"Free cash flow" appraisal ... a better way?

Cash flow is the measure by which investment real estate is valued. Traditionally, appraisers capitalize future net operating income (NOI) - rather than cash flow - for a given property over a holding period into market value. A great deal of attention has been paid to improving the capitalization theories, resulting in a number of techniques for discounting anticipated future cash flows to arrive at present value. Despite this attentiveness to finding a more accurate way to measure the time value of money, risk, appreciation, financing, and the reinvestment of net revenues generated by the real estate investment itself, the traditional income approach should be reconsidered, particularly since anticipating the future of cash flows is hardly a precise science.

Declining markets produce declining returns, which produce declining values. Soon everyone involved is focusing on what went wrong. Lenders, facing huge losses in their real estate portfolios, somewhat rightly, blame the experts who estimated greater value than declining market support. However, the problem should rest not so much with those who perform appraisals for a living, but instead with the generally accepted methodology, the income approach. More specifically, the problem is not so much with the theory that market value is estimated by capitalizing expected future income streams but in not having paid enough attention to the detail of the cash flows themselves. Too much import has been given to somewhat esoteric "present valued" capitalization rates and not enough to the quality, the components, and the direction of the cash flows involved.

This paper is about cash flows, how they are defined, and how they are used in income property appraisal. The objective is to initiate a discussion of "free cash flows," a new term to some, as a better alternative for evaluating the real earning power of real property in use and, at the same time, for providing investors and lenders with a more reliable look at the risks over the next three to five years.

GENERALLY ACCEPTED CASH FLOW STATEMENTS

Cash flow statements have often been confused with income statements. One is often referred to as the other and vice versa. Basically, income statements measure earnings while cash flow statements account for sources and uses of cash funds. The two are very different but related in that both seek to measure the financial health of an enterprise in general terms.

Income statements usually begin with gross revenues and end with net income (operating income or pretax income). Income statements derive much of their input from cash flow activities, hence the confusion. Traditional income statements follow a calculation similar to the following:

Gross projected income
- Vacancy and credit loss
+ Other realized income

Adjusted gross income
- Operating expense            [Real Estate Appraisers
                               estimate market value
Net operating income           primarily from the fi-
- Interest cost                nancial results listed
- Depreciation                 above the dotted line.]

Net income before tax          [Accountants report
- Taxes                        and analyze the present
                               financial condition of
Net income after tax           the enterprise from fi-
- Principal repayment          nancial results listed be-
+ Depreciation                 low the dotted line.]

Net cash flow after tax

Real estate appraisers generally capitalize NOI, calculating an estimate of market value after adjusting for financing conditions and risk within the capitalization rates. Financial analysts and accountants, seeking to value corporate securities or privately held business generally focus on the net income before tax, plus trends in their income and expense components before applying simple income multipliers. For some forms of financial analysis, NOI numbers are more than sufficient. NOI results are revealing for companies operating with few inventories or fixed assets. NOI figures also tell a rather complete story about real estate under long-term lease contracts. However, when asset conditions change, and more in-depth analysis is needed, cash flow results are more revealing since it is important to know where the cash flows are coming from and where they are being spent or distributed.

Figures 1 and 2 recommend guidelines for developing direct cash flow statements for both corporate business- and real estate-related activities. The business guidelines are shown in Figure 1, defining the critical elements of those cash flow statements.

The cash flow statement in Figure 2 emphasizes once again that operating, investing, and financing activities associated with a real estate enterprise must be segregated.

Both sets of guidelines segregate the operating activities, the financing activities and the investing activities because in concept while technically a cash flow, each of the three is very different. Each reflects the condition and, viewed over time, the "direction" of the enterprise from different perspectives. The issue here is not so much the accounting rules or regulations surrounding the development of the cash flow statements themselves, either for businesses in general or for real estate ventures in particular. Rather, the important issue is that each of the three cash flow categories has a significant impact on the short- and long-term abilities of the enterprise to expand, improve market position relative to the competition, and survive during economic downturns. Traditional business income statements, particularly as related to real estate, do not always report that impact.

