Developing Financial Insights

UV5137 Jun. 1, 2011

This case was prepared by Professor Mark Haskins. It was written as a basis for class discussion rather than to illustrate effective or ineffective handling of an administrative situation. Copyright © 2011 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an email to [email protected]. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation.

DEVELOPING FINANCIAL INSIGHTS: USING A FUTURE VALUE (FV) AND A PRESENT VALUE (PV) APPROACH

Base Case Starting Point

Most everyone is familiar with the concept that money placed in a savings account will grow to a larger amount as the years pass if the money in that account earns interest at a specified annual compounded rate. For example, $500 invested in a savings account that earns 10% interest compounded annually will grow to become $550 (i.e., $500 × 1.10) after one year, $605 (i.e., $550 × 1.10) after two years, and $665.50 (i.e., $605 × 1.10) after three years.1

Think about the numerical example just depicted. It took three sequential calculations to

arrive at the $665.50 answer—one calculation for each year involved. If the question had been posed as involving 12 years or even 25 years, a multitude of tedious, repetitive calculations would have been required. Is there a shortcut? Yes. If we take the 1.10 multiplier amount from each of the three parenthetical notations above and simply multiply them together—1.10 × 1.10 × 1.10—we get a numerical factor of 1.331. So, if some reference book could provide us with the 1.331 multiplier as being applicable to a 10% situation over three years, all we would have to do is take the initial $500 amount put into the savings account and multiply it by 1.331 to get the very same answer as above—$665.50.

Are there reference materials that provide such multipliers for a variety of interest and

years combinations? Yes, there are, and they are referred to as future value (FV) factors. Such reference materials are useful because no matter the initial amount invested, a specified combination of time and interest will always have the same multiplier effect. Thus, future value factor tables are readily available, depicting a number of possible different interest rates along one axis and a number of different years along the other.2 In fact, Exhibit 1 presents just such a

1 The 1.10 multiplier comes from the fact that in one year there will be 100% of that year’s starting monetary amount plus an additional 10% due to a year’s worth of interest having been accumulated at a 10% rate. Thus, 100% + 10% = 110%, which converts to an arithmetic multiplier of 1.10.

2 This case explores cash flows on an annual basis. Appendix 1 explains what to do when cash flows occur on a monthly or quarterly basis.

This document is authorized for use only in PGPM - 02252013 by Aniket Khera at Anytime Learning from February 2013 to August 2013.

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table. In it we can find the 10% column and the three-years row and find the multiplier amount at the intersection of those two table coordinates, and it is 1.331—the same as we determined it should be. Moreover, if we were using a financial calculator or Excel, the embedded programs in those tools would derive the exact same multiplier once we entered the data pertaining to years and interest rate. Can We Reverse the Flow of Time?

In business, the basic question posed above often requires us to reverse the focus. For example, the question might be some form of “My customer is willing to pay me $665.50 in three years; what is the equivalent monetary amount, as of today, of that future receipt?” Pause for just a moment—the same basic interplay of time and interest described in the first example must be in play in this scenario also…right? Of course, but instead of a current invested monetary amount growing into a larger future amount due to the compounding of interest, we must now work with a stipulated future monetary amount and in essence, unwind, roll back in time, reverse the compounding of interest phenomenon. In this instance, we are being asked to ascertain the present value (PV)—the value today—of a future monetary amount, using the relevant interest and year information.

To do that, the process is simply the reverse of what we did earlier. So, if the relevant

interest environment is 10% and the number of years is three, we execute the following three calculations:

1. $665.50 ÷ 1.10 = $605

2. $605 ÷ 1.10 = $550

3. $550 ÷ 1.10 = $500 Thus, receiving $665.50 in three years, when the interest rate environment is 10%, is

equivalent to receiving $500 today. In short, this has to be true because as we saw earlier, if we invest $500 today in a savings account that pays 10% interest, that savings account balance will grow to become $665.50 in three years. Another way to state this is this: If, over the next three years, relevant interest rates are 10%, the economic value of $665.50 in three years is exactly equal to $500 today.

