The Mathematics of Finance
Simple Interest
When you deposit money in a bank—for example, in a savings account—you are permitting
the bank to use your money. The bank may lend the deposited money to customers to buy
cars or make renovations on their homes. The bank pays you for the privilege of using your
money. The amount paid to you is called interest.
If you are the one borrowing money from a bank, the amount you pay for the privilege of
using that money is also called interest. The amount deposited in a bank or borrowed from
a bank is called the principal. The amount of interest paid is usually given as a percent of
the principal. The percent used to determine the amount of interest is called the interest
rate. If you deposit $1000 in a savings account paying 5% interest, $1000 is the principal
and the interest rate is 5%. Interest paid on the original principal is called simple interest.
The formula used to calculate simple interest is given below.
The simple interest formula is
I =Prt
where I is the interest, P is the principal, r is the interest rate, and t is the time period.
In the simple interest formula, the time t is expressed in the same period as the rate. For
example, if the rate is given as an annual interest rate, then the time is measured in years; if
the rate is given as a monthly interest rate, then the time must be expressed in months.
Interest rates are most commonly expressed as annual interest rates. Therefore, unless
stated otherwise, we will assume the interest rate is an annual interest rate. Interest rates
are generally given as percents. Before performing calculations involving an interest rate,
write the interest rate as a decimal.
Example:
1. Calculate the simple interest earned in 1 year on a deposit of $1000 if the interest
rate is 5%.
Answer: I =Prt
I =(1000)(0.05)(1)
I=$50
2. Calculate the simple interest due on a three-month loan of $2000 if the interest rate
is 6.5%.
Answer: I =Prt (3 months should be converted to years since the interest is annual)
3
𝐼 = (2000)(0.065) (12)
I=$32.5
3. Calculate the simple interest due on a two-month loan of $500 if the interest rate is
1.5% per month.
Answer: I =Prt (2 months will not be converted to years since the interest is monthly)
𝐼 = (500)(0.015)(2)
I=$15
4. The simple interest charged on a six-month loan of $3000 is $150. Find the simple
interest rate.
Answer: Manipulate the equation since we are looking for the rate.
𝐼
𝑟=
𝑃𝑡
𝑟=
150
6
3000 ( )
12
= 0.1
To convert it to % just multiply 100
150
𝑟=
= 0.1 ∗ 100 = 10%
6
3000 (12)
If the time period of a loan with an annual interest rate is given in days, it is necessary to
convert the time period of the loan to a fractional part of a year. There are two methods for
converting time from days to years: the exact method and the ordinary method. Using the
exact method, the number of days of the loan is divided by 365, the number of days in a
year.
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠
𝐸𝑥𝑎𝑐𝑡 𝑚𝑒𝑡ℎ𝑜𝑑: 𝑡 =
365
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓𝑑𝑎𝑦𝑠
360
The ordinary method is used by most businesses. Therefore, unless otherwise stated, the
ordinary method will be used in this text.
𝑂𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝑚𝑒𝑡ℎ𝑜𝑑:
𝑡=
Example:
1. Calculate the simple interest due on a 45-day loan of $3500 if the annual interest
rate is 8%. Use two method and compare.
Answer (exact method): I =Prt
45
𝐼 = (3500)(0.08)(365)
I=$34.52
Answer (ordinary method): I =Prt
45
𝐼 = (3500)(0.08)(360)
I=$35
It could be seen that ordinary method gives much greater Interest than exact method. This
is the main reason why most businesses used this method.
2. Calculate the simple interest due on a 120-day loan of $7000 if the annual interest
rate is 5.25%.
Answer (ordinary method): I =Prt
120
𝐼 = (7000)(0.0525)(360)
I=$122.50
3. Calculate the simple interest due on a $5000 loan made on September 20 and repaid
on December 9 of the same year. The interest rate is 6%.
Answer (ordinary method): I =Prt
Count how many days are there between Sept 20-Dec 9
80
𝐼 = (5000)(0.06)(360)
I=$66.67
Future Value and Maturity Value
When you borrow money, the total amount to be repaid to the lender is the sum of the
principal and interest. This sum is calculated using the following future value or maturity
value formula for simple interest. The future or maturity value formula for simple interest
is
𝐴= 𝑃+𝐼
where A is the amount after the interest, I, has been added to the principal, P.
This formula can be used for loans or investments. When used for a loan, A is the total
amount to be repaid to the lender; this sum is called the maturity value of the loan. In
Example 5, the simple interest charged on the loan of $3000 was $150. The maturity value
of the loan is therefore $3000 + $150 =$3150.
For an investment, such as a deposit in a bank savings account, A is the total amount on
deposit after the interest earned has been added to the principal. This sum is called the
future value of the investment.
Examples:
1. Calculate the maturity value of a simple interest, eight-month loan of $8000 if the
interest rate is 9.75%.
