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compounded semiannually. If he leaves his money in the account for 24 months
(2 years), he will be paid 4% interest compounded over four periods, each of
154 PART 2 Important Financial Concepts


TABLE 4.5 The Future Value from Investing
$100 at 8% Interest Compounded
Semiannually Over 24 Months
(2 Years)

Beginning Future value Future value at end
principal interest factor of period [(1) (2)]
Period (1) (2) (3)

6 months $100.00 1.04 $104.00
12 months 104.00 1.04 108.16
18 months 108.16 1.04 112.49
24 months 112.49 1.04 116.99




which is 6 months long. Table 4.5 uses interest factors to show that at the end of
12 months (1 year) with 8% semiannual compounding, Fred will have $108.16;
at the end of 24 months (2 years), he will have $116.99.



Quarterly Compounding
Quarterly compounding of interest involves four compounding periods within
quarterly compounding
Compounding of interest over the year. One-fourth of the stated interest rate is paid four times a year.
four periods within the year.
Fred Moreno has found an institution that will pay him 8% interest com-
EXAMPLE
pounded quarterly. If he leaves his money in this account for 24 months (2
years), he will be paid 2% interest compounded over eight periods, each of
which is 3 months long. Table 4.6 uses interest factors to show the amount Fred



TABLE 4.6 The Future Value from Investing
$100 at 8% Interest Compounded
Quarterly Over 24 Months
(2 Years)

Beginning Future value Future value at end
principal interest factor of period [(1) (2)]
Period (1) (2) (3)

3 months $100.00 1.02 $102.00
6 months 102.00 1.02 104.04
9 months 104.04 1.02 106.12
12 months 106.12 1.02 108.24
15 months 108.24 1.02 110.40
18 months 110.40 1.02 112.61
21 months 112.61 1.02 114.86
24 months 114.86 1.02 117.16
155
CHAPTER 4 Time Value of Money


TABLE 4.7 The Future Value at the End of
Years 1 and 2 from Investing $100
at 8% Interest, Given Various
Compounding Periods

Compounding period

End of year Annual Semiannual Quarterly

1 $108.00 $108.16 $108.24
2 116.64 116.99 117.16




will have at the end of each period. At the end of 12 months (1 year), with 8%
quarterly compounding, Fred will have $108.24; at the end of 24 months (2
years), he will have $117.16.

Table 4.7 compares values for Fred Moreno™s $100 at the end of years 1
and 2 given annual, semiannual, and quarterly compounding periods at the 8
percent rate. As shown, the more frequently interest is compounded, the greater
the amount of money accumulated. This is true for any interest rate for any
period of time.



A General Equation for
Compounding More Frequently Than Annually
The formula for annual compounding (Equation 4.4) can be rewritten for use
when compounding takes place more frequently. If m equals the number of times
per year interest is compounded, the formula for annual compounding can be
rewritten as
mn
i
FVn PV 1 (4.18)
m
If m 1, Equation 4.18 reduces to Equation 4.4. Thus, if interest is com-
pounded annually (once a year), Equation 4.18 will provide the same result as
Equation 4.4. The general use of Equation 4.18 can be illustrated with a simple
example.

The preceding examples calculated the amount that Fred Moreno would have at
EXAMPLE
the end of 2 years if he deposited $100 at 8% interest compounded semiannually
and compounded quarterly. For semiannual compounding, m would equal 2 in
Equation 4.18; for quarterly compounding, m would equal 4. Substituting the
appropriate values for semiannual and quarterly compounding into Equation
4.18, we find that

1. For semiannual compounding:
22
0.08
0.04)4
FV2 $100 1 $100 (1 $116.99
2
156 PART 2 Important Financial Concepts


2. For quarterly compounding:
42
0.08
0.02)8
FV2 $100 1 $100 (1 $117.16
4
These results agree with the values for FV2 in Tables 4.5 and 4.6.


If the interest were compounded monthly, weekly, or daily, m would equal 12,
52, or 365, respectively.


Using Computational Tools
for Compounding More Frequently Than Annually
We can use the future value interest factors for one dollar, given in Table A“1,
when interest is compounded m times each year. Instead of indexing the table for
i percent and n years, as we do when interest is compounded annually, we index
it for (i m) percent and (m n) periods. However, the table is less useful,
because it includes only selected rates for a limited number of periods. Instead, a
financial calculator or a computer and spreadsheet is typically required.


Fred Moreno wished to find the future value of $100 invested at 8% interest
EXAMPLE
compounded both semiannually and quarterly for 2 years. The number of com-
pounding periods, m, the interest rate, and the number of periods used in each
case, along with the future value interest factor, are as follows:


Compounding Interest rate Periods Future value interest factor
period m (i · m) (m n) from Table A“1
Input Function
Semiannual 2 8% 2 4% 2 2 4 1.170
100 PV
Quarterly 4 8% 4 2% 4 2 8 1.172
4 N
4 I
CPT
Table Use Multiplying each of the future value interest factors by the initial
FV
$100 deposit results in a value of $117.00 (1.170 $100) for semiannual com-
pounding and a value of $117.20 (1.172 $100) for quarterly compounding.
Solution
116.99

Calculator Use If the calculator were used for the semiannual compounding
calculation, the number of periods would be 4 and the interest rate would be
Input Function
4%. The future value of $116.99 will appear on the calculator display as
100 PV
shown in the first display at the left.
8 N
For the quarterly compounding case, the number of periods would be 8 and
2 I
the interest rate would be 2%. The future value of $117.17 will appear on the calcu-
CPT
lator display as shown in the second display at the left.
FV
Spreadsheet Use The future value of the single amount with semiannual and
Solution
quarterly compounding also can be calculated as shown on the following Excel
117.17
spreadsheet.
157
CHAPTER 4 Time Value of Money




Comparing the calculator, table, and spreadsheet values, we can see that the
calculator and spreadsheet values agree generally with the values in Table 4.7 but
are more precise because the table factors have been rounded.