As before, this can become a laborious series of calculations if the number of years

involved is substantial. And, just as before, there is a shortcut. Mathematically, whereas the compounding of interest phenomenon was a multiplicative mathematical task, the unwinding of a compounded interest phenomenon must be a divisive mathematical task. Specifically, we find


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that: 1 ÷ 1.10 = 0.90909, and then 0.90909 ÷ 1.10 = 0.82644, and finally, 0.82644 ÷ 1.10 = 0.7513.3 And, if we take 0.7513 × $665.50, we get $500 (with a bit of rounding).

Are there reference materials that provide such factor multipliers for a variety of interest

and year combinations when we are seeking to derive the present value of a future monetary amount? Yes, and they contain a host of PV factors. Such reference materials are useful because no matter the future amount to be received, a specified combination of time and interest will always have the same multiplier effect. Exhibit 2 provides an example of such a reference table. In it, we can find the 10% column and the three-years row and extract the multiplier factor at the intersection of those two coordinates and it is, just as we thought it should be, 0.7513. (Please note that the example depicted at the bottom of the table in Exhibit 2 is an additional one, different from the one discussed here.) Moreover, if we are using a financial calculator or Excel, the embedded algorithms in those tools use the same factor once we have entered into those tools the relevant number of years (three) and interest rate (10%). Moving Beyond Single Ending or Starting Monetary Amounts

The two scenarios described above are emblematic of the simplest of situations—they both started with single monetary amounts to be compounded (the Exhibit 1-related example) or discounted (the Exhibit 2-related example). In many personal and business financial situations the reality is that there are multiple cash amounts coming in or going out over the course of a stipulated time period that are pertinent to the sought-after FV or PV. Let’s lay the foundation for those sorts of scenarios.

Assume you invest $80 today and at the beginning of each of the next two years for a

total of three such deposits, in a 6% savings account. At the end of three years, what will that account have in it? Clearly, we could answer that question by applying the technique and Exhibit 1 factors we described and used in the very first example. That is, we could find the future value of three lump sums—one of which is invested for three years, one of which is in the account for two years, and one of which is invested for only one year. In fact, at the bottom of Exhibit 3, this approach is depicted. But there is an easier, quicker way. Since this scenario involves three applications of the Exhibit 1 factors, we can develop reference materials that accumulate the effects of a variety of multiple applications of Exhibit 1’s factors. Indeed, the Exhibit 3 factors portray such accumulations. Please note that the Exhibit 3 factors are various summations of sequential Exhibit 1 factor amounts for a given interest rate. For this scenario’s three deposits in a 6% savings account, the Exhibit 3 factor is 3.375, which is the sum of the Exhibit 1 6% factors associated with one year, two years, and three years (1.06 + 1.1236 + 1.191, subject to minimal rounding). So 3.375 × $80 = $270, the amount to which three (starting today) annual invested amounts of $80 each grow to in a 6% account at the end of three years.

3 Some readers may have anticipated, or be interested to note, that the 0.7513 present value multiplier figure can

also be derived by the following: 1 ÷ (1.10)³, which is the same as 1 ÷ 1.331, which indeed equals 0.7513.


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As before, let’s reverse the direction of the time frame. Assume, for example, you are to receive $70 at the end of each of the next three years. The natural question to arise is, what is the single present value monetary amount, as of today, that is equivalent to that series of three receipts? Assuming a 12% interest rate environment, this question can be answered by applying the technique and Exhibit 2 factors that we used in an earlier example. If we were to pursue that approach, we would have to execute three separate calculations for each of the three $70 receipts—that approach is depicted at the bottom of Exhibit 4.4 But as we were able to do in using Exhibit 1 to develop Exhibit 3, we can use Exhibit 2 to develop Exhibit 4, which in turn can then be used as a shortcut for PV situations with multiple cash flows in the future. Indeed, Exhibit 4 is simply the summation of Exhibit 2 factors for a variety of time periods within an interest rate column. So, in this example, we can easily go to Exhibit 4’s 12% column, three-years row, and find the factor of 2.402, which is the sum (with a bit of rounding) of the pertinent Exhibit 2 factors (0.8929 + 0.7972 + 0.7118). And 2.402 × $70 = $168.14. The interpretation of this present value is as follows: In a 12% interest rate environment, receiving $168.14 today is equivalent to receiving three payments of $70 at the end of each of the next three years. Practice Your FV and PV Skills