Answer: (substitute the formula for I)
𝐴=𝑃+𝐼
𝐴 = 𝑃 + 𝑃𝑟𝑡
𝐴 = 𝑃(1 + 𝑟𝑡)
8
𝐴 = 8000[1 + (0.0975) ( )]
12
𝐴 =$8520
2. Calculate the maturity value of a simple interest, three-month loan of $3800. The
interest rate is 6%.
Answer:
𝐴 = 𝑃(1 + 𝑟𝑡)
3
𝐴 = 3800[1 + (0.06) ( )]
12
𝐴 =$3857
3. The maturity value of a three-month loan of $4000 is $4085.What is the simple
interest rate?
Answer: (manipulate the equation since we are looking for r)
𝐴= 𝑃+𝐼
𝐴 = 𝑃 + 𝑃𝑟𝑡
𝐴−𝑃
𝑟=
𝑃𝑡
4085 − 4000
𝑟=
3
(4000) ( )
12
𝑟 = 0.085 𝑜𝑟 8.5%
Compound Interest
Simple interest is generally used for loans of 1 year or less. For loans of more than 1 year,
the interest paid on the money borrowed is called compound interest. Compound interest is
interest calculated not only on the original principal, but also on any interest that has
already been earned.
To illustrate compound interest, suppose you deposit $1000 in a savings account earning
5% interest, compounded annually (once a year).
Note that the interest earned during the second year ($52.50) is greater than the interest
earned during the first year ($50). This is because the interest earned during the first year
was added to the original principal and the interest for the second year was calculated
using this sum. If the account earned simple interest rather than compound interest, the
interest earned each year would be the same ($50).
At the end of the second year, the total amount in the account is the sum of the amount in the account at
the end of the first year and the interest earned during the second year that is $1102.50.
The interest earned each year keeps increasing. This is the effect of compound interest.
In deriving this equation, interest was compounded annually; therefore, t=1 Applying a
similar argument for more frequent compounding periods, we derive the following
compound amount formula. This formula enables us to calculate the compound amount for
any number of compounding periods per year.
In this example, the interest is compounded annually. However, compound interest can be
compounded semiannually (twice a year), quarterly (four times a year), monthly, or daily.
The frequency with which the interest is compounded is called the compounding period.
Example:
1. You deposit $500 in an account earning 6% interest, compounded semiannually.
How much is in the account at the end of 1 year?
Answer: (substitute the formula for 𝑖 and 𝑁)
𝐴 = 𝑃 (1 + 𝑖 )𝑁
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
0.06 2(1)
)
𝐴 = 500 (1 +
2
𝐴 = $530.45
We use 2 for the value of n because it was stated in the problem that it is
semiannually.
2. How much interest is earned in 2 years on $4000 deposited in an account paying 6%
interest, compounded quarterly?
Answer: (we compute first for the amount after 2 years and then we subtract the
principal from it to get how much is the interest)
𝑟 𝑛𝑡
𝐴 = 𝑃 (1 + )
𝑛
0.06 4(2)
)
𝐴 = 4000 (1 +
4
𝐴 = $4505.97
This is the total amount after 2 years, so to get the interest we subtract that the principal
from it.
𝐼 = $4505.97 − $4000
𝐼 = $505.97
Present Value
The present value of an investment is the original principal invested, or the value of
the investment before it earns any interest. Therefore, it is the principal, P, in the
compound amount formula. Present value is used to determine how much money must be
invested today in order for an investment to have a specific value at a future date.
Example:
1. How much money should be invested in an account that earns 8% interest,
compounded quarterly, in order to have $30,000 in 5 years?
Answer: (substitute the formula for 𝑖 and 𝑁)
𝐴
𝑃=
(1 + 𝑖 )𝑁
𝐴
𝑃=
𝑟 𝑛𝑡
(1 + 𝑛)
30,000
𝑃=
0.08 4(5)
(1 + 4 )
𝑃 = $20,189.14
2. How much money should be invested in an account that earns 9% interest,
compounded semiannually, in order to have $20,000 in 5 years?
𝐴
𝑃=
𝑟 𝑛𝑡
(1 + 𝑛)
20,000
𝑃=
0.09 2(5)
(1 + 2 )
𝑃 = $12,878.55
Credit Cards and Consumer Loans
Credit Cards
When a customer uses a credit card to make a purchase, the customer is actually receiving
a loan. Therefore, there is frequently an added cost to the consumer who purchases on
credit. This added cost may be in the form of an annual fee or interest charges on
purchases. A finance charge is an amount paid in excess of the cash price; it is the cost to
the customer for the use of credit.
Most credit card companies issue monthly bills. The due date on the bill is usually 1 month
after the billing date (the date the bill is prepared and sent to the customer). If the bill is
paid in full by the due date, the customer pays no finance charge. If the bill is not paid in full
by the due date, a finance charge is added to the next bill.
