Formulas for Compound InterestAfter t years, the balance, A, in an account

with principal P and annual interest rate r (in decimal form) is given by the following formulas:

1. For n compoundings per year:

A = P(1 + r/n)nt

2. For continuous compounding:

A = Pert

Date: Topic: Mathematics of Finance (3.6)

ExampleDaysha invests $100 at 8% annual interest compounded continuously. Find the value of her investment at the end of year 7.

A = 100e(.08)(7)

A = 100 e(.56)

A = $175.07

A= Pert

The accumulated value of an investment of $100for 7 years at an interest rate of 8% is $175.07

Suppose that you inherit $30,000. Is it possible to invest $25,000 and have over half a million dollars for early retirement. Basically, how long will it take $25,000 to grow to $500,000 at a current 9% annual interest compounded monthly?

A = P(1 + r/n)nt

500,000 = 25,000(1 + .09/12)12t substitute

500,000 = 25,000(1 + .0075)12t simplify

25,000 25,000 divide both sides by 25,000

20 = (1 .0075)12t simplify

ln 20 = ln (1 .0075)12t take the ln of both sides

ln 20 = 12t ln 1 .0075 rewrite exponent

12 ln 1.0075 12 ln 1.0075 divide by 12 ln 1.0075

ln 20 = t

12 ln 1.0075 33.4 = t approximate

After approximately 33.4 years, the $25,000 will grow to an accumulated value of $500,000.

ln 20 = ln(1 .0075)12t

Computing annual percentage yield (APY)Ben invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY?

40.051

0 12 005

4

equivalent APY: x

1

2000 11

x

2000 2000

40.0515

1 (1 )4

x

40.0515

(4

1 1 ) 11 x

40.0515

1 14

x

0.0525 x

The annual percentage yield is 5.25%. In other words, Ben’s $2000 invested at 5.15% compounded quarterly for 1 year earns the same interest and yields the same as $2000 invested at 5.25% compounded annually for 1 year.

Annuities - Future ValueThe future value FV of an annuity consisting of n equal periodic payments of R dollars at an interest rate i per compounding period (payment interval) is:

(1 ) 1niiFV R

Annuities - Present ValueThe present value PV of an annuity consisting of n equal payments of R dollars earning an interest rate i per period (payment interval) is:

1 (1 ) niiPV R

The net amount of money put into an annuity is its' present value. The net amount returned from the annuity is its’ future value.

At the end of each quarter year, Emily makes a $500 payment into the Lanaghan Mutual Fund. If her investments earn 7.88% annual interest compounded quarterly, what will be the value of Emily’s annuity in 20 years.

20( 4)0.0788

40.0788

4

1 1

500FV

95,483.389FV

(1 ) 1niiFV R

The value of Emily’s annuity in 20 years will be $95,483.39.

Sergio purchases a new pickup truck for $18,500. What are the monthly payments for a 4-year loan with a $2000 down payment if the annual interest rate (APR) is 2.9%?

4(12)0.029

120.029

2

1

1

1

16,500 R

1 (1 ) niiPV R

480.029 0.029

12 12116,500 1R

48

1

1

1

6,50

0.

0 0.029

029

/12

/12R

Sergio purchases a new pickup truck for $18,500. What are the monthly payments for a 4-year loan with a $2000 down payment if the annual interest rate (APR) is 2.9%?

364.487 R

Sergio will have to pay $364.49 per month for 47 months, and slightly less the last month.

48

1

1

1

6,50

0.

0 0.029

029

/12

/12R

Mathematics of Finance