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Math 212 – Week 2 – Finance
Quick Reference Card
This Quick Reference Card (QRC) contains information about the concepts presented in Math 212 Week 2 (Chapter 3 of the
course textbook). For more information about any of these concepts, visit the Center for Math Excellence (CME). Even more
information about Math 212 Week 2 concepts can be found on the Web at sites such as www.JoleneMorris.com and
www.youtube.com. As soon as your Math 212 class begins, you will also have access to FREE tutoring 24-hours a day at the
CME. Ask for tutoring in any of the Week 2 concepts that are not clear to you.
The Simple Interest Formula
The simple interest formula is: 𝑰 = 𝑷𝒓𝒕 where I = interest,
P = principal, r = annual simple interest rate (written as a
decimal), and t = time in years.
EXAMPLE: Find the interest on a loan of $1,000 at 7.5% for 6
months.
𝐼 = 𝑃𝑟𝑡 = (1000)(0.075)(0.5) = $37.50
To find the present value of an investment or the total amount
due on a loan, use this formula (based on the simple interest
formula--the sum of the principal and the interest):
𝑨 = 𝑷 + 𝑷𝒓𝒕 = 𝑷(𝟏 + 𝒓𝒕)
EXAMPLE: Find the total amount due on a car loan of $9,500
at 4% simple interest at the end of 9 months.
𝐴 = 𝑃(1 + 𝑟𝑡)
= 9500(1 + [0.04][0.75])
= 9500(1 + 0.03) = 9500(1.03) = $97.85
Compound Interest
Compound interest is where interest is paid at the end of a
period (monthly, quarterly, etc.). Then the next time interest is
paid, it is paid on the principal and the interest earned. Thus,
interest is compounded each period. To find compound
𝒓 𝒎𝒕
interest, we use this formula: 𝑨 = 𝑷 �𝟏 + � where A =
𝒎
amount at the end of time, P = principal amount, r = interest
rate expressed as a decimal number, m is the number of
compounding periods per year, and t is the time expressed in
years.
EXAMPLE: Find the amount to which $1500 will grow if
compounded quarterly at 6.75% interest for 10 years.
𝑨 = 𝑷(𝟏 + 𝒊)𝒏 = 1500 �1 +
10(4)
0.0675
�
4
Growth and Time
= $2,929.50
How much should you invest now to have a given amount at a
future date? What annual rate of return have your investments
earned? How long will it take your investment to double in
value? The formulas for compound interest and continuous
compound interest can be used to answer such questions.
EXAMPLE: How much should you invest now at 10% to have
$8,000 toward the purchase of a car in 5 years if interest is
(a) compounded quarterly? (b) compounded continuously?
(𝑎) 𝐴 = 𝑃(1 + 𝑖)𝑛
(𝑏) 𝐴 = 𝑃𝑒 𝑟𝑡
8000
→ 𝑃=
= $4,882.17
(1 + 0.025)20
8000
→ 𝑃 = 0.10(5) = $4,852.25
𝑒
Simple Interest and Investments
To find the total value of an investment, use the same formula:
𝑨 = 𝑷(𝟏 + 𝒓𝒕)
EXAMPLE: Find the total amount due on a loan of $600 at 16%
interest at the end of 15 months.
𝐴 = 𝑃(1 + 𝑟𝑡) = 600�1 + 0.16(1.25)� = $720
To find the interest rate, use the above formula but solve it for r:
𝐴−𝑃
𝑟=
𝑃𝑡
EXAMPLE: What is the annual interest rate earned by a 33-day T-
bill with a maturity value of $1,000 that sells for $996.16?
NOTE: We normally use 360 days for a financial year
𝑟=
𝐴 − 𝑃 1000 − 996.16
=
= 0.042 = 4.2%
33
𝑃𝑡
(996.16) �
�
360
Continuous Compound Interest
As the number of compounding periods per year increases without
bound, the compounded amount approaches a limiting value. This
value is given by the following formula: 𝑨 = 𝑷𝒆𝒓𝒕 where r = annual
interest rate compounded continuously and t = number of years.
EXAMPLE: What amount will an account have after 10 years if
$1,500 is invested at an annual rate of 6.75% compounded
continuously?
A = 1500e(0.0675)(10) = $2,946.05
Caution: Do not use the approximation 2.7183 for e; it is not
accurate enough to compute the correct amount to the nearest
cent. Instead, use your calculator’s built-in 𝑒 𝑥 .
In order to solve for rate or time, you will need to take the natural
logarithm of both sides of the equation formula.
Annual Percentage Yield
The simple interest rate that will produce the same amount as a
given compound interest rate in 1 year is called the annual
percentage yield (APY). This is also called the effective rate
𝑨𝑷𝒀 = �𝟏 +
𝒓 𝒎
� −𝟏
𝒎
where r = rate of interest and m = number of periods in a year.
EXAMPLE: What is the annual percentage yield for money that is
invested at 6% compounded monthly?
𝐴𝑃𝑌 = �1 +
𝑟 𝑚
0.06 12
� − 1 = �1 +
� − 1 = 0.06168
𝑚
12
This Quick Reference Card was prepared by Jolene M. Morris ([email protected]) using the course textbook.
