Calculate Present or Future Value
of Cash Flows
Intermediate Cost Analysis
and Management
© 2011
1
Time Value of Money Concepts
• Is $1 received today worth the same as $1 to
be received one year from today?
• Is $1 received today worth the same as $1 to
be received one hundred years from today?
• Why or why not?
© 2011
2
Terminal Learning Objective
• Action: Calculate Present or Future Value of a
Variety of Cash Flow Scenarios
• Condition: You are training to become an ACE
with access to ICAM course handouts, readings,
and spreadsheet tools and awareness of
Operational Environment (OE)/Contemporary
Operational Environment (COE) variables and
actors
• Standard: with at least 80% accuracy
• Identify and enter relevant report data to solve
Present and Future Value equations using macro
enabled cash flow templates
© 2011
3
Time Value of Money Concepts
Money received Today:
• Can be invested Today to
earn interest
Money received in the Future:
• Has not yet begun to earn
interest
• Can be spent Today at
Today’s prices
• Can be spent in the Future
at inflated prices
© 2011
4
Simple Interest
• Interest earned on Principal only
Principal * Annual Interest Rate * Time in Years
• Invest $1 today at 10% interest for 3 years
Interest = $1 * .10 * 3 = $.30
• $1 grows to $1.30 over 3 years
© 2011
5
Compound Interest or Future Value
• Invest $1 today at 10% Interest for 3 years
Principal
* 10% (1 year)
= Interest
New Balance
$1.00
* .10
= $.10
$1.10
$1.10
$1.21
* .10
* .10
= $.11
= $.12
$1.21
$1.33
• This relationship can be expressed as:
Principal * (1 + Annual Interest Rate)Time in Years
$1*(1+.10)3 = $1.33
© 2011
6
Compound Interest or Future Value
• Invest $1 today at 10% Interest for 3 years
Principal
* 10% (1 year)
= Interest
New Balance
$1.00
* .10
= $.10
$1.10
$1.10
$1.21
* .10
* .10
= $.11
= $.12
$1.21
$1.33
• This relationship can be expressed as:
Principal * (1 + Annual Interest Rate)Time in Years
$1*(1+.10)3 = $1.33
© 2011
7
Compound Interest or Future Value
• Invest $1 today at 10% Interest for 3 years
Principal
* 10% (1 year)
= Interest
New Balance
$1.00
* .10
= $.10
$1.10
$1.10
$1.21
* .10
* .10
= $.11
= $.12
$1.21
$1.33
• This relationship can be expressed as:
Principal * (1 + Annual Interest Rate)Time in Years
$1*(1+.10)3 = $1.33
© 2011
8
Compound Interest or Future Value
• Invest $1 today at 10% Interest for 3 years
Principal
* 10% (1 year)
= Interest
New Balance
$1.00
* .10
= $.10
$1.10
$1.10
$1.21
* .10
* .10
= $.11
= $.12
$1.21
$1.33
• This relationship can be expressed as:
Principal * (1 + Annual Interest Rate)Time in Years
$1*(1+.10)3 = $1.33
© 2011
9
Compound Interest or Future Value
• Invest $1 today at 10% Interest for 3 years
Principal
* 10% (1 year)
= Interest
New Balance
$1.00
* .10
= $.10
$1.10
$1.10
$1.21
* .10
* .10
= $.11
= $.12
$1.21
$1.33
• This relationship can be expressed as:
Principal * (1 + Annual Interest Rate)Time in Years
$1*(1+.10)3 = $1.33
© 2011
10
Effect of Interest Rate and Time
$4.00
$3.00
$2.14
$2.00
10%
$1.21
$1.00
After 2 years at 10% …..and after 8 years at 10%
$0
2
4
6
X-Axis = Time in Years
As Time increases, Future Value of $1 Increases
© 2011
8
10
11
Effect of Interest Rate and Time
$4.00
A higher interest rate causes the
future value to increase more in
the same 8 years.
