Chapter 2 – Time Value of Money
[ad_1]A – MULTIPLE CHOICE QUESTIONS
- What do we mean when we say that money has “time value”?
(a) A dollar tomorrow is worth more than a dollar today.
(b) Money is more valuable if you receive it at the time you need to spend it.
(c) Both (a) and (b)
(d) None of the above - If interest compounds more than once a year, which of the following will be greater?
(a) The APR
(b) The EAR
(c) The interest rate per compounding period
(d) Impossible to say without more information
La Trobe University 4
SECTION D – CALCULATION QUESTIONS - What is the future value of $100 invested for 5 years at an interest rate of 7%? *
- You win a $3000 lottery prize, and decide to put it away for your retirement. You invest it in a term deposit
paying 7% p.a., compounded semi‐annually? How much will be in the account after 40 years? * - What is the present value of $5000 to be received in 3 years if the interest rate is 11% p.a.? *
- You would like to buy a car in 5 years, which will cost you $15,000. How much do you need to deposit today
into an account paying an interest rate of 6% p.a., compounded monthly, in order to have enough money to
buy the car? *
La Trobe University 5 - You would like to buy a car in 5 years. The current price of the car is $15,000. You expect the price of the car
to increase at an annual inflation rate of 2.5%. How much do you need to deposit today into an account paying
an interest rate of 6% p.a., compounded daily, in order to have enough money to buy the car? * - What is the future value of $250 if it grows at a continuously compounded interest rate of 4% p.a. for 5 years?
- What is the present value of $470 to be received in 12 years, based on a continuously compounded interest
rate of 10% p.a.? - What is the present value of the series of
cash flows shown at right, if the interest
rate is 7.5% p.a., compounded annually? *
Year Cash flow
1 100
2 200
3 400
4 300
La Trobe University 6 - What is the present value of the series of
cash flows shown at right, if the interest
rate is 8% p.a., compounded quarterly? *
Year Cash flow
0 100
1 200
2 400
3 300 - What is the future value of the series of
cash flows shown at right, as at year 5,
if the interest rate is 9% p.a.,
compounded every three months? *
Year Cash flow
1 100
2 200
3 400
4 300 - What is the future value of the series of
cash flows shown at right, as at year 4,
if the interest rate is 9% p.a.,
compounded monthly? *
Year Cash flow
1 100
2 200
3 400
4 300 - What is the future value of $100 to be received every year in perpetuity, if the interest rate is 10% p.a.? *
La Trobe University 7 - What is the present value of $200 to be received every year in perpetuity, if the interest rate is 10% p.a.? *
- What is the present value of $100 to be received every six months in perpetuity, if the interest rate is 14% p.a.,
compounded semi‐annually? * - What is the present value of $500 to be received every 4 months in perpetuity, with the first payment to be
received immediately, if the interest rate is 6% p.a., compounded quarterly? * - What is the present value of $400 to be received every year in perpetuity, with the first payment to be received
in 4 years, if the interest rate is 8.75% p.a.? *
La Trobe University 8 - You decide to fund a perpetual scholarship at your local University. The prize will be a payment of $500 to
cover the cost of textbooks. You will set up the find on 1 July 2014 and the prize will be paid every year on 1
January and 1 July, beginning on 1 January 2015. If the interest rate is 9% p.a., compounding semi‐annually,
how much will be required to set up the scholarship? * - What is the present value of a growing perpetuity, where the first payment is $11,000 per year, the interest
rate is 6% p.a. and the growth rate is 2% p.a.? * - You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July. The first prize will be $500, paid on 1 January 2015.
The prize is designed to cover the cost of textbooks, which is expected to grow at the inflation rate of 3% p.a.,
and the prize will need to grow at the same rate after the first payment. If the interest rate is 9% p.a.,
compounding semi‐annually, how much will be required to set up the scholarship? * - You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July, beginning on 1 January 2015. The prize is designed
to cover the cost of textbooks, the cost of which is currently $500 (as at 1 July 2014) and which is expected to
grow at the inflation rate of 3% p.a. The prize will need to grow at the same rate. If the interest rate is 9%
p.a., compounding semi‐annually, how much will be required to set up the scholarship? *
La Trobe University 9
Hint, tips, advice and guidance
SECTION D – CALCULATION QUESTIONS - What is the future value of $100 invested for 5 years at an interest rate of 7%?
Interest rates are almost always quoted on a “per annum” basis. Even if the question doesn’t say “p.a.”, you
should assume that it is per annum unless specifically told otherwise. Also, if the question doesn’t say how
often interest is compounded, you should assume that it is compounded annually. - You win a $3000 lottery prize, and decide to put it away for your retirement. You invest it in a term deposit
paying 7% p.a., compounded semi‐annually? How much will be in the account after 40 years?