As mentioned, each of the three types of cash flow activities tells a story. Operating cash flows tell the actual ongoing "operating the present enterprise" story. Financing cash flows speak to the condition and changes in the debt and equity makeup of the ownership of the enterprise. Investing cash flows measure the actual or accrued additions to the physical and economic assets that are part of the enterprise and that, in real estate, determine - along with changing locational preference - its future.

The analysis and management of each of the three types of cash flow activities is also the same, yet different. Jointly, all seek to maximize profitability. The objective in managing the operating cash flows is to maximize the current and future profitability of the business itself. Financing activities seek that perfect combination of debt [TABULAR DATA FOR FIGURE 1 OMITTED] and equity financing for the enterprise, again to maximize current and future profitability. Investing activities attempt to add some kind of value through modification, which creates a greater current and future profitability than the marginal cost of the investment (an investment which may be necessary or discretionary).

ACCOUNTING RATIOS

Batting averages in baseball, yards gained in football, and points per game in basketball each give quick reference benchmarks of specific performance. Similarly, the American Institute of Certified Public Accountants (AICPA), securities analysts, financial institutions, and other accounting and finance professionals use ratios like those shown in Figure 3 to measure the current cash flow position of an enterprise. Some real estate appraisers use like ratios to measure the financial condition of businesses situated in single-use facilities, since the business future directly determines the worth of that real property.

The efficiency ratios provide a reasonably good picture of the profitability of [TABULAR DATA FOR FIGURE 2 OMITTED] business operations while the sufficiency ratios address the profitability of the investment itself, including its ability to sustain the current operation. In other words, the efficiency ratios address issues in the current state of affairs such as the profitability of sales, the cash generated by ongoing operations, and the return on assets. The sufficiency ratios indicate how long that current state is likely to continue, including a look at the adequacy of cash flow to meet both investment, dividend, and debt service needs. Calculated over a four- to five-year period, these nine ratios provide a great deal of insight into the direction of the enterprise from a profit margin, a debt, and an investment viewpoint, at least from a traditional income statement perspective.

Adding efficiency and sufficiency ratio analyses to the current real property appraisal process would lead to a more accurate prediction of anticipated income levels. Traditional income statements, augmented by a presentation of the nine ratios - other ratios may be added as well - would then offer a look at trends and the current income position of the property in aggregate terms. Comparisons with like property average ratios permits lenders and purchasers, as well as management, to get a feel for the competitive position of the subject property, leading to further analyses of the condition of the enterprise. Free cash flow analysis, however, may offer an even better approach.

FREE CASH FLOWS DEFINED

Free cash flows are not the same as traditional cash flows. Free cash flows are cash flows generated by the current operation of a business or real estate venture and are available to be distributed back to the owners/shareholders without affecting current levels of growth. The essential phrase is [TABULAR DATA FOR FIGURE 3 OMITTED] "available . . . without affecting current levels of growth." In theory, free (excess) cash flow amounts can either increase the value of the enterprise or property when they are reinvested or distributed, increasing the current return to the equity owners. Free cash flows address the issue of cash flow adequacy. This fact is coupled with the underlying assumption that investment occurs only when the marginal revenue produced by the reinvestment is greater than the marginal cost of that investment. Of course, in practice that assumption depends on an intelligent, informed management making the right decision.

The formula in Figure 4 describes a basic three-step method for calculating free cash flows. Free cash flow analysis differentiates the discretionary activities and expenditures from the nondiscretionary activities and expenditures for each of the operating investing and financing categories identified by the AICPA. As a result, the needs of an ongoing enterprise are separated from the wants. The quantification of the cash impact of the positive and negative moves by management reveals the underlying earning power of the real property assets. Also, the consensual or forced direction of the future of the income-producing assets becomes more apparent.

