Chapter 9 | Compound Interest
22. Thomas, an architect, is expected to make a first payment of $4000 in one year and a second payment in four
years to a designer. Calculate his second payment if both these payments are economically equivalent to a single
payment of $8000 today. Assume money can earn 6% compounded monthly.
23. Calculate the two equal installments one at the end of one year and the other two years from now, that would replace a
payment of $2000 today and a payment of $8500 in five years. Assume money is worth 4% compounded semi-annually.
24. What equal payments, one in one year and the other in five years, would replace payments of $8000 and $10,000
in four and six years, respectively? Assume money can earn 8% compounded annually.
25. Niranjani was offered to settle her college tuition fee with two equal payments of $2500 in three months and nine
months. She wanted to reschedule her payments with a payment of $1500 now and the balance in six months.
Calculate the payment required in six months to settle the loan. Assume that money earns 3% compounded monthly.
26. Randy was offered the following equivalent payment options to settle his car loan:
Option a.$7500 in 9 months and 10,000 in 18 months.
Option b.Two equal payments: one in 6 months and another in 1 year.
Calculate the value of the equal payments under Option b. assuming that money earns 4.2% compounded quarterly.
9.5 |
Calculating Periodic Interest Rate (i)
and Nominal Interest Rate ( j)
We can calculate the periodic interest rate 'i' by rearranging Formula 9.1(a),
FV = PV(1 + i)n FV
(1 + i)n = PV
Dividing by 'PV' on both sides, we obtain,
Taking the nth root on both sides, we obtain,
1
(1 + i) = ` FV j n
PV
Rearranging to isolate 'i' results in the following formula for periodic interest rate:
Formula 9.5(a)
Periodic Interest Rate
i=
-1
We can calculate the nominal interest rate 'j' by rearranging Formula 9.1(d): i =
j
m
Rearranging to isolate 'j' results in the following formula for nominal interest rate:
Formula 9.5(b)
Nominal Interest Rate
j=m#i
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Chapter 9 | Compound Interest
Example 9.5(a)
Calculating the Nominal Interest Rate When 'FV' and 'PV' are Known
At what rate compounded monthly will an investment of $4500 grow
to $10,000 in four years?
Solution
Do not round the
value of i while
calculating j as it
is an intermediate
step. Use the store
and recall function
in your calculator.
Example 9.5(b)
FV
1
n
1
10, 000 48
m
- 1 = 0.016774...
4500
i = ` PV j
j = m # i = 12 # 0.016774... = 0.201296... = 20.13%
-1=c
Therefore, the investment will grow to $10,000 in four years at an interest
rate of 20.13% compounded monthly.
Calculating the Nominal Interest Rate When 'PV' and 'I' are Known
The interest on a three-year GIC is $8269.17. If the GIC was purchased for $180,000, what is the
nominal interest rate of the GIC if interest is compounded quarterly?
Solution
Note that FV is not directly provided, but the amount of interest is provided.
We know that FV = PV + I
FV = 180,000 + 8269.17
= $188,269.17
FV
i = ` PV j
1
n
-1
1
= ` 188,269.17 j`12j - 1
180,000.00
i = 0.003750...
j = m # i = 4 # 0.003750... = 0.015000... = 1.5%
Therefore, the nominal interest rate of the GIC is 1.5% compounded quarterly.
Example 9.5(c)
Calculating the Nominal Interest Rate When an Investment Doubles
At what nominal interest rate compounded semi-annually will an investment
double in 12 years?
Solution
Assume PV = $1000. If the investment doubles, FV = $2000.
When an investment
doubles,
FV = 2
PV
1
i=`
1
1
FV n
j - 1 = ` 2000 j 24 - 1 = 0.029302...
PV
1000
j = m # i = 2 # 0.029302... = 0.058604... = 5.86%
Therefore, the investment will double at 5.86% compounded semi-annually.
Chapter 9 | Compound Interest
9.5 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
For the following problems, express the answers rounded to two decimal places, wherever applicable.
Calculate the missing values for Problems 1 and 2:
1.
Principal
Maturity Value
a.
$7000
$7975
2 years 9 months
?% compounded quarterly
b.
$2500
$2850
3 years 5 months
?% compounded monthly
c.
$3250
$3925
4 years 6 months
?% compounded semi-annually
d.
