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Future Values

Future Values. Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? Interest = 1000(.05) = 50 Value in one year = principal + interest = 1000 + 50 = 1050 Future Value (FV) = 1000(1 + .05) = 1050

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Future Values

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  1. Future Values • Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? • Interest = 1000(.05) = 50 • Value in one year = principal + interest = 1000 + 50 = 1050 • Future Value (FV) = 1000(1 + .05) = 1050 • Suppose you leave the money in for another year. How much will you have two years from now? • FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50

  2. Effects of Compounding • Simple interest • Compound interest • Consider the previous example • FV with simple interest = 1000 + 50 + 50 = 1100 • FV with compound interest = 1102.50 • The extra 2.50 comes from the interest of .05(50) = 2.50 earned on the first interest payment

  3. Future Values – Example 3 • Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? • FV = 10(1.055)200 = 447,189.84 • What is the effect of compounding? • Simple interest = 10 + 200(10)(.055) = 210.55 • Compounding added $446,979.29 to the value of the investment

  4. Present Values • How much do I have to invest today to have some amount in the future? • FV = PV(1 + r)t • Rearrange to solve for PV = FV / (1 + r)t • There are four parts to this equation • PV, FV, r and t • If we know any three, we can solve for the fourth • When we talk about discounting, we mean finding the present value of some future amount.

  5. PV – One Period Example • Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? • PV = 10,000 / (1.07)1 = 9345.79 • Calculator • 1 N • 7 I/Y • 10,000 FV • PV = -9345.79

  6. Present Values – Example 1 • You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? • PV = 150,000 / (1.08)17 = 40,540.34 • Calculator • 17 N • 8 I/Y • 150,000 FV • PV • -40,540.34

  7. Present Values – Example 2 • Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? • PV = 19,671.51 / (1.07)10 = 10,000 Financial Calculator • 19,671.51 FV • 7 I/YR • 10 N • PV

  8. Discount Rate – Example 1 • You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? • r = (1200 / 1000)1/5 – 1 = .03714 = 3.714% • Calculator – the sign convention matters!!! • 5 N • 1000 +/- PV (you pay 1000 today) • 1200 FV (you receive 1200 in 5 years) • I/Y • Answer 3.714%

  9. Discount Rate – Example 2 • Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? • r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25% • Calculator • 6 N • 1000 +/- PV • 20,000 FV • I/Y

  10. Discount Rate – Example 3 • Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it? • r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27% • Calculator • 17 N • 5,000 +/- PV • 75,000 FV • I/Y

  11. Number of Periods – Example 1 • You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? • t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years • Calculator • 10 I/Y • 15,000 +/- PV • 20,000 FV • N

  12. Multiple Cash Flows – FV Example 1 • Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? • FV = 500(1.09)2 + 600(1.09) = 1248.05 • How much will you have in 5 years if you make no further deposits? • First way: • FV = 500(1.09)5 + 600(1.09)4 = 1616.26 • Second way – use value at year 2: • FV = 1248.05(1.09)3 = 1616.26

  13. Multiple Cash Flows – PV Another Example • You are considering an investment that will pay you $1000 in one year, $2000 in two years and $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? • PV = 1000 / (1.1)1 = 909.09 • PV = 2000 / (1.1)2 = 1652.89 • PV = 3000 / (1.1)3 = 2253.94 • PV = 909.09 + 1652.89 + 2253.94 = 4815.93

  14. Annuity – Sweepstakes Example • Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? • PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29 Financial Calculator 333,333.33 PMT; 5 I/YR; 30 N; PV

  15. Finding the Payment • Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). If you take a 4 year loan, what is your monthly payment? • 20,000 = C[1 – 1 / 1.006666748] / .0066667 • C = 488.26 Financial Calculator • 20,000 PV; 48 N; .6666 I/YR; PMT

  16. Finding the Number of Payments – Another Example • Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan? • 2000 = 734.42(1 – 1/1.05t) / .05 • .136161869 = 1 – 1/1.05t • 1/1.05t = .863838131 • 1.157624287 = 1.05t • t = ln(1.157624287) / ln(1.05) = 3 years Financial Calculator $2000 PV; 734.42 +/- PMT; 5 I/YR; N

  17. Finding the Rate • Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? • Sign convention matters!!! • 60 N • 10,000 PV • -207.58 PMT • I/Y = .75%

  18. Future Values for Annuities • Suppose you begin saving for your retirement by depositing $2000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years? • FV = 2000(1.07540 – 1)/.075 = 454,513.04 Financial Calculator • -2000 PMT (use +/- key to change sign) • 40 N • 7.5 I/YR • FV = $454,513.03

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