If the bill is paid in full before due date, no finance charge is added. However, if the bill is
not paid in full, interest charges on the outstanding balance will start to accrue (be added),
and any purchase made after due date will immediately start accruing interest.
The most common method of determining finance charges is the average daily balance
method. Interest charges are based on the credit card’s average daily balance, which is
calculated by dividing the sum of the total amounts owed each day of the month by the
number of days in the billing period.
𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡𝑠 𝑜𝑤𝑒𝑑 𝑒𝑎𝑐ℎ 𝑑𝑎𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑜𝑛𝑡ℎ
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑖𝑙𝑙𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑
An example of calculating the average daily balance follows.
Suppose an unpaid bill for $315 had a due date of April 10. A purchase of $28 was
made on April 12, and $123 was charged on April 24. A payment of $50 was made on April
15. The next billing date is May 10. The interest on the average daily balance is 1.5% per
month. Find the finance charge on the May 10 bill.
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑎𝑖𝑙𝑦 𝑏𝑎𝑙𝑎𝑛𝑐𝑒 =
To find the finance charge, first prepare a table showing the unpaid balance for each
purchase, the number of days the balance is owed, and the product of these numbers. A
negative sign in the Payments or Purchases column of the table indicates that a payment
was made on that date.
Suppose an unpaid bill for $315 had a due date of April 10. A purchase of $28 was
made on April 12, and $123 was charged on April 24. A payment of $50 was made on April
15. The next billing date is May 10. The interest on the average daily balance is 1.5% per
month. Find the finance charge on the May 10 bill.
Example:
1. An unpaid bill for $620 had a due date of March 10. A purchase of $214 was made
on March 15, and $67 was charged on March 30. A payment of $200 was made on
March 22. The interest on the average daily balance is 2.5% per month. Find the
finance charge on the April 10 bill.
Answer: (make sure sunod-sunod ang date, start on March 10 and end it on April 9, kapag
purchase positive meaning madadagdag sa balance each day, while negative kapag
payment kasi mababawas sa balance dahil nagbayad ka)
Date
Payment/Puchase Balance Each No of Days Until Unpaid Balance
Day
Balance
x (times)
Changes
Number of Days
(count ilang days sa
March 10-14
$620
(620)(5)$3100
mar 10-14)5
March 15-21
$214
(620+214) $834
7
(834)(7)$5838
March 22-29
-$200
(834-200) $634
8
(634)(8)$5072
March 30-April 9
$67
(634+67)
$701
11
(701)(11)$7711
Total kasi 31 days
Total
$21,721
sa March
$21,721
= $700.68
31
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝐹𝑖𝑛𝑎𝑛𝑐𝑒 𝐶ℎ𝑎𝑟𝑔𝑒: 𝐼 = 𝑃𝑟𝑡
𝐼 = $700.68(0.025)(1)
𝐼 = $17.52
The finance charge is approximately $17.52.
Note: r=2.5% and t=1 since it is for just 1 month and the rate is stated in the problem as
monthly.
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝐷𝑎𝑖𝑙𝑦 𝐵𝑎𝑙𝑎𝑛𝑐𝑒 =
Annual Percentage Rate
Federal law, in the form of the Truth in Lending Act, requires that credit customers
be made aware of the cost of credit. This law, passed by Congress in 1969, requires that a
business issuing credit inform the consumer of all details of a credit transaction, including
the true annual interest rate. The true annual interest rate, also called the annual percentage
rate (APR) or annual percentage yield (APY), is the effective annual interest rate on which
credit payments are based. The idea behind the APR is that interest is owed only on the
unpaid balance of the loan.
The Truth in Lending Act stipulates that the interest rate for a loan be calculated only on
the amount owed at a particular time, not on the original amount borrowed. All loans must
be stated according to this standard, thereby making it possible for a consumer to compare
different loans. We can use the following formula to estimate the annual percentage rate
(APR) on a simple interest rate installment loan.
Example:
1. You purchase a refrigerator for $675. You pay 20% down and agree to repay the
balance in 12 equal monthly payments. The finance charge on the balance is 9%
simple interest.
a. Find the finance charge.
Answer (since you have 20% downpayment we will subtract that first before
computing for the finance charge)
𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = $675 − 𝑑𝑜𝑤𝑛𝑝𝑎𝑦𝑚𝑒𝑛𝑡
𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = $675 − (20%)(𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑎𝑚𝑜𝑢𝑛𝑡)
𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = $675 − (0.20)(675)
𝐵𝑎𝑙𝑎𝑛𝑐𝑒 = $540
Finance Charge: 𝐼 = 𝑃𝑟𝑡, note 12 months is equal to 1 year
𝐼 = (540)(0.09)(1)
𝐼 = $48.6
Therefore, the finance charge on the balance is $48.6
b. Estimate the annual percentage rate.