$5000
$5650
3 years
?% compounded daily
Principal
Maturity Value
Term
a.
$2000
$5000
5 years 2 months
?% compounded monthly
b.
$5250
$6000
3 years 6 months
?% compounded semi-annually
c.
$7500
$8600
2 years 3 months
?% compounded quarterly
d.
$4500
$5550
5 years
?% compounded daily
2.
Term
Nominal Interest Rate
Nominal Interest Rate
3. Calculate the nominal interest rate and periodic interest rate for an investment of $100,000 that matures to
$200,000 in 10 years if interest is compounded quarterly.
4. Canary Calendars Inc. invested this year's profits of $64,000 in a fund that matures to $84,500 in two years.
Calculate the nominal interest rate compounded quarterly and the periodic interest rate of the fund.
5. A calculator distributor invested its net income of $35,000 in a mutual fund. Calculate the nominal interest rate
compounded semi-annually if the accumulated value in 2 years and 7 months is $44,650.
6. Rose received an excellent interest rate for her car loan of $11,000. Calculate the nominal interest rate and periodic
interest rate of her loan if it accumulated to $11,170 in five months and interest is compounded quarterly.
7. A computer assembling company took a loan of $30,000 to purchase a conveyer belt. If the debt accumulated to
$45,850 in two years, calculate the nominal interest rate compounded quarterly.
8. If you have $250,000 in your savings account and wish to grow it to a million dollars in 40 years, calculate the
nominal interest rate compounded semi-annually.
9. Mark heard that he could triple his money in 15 years if he invested it in his friend's telecommunications business.
What nominal interest rate compounded monthly does the business offer?
10. Anish wants to double his money in 15 years in a low-risk savings account. What nominal interest rate compounded
monthly do you suggest that he look for? What nominal interest rate compounded monthly would allow him to
double his money in 10 years?
11. The nominal interest rate on a car loan of $8000 that was compounded semi-annually changed at the end of one
year. If the accumulated balance was $8656 at the end of the first year and $9100.50 at the end of the second year,
calculate the nominal interest rates for each year.
12. An investment of $5000 in a TFSA (Tax Free Savings Account) accumulated to $5280 at the end of one year at a
monthly compounding interest rate. However, the monthly compounding interest rate for the second year changed
and the balance in the account at the end of the second year was $5875. Calculate the nominal interest rate for the
first year and the nominal interest rate for the second year.
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Chapter 9 | Compound Interest
13. If an investment grew to $12,000 in two years and the interest amount earned was $800, calculate the nominal
interest rate compounded monthly.
14. Amex Industries Ltd. invested $30,000 in a mutual fund that earned an interest amount of $12,000 in 14 years.
Calculate the nominal interest rate compounded quarterly for the fund.
9.6 |
Calculating Number of Compounding
Periods (n) and Time Period (t)
15.
We can calculate the number of compounding periods 'n' from Formula 9.1(a) as follows:
Refer to
Table 2.4 in
Chapter 2
for Rules of
Logarithms.
Formula 9.6(a)
FV = PV(1 + i)n
Dividing by 'PV' on both sides, we obtain,
FV
n
PV = (1 + i) Taking the natural logarithm on both sides, we obtain,
ln ` FV j = ln(1 + i)n
PV
Using the laws of logarithms, we obtain, (log (A)n = nlnA)
ln ` FV j = n # ln(1+i)
PV
Rearranging to isolate 'n' results in the following formula for calculating the number of compounding periods:
Number of Compounding Periods
n=
We can calculate term 't' in years by rearranging Formula 9.1(e), n = m # t
Rearranging to isolate 't' results in the following formula for calculating the time period in years:
Formula 9.6(b)
Time Period in Years
t=
Note: In determining the time period, 't', in years and months, the value of 'n' is not to be rounded to
the nearest (or rounded up to the next) compounding period.
Example 9.6(a)
Calculating the Time Period When 'PV' and 'FV' are Known
How long (in years and months) will it take for an investment of $30,000
in a mutual fund to mature to $100,025 if it is growing at the rate of 12%
compounded semi-annually?
Solution
ln c FV m ln c 100,025 m
PV
30, 000 = 20.666643... compounding periods
n=
=
ln (1 + i) ln ^1 + 0.06h
t = n = 20.666643... = 10.333321... years
2
m