$3.00
$3.06
15%
$2.14
$2.00
10%
5%
$1.48
$1.00
$0
2
4
6
X-Axis = Time in Years
As interest rate increases, Future Value of $1 Increases
© 2011
8
10
12
The Future Value Table
Future Value of $1 (Compound Interest)
Years
2%
4%
6%
8%
1
1.020
1.040
1.060
1.080
2
1.040
1.082
1.124
1.166
3
1.061
1.125
1.191
1.260
4
1.082
1.170
1.262
1.360
5
1.104
1.217
1.338
1.469
6
1.126
1.265
1.419
1.587
7
1.149
1.316
1.504
1.714
8
1.172
1.369
1.594
1.851
9
1.195
1.423
1.689
1.999
10
1.219
1.480
1.791
2.159
11
1.243
1.539
1.898
2.332
12
1.268
1.601
2.012
2.518
13
1.294
1.665
2.133
2.720
14
1.319
1.732
2.261
2.937
15
1.346
1.801
2.397
3.172
16
1.373
1.873
2.540
3.426
17
1.400
1.948
2.693
3.700
18
1.428
2.026
2.854
3.996
19
1.457
2.107
3.026
4.316
20
1.486
2.191
3.207© 2011 4.661
10%
1.100
1.210
1.331
1.464
1.611
1.772
1.949
2.144
2.358
2.594
2.853
3.138
3.452
3.797
4.177
4.595
5.054
5.560
6.116
6.727
12%
1.120
1.254
1.405
1.574
1.762
1.974
2.211
2.476
2.773
3.106
3.479
3.896
4.363
4.887
5.474
6.130
6.866
7.690
8.613
9.646
14%
1.140
1.300
1.482
1.689
1.925
2.195
2.502
2.853
3.252
3.707
4.226
4.818
5.492
6.261
7.138
8.137
9.276
10.575
12.056
13.743
The Value of $1 at 10% interest after 8 years is $2.14
The Factors are pre-calculated on the FV Table.
16%
1.160
1.346
1.561
1.811
2.100
2.436
2.826
3.278
3.803
4.411
5.117
5.936
6.886
7.988
9.266
10.748
12.468
14.463
13 16.777
19.461
Learning Check
• How does compound interest differ from
simple interest?
• How does number of years affect the future
value of an investment?
© 2011
14
Demonstration Problem
• If I invest $50,000 today at 8%, what will it be
worth in 10 years?
• Steps:
1. Identify the key variables
• Cash flow
• Interest rate
• Time in years
2. Build a timeline
3. Multiply cash flow by FV factor from the Table
© 2011
15
Identify Key Variables
• Cash Flows
• $50,000 to be paid now
• Cash Payments are negative numbers
• Some unknown amount to be received ten years
in the future
• Cash Receipts are positive numbers
• Interest Rate = 8%
• Time in Years = 10
© 2011
16
Build a Timeline
$ 120K
?
100
$50,000 to be
invested now
80
60
Unknown amount
to be received in 10
years
40
20
0
-20
0
1
2
3
4
5
6
7
8
9
10
-40
$ -60K $50K
X-Axis = Time in Years
© 2011
17
Multiply by the FV Factor
Future Value of $1 (Compound Interest)
Years
2%
4%
6%
8%
1
1.020
1.040
1.060
1.080
2
1.040
1.082
1.124
1.166
3
1.061
1.125
1.191
1.260
4
1.082
1.170
1.262
1.360
5
1.104
1.217
1.338
1.469
6
1.126
1.265
1.419
1.587
7
1.149
1.316
1.504
1.714
8
1.172
1.369
1.594
1.851
9
1.195
1.423
1.689
1.999
10
1.219
1.480
1.791
2.159
11
1.243
1.539
1.898
2.332
12
1.268
1.601
2.012
2.518
13
1.294
1.665
2.133
2.720
14
1.319
1.732
2.261
2.937
15
1.346
1.801
2.397
3.172
16
1.373
1.873
2.540
3.426
17
1.400
1.948
2.693
3.700
18
1.428
2.026
2.854
3.996
19
1.457
2.107
3.026
4.316
20
1.486
2.191
3.207© 2011 4.661
10%
1.100
1.210
1.331
1.464
1.611
1.772
1.949
2.144
2.358
2.594
2.853
3.138
3.452
3.797
4.177
4.595
5.054
5.560
6.116
6.727
12%
1.120
1.254
1.405
1.574
1.762
1.974
2.211
2.476
2.773
3.106
3.479
3.896
4.363
4.887
5.474
6.130
6.866
7.690
8.613