1. You just turned 35 and have been saving for an around-the-world vacation. You want to take the trip to celebrate your 40th birthday. You have set aside, as of today, $15,000 for such a trip. You expect the trip will cost $25,000. The financial instruments you have invested the $15,000 in have been earning, on average, about 8%. (You may ignore income taxes.)

a. Will you have enough money in that vacation account on your 40th birthday to take the trip? What will be the surplus, or shortfall, in that account when you turn 40? (Hint: Exhibit 1 will be useful in answering this question.)

b. If you had to, you could further fund the trip by making, starting today, five annual $500 contributions to the account. If you adhered to such a plan, how much will be in the account on your 40th birthday? (Hint: Exhibit 3 and the answer to part (a) above will both be useful in answering this question.)

2. Your company has been offered a contract for the development and delivery of a solar-powered military troop transport vehicle. The request for proposal provides all the necessary technical specifications and it also stipulates that two working, economically feasible prototypes must be delivered in four years, at which time you will receive your only customer payment—a single and final payment of $50 million. Assume a reinvestment interest rate of 18% for all the monies received over the next four years. (You may ignore income taxes.)

4 Appendix 2 shows how to adjust the data in Exhibit 4 for cash flows at the beginning (instead of the end) of

the year.


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a. What lump-sum dollar amount would you be willing to accept today instead of the $50 million in four years? (Hint: Exhibit 2 will be useful in answering this question.)

b. Alternatively, what four yearly receipts, starting a year from now, would you be willing to accept? (Hint: Exhibit 4 and the answer to part (a) above will both be useful in answering this question.)

3. The aged but centrally located golf course you manage does not have an in-ground automated water sprinkling system. Instead, to properly water the course, sprinklers and hoses must be repeatedly set, moved, and put away by some of the grounds crew—a tedious and laborious task. If over the next 12 years you project annual savings of about $40,000 from having an automated system, what is the maximum price you would be willing to pay today for an installed, automated golf course sprinkler system? (Assume an interest rate of 6%, and you may ignore income taxes.)

a. Redo your calculation using a 10-year time period and $48,000 in annual savings.

b. Redo your initial calculation one more time using $50,000 in annual savings for the first six years and $30,000 in annual savings for the next six years.

4. The cafeteria you operate has a regular clientele for all three meals, seven days a week. You want to expand your product line beyond what you are currently able to offer. To do so requires the purchase of some additional specialty equipment costing $45,000, but you project a resultant increase in sales (after deducting the cost of sales) of about $8,000 per year for each of the next eight years with this new equipment. Assuming a required rate of return (i.e., a hurdle rate) of 8%, should you pursue this opportunity? Why or why not? Do the analysis under two conditions:

a. You are part of an income-tax-exempt enterprise.

b. The enterprise you are part of is subject to a 40% corporate income tax rate, and the straight-line, depreciable life of the equipment you are contemplating purchasing is five years.

5. You are contemplating the purchase of a one-half interest in a corporate airplane to facilitate the expansion of your business into two new geographic areas. The acquisition would eliminate about $220,000 in estimated annual expenditures for commercial flights, mileage reimbursements, rental cars, and hotels for each of the next 10 years. The total purchase price for the half-share is $6 million, plus associated annual operating costs of $100,000. Assume the plane can be fully depreciated on a straight-line basis for tax purposes over 10 years. The company’s weighted average cost of capital (commonly referred to as WACC) is 8%, and its corporate tax rate is 40%. Does this endeavor present a positive or negative net present value (NPV)? If positive, how much value is being created for the company through the purchase of this asset? If negative, what additional annual cash flows would be needed for the NPV to equal zero? To what phenomena might those additional positive cash flows be ascribable?