This requires the future value of a single sum, but don’t forget to adjust for compounding more than once a
year. - What is the present value of $5000 to be received in 3 years if the interest rate is 11% p.a.?
This requires the present value of a single sum. - You would like to buy a car in 5 years, which will cost you $15,000. How much do you need to deposit today
into an account paying an interest rate of 6% p.a., compounded monthly, in order to have enough money to
buy the car?
This requires the future value of a single sum, but the interest rate per period and the number of periods need
to be calculated, based on monthly compounding. - You would like to buy a car in 5 years. The current price of the car is $15,000. You expect the price of the car
to increase at an annual inflation rate of 2.5%. How much do you need to deposit today into an account paying
an interest rate of 6% p.a., compounded daily, in order to have enough money to buy the car?
This question requires two steps. First, you need to calculate the value of the car in five years. From a financial
mathematics perspective, inflation works just like an interest rate, and we can use the formula for the future
value of a single sum to determine how much an amount of money will grow to as a result of an inflation rate
of 2.5% p.a. That becomes the future value, and therefore the second step is to use the formula for the present
value of a single sum to calculate the present value of that future value, at an interest rate of 6% p.a.
compounded daily. Because of daily compounding, the interest rate per period and the number of periods need
to be calculated. - What is the present value of the following series of cash flows, if the interest rate is 7.5% p.a., compounded
annually?
Year Cash flow Each cash flow needs to be discounted to a
present value at year 0, and then the four present
values need to be added together.
1 100
2 200
3 400
4 300
La Trobe University 10 - What is the present value of the following series of cash flows, if the interest rate is 8% p.a., compounded
quarterly?
Year Cash flow Each cash flow needs to be discounted to a present
value at year 0, and then the four present values need
to be added together. Look at the year each cash flow
is received to determine how many periods it needs
to be discounted by to get to year 0.
0 100
1 200
2 400
3 300 - What is the future value of the following series of cash flows, as at year 5, if the interest rate is 9% p.a.,
compounded every three months?
Year Cash flow Each cash flow needs to be compounded to a future
value at year 5, and then the four present values need
to be added together. Look at the year each cash flow
is received to determine how many periods it needs
to be grow to get to year 5.
1 100
2 200
3 400
4 300 - What is the future value of the following series of cash flows, as at year 4, if the interest rate is 9% p.a.,
compounded monthly?
Year Cash flow Each cash flow needs to be compounded to a future
value at year 4, and then the four present values need
to be added together. Look at the year each cash flow
is received to determine how many periods it needs
to be grow to get to year 4.
1 100
2 200
3 400
4 300 - What is the future value of $100 to be received every year in perpetuity, if the interest rate is 10% p.a.?
Think carefully about the nature of a perpetuity. This is more a test of your understanding of theory than your
mathematical ability. - What is the present value of $200 to be received every year in perpetuity, if the interest rate is 10% p.a.?
This requires the present value of a perpetuity. - What is the present value of $100 to be received every six months in perpetuity, if the interest rate is 14% p.a.,
compounded semi‐annually?
This requires the present value of a perpetuity, but the interest rate per period needs to be calculated before
you can use the formula. - What is the present value of $500 to be received every 4 months in perpetuity, with the first payment to be
received immediately, if the interest rate is 6% p.a., compounded quarterly?
All payments after the first payment constitute a perpetuity, so you should calculate the present value of that
perpetuity and then add that to the present value of the first payment.
La Trobe University 11 - What is the present value of $400 to be received every year in perpetuity, with the first payment to be received
in 4 years, if the interest rate is 8.75% p.a.?
1 3 400 1.0875 3 $3554.38
0.0875
PV C r
r
This is what is called a deferred perpetuity. You should use the formula for the present value of a perpetuity,
using the information given in the question, but bear in mind that the answer you get applies to the beginning
of the perpetuity period, which is one period before the first payment. If the perpetuity is not deferred, the
beginning of the perpetuity is today, and the first occurs one period in the future. Look at the question, work
out how many periods the perpetuity is deferred, and then treat the present value of the perpetuity as a single
sum occurring that many periods in the future. Discount it to a present value at the interest rate given in the
question.
Any time there is anything slightly unusual happening with the timing of cash flows, a time line is strongly
recommended. - You decide to fund a perpetual scholarship at your local University. The prize will be a payment of $500 to
cover the cost of textbooks. You will set up the find on 1 July 2014 and the prize will be paid every year on 1
January and 1 July, beginning on 1 January 2015. If the interest rate is 9% p.a., compounding semi‐annually,
how much will be required to set up the scholarship?
