WFU Law School
Law & Valuation
5.2.2 Basic DCF Formulas

5.2.3 Applying DCF Formulas

As we have seen, the capitalization of earnings method using the DCF formulas is simplicity itself -- if you know the cash flows and their discount rates. Estimating these inputs (and it is always an estimate) involves significant judgment calls.

Cash flows. You must measure earnings and estimate cash flows. You must decide which particular accounting figures reflect investor returns and adjust book earnings. You must decide whether past trends will continue and what projections are reasonable.

Discount rate. You must estimate a discount rate or capitalization rate to apply to the cash flows. To estimate the discount (capitalization) rate you can use the discount rates of comparable companies, by employing multiples based on their price-earnings ratio and other ratios. You can also use the Capital Asset Pricing Model (CAPM) to estimate the stock's expected rate of return (discount rate) based on historic price volatility compared to the market. To reflect the company's capital structure, you should weight the average cost of capital.

Adjustments. Even if your cash flow and capitalization rates are solid, you must also consider whether there are extraneous risks not captured by the capitalization of earnings. For example, publicly-traded shares often trade at a discount reflecting their lack of control, and minority shares in a closely-held business often include an illiquidity (lack of marketability) discount. On the other hand, a majority block in a closely-held business carries a control premium, since its owner not only sells an investment instrument but the power to control the investment. A business may have special risks, warranting further adjustments. For example, valuators often discount the value of a business that relies on one key person, that has unusually heavy contingent liabilities (such as a manufacturer of dangerous chemicals), or that is being investigated for securities violations.

Cases Applying Income Valuation Method

In Re Pullman Construction Industries Inc., 107 B.R. 909 (1989) (valuation business to determine whether Chapter 11 reorganization plan should be confirmed)

Neal v. Alabama By-Products Corp., 1990 Del. Ch. LEXIS 127 (judicial appraisal of minority shareholder interest)

Laurel Gonsalves v. Straight Arrow Publishers, Inc., 2002 Del. Ch. LEXIS 105. (judicial appraisal of minority shareholder interest)

Cede & Co. v. Technicolor, Inc., 1990 Del. Ch. LEXIS 259 (judicial appraisal of minority shareholder interest)


Estimating cash flows / earnings

Sometimes we can predict a business's cash flows with confidence. For example, the rental income from a shopping center can be computed if there are long-term leases with fixed payment obligations. But the future is rarely this simple. Often shopping center rentals are based on gross receipts, which can rise or fall. Generally, business returns are erratic and uncertain.

Most valuations rely on company income statements, which describe revenues, expenses and income for prior accounting periods. This accounting information presents two difficulties.

  • These numbers are based on accounting conventions that vary between companies (public and private) and countries (US and elsewhere). Thus, these numbers may not accurately reflect the returns that investors may obtain in the future. Often valuators adjust accounting income (or earnings) figures to identify the free cash flow that is available for distribution to investors.
  • Accounting information is historic. It does not purport to provide estimates of future results. Trends must be identified, often using information not available on the company's income statements.

Many valuations also rely on projections by company managers. But these projections, sometimes prepared to entice a lender to provide credit to the business or to comfort an anxious board of directors, must be taken with a grain of salt.


Adjustments of book earnings

Company financial statements reflect information about financial transactions, but not necessarily the company's ability to generate cash flow or income. For example, many small owner-managed businesses show earnings that the owner-manager actually views as personal income for her services. If the owner were asked to work for the business, or if new owners had to hire an outside manager, the company's earnings would be much lower.

Likewise, owner-managers of closely-held businesses often choose to lend money to the business, rather than invest permanent capital. (There are some advantages to this structure: interest on owner loans is tax-deductible by the corporation, while dividends are not; lending may protect the owner from a complete loss in case of bankruptcy; lending can allow different levels of financial participation without affecting voting allocations). This means that sometimes deductible interest payments really reflect payments to investors.

Often professional valuators will adjust book earnings to get a better picture of a company's capacity to present cash flow. Consider the information available from a typical income statement:

ALBEGA CORPORATION
Income Statement
Year ended December 31, 20xx
Net sales
$500,000
Cost of goods sold
$304,000
Gross margin
$196,000
Other expenses
$115,000

Depreciation

$ 15,000

Selling and administrative expenses

$100,000
Operating income
$81,000
Other income
$ 0
1 -Total income
$81,000
Dividends and interest
$13,000

Interest on long-term notes

$ 13,000
2 -Income before income taxes
$68,000
Income taxes
$18,000
3 -Net profit
$50,000
Extraordinary items
$12,000
Net income
$38,000
Net income per share
(1,000 shares outstanding)
$ 38.00

The object of the game is to figure out what cash flow the business is generating for investors -- which requires making adjustments to items in the income statement. To see how important cash flow is, see DeathWatch.

