EXERCISES
EXERCISE 1
In a perfectly competitive market where the interest rate 𝑟 remains constant
over time, security A pays 1200 euro after three years and it costs 1000 euro.
Security B pays 1100 euro after one year. If there is a 20% tax on financial
profits, how much does the investor who buys B earn after one year?
The interest rate is:
1000 =
1200
(1 + 𝑟)3
(1 + 𝑟)3
=
1200
1000
1 + 𝑟 = 1.21 3⁄
𝑟 = 1.21 3⁄
− 1 ≈ 0.0627 = 6.27%
Since the interest rate is 6.27% and we are in a perfect market, the price of B
has to be:
𝑃𝐵 =
1100
1 + r
=
1100
1 + 0.0627
≈ 1035.1
The investor earned 1100 – 1035.1 = 64.9 euro, but has to pay a 20% tax on
it, so after taxes he actually earns:
64.9 − 64.9 ∗ 0.2 = 51.92
EXERCISE 2
A project requires an initial investment of 10000 euro and is expected to
generate a positive cash flow of 5000 euro after one year and one of 8000 euro
after two years. What is the maximum cost of capital with which the project is
profitable?
The project is profitable as long as the cost of capital is lower than the IRR,
i.e., the discount rate that sets the NPV equal to zero. Therefore, we compute
the IRR.
0 = −10000 +
5000
1 + 𝐼𝑅𝑅
+
8000
(1 + 𝐼𝑅𝑅)2
10000 =
5000(1 + 𝐼𝑅𝑅) + 8000
(1 + 𝐼𝑅𝑅)2
10000(1 + 2𝐼𝑅𝑅 + 𝐼𝑅𝑅2) = 5000 + 5000𝐼𝑅𝑅 + 8000
10000 + 20000𝐼𝑅𝑅 + 10000𝐼𝑅𝑅2
− 5000𝐼𝑅𝑅 − 13000 = 0
10000𝐼𝑅𝑅2
+ 15000𝐼𝑅𝑅 − 3000 = 0
10𝐼𝑅𝑅2
+ 15𝐼𝑅𝑅 − 3 = 0
𝐼𝑅𝑅 =
−15 ± √225 + 120
20
≈
−15 ± 18.57
20
The first solution is 0.1785; the second is −1.6785. The second one is clearly
not meaningful, so we consider the first. The project is profitable if the cost of
capital is lower than 17.85%.
EXERCISE 3
A three-year 1000-euro coupon bond pays a 200-euro coupon each year.
Knowing that a bond that pays 100 euro after one-year costs 97 euro, one that
pays 100 euro after two years costs 95 euro, and one that pays 100 euro after
three years costs 90 euro, what is the price at which the coupon bond should
trade in a market in which the law of one price holds?
Due to the law of one price, equal cash flows should have the same price. The
coupon bond pays 200 euro after one year, 200 euro after two years and 1200
(the 1000 euro plus the 200 coupon) after three years.
Therefore, we can replicate its cash flows with two one-year zero-coupon
bonds, two two-year zero-coupon bonds and twelve three-year zero-coupon
bonds. The price of this portfolio of zero-coupon bonds, which must be equal
to the price of the coupon bond, is:
97 ∗ 2 + 95 ∗ 2 + 90 ∗ 12 = 1464
EXERCISE 4
The security S pays 990 euro after one year and it costs 900 euro.
If there is a 20% tax on financial profits and the inflation rate is 2% per year,
what is the real return paid by security S after taxes?
After one year we earn 990 − 900 = 90 euro.
On this we have to pay a 20% tax, so we actually earn 90 − 90 ∗ 0.2 = 72 euro
The nominal return net of taxes is therefore:
𝑅 =
972 − 900
900
= 8%
We now need to account for the inflation. The real return is:
𝑅 𝑟 =
1 + 𝑅
1 + 𝑖
− 1 =
1 + 0.08
1 + 0.02
− 1 ≈ 0.0588
EXERCISE 5
An amortized loan of 100000 euro is paid back in five payments done at
regular intervals. If the interest rate is 5%, what is the total amount paid in
interests?
An amortized loan of this type is an annuity. Therefore, we have:
𝐿 = 𝑃
1
𝑅
(1 −
1
(1 + 𝑅) 𝑁
)
The amount paid at each individual payment is:
𝑃 =
𝐿
1
𝑅
−
1
𝑅(1 + 𝑅) 𝑁
=
100000
1
0.05
−
1
0.05(1 + 0.05)5
≈
100000
20 − 15.67
= 23094.69
So the total payment is 23094.69 ∗ 5 = 115473.4 euro, and the total amount
paid in interests is 115473.4 − 100000 = 15473.4 euro.
EXERCISE 6
A zero-coupon bond with a face value of 10000 euro expires 3 months from now
and costs 9800 euro.
Compute the Modified duration for this bond.
The formula for the Modified duration is:
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑𝐷 =
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷
1 +
𝑌𝑇𝑀
𝑘
Since this is a zero-coupon bond, the Macaulay duration is simply equal to its
maturity expressed in years, i.e.:
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷 =
3
12
= 0.25
The yield to maturity is:
𝑌𝑇𝑀 = (
𝐹𝑉
𝑃
)
1/𝑛
− 1 = (
10000
9800
)
1
0.25
− 1 ≈ 0.0842
We also have 𝑘 = 1 because it is a zero-coupon bond, so the Modified
duration is:
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑𝐷 =
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷
1 +
𝑌𝑇𝑀
𝑘
=
0.25
1 +
0.0842
1
=
0.25
1.0842
≈ 0.23