Future Value of an Annuity
An annuity is any sequence of equal periodic payments. The
future value of an annuity is found by using the following formula
𝑭𝑽 = 𝑷𝑴𝑻
(𝟏+𝒊)𝒏−𝟏
𝒊
where FV = future value, PMT = periodic
payment, i = interest rate per period (r/m), and n = total number of
periods (mt).
EXAMPLE: Suppose a $1000 payment is made at the end of each
quarter and the money in the account is compounded quarterly at
6.5% interest for 15 years. How much is in the account after the 15
year period?
�1 +
(1 + 𝑖)𝑛 − 1
𝐹𝑉 = 𝑃𝑀𝑇
= 1000 �
𝑖
0.065 4(15)
�
−1
4
� = 100,336.68
0.065
4
Approximating Interest Rates
Mr. Ray has deposited $150 per month into an annuity. After 14
years, the annuity is worth $85,000. What annual rate has this
annuity earned (compounded monthly) during the 14 year period?
Solution: Use the FV formula; substitute these values into the
formula; solution is approximated by graphing each side of the
equation separately and finding the point of intersection.
(1 + 𝑖)𝑛 − 1
(1 + 𝑡)14(12) − 1
𝐹𝑉 = 𝑃𝑀𝑇
→ 85,000 = 150 �
�
𝑖
𝑖
(1 + 𝑖)168 − 1
(1 + 𝑖)168 − 1
85,000
=�
� → 566.67 = �
�
150
𝑖
𝑖
Sinking Funds
Any account that is established for accumulating funds to meet
future obligations or debts is called a sinking fund. To derive the
sinking fund payment formula, we use algebraic techniques to
rewrite the formula for the future value of an annuity and solve for
the variable PMT:
𝒊
𝑷𝑴𝑻 = 𝑭𝑽 �
�
(𝟏 + 𝒊)𝒏 − 𝟏
EXAMPLE: How much must Harry save each month in order to buy
a new car for $12,000 in three years if the interest rate is 6%
compounded monthly?
𝑖
𝑃𝑀𝑇 = 𝐹𝑉 �
� = 12000 �
(1 + 𝑖)𝑛 − 1
0.06
12
� = $305.06
0.06 36
�1 +
� −1
12
Present Value of an Annuity
𝟏 − (𝟏 + 𝒊)−𝒏
𝑷𝑽 = 𝑷𝑴𝑻 �
�
𝒊
where PV = present value of all payments, PMT = periodic
payment, i = interest rate per period, and n = number of periods.
EXAMPLE: How much money must you deposit now at 6% interest
compounded quarterly in order to be able to withdraw $3,000 at
the end of each quarter year for two years?
1 − (1 + 𝑖)−𝑛
𝑃𝑉 = 𝑃𝑀𝑇 �
�
𝑖
1 − (1.015)−8
= 3000 �
�
0.015
= $22,457.78
The present value of all payments is $22,457.78. The total amount
of money withdrawn over two years is 3000(4)(2)=$24,000. Thus,
the accrued interest is the difference between the two amounts:
Monthly interest rate ≈1.253%. Annual interest rate ≈ 15%.
$24,000 – $22,457.78 =$1,542.22.
Amortization
Amortization Schedules
Problem: A bank loans a customer $50,000 at 4.5% interest per
year to purchase a house. The customer agrees to make monthly
payments for the next 15 years for a total of 180 payments. How
much should the monthly payment be?
If you borrow $500 that you agree to repay in six equal monthly
payments at 1% interest per month on the unpaid balance, how
much of each monthly payment is used for interest and how much
is used to reduce the unpaid balance?
Use the formula for present value of an annuity and solve for PMT:
𝑖
0.01
𝑃𝑀𝑇 = 𝑃𝑉 �
� = 500 �
� = $86.27
1 − (1 + 𝑖)−𝑛
1 − (1.01)−6
𝒊
1 − (1 + 𝑖)−𝑛
�
� → 𝑷𝑴𝑻 = 𝑷𝑽 �
𝑖
𝟏 − (𝟏 + 𝒊)−𝒏
0.045
12
= 50,000 �
� = $382.50
0.045 −180
1 − �1 +
�
12
If the customer makes a monthly payment of $382.50 to the bank
for 180 payments, then the total amount paid to the bank is the
product of $382.50 and 180 = $68,850. Thus, the interest earned
by the bank is the difference between $68,850 and $50,000
(original loan) = $18,850.
𝑃𝑉 = 𝑃𝑀𝑇 �
In reality, the last payment would be increased by $0.03, so that
the balance is zero.
General Problem-Solving Strategy
Step 1: Determine whether the problem involves a single payment or a sequence of equal periodic payments. Simple and compound
interest problems involve a single present value and a single future value. Ordinary annuities may be concerned with a present
value or a future value but always involve a sequence of equal periodic payments.
Step 2: If a single payment is involved, determine whether simple or compound interest is used. Often simple interest is used for
durations of a year or less and compound interest for longer periods.
Step 3: If a sequence of periodic payments is involved, determine whether the payments are being made into an account that is
increasing in value--a future value problem--or the payments are being made out of an account that is decreasing in value--a
present value problem. Remember that amortization problems always involve the present value of an ordinary annuity.