Chapter 3 Exponential, Logistic, and Logarithmic Functions Pre-Calculus OHHS Mr. J. Focht

3.6 Mathematics of Finance Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities—Future Value Loans and Mortgages— Present Value 3.6

Interest Compounded Annually Invest $1,000 in a savings account at 2.25% compound annually for 10 years. 1000+1000∙0.025 = 1000(1+0.025) = $1025 1025+1025∙0.025 = 1025(1+0.025)= 1000(1+.025)(1+.025) = 1000(1+.025)2 = $1,050.63 It continues in this way for 8 more years. We need an easier and faster way. 3.6

Interest Compounded Annually A = P(1+r)n Principal (Investment) Number of years Annual Interest Rate (decimal) 3.6

Interest Compounded Annually Invest $1,000 in a savings account at 2.25% compound annually for 10 years. A = P(1+r)n A = 1000(1+0.025)10 = $1,280.08 3.6

Your Turn P. 341, #1 3.6

Interest Compounded k Times per Year Annual Interest Rate (decimal) Principal (Investment) Number of years # of compounds per year 3.6

Compounded Monthly Suppose Roberto invests $500 at 9% annual interest compounded monthly. Find the value of his investment 5 years later. 3.6

Your Turn P. 341, #5 3.6

Finding the Time Period Judy has $500 to invest at 9% annual interest compounded monthly. How long will it take for her investment to grow to $3000? Continued on next page 3.6

Finding the Time Period 3.6

Finding the Time Period Verify graphically. 3.6

Your Turn P. 341, #21 3.6

Interest Compounded Continuously Principal (Investment) Number of years A = Pert Annual Interest Rate (decimal) 3.6

Compound Continuously Suppose Larry invests $100 at 8% annual interest compounded continuously. Find the value of his investment at the end of each of the years 1, 2, . . . , 7. 3.6

Your Turn P. 341, #9 3.6

Annual Percentage Yield APY the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period. 3.6

Compounded annually for 1 year Compounded quarterly for 1 year APY Example Ursula invests $2000 with Crab Key Bank at 5.15% annual interest compounded quarterly. What is the equivalent APY? Let x = the APY Compounded annually for 1 year Compounded quarterly for 1 year 3.6

APY 3.6

Your Turn P.342, #41 3.6

Annuities—Future Value An annuity is a sequence of equal periodic payments. Suppose you save $50 a month for 30 years in a savings account compounded monthly. APR 1% 2% 3% 4% 5% Future Value $20,981.41 $24,636.27 $29,136.84 $34,702.47 $41,612.93 Non-interest $18,000.00 3.6

Annuities—Future Value Future Value is the total value of the investment returned from an annuity consists of all the periodic payments together with all the interest. Number of payments Payment Period Rate 3.6

Future Value Example = $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? = $95,483.39 3.6

Your Turn P. 341, #13 3.6

Present Value The net amount of money put into an annuity is its present value. 3.6

Present Value Example Carlos purchases a new pickup truck for $16,500. What are the monthly payments for a 4-year loan if the annual interest rate (APR) is 2.9%? 3.6

Example = $364.49 3.6

Your Turn P. 341, #19 3.6

Home Work P. 341-343, #2, 6, 10, 14, 18, 22, 26, 30, 42, 54, 61-66 3.6