# Dividend Discount Model and Price Earning Model

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Financial theory holds that the value of a share of stock is equal to the sum of the discounted future expected dividends. The Dividend Discount (DD) requires two inputs, firstly a forecast of future dividends and secondly, a rate at which these dividends will be discounted to their present value. The appropriate discount rate that will be used is the rate of return available on risk-free investments plus a risk premium. The Capital Asset Pricing Model is one of the most widely used models for calculating discount rates.

Once the discount rate estimated, all future dividends must be discounted to their present value. Although near term dividends may be estimated with some confidence, to make the DD model operational an assumption regarding long term dividends is necessary. Two common assumptions regarding dividend growth and their associated valuation models are:

(i) Earnings growth as well as dividend growth will be constant with the Gordon Model, and

(ii) Multiple stages of growth can be approximated.

Therefore, it is obvious that the forecasts and the assumptions necessary for operating DD models induce the emergence of significant errors into this theoretically correct approach.

Common Stock Valuation Concepts

The value of a common stock can be defined as the present value of the future dividend stream in perpetuity. This concept is consistent with the assumption that the corporation will indeed have a perpetual life, in accordance with its charter.

If the value of a stock is equivalent to the value for a perpetual annuity with a constant level of payments, the general formula is as follows:

Value per share of stock =

Where = Expected dividend per share

= Cost of equity

The formula shown above for stock valuation treats the firm together with its stock as if they will exist forever. There are two basic inputs to the model. First is the expected dividends and secondly 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.

Zero growth model

In this model it is assumed that the same amount of dividend will be paid for all the time periods up until infinity. The formula is given as follows after it has been simplified by using the formula sum to infinity of geometric progression:

Where V = value,

D = dividends per share

k = percentage discount rate

However, this model is quite restrictive as it is unreasonable to assume that the same amount of dividend will be paid by a stock for an indefinite time period. The model may be useful for determining the value of preferred stock which usually yields a fixed amount of dividend.

Constant (Gordon) growth model

The major drawback of the zero growth model is that it is assumed that a firm will pay the same dividend throughout its lifetime. However, in the real world most companies are expected to grow over time and consequently make more profits leading to more dividends being paid. This model assumes that there is a constant growth rate for the corporation being analysed and it is most suitable for valuation of stable and mature companies. This model was created by Myron Gordon, and thus it was named as the "Gordon Model".

The formula for constant growth model is derived from the zero growth model. If the dividends are assumed to grow at a certain constant rate, the formula becomes:

Where g = annual constant percentage growth in dividends per share

D = next year's dividends.

The Gordon growth model is a simple and powerful approach to valuing equity. In order for the model to work the following assumptions must be held:

Dividends will grow at a constant rate and it will continue for an infinite period. The required rate of return is greater than the steady growth rate.

The required rate of return is constant until infinity.

It is also important to note that the model has some limitations. The Gordon growth model is a simple and convenient way of valuing stocks but it is extremely sensitive to the inputs for the growth rate. Used incorrectly, it can yield misleading or even absurd results, since, as the growth rate converges on the discount rate, the value goes to infinity. As the growth rate approaches the cost of equity, the value per share approaches infinity. If the growth rate exceeds the cost of equity, the value per share becomes negative.

Multistage Dividend Discount Model

The assumption of the Gordon Growth Model that there is a stable dividend growth rate from now on to the indefinite future is not realistic for many or even most companies. The studies of Sharpe, Alexander and Bailey (1999) state that the growth fall into three stages namely the growth phase, transition phase and the mature phase.

Two-stage Dividend Discount Model

The two-stage growth model allows for two stages of growth - 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 is 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 model is based upon two stages of growth, an extraordinary growth phase that lasts n years and a stable growth phase that lasts forever afterwards.

Value of the Stock = PV of Dividends during extraordinary phase + PV of terminal price

Where DPSt = Expected dividends per share in year t

ke = Cost of Equity (hg: High Growth period; st: Stable growth period)

Pn = Price (terminal value) at the end of year n

g = Extraordinary growth rate for the first n years

gn = Steady state growth rate forever after year n

There are three problems with the two-stage dividend discount model. The first two would apply to any two-stage model and the third is specific to the dividend discount model.

The first practical problem is in defining the length of the extraordinary growth period. Since the growth rate is expected to decline to a stable level after this period, the value of an investment will increase as this period is made longer.

The second problem with this model lies in the assumption that the growth rate is high during the initial period and is transformed overnight to a lower stable rate at the end of the period. While these sudden transformations in growth can happen, it is much more realistic to assume that the shift from high growth to stable growth happens gradually over time.

The focus on dividends in this model can lead to skewed estimates of value for firms that are not paying out what they can afford in dividends. In particular, we will under estimate the value of firms that accumulate cash and pay out too little in dividends.

