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• PV = 50000[1 – 1 / 1.07525] / .075 = 557,347 Finding the Payment
• Suppose you want to borrow $20,000 for a
new car. You can borrow at 8% per year. If
you take a 4 year loan, what is your annual
payment?
• Formula Approach
• 20,000 = C[1 – 1 / 1.084] / .08
• C = 6038.42 • Calculator Approach
• 4 = N; 20,000 PV; 8 I/Y; CPT PMT = 6038.42 Finding the Number of Payments –
Example 1
• 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? Finding the Number of Payments –
Example 1 continued
• Formula Approach
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• 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 • Calculator Approach
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• Sign convention matters!!!
5 I/Y
2000 PV
734.42 PMT
CPT N = 3 years Finding the Rate On the
Financial Calculator
• Suppose you borrow $10,000 to buy a car.
You agree to pay $207.58 per month for 60
months. What is the monthly interest rate?
• Calculator Approach
•
•
•
•
• Sign convention matters!!!
60 N
10,000 PV
207.58 PMT
CPT I/Y = .75% Annuity – Finding the Rate Without a
Financial Calculator
• Trial and Error Process
• Choose an interest rate and compute the PV
of the payments based on this rate
• Compare the computed PV with the actual
loan amount
• If the computed PV > loan amount, then the
interest rate is too low
• If the computed PV < loan amount, then the
interest rate is too high
• Adjust the rate and repeat the process until the
computed PV and the loan amount are equal Future Values for Annuities –
Example 1
• Suppose you begin saving for your
retirement by depositing $2000 per year in
an RRSP. If the interest rate is 7.5%, how
much will you have in 40 years?
• Formula Approach
• FV = 2000(1.07540 – 1)/.075 = 454,513.04 • Calculator Approach
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•
•
•
• Remember the sign convention!!!
40 N
7.5 I/Y
2000 PMT
CPT FV = 454,513.04 Annuity Due – Example 1
• You are saving for a new house and you
put $10,000 per year in an account paying
8% compounded annually. The first
payment is made today. How much will
you have at the end of 3 years? Annuity Due – Example 1
Timeline
0 10000 1 10000 2 3 10000
32,464
35,061.12 Annuity Due – Example 1
continued
• Formula Approach
• FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12 • Calculator Approach
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•
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• Reset calculator to BEGIN mode (2nd BGN 2nd Set)
3N
10,000 PMT
8 I/Y
CPT FV = 35,061.12
2nd BGN 2nd Set (be sure to change it back to
an ordinary annuity) Perpetuity – Example 1
• The Home Bank of Canada want to sell
preferred stock at $100 per share. A very
similar issue of preferred stock already
outstanding has a price of $40 per share
and offers a dividend of $1 every year.
What dividend would the Home Bank have
to offer if its preferred stock is going to
sell? Perpetuity – Example 1
continued
• Perpetuity formula: PV = C / r
• First, find the required return for the
comparable issue:
• 40 = 1 / r
• r = .025 • Then, using the required return found above,
find the dividend for new preferred issue:
• 100 = C / .025
• C = 2.50 Growing Perpetuity
• The perpetuities discussed so far are annuities
with constant payments
• Growing perpetuities have cash flows that grow
at a constant rate and continue forever
• Growing perpetuity formula, where C1 is the
payment at time 1:
C1
PV =
r −g Growing Perpetuity – Example 1
• Holmes Corporation is expected to pay a
dividend of $3 per share next year. Investors
anticipate that the annual dividend will rise by
6% per year forever. The required rate of
return is 11%. What is the price of the stock
today?
PV = $3.00
= $60.00
0.11 − 0.06 Growing Annuity
• Growing annuities have a finite number of
growing cash flows
• Growing annuity formula: 1 + g T C1
PV = 1 − r − g 1+ r Growing Annuity – Example 1
• Greg Annuitson has just been offered a job at
$50,000 a year. He anticipates his salary will
increase by 5% a year until his retirement in 40
years. Given an interest rate of 8%, what is the
present value of his lifetime salary?
40
$50,000 1.05 PV = = $1,126,571
1 − 0.08 − 0.05 1.08...
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 Fall '14

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