Chapter 5.3 ~ Compound Interest

Equations:

**Compound Interest (annually):**

A=P(1+r)^t

A= accumulated amount after t years

P=Principal or starting amount to be invested

r= interest rate

t= term (years)

*Use this equation only when interest is being compounded annually.

**Compound Interest (multiple conversions):**

A=P(1+r/m)^(mt)

A= Accumulated amount after t years

P= Principal or starting amount to be invested

r= interest rate

t=term (years)

m=number of conversions or compundings in one year (quarterly, daily, etc.)

*Use this equation if interest is not being compounded annually

**Effective Rate of Interest:**

reff= (1+r/m)^m-1

reff= Effective rate of interest or true rate

r= nominal interest per year

m= number of conversions or compoundings in one year (quarterly, daily, etc.)

*Use this equation if interest is being compounded annually or for other conversion periods (quarterly, daily, etc.)

**Present Value for Compound Interest:**

P=A(1+r/m)^(-mt)

P= present value

A= future value

r= interest rate

m= number of conversions or compoundings in one year (quarterly, daily, etc.)

t= term (years)

**Present Value for Continuous Compounding:**

P=A

*e*^(-rt)

P= present value

A= future value

r= interest rate

t= term (years)

**Continuous Compound Interest Formula**

A=P

*e*^(rt)

A= Accumulated amount after t years

P= Principal or starting amount to be invested

r= interest rate

t= term (years)

Vocabulary Necessary for Understanding Compound Interest:

**compound interest**- interest is added to the principal and then earns itself that same rate which is then added to the new principle and so on and so forth

**accumulated amount**- sum of the principal and the interest after t- years

**term**- how long the money is gaining interest

**nominal or stated rate**- interest rate on the investment

**conversion period**- the time intervals between when interest is taken(such as quarterly, daily, etc.)

**effective or true rate**- the interest rate which produces the equal to the accumulated amount in one year that it would take the nominal rate compounded m- times a year

**present value**- amount which you have now

**future value**- amount which is gotten in the future (such as: at 9%, and at the end of nine years, the accumulated amount will be 20,000)

*****present and future value are used when a question such as "how much money should I invest at a fixed rate, so that I will end up with x amount of money?"

**Extra Notes**:

Effective annual rate is used to compare to investment opportunities to see which would be the better deal. You plug in $1 as your principle, for one year to see which gives the most interest.

Present Value is needed when you need a certain amount of money for the future, so you need to find how much you need to start with at what interest rate to your future value.

**Sample Question:**

Find the accumulated amount after 12 years if $7000 is invested at 5.7% per year compounded annually and daily.

Plug in the quantities into the Annual Compound Interest Formula for one year A=P(1+r)^t, you get A= 7,000(1+.057)^12 = $13614.4 which is how much you would make after 12 years at that interest rate.

Plug in the quantities into the Multiple Conversion Compound Interest Formula for multiple conversions A=P(1+r/m)^(mt) to get A= 7000(1+(.057/365))^(12*365)= $13871.80

**Internet Links for Further Information:**

http://www.math.hawaii.edu/~ramsey/CompoundInterest.html

http://en.wikipedia.org/wiki/Compound_interest

http://mathworld.wolfram.com/CompoundInterest.html

**Reminder**to Danika for your post on Tuesday October 3rd due October 4th about Chapter 2.3 Functions and their Mathematical Models.

**Personalization**-

If you owe the bank $100, that's your problem. If you owe the bank $100 million, that's the bank's problem.

~ John Paul Getty