Time Value of Money AssignmentResource & Review The
time value of money is one of the most important concepts in finance.
So many decisions are made based upon this concept. This assignment
provides an opportunity for learning and practicing some of the basic
time value of money formulas. The time value of money is based
upon the premise that a dollar today is not worth the same as a dollar
in the future or in the past. The following examples use a
HP-12C financial calculator. You might have to modify for different
calculators. However, you can get the same answer by using tables or
Excel. Note: Excel is required for this
assignment. There are two Excel help files provided for your
convenience. Please use the attached spreadsheet provide by my
instructor. If you put $500 in a savings account today at 12% interest, what will it be worth in 5 years?Try the same problem compounding interest on a quarterly basis.You
want $10,000 in your account at the end of 10 years. What do you have
to put into your account today in order to achieve your goal if your
account pays 8% interest?Let’s try the number 3. Take your answer (PV), FV, and n — and try to solve for i. You should get the answer of 8.If you put $500 in a savings account at the end of each year at 12% interest, what will it be worth in 5 years?If you put $10,000 in your account at the end of every year for 10 years, what is it worth now if your account pays 8% interest? Hint: This is a future value problem.Calculator keystrokes: 500 press PV 12 press i 5 press n to solve ___ press FVHint: This is a present value problem.Calculator keystrokes: 10,000 press FV 8 press i 10 press n to solve ___ press PVAnother
Hint: Be sure to clear everything out of financial memory before
continuing. (f key, then the fin key; note the clx key only clears the
register, not the memory.)Hint: When solving for i or n, be sure to enter either the FV or the PV as a negative or you will get an error.With the above items, we have computed two of the four basic time value of money problemsfuture value and present value. The
next two problems are similarthey are called annuities. In a story
problem, it is easy to detect an annuity. You should look for the words
“each” or “every.” See if you can detect them in problems 5 and 6, which
are similar to Problems 1 through 4.Hint: This is a future value of an annuity problem.Calculator keystrokes: 500 press PMT 12 press i 5 press n to solve ___ press FVHint: This is a present value of an annuity problem.Calculator keystrokes: 10,000 press PMT 8 press i 10 press n to solve ___ press PVThere are different ways to solve for these types of problems.Calculation AnalysisThe
above examples were given in a calculator format, where it is easy to
see what is required. Now, calculate the above 6 items in Microsoft
Excel. Note: Use the two Excel help files to assist you in understanding the Time Value of Money Process.Written AnalysisAnswer the following questions in the Excel Spreadsheet after completing the computations: Why is the time value of money concept important to a nonprofits fiscal health?If
you were the executive director of a small nonprofit offering
programmatic technical assistance to other nonprofits, when would you
use cost-benefit analysis?
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Model for Time Value of Money Analysis
1. First, recognize that you do not have to use this model at all to understand time value of money concepts. However, if you do use
the model and experiment with it, this will increase your understanding of the concepts.
2. Start by reading the chapter and working the examples as you come to them with a calculator.
3. When you come to an explanation of spreadsheet calculations in the chapter, access this file, 07model.xls, which was used to
produce the information printed in the chapter.
4. We assume that you know the basics of Excel, but that you have not encountered some of its features and that you may need a
refresher on others. So, we have built in explanations of how to do some of the functions in the model. As a result, you will learn
more about Excel at the same time you learn about the time value of money.
5. Throughout this model, page numbers are provided, in green, indicating about how many pages into the chapter that particular
problem can be found. The models were completed before the text was set in final form, so we could not give actual text page
numbers. In addition, some data will appear in blue, which signifies that it is input data used in formulas. You are encouraged to
play with and change this input data to see the effect on the result. All solutions generated by function wizards appear in red.
FUTURE VALUE
In this section we show the model used to obtain the spreadsheet information printed in the text.
PROBLEM
Find the FV of $100 after five years at an interest rate of 5%.
