

𝑆𝑆 = 55

𝑆𝑆𝑑𝑑 = 46,09 − 4 = 42,09 In part 1 we had

2

𝑆𝑆𝑢𝑢𝑢𝑢 = 73,507 𝐶𝐶𝑢𝑢𝑢𝑢 = 20,972 𝑆𝑆𝑢𝑢𝑢𝑢 = 51,633 𝐶𝐶𝑢𝑢𝑢𝑢 = 0 𝑆𝑆𝑑𝑑𝑑𝑑 = 50,213 𝐶𝐶𝑑𝑑𝑑𝑑 = 0 𝑆𝑆𝑑𝑑𝑑𝑑 = 35,271 𝐶𝐶𝑑𝑑𝑑𝑑 = 0

𝐸𝐸 = 𝐹𝐹 = 𝑆𝑆 ∗ 𝑒𝑒 𝑟𝑟𝑟𝑟 = 55 ∗ 𝑒𝑒 0,04∗3 Then here we have also the dividend which impact the price of the stock then the E 2

1

𝐸𝐸 = 𝑆𝑆 ∗ 𝑒𝑒 𝑟𝑟𝑟𝑟 − 𝑑𝑑𝑑𝑑𝑑𝑑 ∗ 𝑒𝑒 𝑟𝑟𝑟𝑟 = 55 ∗ 𝑒𝑒 0,04∗3 − 4 ∗ 𝑒𝑒 0,04∗3 𝐸𝐸 = 52,432 Be careful, the dividend is in 4 months, so it’s 1/3 and not 2/3 (for the 8 months) 𝐶𝐶𝑢𝑢 =

20,972 ∗ 0,4936 1 𝑒𝑒 0,04∗3

= 10,215

𝐶𝐶𝑢𝑢𝑒𝑒𝑒𝑒 = 13,205 => 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 13,205 ∗ 0,4936 𝐶𝐶𝑈𝑈𝑈𝑈𝑈𝑈 = 1 𝑒𝑒 0,04∗3 𝐶𝐶𝑈𝑈𝑈𝑈𝑈𝑈 = 6,432

e) American call without dividend is equal to European Call.

90

Maths

Exercises

Problem 5 (Reex09) a) 5000 4000 Call E=5000 S=4000 p=0,8

3000 2000

Call E=5000 S=4000 p=0,2

1000 0 0

2000

4000

6000

8000

(don’t forget the axes : PV(CF) and Value !) b) 𝑃𝑃𝑃𝑃(𝐶𝐶𝐶𝐶) = 4𝑀𝑀 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 5𝑀𝑀 𝜎𝜎𝐶𝐶𝐶𝐶 = 0,33 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 3 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑠𝑠 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 4,3% 𝑟𝑟𝑐𝑐 = 𝑙𝑙𝑙𝑙1,043 = 4,21% 0,332 4 ln � � + �𝑙𝑙𝑙𝑙1,043 + 2 � ∗ 3 5 𝑑𝑑1 = = 0,1164 0,33 ∗ √3 0,332 4 ln � � + �𝑙𝑙𝑙𝑙1,043 − 2 � ∗ 3 5 𝑑𝑑2 = = −0,4552 0,33 ∗ √3 𝑁𝑁(𝑑𝑑1 ) = 0,5438 + 0,64 ∗ (0,5478 − 0,5438) = 0,5464 𝑁𝑁(𝑑𝑑2 ) = 0,3264 + 0,52 ∗ (0,3228 − 0,3264) = 0,3245

𝐶𝐶 = 0,5464 ∗ 4𝑀𝑀 − 5𝑀𝑀 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,043∗3 ∗ 0,3245 = 755614,02

What is the value ? We have 20% probability to have a double payoff so we have 2 call ! So we have 20% probability to have 2 call and then 80% to have only one 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 0,2 ∗ 2 ∗ 755614,02 + 0,8 ∗ 755614,02 = 906736,82 c)

3000 2000 Return max 2000

1000 0 -1000 0

5000

10000

Call E=7000

-2000 -3000

d) Notice that a year has passed ! Value of the call with E=5000 𝐸𝐸 = 5000 91

Maths

Exercises

0,332 4 ln � � + �𝑙𝑙𝑙𝑙1,043 + 2 � ∗ 2 5 𝑑𝑑1 = = −0,0644 0,33 ∗ √2 0,332 4 ln � � + �𝑙𝑙𝑙𝑙1,043 − 2 � ∗ 2 5 𝑑𝑑2 = = −0,5311 0,33 ∗ √2 𝑁𝑁(𝑑𝑑1 ) = 0,4761 + 0,44 ∗ (0,4721 − 0,4761) = 0,4743 𝑁𝑁(𝑑𝑑2 ) = 0,2981 + 0,11 ∗ (0,2946 − 0,2981) = 0,2977

𝐶𝐶 = 0,4743 ∗ 4𝑀𝑀 − 5𝑀𝑀 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,043∗2 ∗ 0,2977 = 528903,48 Value of the call with E=7000 𝐸𝐸 = 7000 0,332 4 ln �7� + �𝑙𝑙𝑙𝑙1,043 + 2 � ∗ 2 𝑑𝑑1 = = −0,7853 0,33 ∗ √2 0,332 4 ln �7� + �𝑙𝑙𝑙𝑙1,043 − 2 � ∗ 2 𝑑𝑑2 = = −1,2520 0,33 ∗ √2 𝑁𝑁(𝑑𝑑1 ) = 0,2177 + 0,53 ∗ (0,2148 − 0,2177) = 0,2162 𝑁𝑁(𝑑𝑑2 ) = 0,1056 + 0,2 ∗ (0,1038 − 0,1056) = 0,1052

𝐶𝐶 = 0,2162 ∗ 4𝑀𝑀 − 7𝑀𝑀 ∗ 𝑒𝑒 −𝑙𝑙𝑙𝑙 1,043∗2 ∗ 0,1052 = 187867,82

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 528903,48 − 187867,82 = 341035,66 e)

2,5 2 1,5 1 0,5 0 0

5 10 Binary Call E=5M - BC E=7M

15

2𝑀𝑀 ∗ 𝑁𝑁(𝑑𝑑2 ) 𝑟𝑟 2 In fact 𝑁𝑁(𝑑𝑑2 ) is the probability of the call to end up in the money 𝐵𝐵𝐵𝐵(𝐸𝐸 = 5) =

2𝑀𝑀 ∗ 0,2977 = 547318,6 1,0432 2𝑀𝑀 ∗ 0,1052 𝐵𝐵𝐵𝐵(𝐸𝐸 = 7) = = 193409,2 1,0432 𝑉𝑉𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 𝐵𝐵𝐵𝐵 = 547318,6 − 193409,2 = 353,7256

𝐵𝐵𝐵𝐵(𝐸𝐸 = 5) =

We should take this project because value >

92