Introduction to Financial Management

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
15
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
18
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
20
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
21
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?
22
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
23
1-24
5-24
Annuity Due
Calculate ordinary annuity and
multiply by 1 + r
24
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?
25
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
29
Effective Annual Rate (EAR)
1-30
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.
30
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
31
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
32
1-33
5-33
Calculate EAR
EAR =
FV of investment / PV of investment
33
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
34
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?
35
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
36
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?
37
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
38
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


39
1-40
5-40
Loan Types
• Pure Discount
• Interest Only
• Amortized
40
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
41
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?
42
1-43
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
43
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?
44
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
45
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
46
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
1-49
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
49
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?
50
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
51