When you open a bank account or invest money in a bank or financial institution the bank/financial institution pays you interest for the use of your money. The
bank/financial institution uses your money (e.g. for mortgages) to earn itself more money. When you borrow money, you pay interest for the loan.
There are 2 types of interest: 1. Simple Interest 2. Compound Interest
For simple interest, you only earn interest on the money originally invested. The interest accumulated does NOT earn interest.
The formula for simple interest is:
The amount, A is the sum of the principal (P dollars) and the interest (I dollars):
8.1 Simple Interest and 8.2 Compound Interest
I = (P)(r)(t)
Where,
I is the total interest to be paid or earned in dollars
P is the principal (the money you start off with; the money that is borrowed or originally invested) in dollars
r is the annual interest rate as a decimal
t is the length of time in years. If time is given in months, divide those months by 12 If time is given in days, divide those days by 365 If time is given in years, leave value as is
Amount = Principal + Interest or
A = P + I
You can use this diagram to generate variations of the formula, I= Prt
e.g. I = Prt, r = I , P = I , t = I . Pt rt Pr
1. Determine the amount earned when $458 is invested at 6.5% annually for 4 months.
2. Sara’s account pays annual interest of 3%. On April 30, her balance was $578.24. Twenty-five days
later the account was closed.
a) Calculate the interest earned
b) Then find the total amount she will have in her account on the 25th day.
Example:
You put $1000 in an investment account, which pays 5% annual interest per year once a year. How much money would you have at the end of 5 years if you do not touch the money?
Simple Interest:
Year Principal ($)
Annual Interest
Amount ($) 1
2 3 4 5
Compound Interest:
Year Principal ($)
Annual Interest
Amount ($) 1
2 3 4 5
Look at the Amount Earned Type of sequence:
Look at the Amount Earned Type of sequence:
Graph Graph
Describe the relationship Describe the relationship
SIMPLE INTEREST COMPOUND INTEREST
For simple interest, you only earn interest on the money originally invested.
For compound interest, you earn interest on BOTH the original investment and the interest previously earned. The interest is added to the principal to earn interest for the next interval of time or compounding period. Interest is reinvested at regular intervals. Your principal earns “interest on the interest”.
NOTE:
Use the table below to help you determine the value of “i” and “n” for the compound interest questions below
.
Annually
Once a year
i =annual interest rate
n =# of years
Semi- Annually
2 times per year
i =annual interest rate ÷ ÷ ÷ ÷ 2
n =# of years × × × × 2
Quarterly
4 times per year
i =annual interest rate ÷ ÷ ÷ ÷ 4
n =# of years × × × × 4
Monthly 12 times per year
i =annual interest rate ÷ ÷ ÷ ÷ 12
n =# of years × × × × 12
To find the AMOUNT To find PRESENT VALUE earned use: (principal) use:
A = P (1 + i) n P = A .
(1 + i) n
P is the money invested (the “principal” or “present value”), in dollars.
A is the amount, in dollars, at the end of the time for the investment or loan.
i is the interest rate per the compounding period, expressed as a decimal.
n is the number of compounding periods.
1. Rabia deposited $1000 in an investment savings account. The annual interest rate is 4.25%
compounded annually. Determine the amount after 7 years.
2. Calculate the amount if $570 is invested for 6 years at 4.5% compounded quarterly.
3. Lisa deposited $10900 into an investment savings account. The annual interest rate is 5.7%
compounded monthly. Determine the amount after 4 years.
4. You win $25 000 in a lottery at age 18. You decide to put it into an account, which pays 5% per annum compounded semi-annually. How much will you have when you turn 40?
1. Peter’s grandparents would like to have $24 000 in 5 years, how much money must be invested today at 4.5% per annum, compounded monthly?
