The Time Value of Money

Lesson 16.1

Starter5

1

Evaluate the expression: 2 1n

n

[2(1) 1] [2(2) 1] [2(3) 1] [2(4) 1] [2(5) 1]

3 5 7 9 11 35

1

Remember the Gaussian Pairs formula:

( )2n n

nS a a

5

5(3 11) 35

2S

Objectives

• Students will be able to– Use a formula to find the future value of a

series of payments– Use the TVM Solver on the calculator to

• Find the future value of a series of payments• Find the present value of a future series of

payments

Now let’s do a geometric series:5

1

Evaluate the expression: 3 2n

n

1 2 3 4 53(2) 3(2) 3(2) 3(2) 3(2)

6 12 24 48 96 186

1

Remember the formula for geometric series:

(1 )

1

n

n

a rS

r

5

5

6(1 2 ) 6(1 32) 186186

1 2 1 1S

First Investment Question

• If you invest $100 at 10% annual interest, how much will it be worth in 40 years?

• Simple interest:– I = Prt = $100 x .1 x 40 = $400– Add the original $100 investment to get $500

• Compound interest:– After one year you have $100(1.1)– After two years you have $100(1.1)(1.1)– After 40 years you have $100(1.1)40 = $4525.93

Second Investment Question

• If you invest $100 at 10% annual interest every year for 40 years, how much will you have at the end of 40 years?

• The first investment will be worth $100(1.1)40 = $4525.93

• The next investment will be worth $100(1.1)39 = $4114.48

• Do this for 38 more investments and add them all up to get the answer.

• OR:

Use a formula40 39 1100(1.1) 100(1.1) ... 100(1.1)

40 39 1100(1.1 1.1 ... 1.1 )

40

1

100 1.1n

n

401.1(1 1.1 )100 100(486.8518) $48,685.18

1 1.1

Now let’s make it realistic• If you invest $100 at 10% annual interest

every month, how much will it be worth in 40 years?

• Notice this is the same question, but with two important modifications:– There are 480 months in 40 years, so use

n=480– The 10% annual interest rate must be divided

by 12 to get a monthly rate• 0.1 / 12 = 0.008333… so use r = 1.0083333…

Answer:

0.1Store 1 into the R memory in your calculator.

12

480 479 478 1100 100 100 ... 100R R R R

480480

1

(1 )100 100 $637,678.02

1n

n

R RR

R

• Find the Finance Screen on your TI-83 and go to the TVM Solver

• Here are the meaning of each line:– N is the number of years or months involved– I% is the ANNUAL interest rate– PV is the present value of the account

• For savings, start with zero• For loan payoff, start with the loan amount

– PMT is the annual or monthly payment• Enter the amount as a NEGATIVE number

– FV is the future value of the account– P/Y (payments per year) and C/Y (number of

compoundings per year) should agree• For monthly savings or payments use 12• For annual savings or payments use 1

– PMT: END/BEGIN• Choose BEGIN for savings• Choose END for loan payments

Set up the calculator to answer the last question

• N = 480

• I% = 10

• PV = 0

• PMT = -100

• FV Skip this for now

• P/Y = 12 (Note that C/Y changed to 12)

• PMT: BEGIN

• Go back to FV and tap ALPHA : SOLVE– FV tells you the future value– Note it is the same as we calculated by formula

Practice with the TVM Solver• How much would $200 per month at 10%

grow to in 40 years?• Change the interest to 12%. FV?• How much per month is needed at 12% to

grow to $5,000,000 in 40 years?• If you borrow $20,000 to buy a car at 6%

for 5 years how much will be the monthly payment?

• Borrow $400,000 at 5.5% for 30 years to buy a $500,000 starter house. How much are your monthly payments?

California Lottery Payments• If you win $1,000,000 in the lottery, they pay

$50,000 per year for 20 years.• Use the calculator to find the present value of

this series of payments assuming 5% annual interest.– Note that FV is 0: That’s what they owe you when the

payments are done– Note also that there is 1 payment per year, not 12

• You should get an answer of $654,266.04– This is the amount they will offer you if you want a

lump sum payment– Try it at different interest rates and see what you get

Objectives

• Students will be able to– Use a formula to find the future value of a

series of payments– Use the TVM Solver on the calculator to

• Find the future value of a series of payments• Find the present value of a future series of

payments