9.646
The Factor of $1 at 8% interest for 10 years is 2.159
$50,000 * 2.159 = $107,950
14%
1.140
1.300
1.482
1.689
1.925
2.195
2.502
2.853
3.252
3.707
4.226
4.818
5.492
6.261
7.138
8.137
9.276
10.575
12.056
13.743
16%
1.160
1.346
1.561
1.811
2.100
2.436
2.826
3.278
3.803
4.411
5.117
5.936
6.886
7.988
9.266
10.748
12.468
14.463
18 16.777
19.461
Using the Formula
• The formula proves that the answer from the
table is correct:
$50,000 * (1 + .08)10 = $107,946
• The difference of $4 is caused by rounding in
the table
© 2011
19
Proof
Year
1
2
3
4
5
6
7
8
9
10
Principal
*8%
= Interest
$50,000
$54.000
$58,320
$62,986
$68,024
$73,466
$79,343
$85,690
$92,545
$99,949
* .08
* .08
* .08
* .08
* .08
* .08
* .08
* .08
* .08
* .08
= $4,000
= $4,320
= $4,666
= $5,039
= $5,442
= $5,877
= $6,347
= $6,855
= $7,404
= $7,996
© 2011
New Balance
$54,000
$58,320
$62,986
$68,024
$73,466
$79,343
$85,690
$92,545
$99,949
$107,945
20
Learning Check
• What is the first step in solving a future value
problem?
• How are cash payments represented in the
timeline?
© 2011
21
Future Value vs. Present Value
• Future Value answers the question:
• To what value will $1 grow in the Future?
• Present Value answers the question:
• What is the value Today of $1 to be received in the
Future?
-or• How much must be invested today to achieve $1
in the Future?
© 2011
22
Future Value vs. Present Value
Present Value of $1 at 10%
Future Value of $1 at 10%
$8.00
$7.00
$6.00
$5.00
$4.00
$3.00
$2.00
$1.00
$0.00
$1.00
$0.90
$0.80
$0.70
$0.60
$0.50
$0.40
$0.30
$0.20
$0.10
$0.00
1
3
5
7
9
11
13
15
17
1
19
3
5
7
9
11
13
15
17
19
Periods
Periods
The value of a dollar received today will
increase in the future
A dollar to be received in the future is
worth less than a dollar received today
© 2011
23
Present Value Concepts
• What is the value Today of $1 to be received
one year in the Future?
• How much must be invested Today to grow to
$1 one year from Today?
• The answer to these two questions is the
same!
© 2011
24
Present Value Concepts
• Discount Rate represents interest or inflation
• Assume a rate of 10%
• What is the cost expression for this relationship?
$Investment Today + Interest = $1.00
-or$Investment + ($Investment * .10) = $1.00
$Investment * (1+ .10) = $1.00
$Investment = $1/(1.10)
$Investment = $.91
© 2011
25
Present Value Concepts
• Discount Rate represents interest or inflation
• Assume a rate of 10%
• What is the cost expression for this relationship?
$Investment Today + Interest = $1.00
-or$Investment + ($Investment * .10) = $1.00
$Investment * (1+ .10) = $1.00
$Investment = $1/(1.10)
$Investment = $.91
© 2011
26
Present Value Concepts
• Discount Rate represents interest or inflation
• Assume a rate of 10%
• What is the cost expression for this relationship?
$Investment Today + Interest = $1.00
-or$Investment + ($Investment * .10) = $1.00
$Investment * (1+ .10) = $1.00
$Investment = $1/(1.10)
$Investment = $.91
© 2011
27
Present Value Concepts
• Discount Rate represents interest or inflation
• Assume a rate of 10%
• What is the cost expression for this relationship?
