...section, we will look at the foundations of the approach and some of the preliminary details on how we estimate its inputs.

2.1 Essence of Discounted Cashflow Valuation

We buy most assets because we expect them to generate cash flows for us in the future. In discounted cash flow valuation, we begin with a simple proposition. The value of an asset is not what someone perceives it to be worth but it is a function of the expected cash flows on that asset. Put simply, assets with high and predictable cash flows should have higher values than assets with low and volatile cash flows.

The notion that the value of an asset is the present value of the cash flows that you expect to generate by holding it is neither new nor revolutionary. While knowledge of compound interest goes back thousands of years, (1) the concrete analysis of present value was stymied for centuries by religious bans on charging interest on loans, which was treated as usury. In a survey article on the use of discounted cash flow in history, Parker (1968) notes that detailed interest rate tables date back to 1340 and were prepared by Francesco Balducci Pegolotti, a Florentine merchant and politician, as part of his manuscript titled Practica della Mercatura, which was not officially published until 1766. The development of insurance and actuarial sciences in the next few centuries provided an impetus for a more thorough study of present value. Simon Stevin (1582), a Flemish mathematician, wrote one of the first textbooks on financial mathematics and laid out the basis for the present value rule in an appendix.

The extension of present value from insurance and lending to corporate finance and valuation can be traced to both commercial and intellectual impulses. On the commercial side, the growth of railroads in the United States in the second half of the 19th century created a demand for new tools to analyze long-term investments with significant cash outflows in the earlier years being offset by positive cash flows in the later years. A civil engineer, A.M. Wellington, noted not only the importance of the time value of money but argued that the present value of future cash flows should be compared to the cost of up-front investment. (2) He was followed by Walter O. Pennell, an engineer of Southwestern Bell, who developed present value equations for annuities, to examine whether to install new machinery or retain old equipment. (3)

The intellectual foundations for discounted cash flow valuation were laid by Alfred Marshall and Eugen von Bohm-Bawerk, who discussed the concept of present value in their works in the early part of the 20th century. (4) In fact, Bohm-Bawerk (1903) provided an explicit example of present value calculations using the example of a house purchase with 20 annual installment payments. However, the principles of modern valuation were developed by Irving Fisher in two books that he published--The Rate of Interest (1907) and The Theory of Interest (1930). In these books, he suggested four alternative approaches for analyzing investments, that he claimed would yield the same results. He argued that when confronted with multiple investments, you should pick the investment (a) that has the highest present value at the market interest rate; (b) where the present value of the benefits exceeded the present value of the costs the most; (c) with the "rate of return on sacrifice" that most exceeds the market interest rate or (d) that, when compared to the next most costly investment, yields a rate of return over cost that exceeds the market interest rate. Note that the fist two approaches represent the net present value rule, the third is a variant of the IRR approach and the last is the marginal rate of return approach. While Fisher did not delve too deeply into the notion of the rate of return, other economists did. Looking at a single investment, Boulding (1935) derived the internal rate of return for an investment from its expected cash flows and an initial investment. Keynes (1936) argued that the "marginal efficiency of capital" could be computed as the discount rate that makes the present value of the returns on an asset equal to its current price and that it was equivalent to Fisher's rate of return on an investment. Samuelson (1937) examined the differences between the internal rate of return and net present value approaches and argued that rational investors should maximize the latter and not the former. In the last 50 years, we have seen discounted cash flow models extend their reach into security and business valuation, and the growth has been aided and abetted by developments in portfolio theory.

Using discounted cash flow models is in some sense an act of faith. We believe that every asset has an intrinsic value and we try to estimate that intrinsic value by looking at an asset's fundamentals. What is intrinsic value? Consider it the value that would be attached to an asset by an all-knowing analyst with access to all information available right now and a perfect valuation model. No such analyst exists, of course, but we all aspire to be as close as we can to this perfect analyst. The problem lies in the fact that none of us ever gets to see what the true intrinsic value of an asset is and we therefore have no way of knowing whether our discounted cash flow valuations are close to the mark or not.

There are four variants of discounted cash flow models in practice, and theorists have long argued about the advantages and disadvantages of each. In the fist, we discount expected cash flows on an asset (or a business) at a risk-adjusted discount rate to arrive at the value of the asset. In the second, we adjust the expected cash flows for risk to arrive at what are termed risk-adjusted or certainty equivalent cash flows which we discount at the riskfree rate to estimate the value of a risky asset. In the third, we value a business fist, without the effects of debt, and then consider the marginal effects on value, positive and negative, of borrowing money. This approach is termed the adjusted present value (APV) approach. Finally, we can value a business as a function of the excess returns we expect it to generate on its investments. As we will show in the following section, there are common assumptions that bind these approaches together, but there are variants in assumptions in practice that result in different values.

2.2 Discount Rate Adjustment Models

Of the approaches for adjusting for risk in discounted cash flow valuation, the most common one is the risk adjusted discount rate approach, where we use higher discount rates to discount expected cash flows when valuing riskier assets, and lower discount rates when valuing safer assets. There are two ways in which we can approach discounted cash flow valuation. The fist is to value the entire business, with both assets-in-place and growth assets; this is often termed firm or enterprise valuation.

[ILLUSTRATION OMITTED]

The cash flows before debt payments and after reinvestment needs are termed free cash flows to the firm, and the discount rate that reflects the composite cost of financing from all sources of capital is the cost of capital.

The second way is to just value the equity stake in the business, and this is called equity valuation.

[ILLUSTRATION OMITTED]

The cash flows after debt payments and reinvestment needs are free cash flows to equity, and the discount rate that reflects just the cost of equity financing is the cost of equity.