Note that others in the securities analysis business define free cash flows a little differently, maintaining that free cash flows are derived by subtracting dividends and capital expenditures from traditional cash flow. Theoretically, that calculation results in a measurement of the asset's (company stock or real estate) current "earning power." The dividend and capital expenditure numbers are readily available as are the traditional cash flows. There is no potential for dispute over the meaning and measurement of discretionary outlays whatever the purpose. As you will see, that is no mean issue.

The term "discretionary" refers to lavish landscaping, steel I beams as floor joists in residential construction, and other cases of extra or overimprovement. "Discretionary" implies something greater than what is needed for functional or economic utility purposes or to cure obsolescence of any kind. Discretionary capital expenditures are those excess funds invested in the real estate project greater than the investments normally associated with a project of this type and size as measured on average over the last four to five years. Likewise, discretionary operating cash outlays are those cash outlays spent for operations greater than the expenditures normally associated with the operation of a project of this type and size as measured on average over the last four to five years. A four- to five-year observation period is key because history provides insight into the direction of the assets.

For definitional purposes, operating cash flow calculations require that property and equipment be recorded at cost. Depreciation is calculated on a straight-line method over the useful lives of respective assets. Inventories are stated at the lower of cost or market. Intangible assets are valued at cost, amortized on a straight-line basis. Taxes are assumed to be paid currently. Standardizing the accounting for all of these items permits the free cash flow process to pursue its main objective: determining the condition of the enterprise after extraneous items, nonrepetitive items, and accounting variations are removed.

The case for using free cash flows as the standard for financial health and well-being is restated in Figure 5. Beginning with the appraiser's traditional NOI figure, a host of adjustments may be made, depending on the subject property's business circumstance, resulting finally in net free cash flow before debt service and tax. Net free cash flow after debt service and taxes can and should be calculated as well.

Focusing significantly greater attention on the cash flows (to obtain a free cash flow number) puts a much greater burden on the appraiser than with past practice. The contention here is that net cash flows and, particularly free cash flows, are substantively more important than NOI numbers for estimating the true worth of an enterprise. Each of the above listed items could have a real impact on those cash flows for any of three reasons. First, an expenditure may have been made which is deducted from operating income in part only. Second, revenues or expenses may have occurred which have little relationship to the ongoing enterprise, but are included in the operating income statement. Third, non-income statement issues (assets or liabilities) may have significantly added to or reduced the cash position of the enterprise without appearing on the income statement at all. Knowing exactly what has occurred is very important to the appraisal process, overshadowing the added burden thrust on the appraiser.

It may be obvious to most people that rising free cash flows (over a four- to five-year period) point to a healthy operation, a tenable current debt position, and a property that is probably not in a state of decline (although deferred maintenance could by strategy be increasing). In contrast, falling free cash flows suggest a short- or long-term question of the property's viability, some unusual investing activity, or a management or general economic problem. Comparing investing activities associated with a specific real property with norms of similar properties separates the good from the bad. There is really no magic to the analysis. Too much or too little investing can be seen. Similarly, there is no magic in making comparisons between the operating activities of specific properties and industry norms. Local and regional operating ratios are generally available. The important thing is that a true picture of the opportunity and ability to earn income from the subject property be defined. While the appraiser cannot be responsible for future management or conditions, that appraiser can better focus interested parties on the present cash flows and their likely future. As a side benefit, that appraiser can, when pressed after the fact, say, "I told you so."

Segregating income, expense, and investment and financing activities is not unusual business practice. Owners and prospective purchasers of businesses of all types have long removed unusual, excess, family - related, and other discretionary expenditures and sources of income from their cash flow projections when determining purchase price. Projected growth or decline is forecast from that point. Reserves are used to put aside monies for necessary future investment, but typically only when obviously needed or when required for risk or insurance purposes. The major difference between what has been and what is proposed herein with free cash flow analysis for real estate purposes, aside from the calculations themselves, is the determination of specific free cash flow levels that are required to continue existing operations plus growth. Free cash flow management then demands a certain ongoing, future-thinking analysis of the enterprise. It must be done with a second ongoing, future-thinking objective in mind: keeping the cost and balance of debt and equity financing at optimal levels.