$Investment Today + Interest = $1.00
-or$Investment + ($Investment * .10) = $1.00
$Investment * (1+ .10) = $1.00
$Investment = $1/(1.10)
$Investment = $.91
© 2011
28
Present Value Concepts
• Discount Rate represents interest or inflation
• Assume a rate of 10%
• What is the cost expression for this relationship?
$Investment Today + Interest = $1.00
-or$Investment + ($Investment * .10) = $1.00
$Investment * (1+ .10) = $1.00
$Investment = $1/(1.10)
$Investment = $.91
© 2011
29
Proof
• Plug $.91 in to the original equation:
$.91 + ($.91 * .10) = $1.00
$.91 + .09 = $1.00
• This relationship is fairly simple for one
period, but what about multiple periods?
© 2011
30
Present Value Concepts
• How much must be invested today to achieve
$1.00 three years from today?
• What is the cost expression for this relationship?
$Investment * (1 + Rate) #Years = $Future Value
$Investment = $Future Value / (1 + Rate) #Years
-or$Investment * (1+.10) 3 = $1.00
$Investment = $1.00 / (1+.10) 3
$Investment = $.75
© 2011
31
Present Value Concepts
• How much must be invested today to achieve
$1.00 three years from today?
• What is the cost expression for this relationship?
$Investment * (1 + Rate) #Years = $Future Value
$Investment = $Future Value / (1 + Rate) #Years
-or$Investment * (1+.10) 3 = $1.00
$Investment = $1.00 / (1+.10) 3
$Investment = $.75
© 2011
32
Present Value Concepts
• How much must be invested today to achieve
$1.00 three years from today?
• What is the cost expression for this relationship?
$Investment * (1 + Rate) #Years = $Future Value
$Investment = $Future Value / (1 + Rate) #Years
-or$Investment * (1+.10) 3 = $1.00
$Investment = $1.00 / (1+.10) 3
$Investment = $.75
© 2011
33
Present Value Concepts
• The Investment amount is known as the
Present Value
• The Present Value relationship is expressed in
the formula:
Future Cash Flow * 1/(1 + Rate) #Years
-or$1 * 1/(1.10)3 = $.75
© 2011
34
Proof
Principal
* 10% (1 year)
= Interest
$.75
$.83
$.91
* .10
* .10
* .10
= $.075
= $.083
= $.091
New Balance
$.83
$.91
$1.00
• There is also a table shortcut for Present Value
© 2011
35
The Present Value Table
Present Value of $1
Years
2%
1
0.980
2
0.961
3
0.942
4
0.924
5
0.906
6
0.888
7
0.871
8
0.853
9
0.837
10
0.820
11
0.804
12
0.788
13
0.773
14
0.758
15
0.743
16
0.728
17
0.714
18
0.700
4%
0.962
0.925
0.889
0.855
0.822
0.790
0.760
0.731
0.703
0.676
0.650
0.625
0.601
0.577
0.555
0.534
0.513
0.494
6%
0.943
0.890
0.840
0.792
0.747
0.705
0.665
0.627
0.592
0.558
0.527
0.497
0.469
0.442
0.417
0.394
0.371
© 2011
0.350
8%
0.926
0.857
0.794
0.735
0.681
0.630
0.583
0.540
0.500
0.463
0.429
0.397
0.368
0.340
0.315
0.292
0.270
0.250
10%
0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
0.386
0.350
0.319
0.290
0.263
0.239
0.218
0.198
0.180
12%
0.893
0.797
0.712
0.636
0.567
0.507
0.452
0.404
0.361
0.322
0.287
0.257
0.229
0.205
0.183
0.163
0.146
0.130
14%
0.877
0.769
0.675
0.592
0.519
0.456
0.400
0.351
0.308
0.270
0.237
0.208
0.182
0.160
0.140
0.123
0.108
0.09536
The Present Value of $1 at 10% to be received in 3 years is $.75
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
Effect of Interest Rate and Time
$1.20
$1.00
$0.83
$0.80
$0.60
10%
$0.47
$0.40
$0.20
$1 to be received in 2 years at 10% …..and in 8 years at 10%
$0
2
4
6
X-Axis = Time in Years
As Time increases, Present Value of $1 Decreases
© 2011
8
10
37
Effect of Interest Rate and Time
$1.20
A higher discount rate causes the
present value to decrease more
in the same 8 years.