Continuous Compounding
In the extreme case, interest can be compounded continuously. Continuous
compounding involves compounding over every microsecond”the smallest time
continuous compounding
Compounding of interest an period imaginable. In this case, m in Equation 4.18 would approach infinity.
infinite number of times per year Through the use of calculus, we know that as m approaches infinity, the equation
at intervals of microseconds.
becomes

(ei n)
FVn (continuous compounding) PV (4.19)
where e is the exponential function10, which has a value of 2.7183. The future
value interest factor for continuous compounding is therefore

ei n
FVIFi,n (continuous compounding) (4.20)


To find the value at the end of 2 years (n 2) of Fred Moreno™s $100 deposit
EXAMPLE
(PV $100) in an account paying 8% annual interest (i 0.08) compounded
continuously, we can substitute into Equation 4.19:
e0.08 2
FV2 (continuous compounding) $100
2.71830.16
$100
$100 1.1735 $117.35

Calculator Use To find this value using the calculator, you need first to find the
value of e0.16 by punching in 0.16 and then pressing 2nd and then ex to get 1.1735.



10. Most calculators have the exponential function, typically noted by ex, built into them. The use of this key is espe-
cially helpful in calculating future value when interest is compounded continuously.
158 PART 2 Important Financial Concepts


Next multiply this value by $100 to get the future value of $117.35 as shown at the
Input Function
left. (Note: On some calculators, you may not have to press 2nd before pressing ex.)
2nd
0.16
ex Spreadsheet Use The future value of the single amount with continuous com-
pounding also can be calculated as shown on the following Excel spreadsheet.
1.1735

100


Solution
117.35




The future value with continuous compounding therefore equals $117.35. As
expected, the continuously compounded value is larger than the future value of
interest compounded semiannually ($116.99) or quarterly ($117.16). Continu-
ous compounding offers the largest amount that would result from compounding
interest more frequently than annually.


Nominal and Effective Annual Rates of Interest
Both businesses and investors need to make objective comparisons of loan costs
or investment returns over different compounding periods. In order to put inter-
est rates on a common basis, to allow comparison, we distinguish between nomi-
nal and effective annual rates. The nominal, or stated, annual rate is the contrac-
nominal (stated) annual rate
Contractual annual rate of tual annual rate of interest charged by a lender or promised by a borrower. The
interest charged by a lender or effective, or true, annual rate (EAR) is the annual rate of interest actually paid or
promised by a borrower.
earned. The effective annual rate reflects the impact of compounding frequency,
whereas the nominal annual rate does not.
effective (true) annual rate (EAR)
The annual rate of interest Using the notation introduced earlier, we can calculate the effective annual
actually paid or earned.
rate, EAR, by substituting values for the nominal annual rate, i, and the com-
pounding frequency, m, into Equation 4.21:
im
EAR 1 1 (4.21)
m

We can apply this equation using data from preceding examples.

Fred Moreno wishes to find the effective annual rate associated with an 8% nom-
EXAMPLE
inal annual rate (i 0.08) when interest is compounded (1) annually (m 1);
(2) semiannually (m 2); and (3) quarterly (m 4). Substituting these values into
Equation 4.21, we get

1. For annual compounding:
1
0.08
0.08)1
EAR 1 1 (1 1 1 0.08 1 0.08 8%
1
159
CHAPTER 4 Time Value of Money


2. For semiannual compounding:
2
0.08
0.04)2
EAR 1 1 (1 1 1.0816 1 0.0816 8.16%
2

3. For quarterly compounding:
0.08 4
0.02)4
EAR 1 1 (1 1 1.0824 1 0.0824 8.24%
4

These values demonstrate two important points: The first is that nominal and
effective annual rates are equivalent for annual compounding. The second is that
the effective annual rate increases with increasing compounding frequency, up to
a limit that occurs with continuous compounding.11

At the consumer level, “truth-in-lending laws” require disclosure on credit
card and loan agreements of the annual percentage rate (APR). The APR is the
annual percentage rate (APR)
The nominal annual rate of nominal annual rate found by multiplying the periodic rate by the number of
interest, found by multiplying the
periods in one year. For example, a bank credit card that charges 1 1/2 percent per
periodic rate by the number of
month (the periodic rate) would have an APR of 18% (1.5% per month 12
periods in 1 year, that must be
months per year).
disclosed to consumers on credit
“Truth-in-savings laws,” on the other hand, require banks to quote the
cards and loans as a result of
“truth-in-lending laws.” annual percentage yield (APY) on their savings products. The APY is the effective
annual rate a savings product pays. For example, a savings account that pays 0.5
annual percentage yield (APY)
percent per month would have an APY of 6.17 percent [(1.005)12 1].
The effective annual rate of
Quoting loan interest rates at their lower nominal annual rate (the APR) and
interest that must be disclosed to
consumers by banks on their savings interest rates at the higher effective annual rate (the APY) offers two
savings products as a result of
advantages: It tends to standardize disclosure to consumers, and it enables finan-
“truth-in-savings laws.”
cial institutions to quote the most attractive interest rates: low loan rates and high
savings rates.


Review Questions

4“12 What effect does compounding interest more frequently than annually
have on (a) future value and (b) the effective annual rate (EAR)? Why?
4“13 How does the future value of a deposit subject to continuous compound-
ing compare to the value obtained by annual compounding?
4“14 Differentiate between a nominal annual rate and an effective annual rate
(EAR). Define annual percentage rate (APR) and annual percentage yield
(APY).


11. The effective annual rate for this extreme case can be found by using the following equation:
ek
EAR (continuous compounding) 1 (4.21a)

For the 8% nominal annual rate (k 0.08), substitution into Equation 4.21a results in an effective
annual rate of
e0.08 1 1.0833 1 0.0833 8.33%

in the case of continuous compounding. This is the highest effective annual rate attainable with an
8% nominal rate.
160 PART 2 Important Financial Concepts


Special Applications of Time Value
LG6

Future value and present value techniques have a number of important applica-
tions in finance. We™ll study four of them in this section: (1) deposits needed to
accumulate a future sum, (2) loan amortization, (3) interest or growth rates, and
(4) finding an unknown number of periods.



