1. Financial Applications -Compound Interest Present Value
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2. What is Compound Interest?
Recall:
Simple Interest – Interest is earned on the original sum of money invested. Any interest previously earned does not earn interest.
Compound Interest – Interest is reinvested at regular intervals. The interest is added to the principal to earn interest for the next interval of time, or compounding period.

3. Compound Interest formula
The formula used in compound interest is
Amount (A) .
Principal (P)
Interest rate per period (i)
Number of compounding periods involved (n)

4. Compound periods
Number of compounding periods depends on how many times per year the interest is compounded.
How often interest is compounded
Effective Rate
r = annual interest rate
# of compounding periods in t years
Annually
Once / year
i = r
n = t
Semi-annually
Twice / year
i = r / 2
n = 2t
Quarterly
4 times / year
i = r / 4
n = 4t
Monthly
12 times / year
i = r / 12
n = 12t
Daily
365 times / year
i = r / 365
n = 365t

5. Present Value (P or PV)
The principal that is invested or borrowed is called the present value of the investment or loan. The present value that will result in a specific amount, with accumulated interest, can be calculated when the interest rate, the compounding period, and the time that interest is earned or paid are known.
Recall:

6. Example 1 – Compound Interest-Present Value
Ray’s grandparents would like to have $ when they retire in 9 years. How much should they invest now, at an interest rate of 5.75% per annum, compounded annually?
Compounded annually
Therefore, the original investing amount was $

7. Example 2 – Compound Interest-Present Value
Five years ago a sum of money was invested at 7% per annum compounded quarterly. If the amount of the investment is now $9000, how much money was deposited five years ago?
Compounded quarterly
Therefore, the original investing amount was $
20 quarters
in 5 years

8. Example 3 – Compound Interest-Present Value
Olivia is making arrangements to start her own business in 4 years, and estimates she will need $ She compares investments: 5.1% per annum, compounded semi-annually, or 4.9% per annum, compounded quarterly. How much does Olivia need to invest for the better deal?
Option 1: r = 5.1%
Compounded semi-annually
8 semi-annuals
in 4 years
Option 2: r = 4.9%
Compounded quarterly
16 quarters
in 4 years
Therefore, Option 1 is a better deal since it requires less amount to reach to $26000 in 4 years.

9. Best Investment Returns

10. Example 4 – Can you become a Millionaire?
How much you need to invest right now in order to become a Millionaire when you are 40 years old (say you are 16 years old now)? Check out the previous slide for a suitable annual interest rate you agree on.
Let’s conservatively pick 9%/A
Compounded monthly (Varies)
Therefore, the original investing amount is $
288 months
in 24 years

11. Homework:
WS: Compound Interest