2𝑁𝑟
𝑁+1
Note, N is the number of payment and based on the problem that is 12 monthly
payments.
2(12)(0.09)
𝐴𝑃𝑅 ≈
= 0.166 𝑜𝑟 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 17%
12 + 1
𝐴𝑛𝑠𝑤𝑒𝑟: 𝐴𝑃𝑅 ≈
Consumer Loans: Calculating Monthly Payments
The stated interest rate for most consumer loans, such as a car loan, is normally the annual
percentage rate, APR, as required by the Truth in Lending Act. The payment amount for
these loans is given by the following formula.
Example:
1. Integrated Visual Technologies is offering anyone who purchases a television an
annual interest rate of 9.5% for 4 years. If Andrea Smyer purchases a 50-inch, rear
projection television for $5995 from Integrated Visual Technologies, find her
monthly payment.
𝑖
0.095
) 𝑤ℎ𝑒𝑟𝑒 𝑖 =
𝑃𝑀𝑇 = 𝐴 (
, 𝑡 = 48 𝑚𝑜𝑠 𝑠𝑖𝑛𝑐𝑒 4 𝑦𝑒𝑎𝑟𝑠
−𝑛
1 − (1 + 𝑖 )
12
0.095
12
)
𝑃𝑀𝑇 = 5995 (
0.095 −48
1 − (1 + 12 )
𝑃𝑀𝑇 = $150.61
2. A web page designer purchases a car for $18,395.
a. If the sales tax is 6.5% of the purchase price, find the amount of the sales tax.
Answer:
𝑆𝑎𝑙𝑒𝑠 𝑡𝑎𝑥 = 0.065(18,395) = $1,195.68
b. If the car license fee is 1.2% of the purchase price, find the amount of the license
fee.
Answer:
𝐿𝑖𝑐𝑒𝑛𝑠𝑒 𝑓𝑒𝑒 = 0.012(18,395) = $220.74
c. If the designer makes a $2500 down payment, find the amount of the loan the
designer needs.
Answer:
𝐿𝑜𝑎𝑛 𝑎𝑚𝑜𝑢𝑛𝑡 = 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑝𝑟𝑖𝑐𝑒 + 𝑠𝑎𝑙𝑒𝑠 𝑡𝑎𝑥 + 𝑙𝑖𝑐𝑒𝑛𝑠𝑒 𝑓𝑒𝑒 − 𝑑𝑜𝑤𝑛 𝑝𝑎𝑚𝑒𝑛𝑡
𝐿𝑜𝑎𝑛 𝑎𝑚𝑜𝑢𝑛𝑡 = 18,395 + 1,195.68 + 220.74 − 2500 = $17,311.42
d. Assuming the designer gets the loan in part c at an annual interest rate of 7.5% for
4 years, determine the monthly car payment.
0.075
12
)
𝑃𝑀𝑇 = 17,311.42 (
0.075 −48
1 − (1 + 12 )
𝑃𝑀𝑇 = $418.57
Consumer Loans: Calculating Loan Payoffs
Sometimes a consumer wants to pay off a loan before the end of the loan term. For
instance, suppose you have a five-year car loan but would like to purchase a new car after
owning your car for 4 years. Because there is still 1 year remaining on the loan, you must
pay off the remaining loan before purchasing another car.
This is not as simple as just multiplying the monthly car payment by 12 to arrive at
the payoff amount. The reason, as we mentioned earlier, is that each payment includes both
interest and principal. By solving the Payment Formula for an APR Loan for A, the amount
of the loan, we can calculate the payoff amount, which is just the remaining principal.
Example:
1. Allison Werke wants to pay off the loan on her jet ski that she has owned for 18
months. Allison’s monthly payment is $284.67 on a two-year loan at an annual
percentage rate of 8.7%. Find the payoff amount.
Answer: take note “n” is the number of remaining payments
1 − (1 + 𝑖 )−𝑛
]
𝐴 = 𝑃𝑀𝑇 [
𝑖
0.087 −6
1 − (1 + 12 )
]
𝐴 = 284.67 [
0.087
12
𝐴 = $1,665.50
The amount of loan payoff is $1,665.50.
2. Aaron Jefferson has a five-year car loan based on an annual percentage rate of 8.4%.
The monthly payment is $592.57. After 3 years, Aaron decides to purchase a new car
and must pay off his car loan. Find the payoff amount.
Answer: take note “n” is the number of remaining payments
1 − (1 + 𝑖 )−𝑛
]
𝐴 = 𝑃𝑀𝑇 [
𝑖
0.084 −24
1 − (1 + 12 )
]
𝐴 = 592.57 [
0.084
12
𝐴 = $13,049.34
The amount of loan payoff is $13,049.34.