Financial Maths – Extra Questions
1) Company A is offering €12700 in 5 years in return for an investment of €9950. What is the compound annual rate of
return (CAR) to 2 d.p.?
Hint: The unknown (𝑖) is inside the indexed bracket. So, this involves getting the 5 th root on your calculator (fractional
power).
𝑃(1 + 𝑖)𝑡 = 𝐹
9950(1 + 𝑖)5 = 12700
(1 + 𝑖)5 =
1
5 ]5
[(1 + 𝑖)
12700
9950
1
12700 5
=[
]
9950
1 + 𝑖 = 1.050016511
𝑖 = 0.050016511 = 5.0016511%
𝑖 ≈ 5.00%
2) Company B wants to offer €13000 in return for an investment of €10000 using a CAR of 5%. When (to the nearest
month) should they return the money?
Hint: The unknown (𝑡) is the exponent. So, this involves using logs. Natural logs are easiest on the calculator.
𝑃(1 + 𝑖)𝑡 = 𝐹
10000(1.05)𝑡 = 13000
1.05𝑡 =
13000
= 1.3
10000
ln(1.05𝑡 ) = ln 1.3
𝑡 ln 1.05 = ln 1.3
𝑡=
ln 1.3
= 5.377400731 𝑦𝑒𝑎𝑟𝑠
ln 1.05
𝑡 = 5.377400731 × 12 = 64.52880877 𝑚𝑜𝑛𝑡ℎ𝑠
𝑡 ≈ 65 𝑚𝑜𝑛𝑡ℎ𝑠
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Financial Maths – Extra Questions
3) Explain (in English) what the APR is and what it is used for.
Hint: The formula appears on page 31 of your tables.
Note: Each person’s use of English is different so everyone will have a different answer to this question. However,
there are 5 terms underlined in the answer below which should appear somewhere in your answer.






APR stands for Annual Percentage Rate.
Financial institutions must quote the APR when they are offering loans.
The customer can use the APR to help choose between the loans on offer.
The APR is the interest rate which makes the present value of the loan match with the present value of the
repayments.
APR takes account of the possible different compounding periods in different products and equalises them all
to the equivalent rate compounded annually.
Different countries have different rules around how the APR is calculated. In Ireland:
o The calculation must include all the money which the customer has to pay including any fees or
‘hidden’ charges.
o Time is measured in years from the date the loan is drawn down.
o The APR must be expressed as a percentage, to at least one decimal place.
4) Sonya deposits €300 at the end of each quarter into her savings account. If the money earns 5.75% (EAR), how much
will this investment be worth at the end of four years? EAR stands for Equivalent Annual Rate.
(1 + 𝑖)4 = (1 +
5.75 1
)
100
1
1 + 𝑖 = 1.05754
𝑆𝑛 =
𝑎(1 − 𝑟 𝑛 )
1−𝑟
𝑎=1
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Financial Maths – Extra Questions
𝑟 =1+𝑖
1
So, from above, 𝑟 = 1.05754
1 16
1(1−(1.05754 )
Hence, 𝑆16 =
)
=
1
1−1.05754
1−1.05754
1−1.05750.25
= 17.80519556
Investment is worth 300𝑆16 = 300 × 17.80519556 = 5341.558669
≈ €5341.56
(Sense check: €300 𝑝𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑒𝑟 = €1200 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟 = €4800 𝑖𝑛 4 𝑦𝑒𝑎𝑟𝑠, plus interest seems ok.)
5) A graduate is setting up his own company. He borrows the €5000 for set-up costs for 6 months at a flat rate of 1% per
month (compounded monthly). He wants to arrange to pay this off in equal monthly instalments.
a)
Calculate the monthly repayment amount.
b) Make a schedule showing the monthly payment, the monthly interest on the outstanding balance, the
portion of the payment contributing towards reducing the debt, and the outstanding balance.
c)
After 5 years the company needs to raise money to expand. It proposes to issue a 10-year €2000 bond that
will pay €100 every year. If the current market interest rate is 5% per annum, what is the fair market value
of this bond? Explain your answer and justify any assumptions you make. a)
We can use the formula on page 31 as the interest rate and the compounding periods match; i.e. both
monthly. (The formula book says “Annual” but we can replace this with “Monthly”)
𝐴=𝑃
𝑖(1 + 𝑖)𝑡
(1 + 𝑖)𝑡 − 1
𝐴 = 5000
(0.01)(1.01)6
(1.01)6 − 1
𝐴 = 862.7418336 ≈ €862.74 p.m.
b) Monthly payment = €862.74 = (portion to pay interest) + (portion to reduce debt)
A
B
C
D = 1% x B
E = C-D
F=B-E
Month
Outstanding Balance at
start month
Monthly
Payment
Interest
Reduction in
Debt
Outstanding Balance at
end month
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1
5000
862.74
50.00
812.74
4187.26
2
4187.26
862.74
41.87
820.87
3366.39
3
3366.39
862.74
33.66
829.08
2537.31
4
2537.31
862.74
25.37
837.37
1699.94
5
1699.94
862.74
17.00
845.74
854.20
6
854.20
862.74
8.54
854.20
0.00
Financial Maths – Extra Questions
c)
Assumptions:
 The company want the value at the time the bond is issued; i.e. 5 years from now.
 The current market rate of 5% will apply throughout the next 15 years; i.e. until the bond matures
 The €100 payments are made at the end of each year.
 The €2000 is repaid at the end of the 10 years along with the €100 due that year.
𝑎=𝑟=
𝑆10
1
= 0.952380952
1.05
1
1 10
) (1 − (
) )
1.05
1.05
=
= 7.721734929
1
1−(
)
1.05
(
Hint: Use your calculator to find r, then use the ANS button,
𝑆10 =
(𝐴𝑁𝑆)(1 − (𝐴𝑁𝑆)10 )
1 − (𝐴𝑁𝑆)
2000
= 1227.826507
1.0510
𝐹𝑎𝑖𝑟 𝑉𝑎𝑙𝑢𝑒 = 100𝑆10 + 1227.826507
= 100 × 7.721734929 + 1227.826507
= €2000
Note: We could have got this result very quickly since if €2000 is invested at 5% per annum, interest of €100 will be available to
be paid out at the end of each year and the capital of €2000 will be intact and can be paid out after 10 years.
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