6. The final tally is in: This year’s operating costs were down $100,000, a decrease directly attributable to the $520,000 investment in the automated materials handling system put in


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place at the beginning of the year. If this level of annual savings continues for five more years, resulting in six total years of annual savings, what compounded annual rate of return will that represent? If these annual savings continue for nine more years, what compounded annual rate of return will that represent? (You may ignore income taxes.)


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Exhibit 1


Future Value Factors for a Single Lump Sum Invested Today for n Years: Exhibit 1 Factors = (1 + Interest)years

Annual Interest Rates

2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Years

1 1.0200 1.0400 1.0600 1.0800 1.1000 1.1200 1.1400 1.1600 1.1800 1.2000

2 1.0404 1.0816 1.1236 1.1664 1.2100 1.2544 1.2996 1.3456 1.3924 1.4400

3 1.0612 1.1249 1.1910 1.2597 1.3310 1.4049 1.4815 1.5609 1.6430 1.7280

4 1.0824 1.1699 1.2625 1.3605 1.4641 1.5735 1.6890 1.8106 1.9388 2.0736

5 1.1041 1.2167 1.3382 1.4693 1.6105 1.7623 1.9254 2.1003 2.2878 2.4883

6 1.1262 1.2653 1.4185 1.5869 1.7716 1.9738 2.1950 2.4364 2.6996 2.9860

7 1.1487 1.3159 1.5036 1.7138 1.9487 2.2107 2.5023 2.8262 3.1855 3.5832

8 1.1717 1.3686 1.5938 1.8509 2.1436 2.4760 2.8526 3.2784 3.7589 4.2998

9 1.1951 1.4233 1.6895 1.9990 2.3579 2.7731 3.2519 3.8030 4.4355 5.1598

10 1.2190 1.4802 1.7908 2.1589 2.5937 3.1058 3.7072 4.4114 5.2338 6.1917

11 1.2434 1.5395 1.8983 2.3316 2.8531 3.4785 4.2262 5.1173 6.1759 7.4301

12 1.2682 1.6010 2.0122 2.5182 3.1384 3.8960 4.8179 5.9360 7.2876 8.9161

13 1.2936 1.6651 2.1329 2.7196 3.4523 4.3635 5.4924 6.8858 8.5994 10.6993

14 1.3195 1.7317 2.2609 2.9372 3.7975 4.8871 6.2613 7.9875 10.1472 12.8392

15 1.3459 1.8009 2.3966 3.1722 4.1772 5.4736 7.1379 9.2655 11.9737 15.4070

16 1.3728 1.8730 2.5404 3.4259 4.5950 6.1304 8.1372 10.7480 14.1290 18.4884

17 1.4002 1.9479 2.6928 3.7000 5.0545 6.8660 9.2765 12.4677 16.6722 22.1861

18 1.4282 2.0258 2.8543 3.9960 5.5599 7.6900 10.5752 14.4625 19.6733 26.6233

19 1.4568 2.1068 3.0256 4.3157 6.1159 8.6128 12.0557 16.7765 23.2144 31.9480

20 1.4859 2.1911 3.2071 4.6610 6.7275 9.6463 13.7435 19.4608 27.3930 38.3376

example (assuming 10%):

Today 1 2 3

Start here

$100

× 1.10 $110 × 1.10 $121 × 1.10 $133.10

OR: $100.00 × 1.331 = $133.10

(so: PV amount × Exhibit 1 factor = FV amount)


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Exhibit 2


Present Value Factors for a Single Amount n Years in the Future: Exhibit 2 Factors = 1 ÷ Exhibit 1 Table Factor in the Same Cell


