1-0 5-0
5-0
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
1-1 5-1
Multiple cash flows
• Deposit an amount in one year, then
deposit an amount in two years, then
deposit an amount in three years
1
1-2 5-2
Future Value of Multiple Cash
Flows
1. Compound accumulated balance forward one
year at a time
Deposit x (1+ r ) + New Deposit = New Balance
New Balance x (1+ r ) + New Deposit = New Balance
2. Calculate FV of each cash flow, and add cash
flows together
FV = Principal x (1 + r)t
2
1-3 5-3
Future Value of Multiple Cash
Flows Example
Bank account has $7,000 and pays 8% interest.
You will deposit $4,000 at the end of each year for
the next 3 years. How much will you have in 3
years?
1.Compound accumulated balance forward one
year at a time
2. Calculate FV of each cash flow, and add cash
3
Multiple Cash Flows –
FV Example 2
1-4 5-4
• 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?
4
1-5 5-5
Example 2 Continued
• How much will you have in 5 years if
you make no further deposits?
5
Multiple Cash Flows –
FV Example 3
1-6 5-6
• Suppose you plan to deposit $100 into
an account in one year and $300 into
the account in three years. How much
will be in the account in five years if the
interest rate is 8%?
6
1-7 5-7
Present Value of Multiple Cash
Flows
• How much do you need today in order to
make a payment in one year and another
payment in two years?
7
1-8 5-8
Present Value of Multiple Cash
Flows
1. Discount amount back one period at a
time
2. Calculate the PV of each cash flow and
add together
8
1-9 5-9
PV of Multiple Cash Flows
Examples
You are offered an investment that will pay $200 in
one year, $400 the next year, and $800 at the end
of the next year. Assume an interest rate of 12%.
What is the most you should pay for this
investment?
1. Discount amount back one period at a
time
2. Calculate the PV of each cash flow and
add together
9
Multiple Cash Flows –
PV Another Example
1-10
5-10
• You are considering an investment
that will pay you $1,000 in one year,
$2,000 in two years and $3,000 in
three years. If you want to earn 10%
on your money, how much would you
be willing to pay?
10
Multiple Uneven Cash Flows –
Using the Calculator
1-11
5-11
• Another way to use the financial calculator for
uneven cash flows is to use the cash flow keys
– Texas Instruments BA-II Plus
• Clear the cash flow keys by pressing CF and then 2nd CLR
Work
• Press CF and enter the cash flows beginning with year 0.
• You have to press the “Enter” key for each cash flow
• Use the down arrow key to move to the next cash flow
• The “F” is the number of times a given cash flow occurs in
consecutive years
• Use the NPV key to compute the present value by
[ENTER]ing the interest rate for I, pressing the down arrow,
and then computing NPV
11
1-13
5-13
Annuities
• Multiple cash flows in the same amount
• Ex.: Loan with equal payments
13
1-14
5-14
Ordinary Annuity
• Series of constant cash flows that occur at
the end of the period for a fixed # of
periods
Ex.: Cash flow from asset: $500 at the end
of the next years for 3 years, 10% rate
What is the present value?
14
1-15
5-15
Ordinary Annuity
• Need a formula when there are a large
number of payments
Annuity PV = Cash flow X (1 – PV factor r,t
r
)
Remember, PV factor =
1
(1 + r)t
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1-16
5-16
Find the Cash Flow
Cash flow =
PV of annuity / PV annuity factor
16
1-17
5-17
Find the Cash Flow
Bath’s Bank offers you a $60,000, 7 year
term loan at 9% annual interest. What will
your annual loan payment be?
17
1-18
5-18
Find the # of payments
Find the PV of annuity factor
Annuity PV / Cash Flow = PV annuity factor
Look on table under the rate column for the
factor, see what # of periods corresponds
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1-19
5-19
Find the # of payments
You are investing $30,000 in order to
receive payment at the end of each year of
$10,000. If the rate is 20%, how many
payments will you receive?
19
1-20
5-20
Find the rate
Find PV annuity factor
Annuity PV / Cash Flow = PV annuity factor
Go to # of periods on the table and go
across until you find the factor, then look
to see what % column it falls in
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1-21
5-21
Find the rate
When you see the factor amount falls in
between 2 % columns, can plug #s into
formula to pinpoint the rate
Cash flow x [1 – [1/(1 + r)t ]]
r
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1-22
5-22
Find the rate
You are investing $32,000 in order to
receive payment at the end of each year
for 5 years of $8,000. What is the rate on
this investment?
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1-23
5-23
Annuity Due
• Series of equal payments where the
payment occurs at the beginning of the
period
• Ex.: When you lease an apartment, the
first payment is due immediately,
subsequent payments are due at the
beginning of the month
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1-24
5-24
Annuity Due
Calculate ordinary annuity and
multiply by 1 + r
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1-25
5-25
Annuity Due
• You are saving for a new house and
you put $10,000 per year in an account
paying 8%. The first payment is made
today. How much will you have at the
end of 3 years?
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1-26
5-26
Perpetuities
• Infinite series of equal payments
• Value = Cash flow / r
26
Annuities and Perpetuities –
Basic Formulas
1-27
5-27
• Perpetuity: PV = C / r
• Annuities:
1