This requires the present value of a perpetuity. Time 0 – the beginning of the perpetuity period – is 1 July 2014,
and the first payment is on 1 January 2015. Since each period is six months, this is a normal perpetuity where
the first payment occurs one period in the future. - What is the present value of a growing perpetuity, where the first payment is $11,000 per year, the interest
rate is 6% p.a. and the growth rate is 2% p.a.?
This requires the present value of a growing perpetuity. - You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July. The first prize will be $500, paid on 1 January 2015.
The prize is designed to cover the cost of textbooks, which is expected to grow at the inflation rate of 3% p.a.,
and the prize will need to grow at the same rate after the first payment. If the interest rate is 9% p.a.,
compounding semi‐annually, how much will be required to set up the scholarship?
This requires the present value of a growing perpetuity. Time 0 – the beginning of the perpetuity period – is 1
July 2014, and the first payment is on 1 January 2015. Since each period is six months, this is a normal growing
perpetuity where the first payment occurs one period in the future.
Note that if the cost of textbooks grows at 3% p.a., it grows at 1.5% every six months, so this is your growth
rate in the formula. - You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July, beginning on 1 January 2015. The prize is designed
to cover the cost of textbooks, the cost of which is currently $500 (as at 1 July 2014) and which is expected to
grow at the inflation rate of 3% p.a. The prize will need to grow at the same rate. If the interest rate is 9%
p.a., compounding semi‐annually, how much will be required to set up the scholarship?
This similar to the previous question, except that the formula for the present value of a growing perpetuity
requires the first payment in the numerator, and we are not given the first payment. We are given the cost of
textbooks at the beginning of the perpetuity period (1 July 2014) and we are told that this cost will grow at the
inflation rate (3% p.a. or 1.5% every six months). You need to calculate the first payment, six months in the
future, based on today’s cost and the growth rate, before you can use the formula.
La Trobe University 12
Solutions
SECTION A – MULTIPLE CHOICE QUESTIONS - What do we mean when we say that money has “time value”?
(a) A dollar tomorrow is worth more than a dollar today.
(b) Money is more valuable if you receive it at the time you need to spend it.
(c) Both (a) and (b)
(d) None of the above - If interest compounds more than once a year, which of the following will be greater?
(a) The APR
(b) The EAR
(c) The interest rate per compounding period
(d) Impossible to say without more information
La Trobe University 13
SECTION D – CALCULATION QUESTIONS - What is the future value of $100 invested for 5 years at an interest rate of 7%?
Note: Interest rates are almost always quoted on a “per annum” basis. Even if the question doesn’t say “p.a.”,
you should assume that it is per annum unless specifically told otherwise. Also, if the question doesn’t say how
often interest is compounded, you should assume that it is compounded annually.
5 1 100 1.07 $140.26 n
FVn C r
Note: Dollar values should always be shown to the nearest cent, unless the answer is very large (say, over $1
million). - You win a $3000 lottery prize, and decide to put it away for your retirement. You invest it in a term deposit
paying 7% p.a., compounded semi‐annually? How much will be in the account after 40 years?
80 1 3000 1.035 $47,027.21 n
n FV C r - What is the present value of $5000 to be received in 3 years if the interest rate is 11% p.a.?
3
5000 $3,655.96
1 1.11 n
PV C
r
- You would like to buy a car in 5 years, which will cost you $15,000. How much do you need to deposit today
into an account paying an interest rate of 6% p.a., compounded monthly, in order to have enough money to
buy the car?
60
15,000 $11,120.58
1 1.005 n
PV C
r
- You would like to buy a car in 5 years. The current price of the car is $15,000. You expect the price of the car
to increase at an annual inflation rate of 2.5%. How much do you need to deposit today into an account paying
an interest rate of 6% p.a., compounded daily, in order to have enough money to buy the car?
5 1 15,000 1.025 $16,971.12 n
n FV C π
Note: From a financial mathematics perspective, inflation works just like an interest rate, and we can use the
future value formula to determine how much an amount of money will grow to as a result of inflation. The
above formula uses pi (π) to represent inflation.
5 365
16,971.12 $12,572.82
1 1 0.06
365
n
PV C
r
Note: Be careful of rounding errors. If you calculate the interest rate as 0.06/365, and write it down to 4
decimal places (0.0002), you will get $11,781.72, which is so inaccurate it will be marked as incorrect. If you
round the interest rate to 0.00016, you will get $12,673.79. This is too much of a rounding error and you will
only partial credit. If you round the interest rate to 0.000164, you will get $12,581.63, which is close enough
(rounding error of less than $9 or less than 0.1%). However, it might make it hard to pick the correct answer
from a list of multiple choice answers. The bottom line – don’t round your intermediate answers. Keep them
La Trobe University 14
in the calculator. Use memories or brackets (see the Topic 2 Lecture Slides Appendices for further advice on
avoiding rounding errors). Set up the calculation as shown above and calculate it without writing anything
down until you get to the final answer. - What is the future value of $250 if it grows at a continuously compounded interest rate of 4% p.a. for 5 years?