2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Years

1 0.9804 0.9615 0.9434 0.9259 0.9091 0.8929 0.8772 0.8621 0.8475 0.8333

2 0.9612 0.9246 0.8900 0.8573 0.8264 0.7972 0.7695 0.7432 0.7182 0.6944

3 0.9423 0.8890 0.8396 0.7938 0.7513 0.7118 0.6750 0.6407 0.6086 0.5787

4 0.9238 0.8548 0.7921 0.7350 0.6830 0.6355 0.5921 0.5523 0.5158 0.4823

5 0.9057 0.8219 0.7473 0.6806 0.6209 0.5674 0.5194 0.4761 0.4371 0.4019

6 0.8880 0.7903 0.7050 0.6302 0.5645 0.5066 0.4556 0.4104 0.3704 0.3349

7 0.8706 0.7599 0.6651 0.5835 0.5132 0.4523 0.3996 0.3538 0.3139 0.2791

8 0.8535 0.7307 0.6274 0.5403 0.4665 0.4039 0.3506 0.3050 0.2660 0.2326

9 0.8368 0.7026 0.5919 0.5002 0.4241 0.3606 0.3075 0.2630 0.2255 0.1938

10 0.8203 0.6756 0.5584 0.4632 0.3855 0.3220 0.2697 0.2267 0.1911 0.1615

11 0.8043 0.6496 0.5268 0.4289 0.3505 0.2875 0.2366 0.1954 0.1619 0.1346

12 0.7885 0.6246 0.4970 0.3971 0.3186 0.2567 0.2076 0.1685 0.1372 0.1122

13 0.7730 0.6006 0.4688 0.3677 0.2897 0.2292 0.1821 0.1452 0.1163 0.0935

14 0.7579 0.5775 0.4423 0.3405 0.2633 0.2046 0.1597 0.1252 0.0985 0.0779

15 0.7430 0.5553 0.4173 0.3152 0.2394 0.1827 0.1401 0.1079 0.0835 0.0649

16 0.7284 0.5339 0.3936 0.2919 0.2176 0.1631 0.1229 0.0930 0.0708 0.0541

17 0.7142 0.5134 0.3714 0.2703 0.1978 0.1456 0.1078 0.0802 0.0600 0.0451

18 0.7002 0.4936 0.3503 0.2502 0.1799 0.1300 0.0946 0.0691 0.0508 0.0376

19 0.6864 0.4746 0.3305 0.2317 0.1635 0.1161 0.0829 0.0596 0.0431 0.0313

20 0.6730 0.4564 0.3118 0.2145 0.1486 0.1037 0.0728 0.0514 0.0365 0.0261

example (assuming 8%):

Today 1 2 3

$200.00 Start here

$158.77 1.08 ÷ $171.47 1.08 ÷ $185.19 1.08 ÷

OR: $200.00 × 0.7938 = 158.76*

*difference between $158.77 and $158.76 due to rounding

(So: PV amount = Exhibit 2 factor × FV amount)


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Exhibit 3


Future Value Factors for a Series of Invested Amounts at the Beginning of n Years: Table Factors = Sum of Exhibit 1 Factors for Corresponding Cell

