Transcript Chapter 5

Time Value of Money II:
Analyzing Annuity Cash Flows
Chapter 5
Fin 325, Section 04 – Spring 2010
Washington State University
1
Introduction
 The previous chapter involved moving a
single cash flow from one point in time to
another
 Many business situations involve multiple
cash flows
 Annuity problems deal with regular, evenlyspaced cash flows




Car loans and home mortgage loans
Saving for retirement
Companies paying interest on debt
Companies paying dividends
2
Example
 Consider the following cash flows: you make a
$100 deposit today, followed by a $125 deposit next
year and a $150 deposit at the end of the second
year. If interest rates are 7%, what is the future
value of your account at the end of the 3rd year?
-100
-125
-150
...
0
1
FV3 = $122.50 + $143.11 + $160.50
= $426.11
2
3
FV of Level CFs
 Suppose that the cash flows are the same each period
 Level cash flows (annuities) are common in finance
(1  i )  1
FVAN  PMT 
i
N
 For annuities, we can use the PMT key on a financial
calculator to input the annuity payment
4
Calculator Solution
Example: suppose that $100 deposits are made
at the end of each year for five years. If interest
rates are 8 percent per year, the future value of
the annuity is:
INPUT
OUTPUT
5
N
8
I/YR
0
PV
-100
PMT
FV
586.66
5
 Another example: Calculate the future value if a
$50 deposit is made every year for 20 years at a 6
percent interest rate
INPUT
OUTPUT
20
N
6
I/YR
0
PV
-50
PMT
FV
1,839.28
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What-if Analysis
 What if the amount deposited doubles to
$100 per year?
 What if the $100 is deposited every year for
40 years rather than 20 years?
 What if the interest rate is increased from 6
percent to 10 percent?
7
PV of Multiple Cash Flows
 Consider the example we started with: you make a $100
deposit today, followed by a $125 deposit next year and
a $150 deposit at the end of the second year. Interest
rates are 7%
-100
-125
-150
...
0
1
2
3
 We can find their individual present values and
add them up
8
Present Value of Level Cash Flows
 The present value of an annuity concept has many
practical uses:
 Most loans are set up with even payments
throughout the life of the loan
 The general formula for the present value of an
annuity is:
1
1
N
(1  i )
PVA N  PMT 
i
10
 Example: What is the present value of an annuity
consisting of $100 payments made at the end of
the next 5 years if interest rates are 8 percent per
year?
1
1
(1  i ) N
PVAN  PMT 
i
1
1
(1  0.08) 5
 100
i
 $ 399.27
INPUT
OUTPUT
5
N
8
I/YR
PV
399.27
-100
PMT
0
FV
Perpetuities
 Perpetuities represent a special type of
annuity in which the cash flows go on
forever
 The present value of a perpetuity is
calculated as:
PMT
PV of Perpetuity 
i
12
 Example: Find the present value of a
perpetuity that pays $100 per year forever
if the discount rate is 10 percent.
$100
PV 
0.10
 $1,000
13
Ordinary Annuity vs. Annuity Due
 ordinary annuity - the payment occurs
at the end of each period.
 annuitiy due - the annuity payments
occur at the beginning of each period.
14
FV and PV of Annuity Due
FVAN Due  FVAN  (1 i)
(1  i) N 1  (1  i)
FVAN Due  PMT 
i
PVAN Due  PVAN  (1 i)
1
(1  i ) 
(1  i ) N 1
PVA N Due  PMT 
i
 Set financial calculators to BEGIN mode
(BGN) when calculate annuity due problems.
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 Example: Find the present value of an annuity
due that pays 100 per year for 5 years if the
interest rate is 8 percent. Before you begin:
should the PV be larger or smaller than if the
payments occur at the end of each period?
First Step: Place your calculator in BGN mode
INPUT
OUTPUT
5
N
8
I/YR
PV
431.21
-100
PMT
0
FV
16
Compounding Frequency
 So far we have assumed that interest is
compounded once per year
 What happens when interest is compounded
more frequently?
 Example: What if 12 percent interest is
compounded semiannually? Let’s say that we invest
$100. If interest were compounded annually, we
would end up with $112. But, semiannual
compounding means that our $100 would earn 6
percent halfway through the year and the other 6
percent at the end. We would end up with: FV =
$100 x (1+1.06) x (1.06) = $112.36.
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APR and EAR
 The quoted, or nominal rate is called the
annual percentage rate (APR)
 The rate that incorporates compounding
is called the effective annual rate (EAR)
m
APR 

EAR  1

m 

1
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 Example: A bank loan has a quoted rate of 12
percent. Calculate the effective annual rate if the
interest is compounded monthly


0.12 12
EAR  1 
1
12
EAR  12.68%
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Calculator Solution
 On the TI BAII Plus calculator, ICONV (interest
conversion) function converts nominal rates to
effective rates
 Input two of Nominal rate, Effective rate, and
Compounding periods and the calculator solves
for the 3rd variable.
 For the example above:
 NOM = 12
 C/Yr = 12
 EFF = 12.68
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 Example: What is the effective rate if the
quoted rate is 10 percent compounded
daily?
 ICONV
 NOM = 10
 C/Yr = 365
 EFF = 10.5156%
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Annuity Loans
 Finding the interest rate
 Often a business will know the cost of something,
as well as the associated cash flows.
 For example: A piece of equipment costs $100,000
and provides positive cash flows of $25,000 for 6
years. What rate of return does this opportunity
offer?
INPUT
OUTPUT
6
N
-100,000 25,000
0
I/YR
PV
PMT
FV
12.98
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 Finding Payments on an Amortized Loan
 Example: You want a car loan of $10,000. The
loan is for 4 years and interest rates are 9
percent per year. Calculate your monthly
payment
 Before we work this problem, we need to
discuss how to set our calculator to solve
problems that involve payments that are not
annual. We typically do this by adjusting the N,
I, and PMT to reflect the relevant period
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 For the above problem:
Problem Data
Calculator input
4 years
N = 4 x 12 = 48 months
9% loan
I = 9/12 = 0.75% per month
Loan amount = $10,000
PV = 10,000
 Solution:
INPUT
OUTPUT
48
N
0.75 10,000
I/YR
PV
PMT
248.85
0
FV
 Since N and I are monthly, we know that the PMT is the monthly
payment
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Amortized Loan Schedules
• Amortized loans are characterized by
level payments, with an increasing
portion of the payment consisting of
principal, and a decreasing proportion
of interest
• Example: You have received a $150,00 student
loan that is to be repaid in annual payments
over 3 years. The interest rate is 10%.
Construct an amortization schedule for the
loan.
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 The first step is to calculate the payment.
INPUT
OUTPUT
3
N
10 150,000
0
I/YR
PV
PMT
FV
60,317.22
Beg Bal
Payment
Interest
1
150,000
60,317.22 15,000
2
104,682.78 60,317.22 10,468.28 49,848.94
54,833.84
3
54,833.84
0.00
60,317.22 5,483.38
Principal
End Bal
45,317.22
104,682.78
54,833.84
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 Computing the Time Period
 How long will it take to pay off a loan?
 Example: How long will it take to pay
off a $5,000 loan with a 19 percent APR
which compounds monthly? The
payment is $150 per month
 I = 19/12 = 1.58333
1.58333 5,000
N
I/YR
PV
OUTPUT 47.8
INPUT
-150
PMT
0
FV
27