2. You want to be a millionaire by the age of 40. You are currently 16 years old. How much money should you invest at age 16 if you are offered your choice of two plans:
Plan A: interest rate is 7.1% compounded semi-annually Plan B: interest rate is 6.8% compounded quarterly
8.3 Compound Interest – Present Value
Often, when people invest money, they have a goal for which they want a specific amount of money at a future date.
The PRESENT VALUE of the investment or loan is the principal that is invested or borrowed.
In these examples you be using the formula P = A .
(1+i) n
What is ordinary annuity?
Example:
Example 1: Argin got a part-time job on April 30, 2007. He decided to work hard and save money for a motorbike. At the end of each month he is able to put away $140 into an account that pays 3.6% per annum, compounded monthly. He is planning to continue with this pattern until the end of the summer - August 30, 2007. How much money will he have in his account when he makes the last deposit at that time.
What is the interest rate per compounding period? ________
How many times will each payment compound? May _____ June _____ July _____ Aug _____
Consider the timeline below:
April 30th, 2007 May 30th,2007 June 30th, 2007 July 30th, 2007 August 30th, 2007 8.4 Future Value Annuities
The entries in the last column above (starting from bottom up) can be written as:
And create a GEOMETRIC SEQUENCE. Their sum creates a GEOMETRIC SERIES.
a= r= n=
So using the series formula you get: Now transfer to an annuities formula:
Note that there are two ways to solve this…...
State the formula for the future value of an ordinary annuity and define all of its variables.
Example 1: Michael plans to invest $1000 at the end of each 6-month period in an annuity that earns 4.8% per annum compounded semi-annualy for the next 20 years. What will be the future value of this annuity?
Example 2: Samantha has a well paying job. She decides at the end of every month she will regularly put $50 into her bank account which pays 2% interest compounded monthly. How much money will she have in her account if she does this for 3 years?
Example 3: Suppose Marc makes a deposit at the end of every 3 months for 8 years into an account that pays 7% compounded quarterly. If he has $135,000 at the end, how much did he invest per month.
Future Value vs. Present Value
The key question here will be: when do I use Future Value and when do I use a Present Value approach?
Future Value: "Calculate the value at the end of 25 years"
"How much does he need to invest each month to have $1 000 000 in 40 years?"
"How much interest will she earn after 25 years"
"What will be the total amount of all payments at the end of 3 years?"
Present Value: "...is buying a $500 stereo on credit and will make monthly payments"
"...is investing $10000 and wants to withdraw equal amount at the end of each month"
"...gets a loan of $70 000 to buy a cottage...will make equal monthly payments"
"...how much money needs to be invested now to generate $500 payment every week?"
"what was the selling price of ..."
Example 1: How much would you need to invest now at 8.3% per annum compounded annually to provide $500 per year for the next 6 years?
8.5 Present Value Annuities
Example 2: Kristen borrows $95 000 to buy a cottage. She agrees to pay equal monthly payments for 10 years until the balance is paid off. If Kristen is being charged interest of 5.6% per annum compounded monthly, what is her monthly payment going to be?
What's the difference???
Example 3: If Daniel wishes to have $1000 in the bank in 5 years, how much must he invest today if the rate is 6%/a compounded annually?
OR…….Daniel invests an amount of money with a bank and then withdraws $1000/a for 5 years. If there is no longer any money in Daniel's account and the rate of interest was 6%/a compounded annually, how much did Daniel invest?
Example 4: Amna puts a $50 down payment for a Personal Video Recorder (PVR) and borrows the remaining amount. She is asked to make 10 monthly payments of $40 each. The first payment is due in a month.
a) The interest rate is 18%/a compounded monthly. What was the selling price of the PVR?
b) How much interest will Amna pay over the term of the loan?
Example 5: Tomasz and Justin teamed up to open a Tostin Lounge in the trendy Queen West Village. They borrowed $180 000 and estimated they can afford to make monthly payments between $1000 and $1500 at 4.8%/a compounded monthly. How much sooner will they pay off the loan if they make maximum monthly payment?