Deposits Needed to Accumulate a Future Sum
Suppose you want to buy a house 5 years from now, and you estimate that an ini-
tial down payment of $20,000 will be required at that time. To accumulate the
$20,000, you will wish to make equal annual end-of-year deposits into an account
paying annual interest of 6 percent. The solution to this problem is closely related
to the process of finding the future value of an annuity. You must determine what
size annuity will result in a single amount equal to $20,000 at the end of year 5.
Earlier in the chapter we found the future value of an n-year ordinary annu-
ity, FVAn, by multiplying the annual deposit, PMT, by the appropriate interest
factor, FVIFAi,n. The relationship of the three variables was defined by Equation
4.14, which is repeated here as Equation 4.22:

FVAn PMT (FVIFAi,n) (4.22)

We can find the annual deposit required to accumulate FVAn dollars by solv-
ing Equation 4.22 for PMT. Isolating PMT on the left side of the equation gives us

FVAn
PMT (4.23)
FVIFAi,n

Once this is done, we have only to substitute the known values of FVAn and
FVIFAi,n into the right side of the equation to find the annual deposit required.


As just stated, you want to determine the equal annual end-of-year deposits
EXAMPLE
required to accumulate $20,000 at the end of 5 years, given an interest rate
of 6%.

Table Use Table A“3 indicates that the future value interest factor for an
ordinary annuity at 6% for 5 years (FVIFA6%,5yrs) is 5.637. Substituting
Input Function
20,000 FV
FVA5 $20,000 and FVIFA6%,5yrs 5.637 into Equation 4.23 yields an annual
5 N required deposit, PMT, of $3,547.99. Thus if $3,547.99 is deposited at the end of
each year for 5 years at 6% interest, there will be $20,000 in the account at the
6 I

end of the 5 years.
CPT
PMT
Calculator Use Using the calculator inputs shown at the left, you will find the
Solution
annual deposit amount to be $3,547.93. Note that this value, except for a slight
3547.93
rounding difference, agrees with the value found by using Table A“3.

Spreadsheet Use The annual deposit needed to accumulate the future sum also
can be calculated as shown on the following Excel spreadsheet.
161
CHAPTER 4 Time Value of Money




Loan Amortization
The term loan amortization refers to the computation of equal periodic loan pay-
loan amortization
The determination of the equal ments. These payments provide a lender with a specified interest return and
periodic loan payments repay the loan principal over a specified period. The loan amortization process
necessary to provide a lender
involves finding the future payments, over the term of the loan, whose present
with a specified interest return
value at the loan interest rate equals the amount of initial principal borrowed.
and to repay the loan principal
Lenders use a loan amortization schedule to determine these payment amounts
over a specified period.
and the allocation of each payment to interest and principal. In the case of home
loan amortization schedule
mortgages, these tables are used to find the equal monthly payments necessary to
A schedule of equal payments to
amortize, or pay off, the mortgage at a specified interest rate over a 15- to 30-
repay a loan. It shows the alloca-
year period.
tion of each loan payment to
interest and principal. Amortizing a loan actually involves creating an annuity out of a present
amount. For example, say you borrow $6,000 at 10 percent and agree to make
equal annual end-of-year payments over 4 years. To find the size of the payments,
the lender determines the amount of a 4-year annuity discounted at 10 percent
that has a present value of $6,000. This process is actually the inverse of finding
the present value of an annuity.
Earlier in the chapter, we found the present value, PVAn, of an n-year annu-
ity by multiplying the annual amount, PMT, by the present value interest factor
for an annuity, PVIFAi,n. This relationship, which was originally expressed as
Equation 4.16, is repeated here as Equation 4.24:

PVAn PMT (PVIFAi,n) (4.24)

To find the equal annual payment required to pay off, or amortize, the loan,
PVAn, over a certain number of years at a specified interest rate, we need to solve
Equation 4.24 for PMT. Isolating PMT on the left side of the equation gives us

PVAn
PMT (4.25)
PVIFAi,n

Once this is done, we have only to substitute the known values into the righthand
side of the equation to find the annual payment required.

As just stated, you want to determine the equal annual end-of-year payments nec-
EXAMPLE
essary to amortize fully a $6,000, 10% loan over 4 years.
162 PART 2 Important Financial Concepts


TABLE 4.8 Loan Amortization Schedule ($6,000
Principal, 10% Interest, 4-Year Repayment
Period)

Payments
Beginning- End-of-year
End of-year Loan Interest Principal principal
of principal payment [0.10 (1)] [(2) (3)] [(1) (4)]
year (1) (2) (3) (4) (5)

1 $6,000.00 $1,892.74 $600.00 $1,292.74 $4,707.26
2 4,707.26 1,892.74 470.73 1,422.01 3,285.25
3 3,285.25 1,892.74 328.53 1,564.21 1,721.04
”a
4 1,721.04 1,892.74 172.10 1,720.64
aBecause of rounding, a slight difference ($0.40) exists between the beginning-of-year-4 principal
(in column 1) and the year-4 principal payment (in column 4).




Table Use Table A“4 indicates that the present value interest factor for an
annuity corresponding to 10% and 4 years (PVIFA10%,4yrs) is 3.170. Substituting
PVA4 $6,000 and PVIFA10%,4yrs 3.170 into Equation 4.25 and solving for
PMT yield an annual loan payment of $1,892.74. Thus to repay the interest and
principal on a $6,000, 10%, 4-year loan, equal annual end-of-year payments of
$1,892.74 are necessary.
Calculator Use Using the calculator inputs shown at the left, you will find the
annual payment amount to be $1,892.82. Except for a slight rounding difference,
Input Function
6000 PV
this value agrees with the table solution.
4 N The allocation of each loan payment to interest and principal can be seen in
columns 3 and 4 of the loan amortization schedule in Table 4.8. The portion of
10 I

each payment that represents interest (column 3) declines over the repayment
CPT
period, and the portion going to principal repayment (column 4) increases. This
PMT
pattern is typical of amortized loans; as the principal is reduced, the interest com-
Solution
ponent declines, leaving a larger portion of each subsequent loan payment to
1892.82
repay principal.
Spreadsheet Use The annual payment to repay the loan also can be calculated
as shown on the first Excel spreadsheet. The amortization schedule allocating
each loan payment to interest and principal also can be calculated precisely as
shown on the second spreadsheet.
163
CHAPTER 4 Time Value of Money




