the time value of money lesson 16.1. starter objectives students will be able to –use a formula to...
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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
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