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Future Values

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- 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

- 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

- 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

- 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.

- 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

- 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

- 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

- PV = 19,671.51 / (1.07)10 = 10,000

- 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%

- 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

- 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

- 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

- 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

- 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

- 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

- PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29

- 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

- 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

- 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%

- 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

- FV = 2000(1.07540 – 1)/.075 = 454,513.04