and All Preceding Cells for that Interest Rate


2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Years

1 1.020 1.040 1.060 1.080 1.100 1.120 1.140 1.160 1.180 1.200

2 2.060 2.122 2.184 2.246 2.310 2.374 2.440 2.506 2.572 2.640

3 3.122 3.246 3.375 3.506 3.641 3.779 3.921 4.066 4.215 4.368

4 4.204 4.416 4.637 4.867 5.105 5.353 5.610 5.877 6.154 6.442

5 5.308 5.633 5.975 6.336 6.716 7.115 7.536 7.977 8.442 8.930

6 6.434 6.898 7.394 7.923 8.487 9.089 9.730 10.414 11.142 11.916

7 7.583 8.214 8.897 9.637 10.436 11.300 12.233 13.240 14.327 15.499

8 8.755 9.583 10.491 11.488 12.579 13.776 15.085 16.519 18.086 19.799

9 9.950 11.006 12.181 13.487 14.937 16.549 18.337 20.321 22.521 24.959

10 11.169 12.486 13.972 15.645 17.531 19.655 22.045 24.733 27.755 31.150

11 12.412 14.026 15.870 17.977 20.384 23.133 26.271 29.850 33.931 38.581

12 13.680 15.627 17.882 20.495 23.523 27.029 31.089 35.786 41.219 47.497

13 14.974 17.292 20.015 23.215 26.975 31.393 36.581 42.672 49.818 58.196

14 16.293 19.024 22.276 26.152 30.772 36.280 42.842 50.660 59.965 71.035

15 17.639 20.825 24.673 29.324 34.950 41.753 49.980 59.925 71.939 86.442

16 19.012 22.698 27.213 32.750 39.545 47.884 58.118 70.673 86.068 104.931

17 20.412 24.645 29.906 36.450 44.599 54.750 67.394 83.141 102.740 127.117

18 21.841 26.671 32.760 40.446 50.159 62.440 77.969 97.603 122.414 153.740

19 23.297 28.778 35.786 44.762 56.275 71.052 90.025 114.380 145.628 185.688

20 24.783 30.969 38.993 49.423 63.002 80.699 103.768 133.841 173.021 224.026

example (using 6%):

Today 1 2 3

Start here

$80 $80 $80

× 1.06 = $84.80

× 1.1236 $89.89

× 1.191 = $95.28

$269.97

OR: $80.00 × 3.375 = 270.00*

(So: PV amounts × Exhibit 3 factor = FV amount)


^This exhibit depicts factors for annuities due (where the cash flows occur at the start of each year) as opposed to an

ordinary annuity s i tuation (where the cash flows occur at the end of the year). Most publ ished tables of this sort

are for the l atter.

from Exhibit 1


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Exhibit 4


Present Value Factors for a Series of Amounts n Years in the Future: Exhibit 4 Factors = Sum of Exhibit 2 Factors for Corresponding Cell

and All Preceding Cells for that Interest Rate


2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Years

1 0.980 0.962 0.943 0.926 0.909 0.893 0.877 0.862 0.847 0.833

2 1.942 1.886 1.833 1.783 1.736 1.690 1.647 1.605 1.566 1.528

3 2.884 2.775 2.673 2.577 2.487 2.402 2.322 2.246 2.174 2.106

4 3.808 3.630 3.465 3.312 3.170 3.037 2.914 2.798 2.690 2.589

5 4.713 4.452 4.212 3.993 3.791 3.605 3.433 3.274 3.127 2.991

6 5.601 5.242 4.917 4.623 4.355 4.111 3.889 3.685 3.498 3.326

7 6.472 6.002 5.582 5.206 4.868 4.564 4.288 4.039 3.812 3.605

8 7.325 6.733 6.210 5.747 5.335 4.968 4.639 4.344 4.078 3.837

9 8.162 7.435 6.802 6.247 5.759 5.328 4.946 4.607 4.303 4.031

10 8.983 8.111 7.360 6.710 6.145 5.650 5.216 4.833 4.494 4.192

11 9.787 8.760 7.887 7.139 6.495 5.938 5.453 5.029 4.656 4.327

12 10.575 9.385 8.384 7.536 6.814 6.194 5.660 5.197 4.793 4.439

13 11.348 9.986 8.853 7.904 7.103 6.424 5.842 5.342 4.910 4.533

14 12.106 10.563 9.295 8.244 7.367 6.628 6.002 5.468 5.008 4.611

15 12.849 11.118 9.712 8.559 7.606 6.811 6.142 5.575 5.092 4.675

16 13.578 11.652 10.106 8.851 7.824 6.974 6.265 5.668 5.162 4.730

17 14.292 12.166 10.477 9.122 8.022 7.120 6.373 5.749 5.222 4.775

18 14.992 12.659 10.828 9.372 8.201 7.250 6.467 5.818 5.273 4.812

19 15.678 13.134 11.158 9.604 8.365 7.366 6.550 5.877 5.316 4.843

20 16.351 13.590 11.470 9.818 8.514 7.469 6.623 5.929 5.353 4.870

example (using 12%):

Today 1 2 3

$70 $70 $70 Start here

$62.50 .8929 ×

$55.80 .7972 ×

$49.83 .7118 ×

$168.13

OR: $70.00 × 2.402 = 168.14*

(So: PV amount = Exhibit 4 factor × FV amounts)


