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Unformatted text preview: Ch. 5  The Time Value of Money
Ch. Mind Map
Mind
s Why?: Most longterm project have cash
flows that occur at different points in time.
Given that the passage of time impacts the
value of cash flows, the value of a project is
dependent on how we assess these
differences. The purpose of this chapter is
to develop the tools necessary to evaluate
cash flows over time in order to make
optimal decisions. Mind Map
Mind
s Learning objective:
– Develop an understanding of how time impacts
cash flows
– Quantify the time/value relationship
– Evaluate complex financial contracts/projects Mind Map
Mind
s Key words/concepts:
–
–
–
–
–
–
–
– TVM
PV, FV, PVA, FVA
Lumpsum, annuity
Ordinary annuity vs. Annuity due
Perpetuity
Deferred annuity
Mixedstream cashflow
WORK LOTS OF PROBLEMS! Terminology
Terminology
s Translate $1 today into its equivalent in
Translate
the future (COMPOUNDING).
the
(COMPOUNDING) Today Future ?
Translate $1 in the future into its
Translate
equivalent today (DISCOUNTING).
equivalent
(DISCOUNTING)
Today
s ? Future Note:
Note:
s It’s easiest to use your financial
functions on your calculator to
solve time value problems.
However, you will need a lot of
practice to eliminate mistakes.
s Finance and Accounting Majors:
It will be helpful later to take
extra time now learning to use the
formulas as well as the financial
functions on your calculator! Future Value
Compounding problems
Compounding Future Value  single sums
Future
If you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year? PV = FV =
0 1 Future Value  single sums
Future
If you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year? PV = 100 FV = ??
?? 0
1
Mathematical Solution:
FV = PV (1 + i)n
FV = 100 (1.06)1 = $??
Notice that there is one equation and four
variables. What does this tell you about
how much information you need????? Future Value  single sums
Future
If you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year? PV = 100 FV = ??
?? 0
1
Mathematical Solution:
FV = PV (1 + i)n
“FVIF”
FV = 100 (1.06)1 = $??
Notice that there is one equation and four
variables. What does this tell you about
how much information you need????? HOW ABOUT WITH THE
CALCULATOR?
s There is one equation and 4 variables
– So, how many unknowns can you have? I (or r or I/Y), PV, FV, and n
s For almost all TVM problems, you
simply play a “4 find 3” game
s Three steps: 1) setup your calculator, 2)
enter the 3 known variables, and 3) solve
for the unknown
s Half of you will lose points on
the exam because you ignore
this slide!!
s Two problems
s Pmts per year: I will do all
calculations in 1 P/Yr mode.
s Setup your calculator before
each problem
– HP: 2nd C All; TI: 2nd CLR TVM Future Value  single sums
Future
If you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years? PV = FV =
0 5 Future Value  single sums
Future
If you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years? PV = 100 FV = 0 Calculator Solution:
P/Y = 1
I=6
N=5
PV = 100
FV = $133.82 5 Present Value
Present Present Value  single sums
Present
If you will receive $100 5 years from now, what is
the PV of that $100 if your opportunity cost is 6%? PV = FV =
0 ? Present Value  single sums
If you will receive $100 5 years from now, what is
the PV of that $100 if your opportunity cost is 6%? PV = 74.73
PV FV = 100 0 Calculator Solution:
P/Y = 1
I=6
N=5
FV = 100
PV = 74.73 5 Hint for single sum problems:
Hint
s In every singlesum future value and
present value problem, there are 4
variables:
– FV, PV, i, and n s When doing problems, you will be
given 3 of these variables and asked to
solve for the 4th variable.
s Keeping this in mind makes TVM
problems much easier! The Time Value of Money
The
Compounding and Discounting
Cash Flow Streams 0 1 2 3 4 Annuities
Annuities
s Annuity:
Annuity: an equallyspaced
sequence of equal cash flows.
equal Annuities
Annuities
s Annuity:
Annuity: a equallyspaced sequence
of equal cash flows, as in:
of 0 1 2 3 4 Examples of Annuities:
Examples
s If you buy a bond, you will
receive equal coupon interest
payments over the life of the
bond.
s If you borrow money to buy a
house or a car, you will pay a
stream of equal payments. Future Value  annuity
Future
If you invest $1,000 at the end of the next 3 years, at
8%, how much would you have after 3 years? 0 1 2 3 Future Value  annuity
If you invest $1,000 at the end of the next 3 years,
If
at 8%, how much would you have after 3 years?