Interest rate
Cash flow
0.05
100
Time period
FV at year end
0
100
These are the basic inputs, in blue.
1
105.00
2
110.25
3
115.76
4
121.55
5
127.63
Note: This problem was solved using the formula, FVn = PV (1+I)n. However, there are a number of ways this problem could have
been solved. One of the most valuable features in Excel is the “Function” Wizard. Here is how to access and use the “Function
Wizard”.
First, you must select the Function wizard icon found in the toolbar at the top of the screen, which looks like this
. This button
allows you to enter the function wizard. Upon clicking it, you should be encountered with a dialog box entitled, “Paste Function”.
On the left side of the dialog box is a menu entitled, “Function category”, and on the right is a menu called “Function name”.
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We will be selecting the “FV” function from the “Financial” category, and will be using the following dialog box to input our data.
Notice that we entered a cell reference as the input for the problem instead of the actual value. We do this so that our spreadsheet
can automatically reflect any changes to the input data. This is one of the features that makes the spreadsheet such a valuable tool.
Using the function wizard will yield the following result:
FV =
$127.63
=FV(B35,G38,,-B36)
TABLE
Future Value Interest Factors
With a spreadsheet, calculating FVIF’s is a simple operation, and we can use it to graph the relationship between future value,
growth, interest rates, and time. A similar table can be found in the textbook, along with a corresponding graph.
Period (n)
0
2
4
6
8
10
0%
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
5%
1.0000
1.1025
1.2155
1.3401
1.4775
1.6289
10%
1.0000
1.2100
1.4641
1.7716
2.1436
2.5937
15%
1.0000
1.3225
1.7490
2.3131
3.0590
4.0456
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GRAPH
Relationships among Future Value, Growth, Interest Rates, and Time
Relationships among Future Value, Growth, Interest
Rate, and Time
$5.00
Future Value of $1
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$4.00
$3.00
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$1.00
$0.00
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12
Periods
To create a graph in Excel, you first must access the “Chart Wizard” found in the toolbar near the top of
the screen denoted by
. Upon selecting the chart wizard, the first input dialog box will appear,
and it will ask for the “Chart Type”. In this case, we want a line graph, so we select “Line,” and then we
click one of the sub-types, in this case to first one.
After clicking “Next”, we are presented with the “Source Data” box. We first enter the “Data Range” for the chart. This is the
data that will comprise the lines in our line graph. For our example, this information can be found in the above table. To select the
data range, use the cursor to highlight the cells range from B99 to E104. Note that the data are contained in columns.
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Now, we must select the “Series” for our graph. Essentially, this is the set of data that will comprise each line in the graph. In this
graph, you can name each individual series by selecting it in the Series section, and typing a label into the Name box. When an
individual series is highlighted, the Values section will let you know what data from the data range make up that particular line. To
place appropriate labels on the X axis, you must go to the Category (X) axis labels box and highlight from A109 to A114.
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At this point, all of the necessary data for the chart have been inserted. From here, the chart just needs to be formatted according to
your preferences (e.g. show or hide gridlines, change the numbers on the axes, etc).
PRESENT VALUE
Simply put, the present value is the value today of some future cash flow (or series of cash flows). The interest rate used to
“discount” a given cash flow is the opportunity cost rate, and is equivalent to the next best investment alternative of the same risk.
PROBLEM
Find the PV of $127.63 discounted back five years at an interest rate of 5%.
Interest rate
Cash flow
0.05
127.63
Time period
Present Value
0
127.63
Number of Years Discounted Back
1
2
3
121.55
115.76
110.25
4
105.00
5
100.00
This problem can also be solved using the function wizard using a procedure similar to that for the FV
function. Begin by putting the pointer on the cell in which you want to display the result, in this case
Cell C263. Then, after selecting the “PV” function from the “Paste Function” box, the input data for the
problem must be entered. Then click OK to get the result, $100.
