Chapter 4 Income-Approach Valuation Methods

• Jurisdiction: United States

Chapter 4: Income-Approach Valuation Methods

A. The Foundations of Discounting: Time Value of Money¹⁰

We lost a major government contract because of their lousy components," complained the CEO of Navimetrics, Inc., an electronics manufacturer. His company had, indeed, been dropped from a multimillion-dollar U.S. Navy contract. The cause was repeated failures in its airborne navigation devices, which, through engineering analysis, were directly attributable to components sold to it by a supplier, AZ Circuitry.

The loss of the Navy contract was a financial blow to Navimetrics, and it looked to its supplier to recover its damages. Because of quality assurances made to Navimetrics in its contract, AZ Circuitry thought it best to consider settlement. The only question was the amount of the economic damages. An independent accounting firm analyzed the situation and determined that Navimetrics would lose the following cash flow (in millions) over the five years of the Navy contract:

Year 1	Year 2	Year 3	Year 4	Year 5
$3	$5	$6	$4	$2

"That's $20 million in lost cash flow," said the Navimetrics CEO. "And we won't settle for a penny less."

The attorney for AZ Circuitry had another perspective. "We recognize our culpability in this matter, but a flat settlement now will have to be for less than $20 million, if only because of the time value of money."

This case underscores a common situation involving money and the timing of its receipt. For example, the purchase of an ongoing business may involve the settlement of a current financial obligation by means of a series of payments over several years. Still others aim to settle a present obligation with a single payment at some time in the future. In numerous cases, including bankruptcies involving asbestos claims, an amount has to be set aside today in order to facilitate future claims. Similarly, in a bankruptcy involving an aircraft manufacturer, we were required by the bankruptcy judge to set aside an amount to compensate yet-unknown future claimants of fatalities involving the current fleet of aircraft.

Each of these situations involves a fundamental concept of finance known as the time value of money. It is the same concept the mortgage banker uses when she figures the monthly payment on a home loan. It is also the same concept that makes it possible for you to determine how many dollars you must put away in a lump sum today to meet your son's college expenses 10 years from now.

Of all the tools of modern finance, this tool may be the most useful. And the mathematics of time and money can be applied to an almost endless set of financial problems, just a few of which are examined here.

This section explains the concept of the time value of money and demonstrates the use of its principal analyses:

• the future value of a present single sum;
• the present value of a future single sum;
• the future value of an annuity (equal periodic payments); and
• the present value of an annuity.

These time value of money analyses are extremely useful in cases involving economic damages, valuation methodology [in particular, the discounted cash flow (DCF) valuation method] and bankruptcy. Extensions of these tools are used in business and in the legal field every day: in capital budgeting decisions; in the analysis of the returns from current operations or investments; in calculating the growth rates for a business; whenever mergers, acquisitions, or divestitures are contemplated; in economic damages and lost profits calculations; in valuations for solvency analysis purposes; and in litigation settlement discussions. Such financial analysis extensions include the following:

• net present value; and
• internal rate of return.

These concepts are widely accepted and are used daily in both the practice of corporate law and in litigation. Understanding these concepts will help the analyst to communicate more effectively with business clients and to understand their financial concerns.

This section explains how to perform time value calculations by means of a set of standard tables. This is the best way to learn. Solving the example problems with these tables will help the reader to appreciate the logic underlying time value concepts. Once the logic is explained, the section will describe how to do time value calculations using a financial calculator.

Future Value of a Present Sum

Everyone understands that money left in an interest-paying savings account will compound over time. This is the most commonplace example of the time value of money at work. Compounding means that the principal amount in the account and the interest earned periodically in the account will both earn interest. It is this compounding effect that makes savings accounts increase exponentially instead of at a linear rate, accelerating into an upward curve of increasing value, as shown in Exhibit 4-1.

As the exhibit demonstrates, the higher the rate of interest paid, the greater the increase in value. So, if you put $100 into an account at 5% interest compounded annually, that value will increase over the first 10 years to approximately $163. At 10% interest, that final value is a much larger amount — approximately $259. (The flat line indicates "simple interest" — interest that does not compound over time.)

Another point worth noting is that the frequency of compounding also affects the outcome. For example, a 10% savings account that compounds monthly will produce a larger amount over the years than the same account that compounds annually. The illustrative account in Exhibit 4-1 increased from $100 to $259 by compounding annually at 10% over 10 years. Had this same account been compounded on a monthly basis, its value at the end of the tenth year would have been around $271.