Interest or Growth Rates
It is often necessary to calculate the compound annual interest or growth rate
(that is, the annual rate of change in values) of a series of cash flows. Examples
include finding the interest rate on a loan, the rate of growth in sales, and the rate
of growth in earnings. In doing this, we can use either future value or present
value interest factors. The use of present value interest factors is described in this
section. The simplest situation is one in which a person wishes to find the rate of
interest or growth in a series of cash flows.12

Ray Noble wishes to find the rate of interest or growth reflected in the stream of
EXAMPLE
cash flows he received from a real estate investment over the period 1999 through
2003. The following table lists those cash flows:


Year Cash flow

2003 $1,520
}4
2002 1,440
}3
2001 1,370
}2
2000 1,300
}1
1999 1,250



By using the first year (1999) as a base year, we see that interest has been earned
(or growth experienced) for 4 years.


12. Because the calculations required for finding interest rates and growth rates, given the series of cash flows, are
the same, this section refers to the calculations as those required to find interest or growth rates.
164 PART 2 Important Financial Concepts



In Practice
FOCUS ON PRACTICE Time Is on Your Side
For many years, the 30-year fixed-
rate mortgage was the traditional
Monthly principal Total interest paid
choice of home buyers. In recent
Term Rate and interest over the term of the loan
years, however, more homeown-
ers are choosing fixed-rate mort- 15 years 6.50% $1,742 $113,625
gages with a 15-year term when
30 years 6.85% $1,311 $271,390
they buy a new home or refinance
their current residence. They are
often pleasantly surprised to dis-
cover that they can pay off the Why isn™t everyone rushing to Yet another option is to make
loan in half the time with a monthly take out a shorter mortgage? Many additional principal payments
payment that is only about 25 per- homeowners either can™t afford whenever possible. This shortens
cent higher. Not only will they own the higher monthly payment or the life of the loan without commit-
the home free and clear sooner, would rather have the extra spend- ting you to the higher payments. By
but they pay considerably less in- ing money now. Others hope to do paying just $100 more each month,
terest over the life of the loan. even better by investing the differ- you can shorten the life of a 30-
year mortgage to 24 1/4 years, with
For example, assume you ence themselves. Suppose you in-
need a $200,000 mortgage and can vested $431 each month in a mu- attendant interest savings.
borrow at fixed rates. The shorter tual fund with an average annual
loan would carry a lower rate (be- return of 7 percent. At the end of
Sources: Daniela Deane, “Adding Up Pros,
cause it presents less risk for the 15 years, your $77,580 investment Cons of 15-Year Loans,” Washington Post
lender). The accompanying table would have grown to $136,611, or (October 13, 2001), p. H7; Henry Savage, “Is
15-Year Loan Right for You?” Washington
shows how the two mortgages $59,031 more than you contributed! Times (June 22, 2001), p. F22; Carlos Tejada,
compare: The extra $431 a month, However, many people lack the “Sweet Fifteen: Shorter Mortgages Are Gain-
ing Support,” Wall Street Journal (Septem-
or a total of $77,580, saves $157,765 self-discipline to save rather than
ber 17, 1998), p. C1; Ann Tergesen, “It™s Time
in interest payments over the life spend that money. For them, the to Refinance . . . Again,” Business Week
of the loan, for net savings of 15-year mortgage represents (November 2, 1998), pp. 134“135.
$80,185! forced savings.




Table Use The first step in finding the interest or growth rate is to divide the
amount received in the earliest year (PV) by the amount received in the latest year
(FVn). Looking back at Equation 4.12, we see that this results in the present value
interest factor for a single amount for 4 years, PVIFi,4yrs, which is 0.822
($1,250 $1,520). The interest rate in Table A“2 associated with the factor clos-
est to 0.822 for 4 years is the interest or growth rate of Ray™s cash flows. In the
row for year 4 in Table A“2, the factor for 5 percent is 0.823”almost exactly the
0.822 value. Therefore, the interest or growth rate of the given cash flows is
approximately (to the nearest whole percent) 5%.13

Calculator Use Using the calculator, we treat the earliest value as a present
value, PV, and the latest value as a future value, FVn. (Note: Most calculators


13. To obtain more precise estimates of interest or growth rates, interpolation”a mathematical technique for esti-
WW mating unknown intermediate values”can be applied. For information on how to interpolate a more precise answer
W
in this example, see the book™s home page at www.aw.com/gitman.
165
CHAPTER 4 Time Value of Money


require either the PV or the FV value to be input as a negative number to cal-
Input Function
culate an unknown interest or growth rate. That approach is used here.) Using
1250 PV
the inputs shown at the left, you will find the interest or growth rate to be
1520 FV
5.01%, which is consistent with, but more precise than, the value found using
4 N
Table A“2.
CPT
Spreadsheet Use The interest or growth rate for the series of cash flows also can
I
be calculated as shown on the following Excel spreadsheet.
Solution
5.01




Another type of interest-rate problem involves finding the interest rate asso-
ciated with an annuity, or equal-payment loan.


Jan Jacobs can borrow $2,000 to be repaid in equal annual end-of-year amounts
EXAMPLE
of $514.14 for the next 5 years. She wants to find the interest rate on this loan.

Table Use Substituting PVA5 $2,000 and PMT $514.14 into Equation 4.24
and rearranging the equation to solve for PVIFAi,5yrs, we get

PVA5 $2,000
PVIFAi,5yrs 3.890 (4.26)
PMT $514.14

The interest rate for 5 years associated with the annuity factor closest to 3.890 in
Input Function
514.14 PMT
Table A“4 is 9%. Therefore, the interest rate on the loan is approximately (to the
2000 PV nearest whole percent) 9%.
5 N

Calculator Use (Note: Most calculators require either the PMT or the PV value
CPT
to be input as a negative number in order to calculate an unknown interest rate
I
on an equal-payment loan. That approach is used here.) Using the inputs shown
Solution
at the left, you will find the interest rate to be 9.00%, which is consistent with the
9.00
value found using Table A“4.
166 PART 2 Important Financial Concepts


Spreadsheet Use The interest or growth rate for the annuity also can be calcu-
lated as shown on the following Excel spreadsheet.