at
Mathematical Solution:
Mathematical
FV = PMT (1 + i)n  1
i “FVIFA” Future Value  annuity
If you invest $1,000 at the end of the next 3 years,
at 8%, how much would you have after 3 years? 1000
1000
0 1 Calculator Solution:
P/Y = 1
I=8
PMT = 1,000
FV = $3,246.40 1000 1000
2 N=3 3 Present Value  annuity
Present
What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%? 0 1 2 3 Present Value  annuity
What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%? 1000
1000
0 1 1000 1000
2 Calculator Solution:
Calculator
P/Y = 1
I=8
N=3
PMT = 1,000
PV = $2,577.10 3 Warmup
Question
Question
s Your company has received a $50,000 loan from an
industrial finance company. The annual payments are
$6,202.70. If the company is paying 9% interest per
year, how many loan payments must the company make?
– 15
– 13
– 12
– 19
– None of the above Other Cash Flow Patterns
Other 0 1 2 3 Perpetuities
Perpetuities
s Suppose you will receive a fixed
payment every period (month, year,
etc.) forever. This is an example of
a perpetuity.
s You can think of a perpetuity as an
annuity that goes on forever. Present Value of a Perpetuity
Present
s When we find the PV of an annuity,
we think of the following
relationship: PV = PMT (PVIFA i,, n )
i Mathematically,
(PVIFA i, n ) = 1 1
n
(1 + i) i
We said that a perpetuity is an
annuity where n = infinity. What
happens to this formula when n
gets very, very large? Present Value of a Perpetuity
s So, the PV of a perpetuity is very
So, simple to find:
simple PMT
PV =
i What should you be willing to pay in
order to receive $10,000 annually
forever, if you require 8% per year
on the investment? What should you be willing to pay in
order to receive $10,000 annually
forever, if you require 8% per year
on the investment? PMT
PMT
PV =
i
= $??
$?? = $10,000
$10,000
.08
.08 Ordinary Annuity
Ordinary
vs.
Annuity Due
$1000 4 $1000 $1000 5 6 7 8 Begin Mode vs. End Mode
Begin
1000 4 year
5 5 1000
year
6 6 1000
year
7 7 PV FV in
END
Mode in
END
Mode 8 Begin Mode vs. End Mode
Begin
1000 4 5 1000
year
6 6 1000
year
7 7 year
8 8 PV FV in
BEGIN
Mode in
BEGIN
Mode Earlier, we examined this
“ordinary” annuity:
“ordinary”
1000
1000
0 1000 1000 1 2 Using an interest rate of 8%, we
Using
find that:
find
s The Future Value (at 3)
The Future is $3,246.40.
$3,246.40.
s The Present Value (at 0) is
The Present
$2,577.10.
$2,577.10. 3 What about this annuity?
What
1000
1000 1000 1000 0 1 2 s Same 3year time line,
s Same 3 $1000 cash flows, but...
s The cash flows occur at the
The
beginning of each year, rather
beginning
than at the end of each year.
end
s This is an “annuity due.”
This
“annuity 3 Present Value  annuity due
Present
What is the PV of $1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%? 0 1 2 3 Present Value  annuity due
What is the PV of $1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%? 1000
1000 1000 1000 0 1 2 3 Calculator Solution:
Calculator
Mode = BEGIN P/Y = 1
I=8
Mode
N=3
PMT = 1,000
PV = $2,783.26
$2,783.26 Future Value  annuity due
If you invest $1,000 at the beginning of each of the
next 3 years at 8%, how much would you have at
the end of year 3?
the 1000
1000 1000 1000 0 1 2 3 Calculator Solution:
Calculator
Mode = BEGIN P/Y = 1
I=8
Mode
N=3
PMT = 1,000
FV = $3,506.11
$3,506.11 Before the Exam…
Before
Ordinary annuities vs. annuities due
s Uneven cash flows
s Deferred annuities
s Personal financial plan
s You must practice TVM problems
s HOW DOES IT APPLY TO
YOU?
YOU?
s Suppose you are 22 years old. If you plan
to retire at 65, how much do you need to
save every month to have $2,000,000.
Assume you can earn 12%. ...
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 Fall '11
 JimBrau
 Finance, Time Value Of Money

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