PV =
$100.00
SOLVING FOR I
PROBLEM
What is the interest rate of a security priced at $78.35 that pays $100 after five years?
N
PV
FV
5
78.35
100
Once again, Excel has a special function for this calculation. We suggest using either a financial calculator or the function wizard
to solve this type of problem, because of its complexity. The procedure can be carried out using the function wizard, by selecting the
“Rate” function from the list of financial functions in the “Paste Function” dialog box. Upon entering the time, present value, and
future value, the interest rate can be found. Note that you can either type the data in or else activate the menu slot and then click on
the appropriate cell.
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I =
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5.00%
We noted above the difficulty of solving this problem mathematically. This is because it involves taking the nth root of a value (an
operation which generally requires either a calculator or a computer). However, if you would like to know how to solve the problem
mathematically, the formula is: (FVn/PV)1/ n – 1, which is derived from the FV formula.
N
PV
FV
5
78.35
100
I =
5.00%
SOLVING FOR N
PROBLEM
A security yields 5%, costs $78.35 today, and will return $100 at some future date. What is the security’s term to maturity?
I
PV
FV
0.05
78.35
100
The function wizard to solve for time, or N. This operation can be performed by selecting the “Nper” option from the list of financial
functions, and then entering the input data into the dialog box.
N =
5.00
Solving for N mathematically is very complicated. The formula for finding N involves using natural logarithms, which is a complex
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operation. For this reason, we highly suggest the use of the function wizard or financial calculator to solve this type of problem.
However, here is the formula needed to solve for N is: N = (ln ( |FVn / PV| )/ (ln (1+ I)):
I
PV
FV
0.05
78.35
100
N =
5.00
FUTURE VALUE OF AN ANNUITY
PROBLEM
If the interest rate is 5%, what is the future value of an ordinary annuity that pays $100 at the end of each of the next three years?
As explained below, one way to solve this problem is to find the future value of each of the annuity payments. However, this is
somewhat tedious, especially if a lot of years are involved. In the following example, we use the input data of the interest rate and
time to calculate the future value in time period 3 of each individual cash flow. Lastly, we take the sum of all the future values, which
gives us the future value of the entire annuity.
N
I
PMT
3
0.05
100
Time period
CFn
FV3
0
0
0
1
100
110.25
2
100
105
3
100
100
=
Annuity’s FV:
$315.25
An easier procedure is to solving for the future value of an annuity with the function wizard. This procedure is similar to
that of a lump sum future value. Whereas before we left the “Pmt” field blank, now we insert the annuity payment ($100 in
this case). First, we access the “FV” function box from the list of financial functions. Then, we input our new data. A key
thing to watch is the “Type” input box. Previously, we left this box alone. A “0” or no entry in the box indicates an
ordinary annuity, and a “1” indicates an annuity due. Though we can leave the box blank, it is a good habit to enter a “0”
in the field.
FV =
$315.25
PROBLEM
If the interest rate is 5%, what is the future value of an annuity due that pays $100 at the beginning of each of the next three years?
The procedure for solving this problems follows the previous example with one notable exception. Since, the payments occur at the
beginning of each year, the first annuity payment occurs in time period 0, and the last occurs in time period 2.
N
I
PMT
3
0.05
100
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CFn
FV3
C
0
100
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1
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110.25
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2
100
105
3
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G
=
Annuity FV
$331.01
H
I
Additionally, using the function wizard for this problem is exactly like above, but we enter a “1” instead of a “0” into the “Type”
field.
FV =
$331.01
PRESENT VALUE OF AN ANNUITY
If you were given the option of receiving a lump sum of money today or an annuity that pays $100 at the end of each of the next three
years, at what price should you be indifferent to the two options, if the interest rate is 5%?
The way to solve this problem is to find the PV of the annuity and then compare it with the lump sum. First, we consider each
payment separately .
