Formulas for Compound Interest Date: Topic: Mathematics of Finance (3.6) Formulas for Compound Interest After 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: For n compoundings per year: A = P(1 + r/n)nt For continuous compounding: A = Pert

Example Daysha invests $100 at 8% annual interest compounded continuously. Find the value of her investment at the end of year 7. A= Pert A = 100e(.08)(7) A = 100 e(.56) A = $175.07 The accumulated value of an investment of $100 for 7 years at an interest rate of 8% is $175.07

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

Computing annual percentage yield (APY) Ben invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY? equivalent APY: x 2000 2000 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 Value The 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: Annuities - Present Value The present value PV of an annuity consisting of n equal payments of R dollars earning an interest rate i per period (payment interval) is: The net amount of money put into an annuity is its' present value. The net amount returned from the annuity is its’ future value.

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

Sergio purchases a new pickup truck for $18,500 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%?

Sergio purchases a new pickup truck for $18,500 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%? Sergio will have to pay $364.49 per month for 47 months, and slightly less the last month.

Mathematics of Finance