FV CeAPRY 250e0.045 $305.35
Note: See Slide 34 of the Topic 2 Lecture Slides for tips on how to perform this calculation. - What is the present value of $470 to be received in 12 years, based on a continuously compounded interest
rate of 10% p.a.?
0.10 12
470 $141.56 APR Y
PV C
e e - What is the present value of the following series of cash flows, if the interest rate is 7.5% p.a., compounded
annually?
Year Cash flow
2 3 4
100 200 400 300 $812.71
1.075 1.075 1.075 1.075
1 100 PV
2 200
3 400
4 300 - What is the present value of the following series of cash flows, if the interest rate is 8% p.a., compounded
quarterly?
Year Cash flow
4 8 12
100 200 400 300 $862.71
1.02 1.02 1.02
0 100 PV
1 200
2 400
3 300 - What is the future value of the following series of cash flows, as at year 5, if the interest rate is 9% p.a.,
compounded every three months?
Year Cash flow
16 12
8 4
100 1.0225 200 1.0225
400 1.0225 300 1.0225 $1209.83
FV
1 100
2 200
3 400
4 300 - What is the future value of the following series of cash flows, as at year 4, if the interest rate is 9% p.a.,
compounded monthly?
Year Cash flow
36 24
12
100 1.0075 200 1.0075
400 1.0075 300 $1107.67
FV
1 100
2 200
3 400
4 300
La Trobe University 15 - What is the future value of $100 to be received every year in perpetuity, if the interest rate is 10% p.a.?
This is a trick question. This is impossible to answer. The length of time it took you to realise this fact is a test
of your understanding of the mathematics of perpetuities. - What is the present value of $200 to be received every year in perpetuity, if the interest rate is 10% p.a.?
200 $2000
0.10
PV C
r
- What is the present value of $100 to be received every six months in perpetuity, if the interest rate is 14% p.a.,
compounded semi‐annually?
100 $1428.57
0.07
PV C
r
- What is the present value of $500 to be received every 4 months in perpetuity, with the first payment to be
received immediately, if the interest rate is 6% p.a., compounded quarterly?
500 500 $33,833.33
0.015
PV C C
r
Note: All payments after the first payment constitute a perpetuity. Since the first payment is already in present
value terms, it can be added to the present value of the perpetuity. - What is the present value of $400 to be received every year in perpetuity, with the first payment to be received
in 4 years, if the interest rate is 8.75% p.a.?
1 3 400 1.0875 3 $3554.38
0.0875
PV C r
r
Note: If the first payment occurs in 4 years, instead of one year, the perpetuity is deferred by 3 years. The
present value of the perpetuity (the first term in the above solution) applies to year 3 – one period before the
first payment. It therefore needs to be discounted by 3 periods to find the present value, which is done via the
second term in the above solution.
Any time there is anything slightly unusual happening with the timing of cash flows, a time line is strongly
recommended. - You decide to fund a perpetual scholarship at your local University. The prize will be a payment of $500 to
cover the cost of textbooks. You will set up the find on 1 July 2014 and the prize will be paid every year on 1
January and 1 July, beginning on 1 January 2015. If the interest rate is 9% p.a., compounding semi‐annually,
how much will be required to set up the scholarship?
500 $11,111.11
0.045
PV C
r
- What is the present value of a growing perpetuity, where the first payment is $11,000 per year, the interest
rate is 6% p.a. and the growth rate is 2% p.a.?
1 11,000 $275,000
0.06 0.02
PV C
r g
La Trobe University 16 - You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July. The first prize will be $500, paid on 1 January 2015.
The prize is designed to cover the cost of textbooks, which is expected to grow at the inflation rate of 3% p.a.,
and the prize will need to grow at the same rate after the first payment. If the interest rate is 9% p.a.,
compounding semi‐annually, how much will be required to set up the scholarship?
1 500 $16,666.67
0.045 0.015
PV C
r g
- You decide to fund a perpetual scholarship at your local University. You will set up the find on 1 July 2014 and
the prize will be paid every year on 1 January and 1 July, beginning on 1 January 2015. The prize is designed
to cover the cost of textbooks, the cost of which is currently $500 (as at 1 July 2014) and which is expected to
grow at the inflation rate of 3% p.a. The prize will need to grow at the same rate. If the interest rate is 9%
p.a., compounding semi‐annually, how much will be required to set up the scholarship?
1 0 1 500 1.015
$16,916.67
0.045 0.015
C C g PV
r g r g
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