OPTIMUM FINANCING

The second objective of free cash flow analysis is to seek the optimal balance of debt and equity financing for the real estate venture. Since real estate is capital intensive and long lived, financing is mostly long term in nature. Debt financing is advanced for 10-, 15-, and 25-year periods with ongoing terms and conditions, although lenders have been seeking shorter commitments to reduce their interest rate risks. The current costs to the borrower (principal and interest payments) are reflected in the traditional cash flow statements and in income statements. If no change has occurred for the current period in the long-term debt structure, the current portion of that long-term debt, lease obligation, and off-balance sheet obligations, then the current analysis is usually considered complete. However, from a free cash flow analysis standpoint, the process is just beginning.

Optimal financing depends on the cost of borrowed capital as measured against the desired return on equity capital. Investors, owners, and managers set the desired return on equity rate as they compare the ownership of real property, including its risks, with the returns available from other risk investments. There is really no way to calculate what an equity return should be except in light of the alternatives available. Therefore, that desired equity rate is determined in the minds of those investors, owners, or managers and used as a benchmark for compare-and-contrast purposes with the debt financing alternatives. From that point, the lowest cost of debt capital, and the theoretical optimal amount of debt capital should be sought in the context of steady or growing free cash flows.

Traditional measures of a real estate project's ability to pay its debt service include those ratios listed earlier as sufficiency ratios. One of those ratios, the debt coverage ratio (total annual debt service/cash from operations) speaks directly to the risk associated with the financial risk of that project. In theory, the better the ratio, the greater the cash flow coverage of current debt obligations. But note that the volatility of those cash flows is always at issue, despite what the present ratio might say, at least until the outstanding debt declines to a refinanceable position. Operating revenues, therefore, must be ever sufficient to cover current debt service plus any investing needs before distributions of any kind are made to equity holders, which is guaranteed only if one eye is on the future. Since free cash flows provide a look at real predebt, pretax cash flows after operating, investing, and financing activities positive and negative have been considered, a free cash flow debt coverage ratio should better show how much cushion is in the real financial position in relation to the ever-present requirement to pay back the current and future debt. The best ratio for such an analysis is probably one in which total current debt service is divided by an annual four- to five-year moving average free cash flow number. Four to five years of history smooths out aberrations in the free cash flows while establishing a comparative norm for those free cash flows.

Practically speaking, the optimal financing structure will likely elude even the most accomplished real estate financier. However, the exercise of examining the sources and uses of cash and the relative risks of investing in the property, coupled with the continuing conflict between shareholder dividend demands and the need for more external capital for competitive purposes, keeps that objective clearly in sight. Heretofore, much of the pertinent information for the pursuit has not been formally available. Free cash flow analyses which provide a specific criteria for distribution and retention of profits (for investing or cash reserve purposes) would be helpful to all concerned.

As mentioned, present free cash flow results plus a four- to five-year free cash flow history tells a lot about the enterprise. Free cash flow debt coverage ratios carry the analysis to the next step by answering the question, "How much cushion exists?" But the key issue is volatility. If everything were static, decisions could be made without fear. Since nothing in real estate or in business in general remains static for very long, specific volatility benchmarks should be used. That means, for example, that if free cash flows varied by less than 20% and exhibited a 5% overall growth over the last four to five years as well, then more confidence could be put in the property's operation to cover debt, invest as needed, and return something to the equity position. If, on the other hand, free cash flows exhibited a greater than 20% volatility from year to year and showed no growth, then it might not be wise to distribute anything to the equity holder because free cash would likely be needed for future operating or investing activities.