N
I
PMT
Time period
CFn
PV3
3
0.05
100
0
0
0
1
100
95.24
2
100
90.70
3
100
86.38
=
Annuity PV
$272.32
Or, you could use the function wizard for this ordinary annuity.
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PV =
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$272.32
If you were given the option between a lump sum of money today or an annuity that pays $100 at the beginning of each of the next
three years, at what price would you be indifferent to the two options, if the interest rate is 5%?
This problem is solved just like the previous problem, except that the payments occur in periods 0 through 2.
N
I
PMT
3
0.05
100
Time period
CFn
PV3
0
100
100.00
1
100
95.24
2
100
90.70
3
0
0.00
Annuity PV
$285.94
=
Using the function wizard, we follow the same procedure as above, except remember to enter a “1” to tell Excel that this problem has
payments occurring at the beginning of the periods.
PV =
$285.94
PERPETUITIES
PROBLEM
What is a perpetuity worth if it pays $100 every year and the discount rate is 5%?
PMT
I
100
0.05
PV(Perpetuity)=
$2,000
PV(Perpetuity)=
$1,000
if the discount rate is 10%?
PMT
I
100
0.1
UNEVEN CASH FLOWS
Calculate the present value of the following cash flow stream, discounted at 6%.
1
2
3
4
I =
0.06
5
Time period
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0
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7
0
100
200
200
200
200
0
1000
94.34
178.00
167.92
158.42
149.45
0
665.06
I
Cash Flows
PV of Cash Flows
0
NPV =
$
1,413.19
As we show above, the first way to solve for the present value of this uneven cash flow stream is to use
the time line to find the present value of each of the cash flows in the periods in which they occur, then
sum all the present values. This procedure will yield the correct present value.
This problem could also be set up in a column format; it is a matter of personal preference as to which format is easier to
interpret and use. Once we have converted our data into a data table, we can solve for the present value of each of the
cash flows (like we did previously) and add all of the present values together.
I
0.06
N
0
1
2
3
4
5
6
7
CFn
0
100
200
200
200
200
0
1000
PV0
0
94.34
178.00
167.92
158.42
149.45
0
665.06
PV of CF stream
$1,413.19
With, the financial calculator, we could enter each of these cash flows and the discount rate, and simply press NPV for the present
value of the cash flow stream. In Excel, we can perform a similar calculation by using the “NPV” function. While this function is
very similar, there is a key distinction. In the cash flow register of your calculator, the first entry you make would be the
cash flow to occur in time period zero. However, the “NPV” function interprets the first data entry as being the cash flow in time
period one. Therefore, the initial cash flow must be added seperately. In this particular example, the initial cash flow is zero.
Data from either
the time line or
the columns
could be entered
here.
PV =
$1,413.19
Both methods of entry will yield the same result, so use the one that you are most comfortable with. In the
event that you have a problem consisting of a cash flow at time period zero, you will have to manually add this
value to the NPV of the remaining cashflows. Also note that when using the function wizard, we used
the cash flow data from the data table of cash flows. However, we could have just as easily used the time
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line we previously established.
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PROBLEM
Calculate the future value of the cash flow stream illustrated above in the previous question.
First, we will solve this problem by adding the future values of all the cash flows in time period 7.
N
I
7
0.06
n
0
1
2
3
4
5
6
7
# compoundings
(N-n)
7
6
5
4
3
2
1
0
CFn
0
100
200
200
200
200
0
1000
FV7
0
141.85
267.65
252.50
238.20
224.72
0
1000.00
FV of CF stream
$2,124.92
Excel does not have a net future value function, but the above procedure works for this type of problem.
An alternative is to calculate the NPV and then compound that value out to the end of the cash flow
stream, but that procedure is less easy to follow.
SEMIANNUAL AND OTHER COMPOUNDING PERIODS
PROBLEM
If $100 is invested in an account at an interest rate of 6%, annual compounding, for 3 years, what is the FV?
N
I
PV
3
0.06
100
FV =
$119.10
$0.30
, what is the FV with semiannual compounding?
