FINM1001 Foundations of Finance
1
Financial Mathematics: Solutions to Extra Practice Questions
Note: In attempting these questions, particularly the more complex ones, it may prove
useful to draw a time-line to determine the magnitude and timing of cash flows before
undertaking any calculations.
Question One
Calculate the future value of $1,000 invested today for a period of 20 years at an
interest rate of 10% p.a. compounded daily. Show how and discuss why your answer
would change if interest was compounded quarterly.
The future value of this amount given the interest rate compounds annually is
$7,387.03, and is calculated as follows:
20 365
0.10
$1,000(1 )
365
$7,387.03
FV
=+
=
If the interest rate was compounded quarterly, the future value of the investment
would be lower given the decreased impact of compounding. The future value of the
investment with quarterly compounding is calculated as follows:
20 4
0.10
$1,000(1 )
4
$7,209.57
x
FV =+
=
Question Two
You made a deposit in a bank account exactly 18 months ago today. You have not
made any subsequent deposits, and the balance of your account is now $4,400.
Calculate the value of your initial deposit given you earned an interest rate of 15%
p.a. compounded semi-annually. Show how and discuss why your answer would
change if interest were compounded annually.
The value of the initial deposit is calculated as follows:
1.5 2
3
0.15
$4,400 (1 )
2
$4,400
(1.075)
$3,541.83
F
F
=+
=
=
If the interest rate were compounded annually, the initial deposit you made in the
bank would be greater as the decreased impact of compounding means you would
need to make a larger initial deposit to obtain $4,400 in 18 months’ time. The
required deposit with annual compounding is calculated as follows:
FINM1001 Foundations of Finance
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1.5
$4,400
(1.15)
$3,567.84
P=
=
Question Three
You have just successfully applied for a home loan. Calculate how much you are
borrowing given that the terms of the loan are as follows:
• Your monthly repayments are $1,000;
• The loan is taken over 25 years; and,
• The interest rate you will pay on funds borrowed is fixed at 10% p.a.
compounded quarterly.
Show how and discuss why your answer would change if interest were compounded
annually.
To work how much you borrowed, simply calculate the present value of all
repayments. As payments are evenly spaced and identical in amount, we can
calculate the present value of the cash flows using the ordinary annuity formula:
Note: Here the compounding frequency of the annual nominal rate does not match the
cash flow frequency. You must therefore first convert the annual nominal rate into an
annual effective rate, and then convert the annual effective rate into the monthly
periodic rate:
008264835.01)10381289.01(
10381289.01)
4
1.0
1(
12
1
4
=−+=
=−+=
p
e
r
r
If the interest rate were compounded annually, the amount you borrowed would be
higher as there is less compounding of interest and a larger portion of repayments
pertaining to the principal. The amount borrowed with annual compounding is
calculated as follows:
0079741.01)10.1( 12
1=−=
p
r
4
96.830,113$
00797414.0
)00797414.1(1
000,1$ 300 =
−
=
−
PV
FINM1001 Foundations of Finance
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Question Four
Calculate the present value of an ordinary perpetuity that comprises one cash flow of
$200 at the end of each year given an interest rate of 15% p.a. compounded annually.
Show how and discuss why your answer would change if interest were compounded
weekly.
$200
0.15
$1,333.33
PV =
=
If interest rates were compounded weekly, the present value of the perpetuity would
be lower due to the increased effects of compounding. The present value of the
perpetuity if interest were compounded weekly is:
52
$200
0.15
((1 ) 1)
52
$1,237.75
PV =
+−
=
Question Five
A man invests $500 at 15% p.a. compounded fortnightly and plans to hold this
investment for 10 years. Assuming there are exactly 26 fortnights in a year, how
much will he have at the end of his holding period?
The value of the man’s investment at the end of her holding period is calculated as:
10 26
0.15
$500(1 )
26
$2,231.21
FV
=+
=
Question Six
A business needs $20,000 in 2 years time to replace a piece of equipment. How much
must be invested now at an interest rate of 6% p.a. compounded monthly in order to
provide for this replacement?
To determine the amount the business must invest now, simply calculate the present
value of $20,000 paid in 2 years, bearing in mind that the interest rate is compounded
monthly.
12 2
$20,000
0.06
(1 )
12
$17,743.71
x
PV =
+
=
FINM1001 Foundations of Finance
Question Seven
A woman wants to provide a $10,000 university scholarship every year for 50 years.
The first scholarship is to be awarded one year from now. If the university can earn a
6% p.a. compounded daily as a return on their investments, how much should the
woman give now?
To determine the amount the woman must invest now, simply use the present value of
an ordinary annuity formula, bearing in mind that the interest rate is compounded
daily.
06183131.01
365
06.0
1
365
=−
+=r
=
−
=
−
06183131.0
)06183131.1(1
000,10$50
PV
$153,676.29
Question Eight
How much will $500 grow to if invested for 10 years at an interest rate of 12% p.a.