Every investor wants growing free cash flows while debt and the cost of debt are declining. That offers the ability to make changes in the operation and to invest in physical assets as needed without the need for outside capital infusion. Unfortunately, not all, nor even most, real estate investments reach such an envious financial position, and when they exist in a less-than-optimum condition, investors need to know where they sit on the risk continuum. For a graphic illustration of that risk position, Hackel and Livnat(1) suggest a four-quadrant approach. Figure 6 shows that the volatility of the free cash flow is measured against financial risk as quantified by the free cash flow debt coverage ratio. Specific criteria for inclusion or noninclusion in a particular quadrant depend on the market's tolerance or an investor's tolerance for risk (A 20% volatility and 5% growth rates are a good point of beginning.). In Figure 6, operating risk is charted against the financial risk created by a specific real property in the current year. The key again is to look at these same performances over time, if possible over an economic cycle before deciding which quadrant best applies. Charting each of the last five years' financial results yields that quick analysis. The objective is to seek that point where debt costs are minimized and equity returns are maximized, all within the context of reasonable risk.

Similar analysis could be obtained by setting minimum standards for assessing risk in both the operations and the financial ratios. A 20% volatility in the free cash flow history of a property means more risk than a 5% volatility. If the free cash flow is 1.5 times greater than the annual debt service payment, that suggests less risk than a 1.2 coverage number. Again, there is nothing magical about the application of these standards. Lending institutions have long included statements regarding performance standards in loan documents which force managers and operators to maintain satisfactory coverages or face penalties. On the other hand, using free cash flows instead of traditional income statement numbers really gets to the heart of the earning power of the asset. Generally accepted norms will have to be adjusted accordingly but, once developed, the new free cash flow statistics would be relatively easy to use.

VALUATION USING FREE CASH FLOWS

If free cash flows offer a better way to assess the earning power of an asset in use and if free cash flow analysis may be used to assess risk in light of the cost of capital and the ability to repay that capital, then free cash flows should be better indicators of present and, particularly, future market value than traditional cash flows.

Gross rent multipliers have long been used by sophisticated investors to determine the investment value of income-producing real estate. Those investors are able to translate gross rent into investment value because they know the inner workings of specific property types. They can accurately anticipate the bottom-line results by knowing the costs of operation. They also know how much needs to be invested back into that type of income property to maintain function and appeal. Lastly, they know intimately the costs of capital and the debt service involved. That knowledge makes the income multiplier approach to value as accurate for them as the income capitalization approach for other investors. For some, it may be more accurate.

Since free cash flows are by definition those cash flows generated by the current operation that are "available to be distributed back to the owners/shareholders without affecting current levels of growth," free cash flows are, by definition, measures of the returns to the equity position after all else has been considered. Operating, investing, and current financing activities are considered in the free cash flow numbers, leaving only equity and alternative debt financing to be addressed. The effects of alternative debt financing do affect the property and its ongoing financial viability in the future, but they have little to do with the property operation and ownership in place today, leaving only the equity return to be considered. That equity return as measured by the free cash flows is a key indicator of value and, as such, can be used in comparison with other equity return indicators of that relative worth.

The process of applying income multipliers to free cash flows to calculate market value depends on two assumptions. First, it assumes that the equity rate of return evidenced by the multiplier is indicative of market forces at work. Second, it assumes that the historically determined growth rate of the free cash flows will be constant in the future, growing at a constant rate in each and every future period. Both assumptions are necessary to ensure that the free cash flow projections are viewed over time as they are at present. Obviously, both the equity rate of return and the growth rate of the free cash flows will in practice vary, but it is assumed that those changes will not be substantive in the near future. Ongoing free cash flow analyses can adjust the results accordingly.

Hackel and Livnat propose the following rationale (see Figure 7) to prove the mathematics of using the multiplier process for securities valuations, a rationale which is applicable to income real property values as well.

The first key assumption is that k is larger than r. The equity rate must be greater than the growth rate of the free cash flows in order to sustain investors' interest. That being the case, the free cash flow multiplier will ultimately be a function of the rates of change in the growth rate of the free cash flow in comparison to the growth rate of the difference between the required equity rate and the free cash flow growth rate. If both are assumed constant at the time of valuation, and the growth rate of the free cash flows is assumed to be constant over time, then the increasing or decreasing differences between them determine the relative attractiveness of the income stream. Because those differences are measured over time, they add the time value of money element to the net results. Because those differences include a discounting of the discretionary and nonrecurring incomes and expenses over time, they also add accuracy to the net results.

