MATH 141-501
Section 5.1 Lecture Notes
Overview - Types of Interest
Financial institutions will often give customers interest in exchange for
leaving money in an account.
The initial amount invested in the account is called the principal.
We will look at three different types of interest:
• Simple interest
1. (Interest) I = P rt
2. (Accumulated amount) A = P (1 + rt)
• Compound interest
These problems are solved using the TVM SOLVER calculator program.
• Continuously compounded interest: A = P ert
We will also look at the effective rate of interest for investments.
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Simple Interest
Simple interest is interest which is only paid on the principal.
Alice invests $5000 in an account which pays 5% simple interest annually.
Assuming no withdrawals are made...
• ...what is the interest payment after 1 year?
• ...how much money is in the account after 1 year?
• ...what is the interest payment at the end of year 2 ?
• ...how much total simple interest has been paid after 2 years?
• ...how much money is in the account after 2 years?
• ...how much total simple interest has been paid after 3 years?
• ...how much money is in the account at the end of 3 years?
• ...what is the interest payment after year n? (n is a positive integer)
• ...how much is money in the account at the end of n years?
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Formulas for Simple Interest
If a principal of P is invested in an account at an interest rate of r, then
after t years,
1. the total simple interest is given by
I = P rt.
Interest = Principal x (interest rate) x time.
2. the accumulated amount is
A=P +I
Accumulated Amount = Principal + Interest
A = P + P rt
A = P (1 + rt)(this version of the formula is the one you usually see)
Example 1
Find the accumulated amount at the end of 8 months on a $1000 bank deposit paying simple interest at a rate of 6% per year.
Example 2
Mark made an investment of $3600 in an account paying simple interest.
After 10 months, there was a total of $3748.50 in the account. Find the annual
interest rate.
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Compound Interest
When an account pays compound interest, the interest is periodically
added back to the prinicipal.
After that, this interest itself earns interest.
First, we look at the situation where interest is added back to the principal at
periods of a certain length (annually, semi-annually, quarterly, monthly, weekly,
daily, etc.) In these situations, the accumulated amount is also called the future value, and the initial amount (principal) is also called the present value.
We will use the TVM SOLVER app in the TI-83/84 calculator for compound
interest (and many other problems.)
How to get TVM SOLVER:
On TI-83:
• Press [2nd]
• Press [Finance]
• Select [1: TVM Solver]
On TI-83+ and TI-84:
• Press [APPS]
• Press [1: Finance]
• Select [1: TVM Solver]
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TVM SOLVER
The TVM SOLVER is a screen which has the following variables:
N - number of compounding periods
I% - interest rate (as a percentage)
PV - present value (initial investment)
PMT - payment (not needed yet)
FV - future value (accumulated amount)
P/Y - payments per year
C/Y - conversion periods per year
PMT: whether payments are made at the beginning or end of the period (not
needed yet)
To use TVM SOLVER,
1. Put in relevant information that you know.
2. Move the cursor to the variable that you want to solve for.
3. Hit [ALPHA] [SOLVE].
Example:
An investor puts $4000 in an account where interest is compounded monthly
at a rate of 10% per year. Find the accumulated amount after 8 years.
1. Fill in relevant information that you know.
N:
I%
PV
PMT
FV
P/Y
C/Y
PMT
2. We want to find:
3. Solve.
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Example:
Alicia’s parents have just won the lottery. They want to set aside some of the
prize for an educational fund to pay for Alicia’s college fund. If they estimate
that they will need $110,000 in 8 years, how much should they set aside now if
they can invest the money at 4% per year compounded...
...annually?
...semi-annually?
N:
I%
PV
PMT
FV
P/Y
C/Y
PMT
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...quarterly?
Continuous Compounding
Continuous Compound Interest Formula
A = P ert
P = Principal
r = Nominal interest rate compounded continuously
t = time (in years)
A = accumulated amount at end of t years
Example:
If $4000 is invested in an account compounded continuously at a rate of 3%,
find how much there will be in the account in 9 months.
Example:
You need $8000 in 2 years. Your financial advisor offers an investment which
will pay continuously compounded interest at a nominal rate of 5%. How much
do you need to invest now?
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Effective Rate
The effective rate of interest (also known as annual percentage yield)
is the annual rate of interest that, if compounded annually, will yield the same
accumulated amount as the nominal rate compounded m times per year (over
the same length of time.)
We can find it using the TVM SOLVER function called Eff .
Calculator Syntax: Eff(I,m) gives the effective rate of interest corresponding to a rate of I% compounded m times per year.
Example:
Find the effective rate of interest corresponding to a nominal interest rate
of 9% year compounded semiannually.
Example:
Find the effective rate of interest corresponding to a nominal interest rate
of 5% per year compounded monthly.
Example:
For a one-year investment, a financial company offers you the option of a
10.5% interest rate compounded annually, or a 10% annual interest rate compounded quarterly. Which option should you take?
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