We continue our discussion of statutory fair value with an outline of the discounted cash flow (DCF) model (or method). The DCF valuation method is a core method within the Income Approach to Value (with the other two approaches being the Asset Approach and the Market Approach).
One objective of this series of posts on statutory fair value is to outline sufficient valuation and finance theory so we can begin to examine cases, i.e., judicial interpretations of what fair value means. With the proper background, we will be able to understand and to interpret what the courts have said in the context of valuation theory.
Most judges are not trained in valuation, which is understandable. They make decisions regarding valuation based on economic evidence presented to them by business appraisers. Unfortunately, the valuation evidence presented in courts is often conflicting, unclear and simply wrong from a theoretical viewpoint. The fact is, as we will see, there are a number of “bad” fair value decisions, where “bad” reflects the fact that they do not reflect current valuation theory or practice. Quite often, “bad” fair value decisions are the result of “bad” valuation evidence.
In addition, regardless of the quality of valuation evidence presented, judicial guidance is not always definitive, and reasonable legal interpretations by counsel can lead to differing conclusions about what constitutes fair value in a jurisdiction. For example, I have been criticized for applying a marketability discount in a fair value case in one state where counsel gave me that instruction. I have been criticized in another for not taking minority interest and marketability discounts. It took a decision by that state’s Supreme Court to resolve the issue.
As I mentioned in the first article in this series, it is not the appraiser’s job to determine what kind of value fair value should be. That is for the courts to decide. I hope that this series will, over time, provide assistance to appraisers, counsel for parties in statutory fair value matters, and to the judges in the courts where future decisions will be made.
The following discussion of the discounted cash flow method is excerpted in part and modified in part from Chapter 1 of Business Valuation: An Integrated Theory Second Edition, which I co-authored with Travis W. Harms.
The Value of a Business Defined
The value of a business enterprise can be described as:
- The value today (i.e., in cash-equivalent terms)
- of all expected future cash flows (or benefits) of the business
- forecasted or estimated over an indefinite time period (i.e., into perpetuity)
- that have been discounted to the present (expressed in terms of present value dollars) at an appropriate discount rate (which takes into consideration the riskiness of the projected cash flows of the business relative to alternative investments).
The valuation and finance literature consistently confirm this conceptual definition of the value of a business enterprise. In order to value a business, therefore, we need the following:
- A forecast of all expected future cash flows or benefits to be derived from ownership of the business; and,
- An appropriate discount rate with which to discount the cash flows to the present.
This conceptual definition of business value can be defined symbolically in the following equation:
- Vo is the value of the equity of a business today.
- CF1 to CFn represent the expected cash flows (or benefits) to be derived for periods 1 to n. The discounted cash flow model is based on time periods of time of equal length. Because forecasts are often made on an annual basis in practice, we use the terms “periods” and “years” almost interchangeably for purposes of this theoretical discussion.
- r is the discount rate that converts future dollars of CF into present dollars of value.
The equation above is the basic discounted cash flow (DCF) model. To employ the model in this form, however, the analyst must make a forecast of all the relevant cash flows into the indefinite future. For clarity, the cash flows or earnings discussed in this chapter are the net earnings and net cash flows of the enterprise or the business as a whole. V0 is the value of the equity of the enterprise, or the present value of the expected cash flows to the owners of the equity of the enterprise.
The Gordon Model
In his 1962 finance text, The Investment, Financing, and Valuation of the Corporation, Myron J. Gordon showed that under the appropriate assumptions, the DCF equation is equivalent to the simplified equation shown below:
The Gordon Model initially dealt with dividends, hence it has been called the Gordon Dividend Model, or the Gordon Growth Model. The Gordon Model has become so generalized that it reflects what can be called the generalized valuation model. In practice, CF1 often represents the estimate of earnings for the next period so we can generalize and refer to the cash flow measure as Earnings. The expression (r – g) is known as the capitalization rate (see “Glossary,” ASA Business Valuation Standards (Washington, DC: American Society of Appraisers, 2005), p.21.) And the expression ( 1 /(r – g) ) is a multiple of earnings. So the Gordon Model is consistent with the general valuation model:
These factors are so familiar that appraisers sometimes forget their source. Earnings in the generalized valuation model must be clearly defined and the “multiple” must be appropriate for the defined measure of earnings. These comments could be based on common sense, and they are. However, as will be shown, they are also theoretically sound.
For the DCF model and the Gordon Model to be equivalent, the following conditions must hold:
- CF1 is the measure of expected cash flow for the next period (sometimes derived as (CF0 x (1 + g)) or otherwise derived specifically).
- Cash flows must grow at the constant rate of g into perpetuity.
- All cash flows must be: 1) distributed to owners; or, 2) reinvested in the enterprise at the discount rate, r.
The discount rate, r, must be the appropriate discount rate for the selected measure of cash flow, CF. In the real world, businesses make reinvestments and accept the returns of these investments, some of which will exceed r and some of which may be less than r. This model assumes that all reinvestments will achieve a return of r.
By comparing the DCF model equation with the Gordon model equation, we see two ways to estimate the value of an enterprise. The next equation restates the DCF model to reflect constant growth and relates it to the Gordon Model.
- The left portion of the equation illustrates a forecast of cash flows at a constant rate into perpetuity, discounted to the present at the discount rate r.
- With appropriate algebraic manipulation, the left portion of the equation reduces to the Gordon Model, which is shown at the right above.