Note also that we can always get from the former (firm value) to the latter (equity value) by netting out the value of all non-equity claims from firm value. Done right, the value of equity should be the same whether it is valued directly (by discounting cash flows to equity at the cost of equity) or indirectly (by valuing the firm and subtracting out the value of all non-equity claims).

2.2.1 Equity DCF models

In equity valuation models, we focus our attention of the equity investors in a business and value their stake by discounting the expected cash flows to these investors at a rate of return that is appropriate given the equity risk in the company. The fist set of models examined take a strict view of equity cash flows and consider only dividends to be cash flows to equity. These dividend discount models represent the oldest variant of discounted cash flow models. We then consider broader definitions of cash flows to equity, by fist including stock buybacks in cash flows to equity and by then expanding out analysis to cover potential dividends or free cash flows to equity.

2.2.1.1 Dividend discount model

The oldest discounted cash flow models in practice tend to be dividend discount models. While many analysts have turned away from these models on the premise that they yield estimates of value that are far too conservative, many of the fundamental principles that come through with dividend discount models apply when we look at other discounted cash flow models.

Basis for Dividend Discount Models. When investors buy stock in publicly traded companies, they generally expect to get two types of cashflows--dividends during the holding period and an expected price at the end of the holding period. Since this expected price is itself determined by future dividends, the value of a stock is the present value of dividends through infinity.

Value per share of stock = ([t = [infinity].summation over (t = 1)] E([DPS.sub.t])/[(1 + [k.sub.e]).sup.t],

where

E([DPS.sub.t]) = Expected dividends per share in period t

[k.sub.e] = Cost of equity

The rationale for the model lies in the present value rule--the value of any asset is the present value of expected future cash flows discounted at a rate appropriate to the riskiness of the cash flows. There are two basic inputs to the model--expected dividends and the cost on equity. To obtain the expected dividends, we make assumptions about expected future growth rates in earnings and payout ratios. The required rate of return on a stock is determined by its riskiness, measured differently in different models--the market beta in the CAPM, and the factor betas in the arbitrage and multi-factor models. The model is flexible enough to allow for time-varying discount rates, where the time variation is caused by expected changes in interest rates or risk across time.

While direct mention of dividend discount models did not show up in research until the last few decades, investors and analysts have long linked equity values to dividends. Perhaps the fist book to explicitly connect the present value concept with dividends was The Theory of Investment Value by John Burr Williams (1938), where he stated the following:

A stock is worth the present value of all the dividends ever to be paid upon it, no more, no less ... Present earnings, outlook, financial condition, and capitalization should bear upon the price of a stock only as they assist buyers and sellers in estimating future dividends.

Williams also laid the basis for forecasting pro forma financial statements and drew a distinction between valuing mature and growth companies. While much of their work has become shrouded with myth, Dodd and Graham (1934) also made the connection between dividends and stock values, but not through a discounted cashflow model. They chose to develop instead a series of screening measures, across stocks, that included low PE, high dividend yields, reasonable growth, and low risk that highlighted stocks that would be under valued using a dividend discount model.

Variations on the Dividend Discount Model. Since projections of dollar dividends cannot be made in perpetuity and publicly traded firms, at least in theory, can last forever, several versions of the dividend discount model have been developed based upon different assumptions about future growth. We will begin with the simplest--a model designed to value stock in a stable-growth firm that pays out what it can afford to in dividends. The value of the stock can then be written as a function of its expected dividends in the next time period, the cost of equity and the expected growth rate in dividends.

Value of stock = Expected dividends next period/(Cost of equity - Expected growth rate in perpetuity.

Though this model has made the transition into every valuation textbook, its origins are relatively recent and can be traced to early work by David Durand and Myron Gordon. It was Durand (1957) who noted that valuing a stock with dividends growing at a constant rate forever was a variation of The Petersburg Paradox, a seminal problem in utility theory for which a solution was provided by Bernoulli in the 18th century. It was Gordon (1962), though, who popularized the model in subsequent articles and a book, thus giving it the title of the Gordon growth model. While the Gordon growth model is a simple approach to valuing equity, its use is limited to firms that are growing at stable rates that can be sustained forever. There are two insights worth keeping in mind when estimating a perpetual growth rate. First, since the growth rate in the firm's dividends is expected to last forever, it cannot exceed the growth rate of the economy in which the firm operates. The second is that the firm's other measures of performance (including earnings) can also be expected to grow at the same rate as dividends. To see why, consider the consequences in the long term for a firm whose earnings grow 3% a year forever, while its dividends grow at 4%. Over time, the dividends will exceed earnings. On the other hand, if a firm's earnings grow at a faster rate than dividends in the long term, the payout ratio, in the long term, will converge toward zero, which is also not a steady state. Thus, though the model's requirement is for the expected growth rate in dividends, analysts should be able to substitute in the expected growth rate in earnings and get precisely the same result, if the firm is truly in steady state.

In response to the demand for more flexibility when faced with higher growth companies, a number of variations on the dividend discount model were developed over time in practice. The simplest extension is a two-stage growth model that allows for an initial phase where the growth rate is not a stable growth rate and a subsequent steady state where the growth rate is stable and is expected to remain so for the long term. While, in most cases, the growth rate during the initial phase will be higher than the stable growth rate, the model can be adapted to value companies that are expected to post low or even negative growth rates for a few years and then revert back to stable growth. The value of equity can be written as the present value of expected dividends during the non-stable growth phase and the present value of the price at the end of the high growth...

NOTE: All illustrations and photos have been removed from this article.



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