$1.00
$0.80
$0.68
5%
$0.60
10%
$0.47
15%
$0.40
$0.33
$0.20
$0
2
4
6
X-Axis = Time in Years
As Time increases, Present Value of $1 Decreases
© 2011
8
10
38
Learning Check
• What does Present Value represent?
• How does the Present Value table differ from
the Future Value table?
© 2011
39
Demonstration Problem
• What is the Present Value of a $60,000 cash flow
to be received 6 years from today assuming 12%
discount rate?
• Steps:
1. Identify the key variables
• Cash flow
• Discount rate
• Time in years
2. Build a timeline
3. Multiply cash flow by the Factor from the PV Table
© 2011
40
Identify Key Variables
• Cash Flow
• $60,000 to be received in the Future
• Is equal to some unknown amount Today
• Discount Rate = 12%
• Time in Years = 6
© 2011
41
Build a Timeline
$ 70K
$60K
$60,000 to be
received in 6 years
60
50
Unknown
Present Value
40
30
20
?
10
0
0
1
2
3
4
5
6
X-Axis = Time in Years
© 2011
42
Multiply by the PV Factor
Present Value of $1
Years
2%
1
0.980
2
0.961
3
0.942
4
0.924
5
0.906
6
0.888
7
0.871
8
0.853
9
0.837
10
0.820
11
0.804
12
0.788
13
0.773
14
0.758
15
0.743
16
0.728
17
0.714
4%
0.962
0.925
0.889
0.855
0.822
0.790
0.760
0.731
0.703
0.676
0.650
0.625
0.601
0.577
0.555
0.534
0.513
6%
0.943
0.890
0.840
0.792
0.747
0.705
0.665
0.627
0.592
0.558
0.527
0.497
0.469
0.442
0.417
0.394
0.371
8%
0.926
0.857
0.794
0.735
0.681
0.630
0.583
0.540
0.500
0.463
0.429
0.397
0.368
0.340
0.315
0.292
0.270
10%
0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
0.386
0.350
0.319
0.290
0.263
0.239
0.218
0.198
12%
0.893
0.797
0.712
0.636
0.567
0.507
0.452
0.404
0.361
0.322
0.287
0.257
0.229
0.205
0.183
0.163
0.146
The Factor of $1 at 12% discount for 6 years is 0.507
$60,000 * 0.507 = $30,420
© 2011
14%
0.877
0.769
0.675
0.592
0.519
0.456
0.400
0.351
0.308
0.270
0.237
0.208
0.182
0.160
0.140
0.123
0.108
16%
0.862
0.743
0.641
0.552
0.476
0.410
0.354
0.305
0.263
0.227
0.195
0.168
0.145
0.125
0.108
0.093
0.080
43
Using the Formula
• The formula proves that the answer from the
table is correct:
$60,000 * 1/(1 + .12)6 = $30,398
• The difference of $22 is caused by rounding in
the table
© 2011
44
Proof
Year
Principal
1
2
3
4
5
6
30,420
34,070
38,159
42,738
47,866
53,610
*8%
= Interest
* .12
* .12
* .12
* .12
* .12
* .12
© 2011
= $3,650
= $4,088
= $4,579
= $5,129
= $5,744
= $6,433
New Balance
$34,070
$38,159
$42,738
$47,866
$53,610
$60,044
45
Practical Exercise
© 2011
46
Time Value of Money Worksheet
Enter key variables in the
blank white cells to generate
the graph shown below
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47
Time Value of Money Worksheet
The spreadsheet tool
also calculates
Present Value
© 2011
48
Practical Exercise
© 2011
49