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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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### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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%

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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%

### 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