Finding an Unknown Number of Periods
Sometimes it is necessary to calculate the number of time periods needed to gener-
ate a given amount of cash flow from an initial amount. Here we briefly consider
this calculation for both single amounts and annuities. This simplest case is when
a person wishes to determine the number of periods, n, it will take for an initial de-
posit, PV, to grow to a specified future amount, FVn, given a stated interest rate, i.

Ann Bates wishes to determine the number of years it will take for her initial
EXAMPLE
$1,000 deposit, earning 8% annual interest, to grow to equal $2,500. Simply
stated, at an 8% annual rate of interest, how many years, n, will it take for Ann™s
$1,000, PV, to grow to $2,500, FVn?

Table Use In a manner similar to our approach above to finding an unknown
interest or growth rate in a series of cash flows, we begin by dividing the amount
deposited in the earliest year by the amount received in the latest year. This
results in the present value interest factor for 8% and n years, PVIF8%,n, which is
0.400 ($1,000 $2,500). The number of years (periods) in Table A“2 associated
with the factor closest to 0.400 for an 8% interest rate is the number of years
required for $1,000 to grow into $2,500 at 8%. In the 8% column of Table A“2,
the factor for 12 years is 0.397”almost exactly the 0.400 value. Therefore, the
number of years necessary for the $1,000 to grow to a future value of $2,500 at
8% is approximately (to the nearest year) 12.

Calculator Use Using the calculator, we treat the initial value as the present
Input Function
value, PV, and the latest value as the future value, FVn. (Note: Most calculators
1000 PV
require either the PV or the FV value to be input as a negative number to calcu-
2500 FV
late an unknown number of periods. That approach is used here.) Using the
8 I
inputs shown at the left, we find the number of periods to be 11.91 years,
CPT
which is consistent with, but more precise than, the value found above using
N Table A“2.
Solution
Spreadsheet Use The number of years for the present value to grow to a
11.91
specified future value also can be calculated as shown on the following Excel
spreadsheet.
167
CHAPTER 4 Time Value of Money




Another type of number-of-periods problem involves finding the number of
periods associated with an annuity. Occasionally we wish to find the unknown
life, n, of an annuity, PMT, that is intended to achieve a specific objective, such as
repaying a loan of a given amount, PVAn, with a stated interest rate, i.

Bill Smart can borrow $25,000 at an 11% annual interest rate; equal, annual
EXAMPLE
end-of-year payments of $4,800 are required. He wishes to determine how long it
will take to fully repay the loan. In other words, he wishes to determine how
many years, n, it will take to repay the $25,000, 11% loan, PVAn, if the pay-
ments of $4,800, PMT, are made at the end of each year.

Table Use Substituting PVAn $25,000 and PMT $4,800 into Equation 4.24
and rearranging the equation to solve PVIFA11%,n yrs, we get

PVAn $25,000
PVIFA11%,n yrs 5.208 (4.27)
PMT $4,800

The number of periods for an 11% interest rate associated with the annuity
Input Function
factor closest to 5.208 in Table A“4 is 8 years. Therefore, the number of peri-
4800 PMT
ods necessary to repay the loan fully is approximately (to the nearest year)
25000 PV
8 years.
11 I

Calculator Use (Note: Most calculators require either the PV or the PMT value
CPT
to be input as a negative number in order to calculate an unknown number of
N
periods. That approach is used here.) Using the inputs shown at the left, you will
Solution
find the number of periods to be 8.15, which is consistent with the value found
8.15
using Table A“4.

Spreadsheet Use The number of years to pay off the loan also can be calculated
as shown on the following Excel spreadsheet.
168 PART 2 Important Financial Concepts


Review Questions

4“15 How can you determine the size of the equal annual end-of-period deposits
necessary to accumulate a certain future sum at the end of a specified future
period at a given annual interest rate?
4“16 Describe the procedure used to amortize a loan into a series of equal peri-
odic payments.
4“17 Which present value interest factors would be used to find (a) the growth
rate associated with a series of cash flows and (b) the interest rate associ-
ated with an equal-payment loan?
4“18 How can you determine the unknown number of periods when you know
the present and future values”single amount or annuity”and the applic-
able rate of interest?




SUMMARY
FOCUS ON VALUE
Time value of money is an important tool that financial managers and other market partici-
pants use to assess the impact of proposed actions. Because firms have long lives and their
important decisions affect their long-term cash flows, the effective application of time-
value-of-money techniques is extremely important. Time value techniques enable financial
managers to evaluate cash flows occurring at different times in order to combine, compare,
and evaluate them and link them to the firm™s overall goal of share price maximization. It
will become clear in Chapters 6 and 7 that the application of time value techniques is a key
part of the value determination process. Using them, we can measure the firm™s value and
evaluate the impact that various events and decisions might have on it. Clearly, an under-
standing of time-value-of-money techniques and an ability to apply them are needed in
order to make intelligent value-creating decisions.




REVIEW OF LEARNING GOALS
agers rely primarily on present value techniques.
Discuss the role of time value in finance, the use
LG1
Financial tables, financial calculators, and comput-
of computational tools, and the basic patterns
ers and spreadsheets can streamline the application
of cash flow. Financial managers and investors use
of time value techniques. The cash flow of a firm
time-value-of-money techniques when assessing the
can be described by its pattern”single amount, an-
value of the expected cash flow streams associated
nuity, or mixed stream.
with investment alternatives. Alternatives can be as-
sessed by either compounding to find future value or
Understand the concepts of future and present
discounting to find present value. Because they are
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value, their calculation for single amounts, and
at time zero when making decisions, financial man-
169
CHAPTER 4 Time Value of Money