̂See Appendix 2 for a discussion of how to use this Exhibit when cash flows begin immediately.

from Exhibit 2


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Appendix 1

DEVELOPING FINANCIAL INSIGHTS: USING A FUTURE VALUE (F) AND A PRESENT VALUE (PV) APPROACH

What to Do When Cash Flows Occur on a Monthly or Quarterly Basis

All the examples in this case involve annual time periods. It is not unusual for payments or receipts of cash to occur on a monthly or even a quarterly basis. There is an easy adjustment process to accommodate such alternative time frames. All that is required is to note that, unless stated otherwise, interest rates are always assumed to involve annual compounding. Thus, if the scenario under consideration involves quarterly cash flows, the vertical axes of the tables in Exhibit 1 through Exhibit 4 can be assumed to pertain to the number of quarters (instead of years) and the stated annual interest rate must be divided by 4 (because there are four quarters per year) before picking the appropriate interest rate column to use in the tables. So, if the desire is to ascertain the PV of a series of quarterly payments, beginning at the end of the first quarter, for the next three years, and the pertinent annual interest rate is 16%, these are the two required adjustments for using the tables:

1. The number of periods to use on the tables’ vertical axes is 12 (3 years × 4 quarters per

year).

2. The interest rate to use on the tables’ horizontal axes is 4% (16% annual rate ÷ 4 compounding quarters per year). In short, for a quarterly series of cash flows, we adjust the table axes coordinates by

scaling up the number of periods by a multiple of 4 and scaling down the interest rate by a divisor of 4. Similarly, for a monthly series of cash flows, we adjust the table axes coordinates by scaling up the number of periods by a multiple of 12 and scaling down the interest rate by a divisor of 12.1

1 Note: Technically, an interest rate of 16% compounded annually is not equivalent to a 4% rate compounded

quarterly. The reason is that the interest earned on a quarterly basis is itself subject to the next quarter’s compounding, quarter after quarter. For example, $100 earning interest at a 16% annually compounded rate will grow to $116 at the end of one year. On the other hand, $100 earning interest at the rate of 4% compounded quarterly will grow to $117 at the end of one year. The fact that the two scenarios are not identical is assumed to be immaterial, and thus the adjustments described above are common when using FV and PV tables.

$100 × 1.04 = $104 $104 × 1.04 = $108.16 $108.16 × 1.04 = $112.49 $112.49 × 1.04 = $117


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Appendix 2


Adjusting Exhibit 4 for Cash Flows at the Beginning (Instead of the End) of the Year

At times, the series of cash flows for which a present value amount needs to be calculated begins at the start of each year as opposed to the end of each year. The discussion and example depicted in Exhibit 4 identifies the cash flows according to this latter pattern. It is not unusual, however, for the series of cash flows to begin immediately as depicted in the following revised Exhibit 4 example. Please note there are still three annual cash flows, they simply now begin at the start of their respective years. Exhibit 4 can still be used to ascertain today’s PV of this series of cash flows. The way to do that involves two steps. First, use the appropriate interest rate column (12% in this example) and use the two-years row, instead of the three-years row as originally done. In Exhibit 4, that factor is 1.69, and it will be used to PV all the cash flow amounts except the very first one. Second, because the first cash flow item occurs today, its PV is equivalent to the cash flow amount itself. So, to value it, we simply add 1.0 to the 1.69 factor pertaining to the other cash flow amounts in the example, arriving at an adjusted table factor of 2.69. Using that adjusted factor in the following fashion, $70.00 × 2.69 = $188.30, we get the PV of a series of three annual amounts of $70 each, where the series begins today (immediately), as equaling $188.30. In the above depiction, this is verified by discounting each of the three amounts separately (using Exhibit 2 factors) and obtaining the same total PV amount.

example (using 12%):

Today 1 2 3

$70 $70 $70 Start here

$62.50 .8929 ×

$55.80 .7972 ×

_______

$188.30

OR: $70.00 × (1.69 + 1) = $188.30

$70.00 × 2.69 = $188.30

from Exhibit 2

from Exhibit 4


Developing Financial Insights

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