Section 5.1 - Compound Interest
Simple Interest Formulas
If I denotes the interest on a principal P (in dollars) at an interest rate of r (as a decimal) per year for t years, then we have:
Interest:
Accumulated Amount:
Example 1: Find the simple interest on a $2,000 investment made for 3 months at an interest rate of 6% per year. What is the accumulated amount?
Example 2: An investment paying simple interest at the rate of 5% per year grew to $3,100 in 10 months. Find the principal.
Example 3: Find the accumulated amount after 3 years if $3,500 is invested at 5% interest per year compounded annually.
Example 4: Find the accumulated amount after 3 months if $1,000 is invested at an annual interest rate of 4.5% compounded monthly.
Compound Interest Formula
where A = Accumualated amount at the end of the time period, P =Principal, r =Nominal interest rate per year as a decimal, m = Number of compounding periods per year, and t =number of years
Example 5: Find the accummulated amount after 5 years if $3,500 is invested at 3.8% interest per year com-pounded quarterly.
TVM Solver: We can also use the TVM Solver on our calculator to solve problems involving compound interest. To access the Finance Menu, you need to press APPS− > 1:Finance (Please note that if you have a plain TI-83, you need to press 2nd x−1to access the Finance Menu). Below we define the inputs on the TVM Solver:
N=the total number of compounding periods I% = interest rate (as a percentage)
PV = present value (principal amount). Entered as a negative number if invested, a positive number if borrowed.
PMT = payment amount (0 if no payments are involved) FV =future value (accummulated amount)
P/Y = C/Y =the number of compounding periods per year.
Move the cursor to the value you are solving for and hit ALPHA and then ENTER. In all of the problems we do make sure that END is highlighted at the bottom of the screen. This represents that payments are received at the end of each period.
Example 6: How much is in an account after 10 years if $1000 is invested at 2.4% annual interest compounded a) annually?
b) quarterly?
e) daily?
f) continuously?
Continuous Compound Interest Formula
A= Pert
where P =principal, r=Annual interest rate compounded continously (as a decimal), t =Time in years, A =Accumulated amount at the end of t years.
Definition: The effective rate of interest is the equivalent interest rate if compounding was only done once a year. It allows us to compare different interest rates with different compounding frequencies.
We use the C:Eff( option on the Finance Menu to compute the effective rate of interest. The inputs are as follows: Eff(annual interest rate as a percentage, the number of compounding periods per year)
Example 7: You have been doing some research and have found that you can either invest your money at 3.55% compounded daily or 3.60% compounded quarterly. Which one would you choose?
Example 8: Find the present value of $30,000 due in 6 years at an interest rate of 8%/year compounded monthly.
Example 9: How long will it take an investment of $8,000 to grow to $10,000 if the investment earns interest at the rate of 6%/year compounded daily?
Sections 5.2 and 5.3 - Annuities
Definition: An annuity is a sequence of payments made at regular time intervals. In general, the amounts in the payments need not be equal.
Definition: An Ordinary Annuity is an annuity in which payments are made of at the end of each payment period.
Definition: An Annuity Due is an annuity in which payments are made at the beginning of each payment period. Definition: A Simple Annuity is an annuity in which the payment period coincides with the conversion period. In this course, we will study annuities with the following properties:
1. The terms are given by fixed time intervals. 2. The periodic payments are equal in size.
3. The payments are made at the end of the payment periods.
4. The payment periods coincide with the interest conversion periods.
Example 1: Since you are a poor college student you currently have $10 in your bank account. If you put $50 each month into your bank account that earns 3.45% compounded monthly, how much would you have when you retire? (Let’s assume that is 46 years from now)
Example 2: How much would you need to put into the bank account from Example 1 if you want $1, 000, 000 when you retire? (i.e. 46 years from now) How much money did you actually put into the bank account?
Example 3: If instead you waited 10 years to start putting payments into your bank account, how much would the payments need to be to have $1, 000, 000 when you retire? How much money did you actually put into the bank account? (Use the same information from Example 2)
Example 4: You are searching for a new car and not sure what you can afford. You’ve discovered that you can get a 60 month loan with a 5.24% interest rate compounded monthly. Looking at your current income, you’ve decided that you can afford a $400 monthly car payment. What’s the most expensive car that you can afford?
Example 5: At the beginning of 2000 Jenny and Eddie bought a house for $170,000. They financed it for 30 years at a 6.9% annual interest rate compounded monthly on the unpaid balance.
a) What were their monthly payments?
b) How much total interest would they end up paying?
c) At the beginning of 2004 they decided to refinance their house with a 30 year mortgage that has a 5.325% annual interest rate compounded monthly on the unpaid balance. What are their new monthly payments?
Example 6: Angie has graduated from college and is ready to start paying back her student loans. She has determined that she will need to make monthly payments to pay back her student loans of $30,000 over a 20 year period with a 6.125% annual interest rate compounded monthly on the unpaid balance.
a) What will her monthly payments be?
b) How much total interest will she be paying?
c) Angie has received good advice from her family and friends and has decided to pay $100 extra each month towards the principal. How long will it take her to pay-off the student loans now?
d) How much total interest will she be paying now that she is paying an extra $100 a month?
Example 7: You purchase a $150,000 home and decide to finance it with an 80/15/5. The interest rate of the primary lien is 5.75%/year compounded monthly whereas the interest rate for the second lien is 7.75%/year compounded monthly. If you are going to make monthly payments over a 30 year period,
a) What is your total monthly payment?
b) If you decide to pay $200 extra each month on the second lien, how long will it take you to pay-off the second lien?
c) Create an amortization schedule for the first two payments and the 61st payment of the second lien without the extra payment.
d) Create an amortization schedule for the first two payments and the 61st payment of the second lien with the extra payment.