Some Terminology

1 - Total income This is a misleading measure of cash flow. It does not take into account taxes, nor does it reflect that depreciation is an accounting convention that does not actually diminish cash produced by a business.
2 - Pre-tax profits
(or "Income Before Income Taxes")
This reflects income after payment of interest and dividends, but is also misleading. Often in a closely held business, investors receive interest payments instead of dividends since interest is deductible, but dividends are not. In such a case, cash flow should be seen as including interest and dividends.
3- Net profits after taxes This reflects the effect of income taxes. [Taxes are not inevitable - and sometimes this creates a misimpression about the cash flow available to a purchaser of the business.]
Earnings before interest and taxes (EBIT) This is a common adjustment in a closely-held business where interest payments to owners may actually represent an investment return to the owners. (Remember interest is deductible and reduces the business's tax liability.) By looking at earnings before interest and taxes are deducted, we are assuming that interest is really an element of return to the owners and its deduction reduces taxes.
Earnings before interest, taxes, depreciation and amortization (EBITDA) These adjustments reflect the effect of taxes on cash flows, the cost of replacing aging assets [and so on.] This adjustment adds back depreciation and amortization [links]. The assumption is that these items did not actually diminish cash flow. There should, nonetheless, be a recognition in calculating earnings that aging assets eventually will have to be replaced.
Gross cash flow This is net income, plus depreciation, amortization and other non-cash charges. [more here]
Free cash flow This is the valuator's holy grail -- cash flow after taking account of cash drains. It reflects the actual returns of the investment.

Future projections

Once a valuator has identified (and adjusted) those earnings items that reflect investor returns, the next step is to project these cash flows into the future. This is tricky. Will current earning levels continue indefinitely? Will earnings rise with inflation? Will the rise more than inflation? These questions must be answered in projecting cash flows.

Generally, the projections for the first several years will be critical. Given the time value of money -- particularly when the discount rate is relatively high -- cash flows that occur after five years will carry relatively less valuation weight. Often valuators will focus on the first 5-7 years and then project a stable cash stream afterward to compute a "terminal value." (Why? Because they really can't be sure what cash flows will be that far into the future, so they just make a broadbrush guess.) Summing the present value of the first 5-7 years of income and the present value of the terminal value yields an estimate of firm value.

Example

Consider two companies, whose average earnings are identical, but the record of past earnings is remarkably different. How should earnings be projected into the future?

Year
Company A
Company B
1 (actual)
$100,000
$140,000
2 (actual)
$120,000
$180,000
3 (actual)
$150,000
$100,000
4 (actual)
$170,000
$90,000
5 (actual)
$200,000
$230,000
Average
$148,000
$148,000
Average rate of growth
19.00%
32.42%
Period growth rate
14.87%
10.44%

This leads to a host of questions. For Company B were the two off years (Years 3 and 4) aberrational, or was Year 5 an aberration? Is the steady growth for Company A likely to continue or has it topped off? Is Company B on the verge of explosive growth, or does its period growth rate (from Year 1 to Year 5) tell the story?


Answer: We get dramatically different results depending on how we answer these questions about the companies' growth prospects. For example, consider the valuation of the two companies if we assume that Company A's earnings of $200,000 will continue to grow at a 15% rate for the next five years and then level off, and that Company B's earnings will be $148,000, but will grow at a 33% rate for the next five years before leveling off. Assume a discount rate of 20% -- to reflect the riskiness of both companies.

Year
Company A
Company B
6 (estimate)
$230,000
$196,840
7 (estimate)
$264,500
$261,797
8 (estimate)
$304,175
$348,190
9 (estimate)
$349,801
$463,093
10 (estimate)
$402,271
$615,914
11-on (estimate)
$462,612
$819,165
PV of Years 6-10
$881,731
$1,018.186
PV of Years11-on perpetuity
$905,398
$1,603,224
TOTAL
$1,787,130
$2,621,410

This example is similar to that of Wilson v. Great American [earnings for first five years].


Estimating the discount (capitalization) rate

While earnings projections can be grounded in accounting data on past performance, the expected riskiness of those earnings floats in the air. Determining the discount rate (or capitalization rate) often turns on a comparative analysis of multipliers that have been used in valuing similar businesses. Ultimately, the question is how an investor would discount (or capitalize) an investment with similar earnings potential?

A capitalization rate allows a valuator to calculate the value of an earnings stream. The market determines capitalization rates, like discount rates, based on the duration and risk of the investment. They vary with time and incorporate, long-term inflationary expectations.

Capitalization rates based on market comparisons do not, however, tell us the discount rate. Why is this? A market-based capitalization rate does not tell us what growth rate is built into the market's pricing assumptions. That is, the capitalization rate is really the denominator in the Gordon model --

Capitalization rate = discount rate - growth rate
C = D - g

Capitalization rates assume the principle of substitution, since they are based on the yields available on alternative investments. There are two of principal methods, both based on comparisons to market, for determining a capitalization rate or multiplier.