N (years x 2)
I (I per year/2)
PV
6
0.03
100
FV =
$119.41
What is the PV of an ordinary annuity of $100 per year for three years when the interest rate is 8%, compounded annually?
N
I
PMT
3
0.08
100
PV =
$257.71
What is the PV of an ordinary annuity of $100 per year for three years when the interest rate is 8%, compounded semiannually?
N
I
PMT
6
0.04
50
FV =
$262.11
Remember that in cases of non-annual compounding, all input variables (N, I, PMT) must reflect the number of compounding periods.
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PROBLEM
What is the FV of an investment account if you make eight quarterly payments of $100 and the account pays 12% interest,
compounded quarterly?
N
I
PMT
8
0.03
100
FV =
I
$889.23
AMORTIZED LOANS
What would the required payment be on a $1,000 loan that is to be repaid in three equal installments at the end of each of the next
three years if the interest rate is 6%?
N
I
PV
3
0.06
1000
PMT =
$374.11
Now, construct an amortization table for the loan described above.
N
1
2
3
Loan amount
$1,000.00
$685.89
$352.93
Payment
$374.11
$374.11
$374.11
Interest
$60.00
$41.15
$21.18
Principal
$314.11
$332.96
$352.93
Balance
$685.89
$352.93
$0.00
Total payments Total int. paid Total prin. Paid
$1,122.33
$122.33
$1,000.00
Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.
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ecting276
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g periods.
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Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.
Tables of Time Value Factors
TABLE 1: Future Value of $1 = (1 + interest)^n == FV factor (single sum, i%, n periods)
Shorthand notation for our course for this factor is FVSS(i%, n periods).
interest i
1.0%
2.0%
2.5%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
15.0%
1.01000
1.02010
1.03030
1.04060
1.05101
1.02000
1.04040
1.06121
1.08243
1.10408
1.02500
1.05063
1.07689
1.10381
1.13141
1.03000
1.06090
1.09273
1.12551
1.15927
1.04000
1.08160
1.12486
1.16986
1.21665
1.05000
1.10250
1.15763
1.21551
1.27628
1.06000
1.12360
1.19102
1.26248
1.33823
1.07000
1.14490
1.22504
1.31080
1.40255
1.08000
1.16640
1.25971
1.36049
1.46933
1.09000
1.18810
1.29503
1.41158
1.53862
1.10000
1.21000
1.33100
1.46410
1.61051
1.11000
1.23210
1.36763
1.51807
1.68506
1.12000
1.25440
1.40493
1.57352
1.76234
1.15000
1.32250
1.52088
1.74901
2.01136
1.06152
1.07214
1.08286
1.09369
1.10462
1.12616
1.14869
1.17166
1.19509
1.21899
1.15969
1.18869
1.21840
1.24886
1.28008
1.19405
1.22987
1.26677
1.30477
1.34392
1.26532
1.31593
1.36857
1.42331
1.48024
1.34010
1.40710
1.47746
1.55133
1.62889
1.41852
1.50363
1.59385
1.68948
1.79085
1.50073
1.60578
1.71819
1.83846
1.96715
1.58687
1.71382
1.85093
1.99900
2.15892
1.67710
1.82804
1.99256
2.17189
2.36736
1.77156
1.94872
2.14359
2.35795
2.59374
1.87041
2.07616
2.30454
2.55804
2.83942
