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 Todays 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 $1.10 $1.21 * .10 * .10

* .10 = $.10 = $.11 = $.12 $1.10 $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 $1.10 $1.21 * .10 * .10 * .10 = $.10 = $.11 = $.12 $1.10 $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 $1.10 $1.21 * .10 * .10 * .10 = $.10 = $.11 = $.12 $1.10 $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 $1.10 $1.21 * .10 * .10 * .10 = $.10 = $.11 = $.12 $1.10 $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 $1.10 $1.21 * .10 * .10 * .10 = $.10 = $.11 = $.12 $1.10 $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

1 2 3 4 5 6 X-Axis = Time in Years As Time increases, Future Value of $1 Increases 2011 7 8 9 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 $2.14 15% 10% 5%

$2.00 $1.48 $1.00 $- 0 1 2 3 4 5 6 7 X-Axis = Time in Years As interest rate increases, Future Value of $1 Increases 2011 8 9 10 12 The Future Value Table Future Value of $1 (Compound Interest) Years 2% 4% 6% 1

1.020 1.040 1.060 2 1.040 1.082 1.124 3 1.061 1.125 1.191 4 1.082 1.170 1.262 5 1.104 1.217 1.338 6 1.126 1.265 1.419 7 1.149 1.316 1.504 8 1.172 1.369 1.594 9 1.195 1.423 1.689 10 1.219 1.480 1.791 11 1.243

1.539 1.898 12 1.268 1.601 2.012 13 1.294 1.665 2.133 14 1.319 1.732 2.261 15 1.346 1.801 2.397 16 1.373 1.873 2.540 17 1.400 1.948 2.693 18 1.428 2.026 2.854 19 1.457 2.107 3.026 20 1.486 2.191 3.207 25 1.641 2.666

4.292 30 1.811 3.243 5.743 35 2.000 3.946 7.686 40 2.208 4.801 10.286 8% 1.080 1.166 1.260 1.360 1.469 1.587 1.714 1.851 1.999 2.159 2.332 2.518 2.720 2.937 3.172 3.426 3.700 3.996 4.316 4.661 6.848 10.063 14.785 21.725 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 10.835 17.449 28.102 45.259 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 17.000 29.960 52.800 93.051 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 26.462 50.950 98.100 188.884 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 16.777 19.461 40.874 85.850 180.314 378.721 18% 1.180 1.392 1.643 1.939 2.288 2.700 3.185 3.759 4.435 5.234 6.176 7.288 8.599 10.147 11.974 14.129 16.672 19.673 23.214

27.393 62.669 143.371 327.997 750.378 20% 1.200 1.440 1.728 2.074 2.488 2.986 3.583 4.300 5.160 6.192 7.430 8.916 10.699 12.839 15.407 18.488 22.186 26.623 31.948 38.338 95.396 237.376 590.668 1,469.772 The Value of $1 at 10% interest after 8 years is $2.14 The Factors are pre-calculated on the FV Table. 2011 13 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 0 1 2 3 4 5 6 7 8 9 10 -20 -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% 1 1.020 1.040 1.060 2 1.040 1.082 1.124 3 1.061 1.125 1.191 4 1.082 1.170 1.262 5 1.104 1.217 1.338 6 1.126 1.265 1.419 7 1.149 1.316 1.504 8 1.172 1.369

1.594 9 1.195 1.423 1.689 10 1.219 1.480 1.791 11 1.243 1.539 1.898 12 1.268 1.601 2.012 13 1.294 1.665 2.133 14 1.319 1.732 2.261 15 1.346 1.801 2.397 16 1.373 1.873 2.540 17 1.400 1.948 2.693 18 1.428 2.026 2.854

19 1.457 2.107 3.026 20 1.486 2.191 3.207 25 1.641 2.666 4.292 30 1.811 3.243 5.743 35 2.000 3.946 7.686 40 2.208 4.801 10.286 8% 1.080 1.166 1.260 1.360 1.469 1.587 1.714 1.851 1.999 2.159 2.332 2.518 2.720 2.937 3.172

3.426 3.700 3.996 4.316 4.661 6.848 10.063 14.785 21.725 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 10.835 17.449 28.102 45.259 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 17.000 29.960 52.800 93.051 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 26.462 50.950 98.100 188.884 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 16.777 19.461 40.874 85.850 180.314 378.721 18% 1.180 1.392 1.643 1.939 2.288 2.700 3.185 3.759

4.435 5.234 6.176 7.288 8.599 10.147 11.974 14.129 16.672 19.673 23.214 27.393 62.669 143.371 327.997 750.378 The Factor of $1 at 8% interest for 10 years is 2.159 $50,000 * 2.159 = $107,950 2011 20% 1.200 1.440 1.728 2.074 2.488 2.986 3.583 4.300 5.160 6.192 7.430 8.916 10.699 12.839 15.407 18.488 22.186 26.623 31.948

38.338 95.396 237.376 590.668 1,469.772 18 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 $0.80 $0.60 $0.40 $0.20 $0.00 1 2 3 4 5 6 7 8 9 10111213141516 17181920 1 2 3 4 5 6 7 8 9 1011121314151617181920 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 Years 1 2 3 4

5 6 7 8 9 10 11 12 13 14 15 16 17 18 of $1 2% 0.980 0.961 0.942 0.924 0.906 0.888 0.871 0.853 0.837 0.820 0.804 0.788 0.773 0.758 0.743 0.728 0.714 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 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 0.095 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 0.069 18% 0.847 0.718 0.609 0.516 0.437 0.370 0.314 0.266 0.225 0.191 0.162 0.137 0.116 0.099 0.084 0.071 0.060 0.051 The Present Value of $1 at 10% to be received in 3 years is $.75 2011 36 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 1 2 3 4 5 6 X-Axis = Time in Years As Time increases, Present Value of $1 Decreases 2011 7 8 9 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% 10% 15% $0.60 $0.47 $0.40 $0.33 $0.20 $- 0 1 2 3 4 5 6 X-Axis = Time in Years As Time increases, Present Value of $1 Decreases 2011 7

8 9 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 $ 70 K $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 Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 of $1 2% 0.980 0.961 0.942 0.924 0.906 0.888 0.871 0.853 0.837 0.820 0.804 0.788 0.773

0.758 0.743 0.728 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 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 18% 0.847 0.718 0.609 0.516 0.437 0.370 0.314 0.266 0.225 0.191 0.162 0.137 0.116 0.099 0.084 0.071 0.060 The Factor of $1 at 12% discount for 6 years is 0.507 $60,000 * 0.507 = $30,420 2011 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 2011 47 Time Value of Money Worksheet The spreadsheet tool also calculates Present Value 2011

48 Practical Exercise 2011 49