Chapter 4: The Time Value
of Money
Tsui Ping Chung
Tsui-Ping
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y The
Th objective
bj ti off Chapter
Ch t 4 is
i to
t
explain
ytime value of money calculations
yto illustrate economic equivalence.
equivalence
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Money has a time value
y Capital refers to wealth in the form of money or property
that can be used to produce more wealth.
y Engineering economy studies involve the commitment of
capital for extended periods of time.
y A dollar today is worth more than a dollar one or more years
from now (for several reasons).
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Return to capital in the form
of interest and profit is an
essential part
y Interest and p
profit pay
p y the pproviders of capital
p for
forgoing its use during the time the capital is being used.
y Interest and p
profit are ppayments
y
for the risk the investor
takes in letting another use his or her capital.
y Anyy p
project
j or venture must pprovide a sufficient return
to be financially attractive to the suppliers of money or
property.
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Simple interest
y When the total interest earned or charged is
linearly proportional to the initial amount of
the loan (principal), the interest rate, and the
periods, the interest and
number of interest p
interest rate are said to be simple.
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Computation of simple interest
The total
Th
t t l interest,
i t
t I,
I earnedd or paid
id may be
b computed
t d
using the formula below.
P = principal
i i l amountt lent
l t or borrowed
b
d
N = number of interest pperiods ((e.g.,
g , yyears))
i = interest rate per interest period
The total amount repaid at the end of N interest periods is P + I.
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If $1,000
$1 000 were loaned for 3 years
at a simple interest rate of 10%
per year, the
h interest
i
earned
d
would be
So, the total amount repaid at the end of 3 years would be
the original
g
amount ($
($1,000)
,
) plus
p the interest ($
($300),
), or
$1,300.
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Compound interest reflects both the
remaining principal and any accumulated
interest. For $1,000 at 10%…
Period
1
(1)
(2)=(1)x10%
Amount owed
Interest
at beginning of amount for
period
period
$1,000
$100
(3)=(1)+(2)
Amount
owed at end
of period
$1,100
2
$1 100
$1,100
$110
$1 210
$1,210
3
$1,210
$121
$1,331
Compound interest is commonly used in personal and professional financial
transactions.
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The concept of eguivalence
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Economic equivalence allows us
to compare alternatives on a
common basis.
y Each
E h alternative
l
can bbe reduced
d d to an equivalent
l bbasis
dependent on
y interest rate
y amount of money involved
y timingg of monetaryy receipts
p or expenses.
p
y Using these elements we can “move” cash flows so that we
can compare them at particular points in time.
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We need some tools to find
economic equivalence.
y Notation used in formulas for compound
p
interest
calculations.
y i = effective interest rate per interest period
y N = number of compounding (interest) periods
y P = present sum of money; equivalent value of one or more cash
flows att a reference
fl
f
point
i t iin ti
time; th
the presentt
y F = future sum of money; equivalent value of one or more cash
flows at a reference ppoint in time;; the future
y A = end-of-period cash flows in a uniform series continuing for
a certain number of periods, starting at the end of the first
period
i d andd continuing
i i through
h
h the
h llast
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A cash flow diagram is an
necessary tool for clarifying and
visualizing
i
li i
a series
i
off cash
h flows.
fl
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Example 4.1
y An investment of $10,000
$10 000 can be made that will produce
uniform revenue of $5,310 for five years and then have a
market
a et ((recovery)
ecove y) value
va ue of
o $2,000
$ ,000 at the
t e end
e of
o year
yea (EOY)
( O )
five. Annual expenses will be $3,000 at the end of each year
for operating
p
g and maintainingg the pproject.
j
Draw a cash-flow
diagram for the five-year life of the project.
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Example 4.2
y Alternative A: rebuilt a existing system
y Equipment $18000
y Annual cost of electricity $32000
y Annual maintenance expenses$2400
y Alternative B: install a new system
y Equipment $60000
y Annual maintenance expenses$1600
y A component replacement four years hence $9400
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We can apply compound interest
f
formulas
l
tto “
“move”
” cash
h flows
fl
along the cash flow diagram.
Using the standard notation, we find that a
present amount, P, can grow into a future
amount F,
amount,
F in N time periods at interest rate
i according to the formula below.
In a similar way we can find P given F by
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Itt is
s common
co
o to use standard
sta da d
notation for interest factors.