1.97382
2.21068
2.47596
2.77308
3.10585
2.31306
2.66002
3.05902
3.51788
4.04556
1.11567
1.12683
1.13809
1.14947
1.16097
1.24337
1.26824
1.29361
1.31948
1.34587
1.31209
1.34489
1.37851
1.41297
1.44830
1.38423
1.42576
1.46853
1.51259
1.55797
1.53945
1.60103
1.66507
1.73168
1.80094
1.71034
1.79586
1.88565
1.97993
2.07893
1.89830
2.01220
2.13293
2.26090
2.39656
2.10485
2.25219
2.40985
2.57853
2.75903
2.33164
2.51817
2.71962
2.93719
3.17217
2.58043
2.81266
3.06580
3.34173
3.64248
2.85312
3.13843
3.45227
3.79750
4.17725
3.15176
3.49845
3.88328
4.31044
4.78459
3.47855
3.89598
4.36349
4.88711
5.47357
4.65239
5.35025
6.15279
7.07571
8.13706
1.17258
1.18430
1.19615
1.20811
1.22019
1.37279
1.40024
1.42825
1.45681
1.48595
1.48451
1.52162
1.55966
1.59865
1.63862
1.60471
1.65285
1.70243
1.75351
1.80611
1.87298
1.94790
2.02582
2.10685
2.19112
2.18287
2.29202
2.40662
2.52695
2.65330
2.54035
2.69277
2.85434
3.02560
3.20714
2.95216
3.15882
3.37993
3.61653
3.86968
3.42594
3.70002
3.99602
4.31570
4.66096
3.97031
4.32763
4.71712
5.14166
5.60441
4.59497
5.05447
5.55992
6.11591
6.72750
5.31089
5.89509
6.54355
7.26334
8.06231
6.13039
6.86604
7.68997
8.61276
9.64629
9.35762
10.76126
12.37545
14.23177
16.36654
periods n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Page 1 of 5
Tables of Time Value Factors
TABLE 2:
interest i
1.0%
2.0%
Present Value of $1 = 1/((1 + interest)^n) == PV factor (single sum, i%, n periods)
Shorthand notation for our course for this factor is PVSS(i%, n periods).
2.5%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
15.0%
periods n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.99010
0.98030
0.97059
0.96098
0.95147
0.98039
0.96117
0.94232
0.92385
0.90573
0.97561
0.95181
0.92860
0.90595
0.88385
0.97087
0.94260
0.91514
0.88849
0.86261
0.96154
0.92456
0.88900
0.85480
0.82193
0.95238
0.90703
0.86384
0.82270
0.78353
0.94340
0.89000
0.83962
0.79209
0.74726
0.93458
0.87344
0.81630
0.76290
0.71299
0.92593
0.85734
0.79383
0.73503
0.68058
0.91743
0.84168
0.77218
0.70843
0.64993
0.90909
0.82645
0.75131
0.68301
0.62092
0.90090
0.81162
0.73119
0.65873
0.59345
0.89286
0.79719
0.71178
0.63552
0.56743
0.86957
0.75614
0.65752
0.57175
0.49718
0.94205
0.93272
0.92348
0.91434
0.90529
0.88797
0.87056
0.85349
0.83676
0.82035
0.86230
0.84127
0.82075
0.80073
0.78120
0.83748
0.81309
0.78941
0.76642
0.74409
0.79031
0.75992
0.73069
0.70259
0.67556
0.74622
0.71068
0.67684
0.64461
0.61391
0.70496
0.66506
0.62741
0.59190
0.55839
0.66634
0.62275
0.58201
0.54393
0.50835
0.63017
0.58349
0.54027
0.50025
0.46319
0.59627
0.54703
0.50187
0.46043
0.42241
0.56447
0.51316
0.46651
0.42410
0.38554
0.53464
0.48166
0.43393
0.39092
0.35218
0.50663
0.45235
0.40388
0.36061
0.32197
0.43233
0.37594
0.32690
0.28426
0.24718
0.89632
0.88745
0.87866
0.86996
0.86135
0.80426
0.78849
0.77303
0.75788
0.74301
0.76214
0.74356
0.72542
0.70773
0.69047
0.72242
0.70138