Two-Stage DCF Model
Recall the conditions that must hold for equivalency of the DCF and Gordon Models to be equivalent expressions. In practice, these conditions may limit the strict application of either expression.
- Application of the DCF model as in the first equation requires a discrete forecast to time period n, or effectively into perpetuity. Few forecasts extend reliably beyond five or 10 years in practice.
- Application of the Gordon Model requires that the estimate of next year’s cash flow grow into perpetuity at a constant rate of g. This condition may not be consistent with an analyst’s expectations regarding near-term cash flow growth, which may be significantly different from longer-term expectations for growth.
In practice, these two limitations are overcome by use of a “two-stage” DCF model that combines elements of the perpetuity DCF model and the Gordon Model. The two-stage DCF model is presented below, and consists of the following two sets of forecast cash flows:
- Interim Cash Flows (for finite period ending in Year f). While accurate predictions regarding the future are certainly elusive, diligent analysts can often prepare reasonable forecasts of near-term financial results for most businesses. The left side of the equation depicts the Present Value of Interim Cash Flows (PVICF).
- Terminal Value (all remaining cash flows after Year f). Following the discrete forecast period, the two-stage DCF model reverts to the Gordon Model, as the accuracy of the analyst’s discrete financial forecast wanes, and violation of the constant-growth condition becomes less significant. When discounted to the present from the end of year f, the Present Value of the Terminal Value (PVTV) is obtained.
Appraisers using the two-stage DCF model typically employ discrete forecast periods ranging from about three to 10 years or so, followed by application of the Gordon Model as shown in Equation 1-4. Alternatively, in practice, many appraisers and market participants use a market-based method that applies current market multiples to the forecasted cash flow for Year f or Year f-plus-1. This alternative practice, if employed with reasonable multiples from the public marketplace, should not be considered unusual or incorrect.
DCF, the Gordon Model and Public Security Valuation
Thus far, we have been speaking about the DCF Model and the Gordon Model. Both of these are valuation models employed when using the income approach to valuation. The other commonly used valuation approach used in valuing profitable business enterprises is the market approach. Under the market approach, comparisons are made with valuation metrics of a subject company and the similar metrics of similar, or “guideline companies.”
We know that we can estimate value using a single period income capitalization method, i.e., the Gordon Model, for a public or private company. If expected earnings are $1.00 per share, the (constant) growth in earnings is 5.0%, and the discount rate is 15.0%, then the indicated value is $10.00 per share ($1.00 / (15% minus 5%).
In the context of a publicly traded stock, we can specify the Gordon Model as follows:
The price of a publicly traded stock today reflects the present value of all expected future dividends. Ignoring for a moment the possibility of share repurchases by the company, the receipt of dividends represents the only return the shareholders, will receive from ownership of the stock – other than a sale of stock in the public market, where all expected future dividends are continuously capitalized in the market price. We derive the price/earnings multiple by dividing both sides of the equation by earnings for the coming year (E1).
Recognize that the expression (D1/E1) is the dividend payout ratio, or DPO.
Now, assume that DPO equals 100%, or 1.0. Therefore, the P/E of Equation 1-12 is (1/(r–g)). This should clarify that valuation analysts, who typically derive earnings multiples as (1/(r-g)), are making an implied assumption that all earnings of the company will be distributed, i.e., that the DPO = 100%.
Now, assume that we observe that the market price for a public company is $10.00 per share (for convenience and comparability to the private company example above). Expected earnings are $1.00 per share as indicated by the consensus of analysts’ estimates. The Price/Earnings ratio, or multiple, is 10.0x ($10.00 price / $1.00 expected earnings).
With the Gordon Model and the income approach, we use analyst-derived estimates of expected earnings (CF1 or E1) and the analyst’s estimate of the discount rate (r). Using the equation above and assuming a DPO of 100% (or 1.0), we can now derive the discount rate for the public company, or r, which is 15.0% (P/E = E / (r – g). Given P/E = 10.0, g = 5.0% and DPO = 1.0, solve for r).
At its simplest, in a perfect world, analysts will develop the very same indications of value using income methods (the DCF model or the Gordon Model) and market approach methods (using guideline public companies as the basis for applying multiples to earnings).
Relationship of the DCF Method to Fair Value
The DCF method is a commonly used valuation method, particularly when valuing sizeable companies where management routinely prepares forecasts of future financial performance. For example, the DCF method would appear to be the favored valuation method for valuations presented to the Delaware Court of Chancery.
Even when the Gordon Model is used as a single period income capitalization method, there is an implicit forecast of future performance.
In next post, we will discuss what is called the conceptual levels of value chart. This chart as it has been developed over the last twenty years or so by valuation writers (including the present writer) can be used to illustrate the various “kinds of value” that courts might consider to be fair value as they interpret fair value statutes in the various states.
Following the initial discussion of levels of value, we will then use the Gordon Model, and implicitly, the DCF method, to define what I refer to as the marketable minority level of value. This is the level at which public companies trade in normal and active markets. It is also one level at which appraisers develop valuation indications. Along the way, we’ll discuss the relationship between the DCF method and valuation by reference to what are called guideline public companies.
After a number of additional posts where we address valuation theory, we will begin to look at some fair value cases. I believe that the investment in background will pay dividends in our collective understanding of statutory fair value in any particular jurisdiction as well as similarities and differences among and between the various states.