This iis also
Thi
l kknown as th
the single
i l paymentt
compound amount factor. The term on the
right is read “F given P at i% interest per
period for N interest p
p
periods.”
is called the single payment present worth
factor
factor.
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We can use these to find economically
equivalent values at different points in time.
$8,000 at time zero is equivalent to how much after 4
years if the interest rate is 10% per year?
$10,000
$10
000 at the end of year six is equivalent to how much
today (time zero) if the interest rate is 8% per year?
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Interest rate? P,F and N are
given
y The average price of gasoline in 2005 was $2
$2.31
31 per gasoline.
gasoline
In 1993, the average price was $1.07. what was the average
aannual
ua rate
ate oof increase
c ease in tthee pprice
ce oof gaso
gasolinee ove
over this
t s 12year period?
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N? P,F and i are given
y How long would it take for $500 invested today at 15%
interest per year to be worth $1000?
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Relating a uniform series to its
presentt and
d future
f t
equivalent
i l t
values
y A = end-of-period cash flows in a uniform series
continuing for
f a certain number
b off periods,
d
y Draw a cash flow
F = A( F / P, i %, N − 1) + A( F / P, i %, N − 2) + ... + A( F / P, i %,0)
= A ⎡⎣((1 + i ) N −1 + ((1 + i ) N −2 + ... + (1
( + i )0 ⎤⎦
y The sum of geometric sequence
⎡ (1 + i ) N −1 − (1 + i ) −1 ⎤
⎡ (1 + i ) N − 1 ⎤
F = A⎢
= A⎢
= A( F / A, i %, N )
⎥
⎥
−1
1 − (1 + i )
i
⎣
⎦
⎣
⎦
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Production Scheduling Lab
Example 4-7
y The college degree is worth an extra $23000 per year
income, if the interest rate is 6% per year and you work for
40 years,
yea s, what
w at iss the
t e future
utu e compound
co pou amount
a ou t of
o this
t s extra
e ta
income?
y If your are 20 years of age and save $1 each day for the rest of
life, you can become a millionaire
life
millionaire. Lets assume you live to
age 80 and the annual interest rate is 10%. What is the future
compound amount of this?
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P? Given A
⎡ (1 + i ) N − 1 ⎤
N
F = A⎢
=
P
(1
+
i
)
⎥
i
⎣
⎦
⎡ (1 + i ) N − 1 ⎤
P = A⎢
= A( P / A, i %, N )
N ⎥
⎣ i (1 + i ) ⎦
y If we overhaul a machine and output can be increased 20%, it
can translate into additional cash flow of $20000 for each five
years. If interest rate is 15%, how much can we afford to
invest to overhaul
h l this
h machine?
h
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A? Given F
y The amount to be accumulated after 60 monthly deposits is
$309 million,
illi andd the
h opportunity
i cost off capital
i l is
i 0.5%
0 5% per
month. What is the monthly sinking fund amount?
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A? Given P
y If you had $17000 today in an account earning 1%
each year, how much could you withdraw each
year for 4 years?
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It can be challenging to solve
for N or i.
y We may know P, A, and i and want to find N.
y We may know P, A, and N and want to find i.
y These problems present special challenges that are best
handled on a spreadsheet.
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N? Given A, P, i
y Acme borrowed $100,000
$100 000 from a local bank,
bank which charges
them an interest rate of 8% per year. If Acme pays the bank
$8,800 pper year,
y
now manyy years
y
will it take to ppay
y off the
loan?
y Now if pay $10000, now many years will it take to pay off
the loan?
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i? Given A, P, N
y The car you want to buy will cost $60000 in eight years.
years You
are going to put aside $6000 each for eight years. What the
interest
te est rate
ate must
ust you invest
vest you
your money
o ey to achieve
ac eve you
your
goal?
F = A( F / A, i %, N )
y
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There are specific
p
spreadsheet functions to find
N and i.
i
The Excel function used to solve for N is NPER(rate,
NPER(rate pmt,
pmt
pv), which will compute the number of payments of
magnitude
g
ppmt required
q
to ppayy off a ppresent amount (p
(pv)) at
a fixed interest rate (rate).
One Excel function used to solve for i is RATE(nper, pmt, pv,
fv), which returns a fixed interest rate for an annuity of pmt
that lasts for nper periods to either its present value (pv) or
future value (fv).
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TABLE 4-2 Discrete Cash-Flow Examples Illustrating Equivalence
TABLE 4-2 (continued) Discrete Cash-Flow Examples Illustrating
Equivalence
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