As the concept of compound interest sinks in, let's consider client McDonald, a man who suffered $100,000 in damages in a business transaction that happened five years ago. Assuming that his current claim is valid, we need to determine the amount that his claim should be today. We know intuitively that it should be more than $100,000. Had McDonald received $100,000 when the damages occurred, he could have invested the $100,000 compounded annually at 5% over a period of five years.

But how much would that future value be? We can calculate future value directly using the standard future value table. Here, we are trying to determine the future value of a present single sum.

The future value of a present single sum is what a current sum will increase to over a certain period of time at some compounding rate.

To calculate future value, let us frame the problem within a format that can be used as we move to more complex cases. Here, we have what is called a "present value" (PV), the compounding interest rate (i), the number of compounding periods (n), and a quantity we hope to determine: the future value (FV). Let's consider a $1 initial deposit compounded at a 10% interest rate.

PV = $1 i= 10% n=1 FV= ?

If we multiply the PV times the annual i, we get the amount by which our PV will increase in one period:

PV x i = $1 x .10 = $0.10

Adding this $0.10 of earnings to our original $1 gives us a FV of $1.10. If we left the interest and the original sum in an account to compound at the same rate for another year, we would have:

$1.10 x .10 = $0.11 in earned interest;

$1.10 + .11 = $1.21

So, the FV of $1 compounded at 10% for two years is $1.21.

Now let's look at Exhibit 4-2, "Future Value of $1." The compounding periods (n) are arrayed horizontally across the top, and the different rates of return (here from 1% to 20%) are arrayed vertically down the left-hand side. The numbers in the table are what we call "future value interest factors," or FVIFs.

As we can see from the intersection for one period at 10%, our calculation of the FV of our $1 is $1.10. For two periods, it is $1.21. We can use this table to determine the future value of money of any other sum for any given period.

For practice, we can use it to find the future value of the $100,000 claimed by client McDonald.

Exhibit 4-2: Future Value of $1

INTEREST				YEARS
RATE	1	2	3	4	5	6	7	8	9	10
1%	1.0100	1.0201	1.0303	1.0406	1.0510	1.0615	1.0721	1.0829	1.0937	1.1046
2%	1.0200	1.0404	1.0612	1.0824	1.1041	1.1262	1.1487	1.1717	1.1951	1.2190
3%	1.0300	1.0609	1.0927	1.1255	1.1593	1.1941	1.2299	1.2668	1.3048	1.3439
4%	1.0400	1.0816	1.1249	1.1699	1.2167	1.2653	1.3159	1.3686	1.4233	1.4802
5%	1.0500	1.1025	1.1576	1.2155	1.2763	1.3401	1.4071	1.4775	1.5513	1.6289
6%	1.0600	1.1236	1.1910	1.2625	1.3382	1.4185	1.5036	1.5938	1.6895	1.7908
7%	1.0700	1.1449	1.2250	1.3108	1.4026	1.5007	1.6058	1.7182	1.8385	1.9672
8%	1.0800	1.1664	1.2597	1.3605	1.4693	1.5869	1.7138	1.8509	1.9990	2.1589
9%	1.0900	1.1881	1.2950	1.4116	1.5386	1.6771	1.8280	1.9926	2.1719	2.3674
10%	1.1000	1.2100	1.3310	1.4641	1.6105	1.7716	1.9487	2.1436	2.3579	2.5937
11%	1.1100	1.2321	1.3676	1.5181	1.6851	1.8704	2.0762	2.3045	2.5580	2.8394
12%	1.1200	1.2544	1.4049	1.5735	1.7623	1.9738	2.2107	2.4760	2.7731	3.1058
13%	1.1300	1.2769	1.4429	1.6305	1.8424	2.0820	2.3526	2.6584	3.0040	3.3946
14%	1.1400	1.2996	1.4815	1.6890	1.9254	2.1950	2.5023	2.8526	3.2519	3.7072
15%	1.1500	1.3225	1.5209	1.7490	2.0114	2.3131	2.6600	3.0590	3.5179	4.0456
16%	1.1600	1.3456	1.5609	1.8106	2.1003	2.4364	2.8262	3.2784	3.8030	4.4114
17%	1.1700	1.3689

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