the relationship of present value to future value. future value of a mixed stream of cash flows is the
Future value relies on compound interest to mea- sum of the future values of each individual cash
sure future amounts: The initial principal or deposit flow. Similarly, the present value of a mixed stream
in one period, along with the interest earned on it, of cash flows is the sum of the present values of the
becomes the beginning principal of the following individual cash flows.
period. The present value of a future amount is the
amount of money today that is equivalent to the Understand the effect that compounding inter-
LG5
given future amount, considering the return that est more frequently than annually has on future
can be earned on the current money. Present value value and on the effective annual rate of interest.
is the inverse future value. The interest factor for- Interest can be compounded at intervals ranging
mulas and basic equations for both the future value from annually to daily, and even continuously. The
and the present value of a single amount are given more often interest is compounded, the larger the
in Table 4.9. future amount that will be accumulated, and the
higher the effective, or true, annual rate (EAR). The
Find the future value and the present value of annual percentage rate (APR)”a nominal annual
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an ordinary annuity and find the present value rate”is quoted on credit cards and loans. The an-
of a perpetuity. An annuity is a pattern of equal nual percentage yield (APY)”an effective annual
periodic cash flows. For an ordinary annuity, the rate”is quoted on savings products. The interest
cash flows occur at the end of the period. For an factor formulas for compounding more frequently
annuity due, cash flows occur at the beginning of than annually are given in Table 4.9.
the period. Only ordinary annuities are considered
in this book. The future value of an ordinary annu- Describe the procedures involved in (1) deter-
LG6
ity can be found by using the future value interest mining deposits to accumulate a future sum,
factor for an annuity; the present value of an ordi- (2) loan amortization, (3) finding interest or growth
nary annuity can be found by using the present rates, and (4) finding an unknown number of peri-
value interest factor for an annuity. The present ods. The periodic deposit to accumulate a given fu-
value of a perpetuity”an infinite-lived annuity”is ture sum can be found by solving the equation for
found using 1 divided by the discount rate to rep- the future value of an annuity for the annual pay-
resent the present value interest factor. The interest ment. A loan can be amortized into equal periodic
factor formulas and basic equations for the future payments by solving the equation for the present
value and the present value of an ordinary annuity value of an annuity for the periodic payment. Inter-
and the present value of a perpetuity, are given in est or growth rates can be estimated by finding the
Table 4.9. unknown interest rate in the equation for the pre-
sent value of a single amount or an annuity. Simi-
Calculate both the future value and the present larly, an unknown number of periods can be esti-
LG4
value of a mixed stream of cash flows. A mixed mated by finding the unknown number of periods
stream of cash flows is a stream of unequal periodic in the equation for the present value of a single
cash flows that reflect no particular pattern. The amount or an annuity.




SELF-TEST PROBLEMS (Solutions in Appendix B)
ST 4“1 Future values for various compounding frequencies Delia Martin has $10,000
LG5
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that she can deposit in any of three savings accounts for a 3-year period. Bank A
compounds interest on an annual basis, bank B compounds interest twice each
year, and bank C compounds interest each quarter. All three banks have a stated
annual interest rate of 4%.
170 PART 2 Important Financial Concepts


TABLE 4.9 Summary of Key Definitions, Formulas, and
Equations for Time Value of Money

Definitions of variables

e exponential function 2.7183
EAR effective annual rate
FVn future value or amount at the end of period n
FVAn future value of an n-year annuity
i annual rate of interest
m number of times per year interest is compounded
n number of periods”typically years”over which money earns a return
PMT amount deposited or received annually at the end of each year
PV initial principal or present value
PVAn present value of an n-year annuity
t period number index


Interest factor formulas

Future value of a single amount with annual compounding:
i)n
FVIFi,n (1 [Eq. 4.5; factors in Table A“1]
Present value of a single amount:
1
PVIFi,n [Eq. 4.11; factors in Table A“2]
(1 i)n
Future value of an ordinary annuity:
n
i)t 1
FVIFAi,n (1 [Eq. 4.13; factors in Table A“3]
t=1
Present value of an ordinary annuity:
n
1
PVIFAi,n [Eq. 4.15; factors in Table A“4]
t
t=1 (1 i)
Present value of a perpetuity:
1
PVIFAi,∞ [Eq. 4.17]
i
Future value with compounding more frequently than annually:
i mn
FVIFi,n 1 [Eq. 4.18]
m
∞:
for continuous compounding, m
ei n
FVIFi,n (continuous compounding) [Eq. 4.20]
to find the effective annual rate:
im
EAR 1 1 [Eq. 4.21]
m


Basic equations

Future value (single amount): FVn PV (FVIFi,n) [Eq. 4.6]
Present value (single amount): PV FVn (PVIFi,n) [Eq. 4.12]
Future value (annuity): FVAn PMT (FVIFAi,n) [Eq. 4.14]
Present value (annuity): PVAn PMT (PVIFAi,n) [Eq. 4.16]
171
CHAPTER 4 Time Value of Money


a. What amount would Ms. Martin have at the end of the third year, leaving all
interest paid on deposit, in each bank?
b. What effective annual rate (EAR) would she earn in each of the banks?
c. On the basis of your findings in parts a and b, which bank should Ms.
Martin deal with? Why?
d. If a fourth bank (bank D), also with a 4% stated interest rate, compounds
interest continuously, how much would Ms. Martin have at the end of the
third year? Does this alternative change your recommendation in part c?
Explain why or why not.

ST 4“2 Future values of annuities Ramesh Abdul wishes to choose the better of two
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equally costly cash flow streams: annuity X and annuity Y. X provides a cash
inflow of $9,000 at the end of each of the next 6 years. Y provides a cash inflow
of $10,000 at the end of each of the next 6 years. Assume that Ramesh can earn
15% on annuity X and 11% on annuity Y.
a. On a purely subjective basis, which annuity do you think is more attractive?
Why?
b. Find the future value at the end of year 6, FVA6, for both annuity X and
annuity Y.
c. Use your finding in part b to indicate which annuity is more attractive. Com-
pare your finding to your subjective response in part a.

ST 4“3 Present values of single amounts and streams You have a choice of accepting
LG2 LG3 LG4
either of two 5-year cash flow streams or single amounts. One cash flow stream
is an ordinary annuity, and the other is a mixed stream. You may accept alterna-
tive A or B”either as a cash flow stream or as a single amount. Given the cash
flow stream and single amounts associated with each (see the accompanying
table), and assuming a 9% opportunity cost, which alternative (A or B) and in
which form (cash flow stream or single amount) would you prefer?