1


(1  r ) t
PV  C 
r









 (1  r ) t  1 
FV  C 

r


27
1-28
5-28
Annuities and the Calculator
• You can use the PMT key on the
calculator for the equal payment
• The sign convention still holds
• Ordinary annuity versus annuity due
– You can switch your calculator between
the two types by using the 2nd BGN 2nd Set
on the TI BA-II Plus
– If you see “BGN” or “Begin” in the display
of your calculator, you have it set for an
annuity due
– Most problems are ordinary annuities
28
1-29
5-29
Example: Spreadsheet
Strategies – Annuity PV
• The present value and future value
formulas in a spreadsheet include a
place for annuity payments
• Click on the Excel icon to see an
example
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Effective Annual Rate (EAR)
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5-30
• This is the actual rate paid (or received) after
accounting for compounding that occurs
during the year
• If you want to compare two alternative
investments with different compounding
periods you need to compute the EAR and
use that for comparison.
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1-31
5-31
EAR
• Stated or quoted interest rate
– May be compounded
• Annually
– Then quoted rate = EAR
• Semiannually m = 2
• Quarterly m = 4
• Monthly m = 12
m = # of times interest compounds during
the year
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1-32
5-32
Calculate EAR
1. Divide quoted rate by # of times interest
is being compounded during the year (m)
2. Add 1 to the result in step 1
3. Raise the result in step 2 to the power of
m
4. Subtract 1 from the result in step 3
EAR = (1+ Quoted rate / m)m - 1
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1-33
5-33
Calculate EAR
EAR =
FV of investment / PV of investment
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1-34
5-34
Annual Percentage Rate
• This is the annual rate that is quoted by
law
• By definition APR = period rate times the
number of periods per year
• Consequently, to get the period rate we
rearrange the APR equation:
– Period rate = APR / number of periods per
year
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1-35
5-35
Computing APRs
• What is the APR if the monthly rate is .5%?
• What is the APR if the semiannual rate is
.5%?
• What is the monthly rate if the APR is 12%
with monthly compounding?
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Things to Remember
1-36
5-36
• You ALWAYS need to make sure that the
interest rate and the time period match.
– If you are looking at annual periods, you
need an annual rate.
– If you are looking at monthly periods, you
need a monthly rate.
• If you have an APR based on monthly
compounding, you have to use monthly
periods for lump sums, or adjust the interest
rate appropriately if you have payments
other than monthly
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Computing EARs - Example
1-37
5-37
• Suppose you can earn 1% per month
on $1 invested today.
– What is the APR?
– How much are you effectively earning?
• Suppose if you put it in another account,
you earn 3% per quarter.
– What is the APR?
– How much are you effectively earning?
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1-38
5-38
EAR - Formula
m
 APR 
1

EAR  1 

m 

Remember that the APR is the quoted rate,
and m is the number of compounds per year
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1-39
5-39
Computing APRs from EARs
• If you have an effective rate, how can
you compute the APR? Rearrange the
EAR equation and you get:
1


m
APR  m (1  EAR)
-1


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1-40
5-40
Loan Types
• Pure Discount
• Interest Only
• Amortized
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1-41
5-41
Pure Discount Loans
•
•
•
•
Short term loans
Simplest loan form
Pay a lump sum at the end
Determine how much would you be willing
to lend in order to get a FV given r,t
– Lender collect the present value from the
borrower
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Pure Discount Loans – Example
5.11
1-42
5-42
• Treasury bills are excellent examples of
pure discount loans. The principal
amount is repaid at some future date,
without any periodic interest payments.
• If a T-bill promises to repay $10,000 in
one year, and the market interest rate is
7 percent, how much will the bill sell for
in the market?
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5-43
Interest-Only Loans
• Example: Bonds
• Repay the original loan amount at some
point in the future
• Interest is paid each period (PxRxT)
• At the end of the last period, pay principal
in addition to last interest payment
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1-44
5-44
Interest-Only Loan - Example
• Consider a 5-year, interest-only loan with
a 7% interest rate. The principal amount
is $10,000. Interest is paid annually.
– What would the stream of cash flows be?
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1-45
5-45
Amortized loans
• Used for medium term loans
• Lender requires borrower to make payments
that include a portion of the principal and interest
• Amortizing a loan: make regular principal
payments; pay interest and a fixed amount of
principal
– Total payment will go down each period because as
principal goes down, interest amount each period will
go down
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1-46
5-46
Amortized loans: Fixed principal
repayments
Amortization Schedule
Beg. Total Interest
Year Bal. Pmt
Paid
1
5,000 1,450 450
2
4,000 1,360 360
3
3,000 1,270 270
4
2,000 1,180 180
5
1,000 1,090
90
Princ.
Paid
1,000
1,000
1,000
1,000
1,000
End.
Bal.
4,000
3,000
2,000
1,000
0
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Amortized Loan with Fixed
Payment
1-47
5-47
• Borrower makes a fixed payment every
month
• Each payment covers the interest
expense; plus, it reduces principal
• used for car loans, mortgages
47
Amortized Loan with Fixed
Payment
1-48
5-48
To determine the fixed payment
Loan amount = Payment x [1- 1/(1+r)t ]
r
or
Loan amount = payment x PV annuity factor
Payment = loan amount / PV annuity factor
48
Amortized Loan with Fixed
Payment
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5-49
Interest = Beg of period balance x r
Principal = Payment amount – Interest
End of period balance = Beg. of period balance
– principal repaid
Interest portion goes down each period
Principal portion goes up each period
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Amortized Loan with Fixed
Payment - Example
1-50
5-50
• Each payment covers the interest
expense; plus, it reduces principal
• Consider a 4-year loan with annual
payments. The interest rate is 8% and
the principal amount is $5,000.
– What is the annual payment?
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1-51
5-51
Amortization Table for Example
Year
1
Beg.
Balance
5,000.00
Total
Payment
Interest
Paid
Principal
Paid
End.
Balance
1,509.60
2
3
4
Totals
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