  • Comparison of earnings and prices of comparable business -- establishing a price-earnings (or P/E) ratio.
  • Using the CAPM to determine an investment's required rate of return based on the asset's volatility, compared to market volatility.

Usually, valuators choose a single discount rate for a particular investment or asset -- taking into account both the risk-free time value of money and the particular risk of the investment or asset. But sometimes valuators choose different discount rates for different cash flows.

Note on Terminology

The term capitalization rate is often used in this context rather than discount rate -- though they are not quite the same. The discount rate is the rate of return investors would demand (required rate) for a particular return, compared to other available returns. But the discount rate does not take into account a growing investment. The capitalization rate does -- it is the rate that investors would demand for a particular investment, which is expected to offer growing returns. Thus, the capitalization rate of an investment with growing returns is --

Capitalization rate = Discount rate minus Growth rate

The term capitalization multiplier is similarly the reciprocal of the capitalization rate. As the earlier table showing various capitalization rates and multipliers illustrated, the higher the risk, the smaller the multiplier.

Capitalization Rates in Legal Contexts

Section 6 of Revenue Ruling 59-60:

"A determination of the proper capitalization rate presents one of the most difficult problems in valuation."

Smith v. Smith, 111 N.C. App. 460, 433 S.E.2d 196 (1994):

"A critical element of the excess earnings methodology is the capitalization rate used."

Barth H. Goldberg, Valuation of Divorce Assets, §; 6.6 (1984):

"It is generally true that the higher the risk of creating future earnings, the higher will be the annual rate or return which an investor will seek, and hence, the higher the capitalization rate.

"Determination of the proper percentage rate to use in the capitalization of earnings of a particular business is somewhat speculative.

2 John P. McCahey, Valuation and Distribution of Marital Property, § 22.08(2), 22-103-104 (1993).

"The risk involved refers to the degree of uncertainty that there will be any return on the investment. The greater uncertainty as to a return, the higher the risk."

"Thus, the capitalization rate represents the rate of return a prudent investor would expect annually on his investment given current interest rates and the relative risk involved in the type of business in question."

"In fact, criticism of the excess earnings methodology has been based on the difficulty in selection of an appropriate capitalization rate."

 

Capital Asset Pricing Model (CAPM)

Another short-cut for finding a capitalization rate is to use the CAPM.

Valuators use CAPM as part of a "building up" pricing model to estimate a small company's discount rate.

  • risk-free rate (which includes both a time-value-of-money real interest rate and an inflation factor)
  • risk premium (derived using CAPM for comparable large company common stock)
  • additional risk premium (for small company common stock) -- a somewhat controversial matter
  • specific company risk

Added together these risk rates provide a discount rate for the company's equity stock.

Example

If a publicly-traded company has a beta of 1.7 and the riskless rate for one-year Treasury bills is 4.5% and the expected return for the market is 12.5%. What is the company's capitalization rate?

Suppose the company has current earnings of $5.20 per share that have been growing at a rate of 6.5%. What should be the company's share price using a capitalization of earnings approach?


Answer: Using the CAPM straight-line formula, the capitalization rate (the required return an investor would demand to invest in the company) is

4.8% + 1.7*(12.5% - 4.5%) =
4.8% + 13.6% = 18.4%.

Applying the Gordon model, the price should be

[$5.20*(1+.065] / (.184 - .065) =
$5.54 / .119 = $46.54


Weighted average cost of capital

Many appraisers derive the discount rate from the weighted average cost of capital (WACC). This methodology has its roots in sophisticated financial management theory, but it is premised on basic, easily understood assumptions.

A firm derives capital from debt investors (who demand a fixed return) and equity investors (who accept a variable return). Debt, up to a point, is cheaper than equity—that is, the required rate of return for debt is typically less than for equity. To determine the value of a firm, we must be sensitive to the firm's different costs of capital.

For example, consider a company that has an overall risk (discount rate) of 15.6%, but 40% of its capital is supplied by debt which carries a 9.0% interest rate. Notice that this means its equity capital has a much higher required rate of return.

Proportion
Rate of return
Weighted
Debt
40%
.09
.036
Equity
60%
.20
.120
Weighted
100%
.156

Debt cost is derived by comparing the subject company’s debt with that of comparable issues in the public market. For example, if a AAA-rated company’s debt is currently selling at 6.5%, and the subject company being valued would be AAA rated, then the appraiser would use a 6.5% cost for the debt component.

Deriving equity cost is more complicated.Often, the appraiser starts with the risk-free rate (ordinarily, the U.S. Treasury borrowing rate) and adds elements that account for the risk characteristics of the subject company—such as the volatility of the business related to other firms (beta), the economic outlook for the particular industry, and other factors.

Example

Still to come

 

5.2.2 Basic DCF Formulas

©2003 Professor Alan R. Palmiter

This page was last updated on: April 5, 2004