Math 141: Business Mathematics I
Fall 2015
§5.1 Compound Interest
Instructor: Yeong-Chyuan Chung
Simple Interest
Simple interest is interest that is computed on the original principal only. If we know the
principal P (in dollars), the interest rate r (as a decimal), and the number of years t, then
the simple interest is given by
I = P rt
The accumulated amount A is the sum of the principal and the interest after t years, so
it is given by
A = P + I = P + P rt = P (1 + rt)
Notice that A is a linear function of t (i.e. A can be written in the form mt + c), so this is
an application of linear functions to business/financial situations.
Example (Exercise 2 in the text). Find the simple interest on a $1000 investment made for
3 years at an interest rate of 5%/year. What is the accumulated amount?
Example (Exercise 6 in the text). A bank deposit paying simple interest at the rate of
5%/year grew to a sum of $3100 in 10 months. Find the principal.
Example (Exercise 8 in the text). How many days will it take for a sum of $1500 to earn
$25 interest if it is deposited in a bank paying simple interest at the rate of 5%/year? (Use
a 365-day year.)
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§5.1 Compound Interest
Compound Interest
Compound interest is interest that is periodically added to the principal and thereafter
itself earns interest at the same rate. For example, if $1000 is deposited in a bank, and
the interest rate is 5%/year compounded annually, then after 1 year, we will have 1000 +
1000(.05) = $1050. After 2 years, we will have 1050 + 1050(.05) = $1102.50. After 3 years,
we will have 1102.50 + 1102.50(.05) = $1157.63.
The computation for the first year is exactly the same as the computation for simple interest,
and we could have written it as 1000(1 + .05). Observe that the computation for the second
year can be rewritten as 1050(1+.05) = 1000(1+.05)(1+.05) = 1000(1+.05)2 . Similarly, the
computation for the third year can be rewritten as 1102.50(1+.05) = 1000(1+.05)2 (1+.05) =
1000(1 + .05)3 .
This observation leads us to the formula for computing compound interest. If P is the
principal, r is the nominal interest rate per year (as a decimal), m is the number of conversion/compounding periods per year, and t is the number of years, then the accumulated
amount A is given by
r
A = P (1 + )mt
m
In the example above, we took P = 1000, r = .05, m = 1, and t was 1, 2, or 3. If the interest
is compounded monthly instead of annually, then m will be 12 since there are 12 months in
a year. If the interest is compounded quarterly, then m will be 4.
Example (Exercise 20 in the text). Find the accumulated amount A after 4 years if $200,000
is invested at an interest rate of 8% per year compounded daily.
Continuous Compounding of Interest
Intuitively, the more often interest is compounded, the larger the accumulated amount will
be. A question then arises as to whether the accumulated amount keeps growing without
bound, or whether it approaches a particular number when the interest is compounded more
and more frequently over a fixed period of time. It turns out that the latter case is what
happens.
§5.1 Compound Interest
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If P is the principal, r is the interest rate compounded continuously (as a decimal), and t is
the number of years, then the accumulated amount A is given by
A = P ert
Example (Exercise 64 in the text). Leonard’s current annual salary is $65,000. Ten years
from now, how much will he need to earn to retain his present purchasing power if the rate
of inflation over that period is 3%/year compounded continuously?
Effective Rate of Interest
The effective rate of interest is the interest rate if compounding is done annually (instead
of more than once a year). It allows us to compare different interest rates with different
compounding frequencies. The effective rate of interest is sometimes also called the annual
percentage yield.
We will use the Eff command in the Finance menu to compute effective rates of interest.
The Finance menu can be accessed by pressing APPS. (If you are using a plain TI-83, then
you need to press 2nd x−1 to access the Finance menu.) The syntax is as follows:
Eff(annual interest rate as a percentage, the number of compounding periods per year)
Example (Exercise 56 in the text). Fleet Street Savings Bank pays interest at the rate of
4.25%/year compounded weekly in a savings account, whereas Washington Bank pays interest
at the rate of 4.125%/year compounded daily (assume a 365-day year.) Which bank offers a
better rate of interest?
Present Value
Recall the compound interest formula A = P (1 + mr )mt . The principal P is often referred
to as the present value, and the accumulated amount A is called the future value. In
certain instances, an investor might want to know how much money he/she should invest
§5.1 Compound Interest
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now, at a given rate of interest, so that he/she will accumulate a certain sum at some future
date. We can get a formula by rewriting the compound interest formula to get P in terms
of A. However, we will just use the TVM Solver in the calculator for such computations.
It can be used to compute the future value if the present value is given, or to compute the
present value if the future value is given.
The TVM Solver can be found in the Finance menu, and the required inputs are as follows:
N - the total number of compounding periods
I% - interest rate (as a percentage)
P V - present value (principal). Entered as a negative number if invested, and a positive
number if borrowed.
P M T - payment amount (0 if no payments are involved)
F V - future value (accumulated amount)
P/Y = C/Y - the number of compounding periods per year
Make sure that END is highlighted at the bottom of the screen. This means that payments
are received at the end of each period. Move the cursor to the value you are solving for, and
press ALPHA and ENTER.
Example. How much money will be in an account after 10 years if $1000 is invested at an
interest rate of 2.4%/year compounded
(a) annually?
(b) quarterly?
(c) daily?
§5.1 Compound Interest
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Example (Exercise 54 in the text). Anthony invested a sum of money 5 years ago in a
savings account that has since paid interest at the rate of 4%/year compounded quarterly.
His investment is now worth $22,289.22. How much did he originally invest?
Example (Exercise 80 in the text). How long will it take $12,000 to grow to $15,000 if the
investment earns interest at the rate of 4%/year compounded monthly?