Compound InterestFinance 321Professor D’Arcy

Adam JohariLauren Dufour

Introduction to Compound Interest

Definition: interest that is calculated both on the principal as well as accumulated interest

Where is it used? Loans, mortgages, annuities, etc…Why is it used?-It refers to the interest on interest

principle.

Simple Interest

Definition: Interest calculated on solely the principal, and not off of past earned interest

Formula:I = Prt

(where I = interest, P = principle, r = annual interest rate, t = time in years)

Compound Interest vs. Simple Interest

Simple CompoundSolely earning int. on Principle

Better when borrowing

Earns int. on P and Int.

Better when lending and investing

Compound Interest

Formula:FV = PV(1+r)n

Explanation of Variables:FV = future valuePV = present valuer = annual interest raten = number of compounding periods

Adjusting Interest Rates

Why do we have to adjust? Interest rates are not always given to

us as an annual percentage They are sometimes stated as semi-

annually, monthly, etc...

How?

Adjusting Interest Rates

FV = PV (1 + r(n)/n)nt

r(n) = nominal interest rate n = number of compounding

periods in a year t = time in years

r(n)/n = is the effective interest rate for n periods

Adjusting Interest Rate Example

Find the future value of $500 invested for five years with a nominal interest rate of 8% compounded quarterly.

FV = PV (1 + r (n) /n) nt

FV = 500(1 + 0.08/4) 4*5

FV = $742.97

Timeline

PV FV

n=1n=0 n=2

i = 6%

Timeline

$1000

n=1n=0 n=2

i = 6%

$1123.6

FV = PV(1+i)n

FV = 1000(1.06)2

Compound Interest Example 1

Dyer needs $5000 five years from now to fund his Simpson collection. The current annual interest rate is 6%, and is expected to remain the same. How much would he have to invest today in order to reach his goal?

Solution to Example 1

FV = $5000 R = 6% N = 5 PV = ?

FV = PV(1+r)n

5000 = PV(1.06)5

PV = $3736.29

Compound Interest Example 2

Dyer invests $5000 today. The current nominal interest rate is 6%, which is compounded monthly. How much will he have 5 years from now?

Solution to Example 2

PV = $5000 R = 0.06/12 = 0.005 N = 5*12 = 60 months FV = ?

FV = PV(1+r)n

FV = 5000(1.005)60

FV = $6744.25

Continuous Compounding

Definition: Interest that is compounded on a continuous basis, rather than at fixed intervals

Formula:FV = PVect

where e is approximately 2.718where c is the continuously compounded

interest rate*Note: c = ln(1+r), where r is the annual interest rate

Continuous Compounding Example

You have $400 and it grows at a continuous rate to $500 over 3 years. Find the continuously compounded interest rate.

Solution

FV = PVect 500 = 400e c*3

C = 7.44%

Question for the Class

Uncle Joe wants to purchase a Porsche at the end of the year 2008. Today is January 1, 2007. The Porsche is expected to cost $100000 on December 31, 2008. The current interest rate for 2007 is 7%, and the interest rate is expected to go up to 8% for the year 2008. On January 1, 2008, Aunt Edna promises to give Uncle Joe a $50000 New Years present. How much money would Uncle Joe need today in order to finance his dream car?

Thank You!