FREE CASH FLOW APPRAISAL

The argument for free cash flows is only theoretical at this point. Within the definition of this term, all the traditional appraisal variables are quantified, and many are qualified as well. It would be more convincing if specific criteria could be advanced for businesses and for income-producing real property which would identify optimal debt/equity levels of investment and percentages of growth for each cash flow activity: operating, investing, and financing. Unfortunately, specific criteria are not readily available. For various types of properties, however, a pretty solid history of operating and investing expenditure levels has been established. Nevertheless, proving the practical validity of free cash flow analysis and valuation will take time. It should be worth the effort.

The first item on the agenda needs to be the development of a commonly accepted definition of "discretionary." Remember that we are trying to ascertain the real earning power of the income property. Therefore, any definition should fit within the scope of that objective. The purpose is not to identify and punish those who overpay for goods or services; the purpose is to determine for prospective purchasers and lenders the likely earnings of the property in use.

If the term discretionary reaches a generally accepted accounting practice definition, then dividends of any form must be subtracted from cash flow, or added back, depending on the perspective. There is no argument with the dividend treatment, only that it be uniform. Once dividends are excluded, the free cash flow number should be an accurate indicator of income streams, given a few limitations inherent in free cash flow theory itself.

One limitation of using free cash flow methods for valuation lies in the fact that the value of a future sale and its associated cash flow are not included because it is outside the scope of normal operating, investing, and financing activities. When a contract for purchase exists, that contract can be valued separately and adjusted for its present value.

Another drawback of using free cash flow methods for valuation exists when free cash flows are negative (r [greater than] k in the formula). Value tends to decrease oftentimes more greatly than the cost or market approaches would dictate. Then again, that is what the correlation process is all about. One approach is more applicable than another in a given situation.

A third and previously mentioned limitation lies in the definition (as it now exists) of the free cash flows themselves. Time and experience and the minds of many practitioners will be needed to get everyone on the same page about the definition of "discretionary" expenditures. Nonrecurring incomes and expenses are not so difficult to identify; however, the matter of judgment, as far as typical expenses, necessary salaries, competitive reinvestment, and the like, is not so easy without generally accepted guidelines.

A fourth drawback exists when there is no history for the subject property. Then history must come from similar properties in similar locales. Sometimes history is transferrable and sometimes it is not. History improves the accuracy of the free cash flow method just as it does with traditional appraisal approaches.

With an internal rate of return (IRR) believer, a fifth drawback may exist, depending on the person's perspectives, because the concept of reinvesting the returns generated by an income-producing property is different under the free cash flow model. Those returns are not reinvested at an internal rate.

Despite the reservations and deficiencies, free cash flow analysis has much to recommend itself. The overriding benefit of looking critically at the cash flows themselves is getting to the heart of the matter of value in use. The present and future accuracy of those cash flows has been the single greatest deficiency of current appraisal techniques. As has been said more than once, free cash flow analysis, at its very least, allows the examiner to take a critical look at the current level and at the future direction of those cash flows both in terms of quantity and quality.

Free cash flow analysis is particularly applicable for real estate investment trusts (REITs). The portfolios of properties owned by REITs need to be understood by market analysts and investors. REIT performance is sometimes skewed by short-run events and strategies, leaving the true earning power of the assets themselves unknown. REITs possessing sound income-producing properties want to differentiate themselves from the pack. Publishing free cash flows for knowledgeable investors is a simple yet direct way of making that distinction. The same can be said for any institutional fund that invests in income-producing real property.

Whether you are an investor, an operator, or a lender of income-producing real property, free cash flow analysis yields valuable information and perspective. While the art of calculating, analyzing, and valuing free cash flows is just beginning, gaining proficiency with it is worth the effort. Hopefully, the industry movers will agree, and offer additional and better suggestions for their use.