Cash flow stream
End of year Alternative A Alternative B

1 $700 $1,100
2 700 900
3 700 700
4 700 500
5 700 300

Single amount

At time zero $2,825 $2,800



ST 4“4 Deposits needed to accumulate a future sum Judi Janson wishes to accumulate
LG6
$8,000 by the end of 5 years by making equal annual end-of-year deposits over
the next 5 years. If Judi can earn 7% on her investments, how much must she
deposit at the end of each year to meet this goal?
172 PART 2 Important Financial Concepts


PROBLEMS
4“1 Using a time line The financial manager at Starbuck Industries is considering
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an investment that requires an initial outlay of $25,000 and is expected to result
in cash inflows of $3,000 at the end of year 1, $6,000 at the end of years 2 and
3, $10,000 at the end of year 4, $8,000 at the end of year 5, and $7,000 at the
end of year 6.
a. Draw and label a time line depicting the cash flows associated with Starbuck
Industries™ proposed investment.
b. Use arrows to demonstrate, on the time line in part a, how compounding to
find future value can be used to measure all cash flows at the end of year 6.
c. Use arrows to demonstrate, on the time line in part b, how discounting to
find present value can be used to measure all cash flows at time zero.
d. Which of the approaches”future value or present value”do financial man-
agers rely on most often for decision making? Why?

4“2 Future value calculation Without referring to tables or to the preprogrammed
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function on your financial calculator, use the basic formula for future value
along with the given interest rate, i, and the number of periods, n, to calculate
the future value interest factor in each of the cases shown in the following table.
Compare the calculated value to the value in Appendix Table A“1.

Case Interest rate, i Number of periods, n

A 12% 2
B 6 3
C 9 2
D 3 4



4“3 Future value tables Use the future value interest factors in Appendix Table A“1
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in each of the cases shown in the following table to estimate, to the nearest year,
how long it would take an initial deposit, assuming no withdrawals,
a. To double.
b. To quadruple.

Case Interest rate

A 7%
B 40
C 20
D 10



4“4 Future values For each of the cases shown in the following table, calculate the
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future value of the single cash flow deposited today that will be available at the
end of the deposit period if the interest is compounded annually at the rate speci-
fied over the given period.
173
CHAPTER 4 Time Value of Money


Case Single cash flow Interest rate Deposit period (years)

A $ 200 5% 20
B 4,500 8 7
C 10,000 9 10
D 25,000 10 12
E 37,000 11 5
F 40,000 12 9




4“5 Future value You have $1,500 to invest today at 7% interest compounded
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annually.
a. Find how much you will have accumulated in the account at the end of
(1) 3 years, (2) 6 years, and (3) 9 years.
b. Use your findings in part a to calculate the amount of interest earned in
(1) the first 3 years (years 1 to 3), (2) the second 3 years (years 4 to 6), and
(3) the third 3 years (years 7 to 9).
c. Compare and contrast your findings in part b. Explain why the amount of
interest earned increases in each succeeding 3-year period.


4“6 Inflation and future value As part of your financial planning, you wish to pur-
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chase a new car exactly 5 years from today. The car you wish to purchase costs
$14,000 today, and your research indicates that its price will increase by 2% to
4% per year over the next 5 years.
a. Estimate the price of the car at the end of 5 years if inflation is (1) 2% per
year, and (2) 4% per year.
b. How much more expensive will the car be if the rate of inflation is 4% rather
than 2%?

4“7 Future value and time You can deposit $10,000 into an account paying 9%
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annual interest either today or exactly 10 years from today. How much better
off will you be at the end of 40 years if you decide to make the initial deposit
today rather than 10 years from today?


4“8 Single-payment loan repayment A person borrows $200 to be repaid in 8 years
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with 14% annually compounded interest. The loan may be repaid at the end of
any earlier year with no prepayment penalty.
a. What amount will be due if the loan is repaid at the end of year 1?
b. What is the repayment at the end of year 4?
c. What amount is due at the end of the eighth year?


4“9 Present value calculation Without referring to tables or to the preprogrammed
LG2
function on your financial calculator, use the basic formula for present value,
along with the given opportunity cost, i, and the number of periods, n, to calcu-
late the present value interest factor in each of the cases shown in the accompa-
nying table. Compare the calculated value to the table value.
174 PART 2 Important Financial Concepts


Opportunity Number of
Case cost, i periods, n

A 2% 4
B 10 2
C 5 3
D 13 2



4“10 Present values For each of the cases shown in the following table, calculate the
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present value of the cash flow, discounting at the rate given and assuming that
the cash flow is received at the end of the period noted.

Single cash End of
Case flow Discount rate period (years)

A $ 7,000 12% 4
B 28,000 8 20
C 10,000 14 12
D 150,000 11 6
E 45,000 20 8



4“11 Present value concept Answer each of the following questions.
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a. What single investment made today, earning 12% annual interest, will be
worth $6,000 at the end of 6 years?
b. What is the present value of $6,000 to be received at the end of 6 years if the
discount rate is 12%?
c. What is the most you would pay today for a promise to repay you $6,000 at
the end of 6 years if your opportunity cost is 12%?
d. Compare, contrast, and discuss your findings in parts a through c.

4“12 Present value Jim Nance has been offered a future payment of $500 three years
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from today. If his opportunity cost is 7% compounded annually, what value
should he place on this opportunity today? What is the most he should pay to
purchase this payment today?

4“13 Present value An Iowa state savings bond can be converted to $100 at maturity
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6 years from purchase. If the state bonds are to be competitive with U.S. Savings
Bonds, which pay 8% annual interest (compounded annually), at what price
must the state sell its bonds? Assume no cash payments on savings bonds prior
to redemption.

4“14 Present value and discount rates You just won a lottery that promises to pay
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you $1,000,000 exactly 10 years from today. Because the $1,000,000 payment
is guaranteed by the state in which you live, opportunities exist to sell the claim
today for an immediate single cash payment.
a. What is the least you will sell your claim for if you can earn the following
rates of return on similar-risk investments during the 10-year period?
(1) 6% (2) 9% (3) 12%
175
CHAPTER 4 Time Value of Money


b. Rework part a under the assumption that the $1,000,000 payment will be
received in 15 rather than 10 years.
c. On the basis of your findings in parts a and b, discuss the effect of both the
size of the rate of return and the time until receipt of payment on the present
value of a future sum.