0.68095
0.66112
0.64186
0.64958
0.62460
0.60057
0.57748
0.55526
0.58468
0.55684
0.53032
0.50507
0.48102
0.52679
0.49697
0.46884
0.44230
0.41727
0.47509
0.44401
0.41496
0.38782
0.36245
0.42888
0.39711
0.36770
0.34046
0.31524
0.38753
0.35553
0.32618
0.29925
0.27454
0.35049
0.31863
0.28966
0.26333
0.23939
0.31728
0.28584
0.25751
0.23199
0.20900
0.28748
0.25668
0.22917
0.20462
0.18270
0.21494
0.18691
0.16253
0.14133
0.12289
0.85282
0.84438
0.83602
0.82774
0.81954
0.72845
0.71416
0.70016
0.68643
0.67297
0.67362
0.65720
0.64117
0.62553
0.61027
0.62317
0.60502
0.58739
0.57029
0.55368
0.53391
0.51337
0.49363
0.47464
0.45639
0.45811
0.43630
0.41552
0.39573
0.37689
0.39365
0.37136
0.35034
0.33051
0.31180
0.33873
0.31657
0.29586
0.27651
0.25842
0.29189
0.27027
0.25025
0.23171
0.21455
0.25187
0.23107
0.21199
0.19449
0.17843
0.21763
0.19784
0.17986
0.16351
0.14864
0.18829
0.16963
0.15282
0.13768
0.12403
0.16312
0.14564
0.13004
0.11611
0.10367
0.10686
0.09293
0.08081
0.07027
0.06110
Page 2 of 5
Tables of Time Value Factors
TABLE 3: Future Value of an Ordinary Annuity of $1 = (((1 + i%)^n) -1)/(i%)
Shorthand notation for our course for this factor is FVOA(i%, n periods).
interest i
1.0%
2.0%
2.5%
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
15.0%
periods n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1.00000
2.01000
3.03010
4.06040
5.10101
1.00000
2.02000
3.06040
4.12161
5.20404
1.00000
2.02500
3.07563
4.15252
5.25633
1.00000
2.03000
3.09090
4.18363
5.30914
1.00000
2.04000
3.12160
4.24646
5.41632
1.00000
2.05000
3.15250
4.31013
5.52563
1.00000
2.06000
3.18360
4.37462
5.63709
1.00000
2.07000
3.21490
4.43994
5.75074
1.00000
2.08000
3.24640
4.50611
5.86660
1.00000
2.09000
3.27810
4.57313
5.98471
1.00000
2.10000
3.31000
4.64100
6.10510
1.00000
2.11000
3.34210
4.70973
6.22780
1.00000
2.12000
3.37440
4.77933
6.35285
1.00000
2.15000
3.47250
4.99338
6.74238
6.15202
7.21354
8.28567
9.36853
10.46221
6.30812
7.43428
8.58297
9.75463
10.94972
6.38774
7.54743
8.73612
9.95452
11.20338
6.46841
7.66246
8.89234
10.15911
11.46388
6.63298
7.89829
9.21423
10.58280
12.00611
6.80191
8.14201
9.54911
11.02656
12.57789
6.97532
8.39384
9.89747
11.49132
13.18079
7.15329
8.65402
10.25980
11.97799
13.81645
7.33593
8.92280
10.63663
12.48756
14.48656
7.52333
9.20043
11.02847
13.02104
15.19293
7.71561
9.48717
11.43589
13.57948
15.93742
7.91286
9.78327
11.85943
14.16397
16.72201
8.11519
10.08901
12.29969
14.77566
17.54874
8.75374
11.06680
13.72682
16.78584
20.30372
11.56683
12.68250
13.80933
14.94742
16.09690
12.16872
13.41209
14.68033
15.97394
17.29342
12.48347
13.79555
15.14044
16.51895
17.93193
12.80780
14.19203
15.61779
17.08632
18.59891
13.48635
15.02581
16.62684
18.29191
20.02359
14.20679
15.91713
17.71298
19.59863
21.57856
14.97164
16.86994
18.88214
21.01507
23.27597
15.78360
17.88845
20.14064
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