RELATED ARTICLE: FIGURE 4

Free Cash Flow Formula

Free cash flow = Traditional cash flow

[Net operating cash flow - Capital expenditures]

+ Discretionary capital expenditures + Discretionary cash outflows for nondiscretionary purposes

SOURCE: Kenneth S. Hackle and Joshua Livnat, Cash Flow and Security Analysis, Chap. 5 (Burr Ridge, Illinois: Irwin Professional Publications, 1992), 170-224.

RELATED ARTICLE: FIGURE 5

Free Cash Flow Statement Adjustments

Net operating income

+/- Adjustments: to reconcile net pretax, pre-debt income to net cash provided from operating activities

- Depreciation

- Depletion (for oil and gas properties, etc.)

- Deferred income taxes and Investment tax credits

- Earnings from other ventures (less distributions)

- Gain on sale of assets

- Reserves

- Legal settlement costs (such as gas purchase contract settlement cost, etc.)

+/- Accounts Payable changes

+/- Accounts Receivable change

+/- Inventories changes

+/- Other Current Asset and Liabilities changes

+/- Capital Lease changes

+/- Insurance Cash Flow

= Net free cash flow before debt service and tax

RELATED ARTICLE: FIGURE 7

Multiplier Process for Securities Valuations

Let us define the variables that we use in the analysis:

[V.sub.t] = The value of the firm at the end of period t. We assume that period 0 is the current period.

[FCF.sub.t] = The free cash flow generated during period t. For simplicity we assume that the free cash flow is generated at the end of period t.

k = The required rate of return on the security of the firm. The required rate of return is assumed constant across periods.

r = The growth rate of free cash flows from period to period. This rate is assumed constant across periods, i.e., free cash flows grow by a rate of r every period.

M = The free cash flow multiple, that is, [V.sub.t]/[FCF.sub.t].

The current market value of the firm is equal to the present value of the free cash flow in the next period and the market value at the end of the next period. Formally,

[V.sub.0] = [FCF.sub.1]/1 + k + [V.sub.1]/1 + k

Substituting for [V.sub.1] in the above equation the present value of the free cash flow in period two, [FCF.sub.2], plus the value at the end of period 2, [V.sub.2], we get,

[V.sub.0] = [FCF.sub.1]/1 + k + [FCF.sub.2]/[(1 + k).sup.2] + [V.sub.2]/[(1 + k).sup.2].

Similar substitutions for all future values of the firm yield,

[V.sub.0] = [FCF.sub.1]/1 + k + [FCF.sub.2]/[(1 + k).sup.2] + [FCF.sub.3]/[(1 + k).sup.3] + ...

Assuming that free cash flows grow at a constant rate, r, we get the following series:

[V.sub.0] = [FCF.sub.0](1 + r)/(1 + k) + [FCF.sub.0][(1 + r).sup.2]/[(1 + k).sup.2] + [FCF.sub.0][(1 + r).sup.3]/[(1 + k).sup.3] + ...

Using the formula for a sum of a geometric series and assuming that k is larger than r, we get,

[Mathematical Expression Omitted].

After some simplifications, this yields,

[v.sub.0] = [FCF.sub.0] (1 + r)/(k - r).

Thus, we get the following free cash flow multiple:

M = [V.sub.0]/[FCF.sub.0] = (1 + r)/(k - r)

SOURCE: Kenneth S. Hackle and Joshua Livnat, Cash Flow and Security Analysis (Burr Ridge, Illinois: Irwin Professional Publications, 1992): 280-281.

1. Kenneth S. Hackle and Joshua Livnat, Cash Flow and Security Analysis (New York City: Business One, 1992).

Gordon T. Brown, PhD, of Upper St. Clair, Pennsylvania, is adjunct professor of finance at Robert Morris College, Coraopolis, Pennsylvania. He is also a real estate developer and owner and marketer of several patented inventions. He received a PhD in real estate from Georgia State University, Atlanta, and has authored other articles for The Appraisal Journal.