4“15 Present value comparisons of single amounts In exchange for a $20,000 pay-
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ment today, a well-known company will allow you to choose one of the alterna-
tives shown in the following table. Your opportunity cost is 11%.


Alternative Single amount

A $28,500 at end of 3 years
B $54,000 at end of 9 years
C $160,000 at end of 20 years



a. Find the value today of each alternative.
b. Are all the alternatives acceptable, i.e., worth $20,000 today?
c. Which alternative, if any, will you take?

4“16 Cash flow investment decision Tom Alexander has an opportunity to purchase
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any of the investments shown in the following table. The purchase price, the
amount of the single cash inflow, and its year of receipt are given for each invest-
ment. Which purchase recommendations would you make, assuming that Tom
can earn 10% on his investments?


Investment Price Single cash inflow Year of receipt

A $18,000 $30,000 5
B 600 3,000 20
C 3,500 10,000 10
D 1,000 15,000 40



4“17 Future value of an annuity For each of the cases shown in the following
LG3
table, calculate the future value of the annuity at the end of the deposit
period, assuming that the annuity cash flows occur at the end of each
year.


Amount of Interest Deposit period
Case annuity rate (years)

A $ 2,500 8% 10
B 500 12 6
C 30,000 20 5
D 11,500 9 8
E 6,000 14 30
176 PART 2 Important Financial Concepts


4“18 Present value of an annuity For each of the cases shown in the table below, cal-
LG3
culate the present value of the annuity, assuming that the annuity cash flows
occur at the end of each year.



Case Amount of annuity Interest rate Period (years)

A $ 12,000 7% 3
B 55,000 12 15
C 700 20 9
D 140,000 5 7
E 22,500 10 5




4“19 Future value of a retirement annuity Hal Thomas, a 25-year-old college
LG3
graduate, wishes to retire at age 65. To supplement other sources of
retirement income, he can deposit $2,000 each year into a tax-deferred in-
dividual retirement arrangement (IRA). The IRA will be invested to earn an
annual return of 10%, which is assumed to be attainable over the next
40 years.
a. If Hal makes annual end-of-year $2,000 deposits into the IRA, how much
will he have accumulated by the end of his 65th year?
b. If Hal decides to wait until age 35 to begin making annual end-of-year
$2,000 deposits into the IRA, how much will he have accumulated by the end
of his 65th year?
c. Using your findings in parts a and b, discuss the impact of delaying making
deposits into the IRA for 10 years (age 25 to age 35) on the amount accumu-
lated by the end of Hal™s 65th year.


4“20 Present value of a retirement annuity An insurance agent is trying to sell you
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an immediate-retirement annuity, which for a single amount paid today will pro-
vide you with $12,000 at the end of each year for the next 25 years. You cur-
rently earn 9% on low-risk investments comparable to the retirement annuity.
Ignoring taxes, what is the most you would pay for this annuity?


4“21 Funding your retirement You plan to retire in exactly 20 years. Your goal is to
LG3
LG2
create a fund that will allow you to receive $20,000 at the end of each year for
the 30 years between retirement and death (a psychic told you would die after
30 years). You know that you will be able to earn 11% per year during the 30-
year retirement period.
a. How large a fund will you need when you retire in 20 years to provide the
30-year, $20,000 retirement annuity?
b. How much will you need today as a single amount to provide the fund calcu-
lated in part a if you earn only 9% per year during the 20 years preceding
retirement?
177
CHAPTER 4 Time Value of Money


c. What effect would an increase in the rate you can earn both during
and prior to retirement have on the values found in parts a and b?
Explain.


LG3
LG2 4“22 Present value of an annuity versus a single amount Assume that you just won
the state lottery. Your prize can be taken either in the form of $40,000 at the
end of each of the next 25 years (i.e., $1,000,000 over 25 years) or as a single
amount of $500,000 paid immediately.
a. If you expect to be able to earn 5% annually on your investments over the
next 25 years, ignoring taxes and other considerations, which alternative
should you take? Why?
b. Would your decision in part a change if you could earn 7% rather than 5%
on your investments over the next 25 years? Why?
c. On a strictly economic basis, at approximately what earnings rate would you
be indifferent between the two plans?


4“23 Perpetuities Consider the data in the following table.
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Perpetuity Annual amount Discount rate

A $ 20,000 8%
B 100,000 10
C 3,000 6
D 60,000 5




Determine, for each of the perpetuities:
a. The appropriate present value interest factor.
b. The present value.


4“24 Creating an endowment Upon completion of her introductory finance
LG3
course, Marla Lee was so pleased with the amount of useful and interesting
knowledge she gained that she convinced her parents, who were wealthy
alums of the university she was attending, to create an endowment. The endow-
ment is to allow three needy students to take the introductory finance course
each year in perpetuity. The guaranteed annual cost of tuition and books for the
course is $600 per student. The endowment will be created by making a single
payment to the university. The university expects to earn exactly 6% per year
on these funds.
a. How large an initial single payment must Marla™s parents make to the univer-
sity to fund the endowment?
b. What amount would be needed to fund the endowment if the university
could earn 9% rather than 6% per year on the funds?
178 PART 2 Important Financial Concepts


4“25 Future value of a mixed stream For each of the mixed streams of cash flows
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shown in the following table, determine the future value at the end of the final
year if deposits are made at the beginning of each year into an account paying
annual interest of 12%, assuming that no withdrawals are made during the
period.


Cash flow stream
Year A B C

1 $ 900 $30,000 $1,200
2 1,000 25,000 1,200
3 1,200 20,000 1,000
4 10,000 1,900
5 5,000



4“26 Future value of a single amount versus a mixed stream Gina Vitale has just
LG4
contracted to sell a small parcel of land that she inherited a few years ago. The
buyer is willing to pay $24,000 at the closing of the transaction or will pay the

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