The H Model for valuing Growth

Fuller and Hsia (1984) presented the H model is a two-stage model for growth, but unlike the classical two-stage model, the growth rate in the initial growth phase is not constant but declines linearly over time to reach the stable growth rate in steady stage. The model is based upon the assumption that the earnings growth rate starts at a high initial rate and declines linearly over the extraordinary growth period (which is assumed to last 2H periods) to a stable growth rate. It also assumes that the dividend payout and cost of equity are constant over time and are not affected by the shifting growth rates.

However, the limitations of this model is that it avoids the problems associated with the growth rate dropping precipitously from the high growth to the stable growth phase, but it does so at a cost. First, the growth rate is expected to strictly decline linearly. Therefore small deviations from this assumption do not affect the value significantly, but large deviations can cause problems. Another important point is that the assumption that the payout ratio is constant through both phases of growth exposes the analyst to an inconsistency i.e. as growth rates decline the payout ratio usually increases.

Three-stage Dividend Discount Model

The three-stage dividend discount model combines the features of the two-stage model and the H-model. It allows for an initial period of high growth, a transitional period where growth declines and a final stable growth phase. It is the most general of the models because it does not impose any restrictions on the payout ratio. This model assumes an initial period of stable high growth, a second period of declining growth and a third period of stable low growth that lasts forever.

This model removes many of the constraints imposed by other versions of the dividend discount model. In return, however, it requires a much larger number of inputs for instance year specific payout ratios, growth rates and betas. For firms where there is substantial noise in the estimation process, the errors in these inputs can overwhelm any benefits that accrue from the additional flexibility in the model.

## Estimating k and g

Companies with unpredictable or recurring earnings pattern, or rapidly thriving companies, require a more complex dividend capitalisation model framework that can accommodate dissimilar dividend growth patterns.

In practice, applications may require elaborate variations of the dividend capitalisation model, nevertheless this simplified form provides a convenient means of analysing the determinants of stock value. To begin with, the value of the stock should be greater, the greater the earning power and capacity of the corporation to pay out current dividends, D. Correspondingly, the higher the growth rate of the dividends, g, the greater the value of the corporation's stock. Finally, the greater the risk of the corporation (the higher the discount rate, k) the lower will be the value of the stock.

The discount rate is alternatively referred to as a required return. It is composed of two elements-a risk-free return and a risk premium. The risk-free return is, in turn, generally considered to consist of a real return component and an inflation premium. The real return is the basic investment compensation that investors demand for forgoing current consumption or, alternatively, the compensation for saving. Investors also require a premium to compensate for inflation; this premium will be high when the inflation rate is expected to be high and low when the inflation rate is expected to be low. Because the real return and the inflation premium comprise a basic return demanded by all investors, the risk-free return is a component of all securities. The risk premium is made up of the following elements-interest rate risk, purchasing power risk, business risk and financial risk. The risk premium might be considered to be a function of the stock's systematic risk (beta), which is determined by these four fundamental risk factors. As securities differ in their exposure to these risk elements, the premium or return that investors require to compensate for risk will differ across securities.

The constant dividend growth model reveals that the following three factors affect stock prices, ceteris paribus:

1) the higher the dividend, the higher the stock price;

2) the higher the dividend growth rate, the higher the stock price;

3) the lower the required rate of return r, the higher the stock price.

## Empirical Studies on the DDM

Issues of dividend policy range from its puzzle by Black (1976) to its irrelevance by Miller and Modigliani (1961), to its relevance by DeAngelo et al. (1996). Other issues include theories on dividend payment, such as stakeholders' theory, pecking order theory, agency cost, signalling theory, bird-in-hand fallacy and clientele effect. The information asymmetry between managers and shareholders, along with the separation of ownership and control, formed the base for another explanation of why dividend policy has been so popular.

Dividend irrelevance theory

Miller and Modigliani (1961) proposed that dividend policy is irrelevant to the shareholder and that stockholder wealth is unchanged when all aspects of investment policy are fixed and any increase in the current payout is financed by fairly priced stock sales. The main assumption is that there is 100 per cent payout by management in every period. Other assumptions are:

that there exist perfect capital markets; that is, no taxes or transactional cost, the market price cannot be influenced by a single buyer or seller, and free and costless access to information about the market;

that investors are rational and that they value securities based on the value of discounted future cash flow to investors;

that managers act as the best agents of shareholders; and

that there is certainty about the investment policy of the firm, with full knowledge of future cash flows.

Bird-in-hand theory

Al-Malkawi (2007) asserts that in a world of uncertainty and information asymmetry, dividends are valued differently from retained earnings (capital gains): "A bird in hand (dividend) is worth more than two in the bush (capital gains)". Owing to the uncertainty of future cash flow, investors will often tend to prefer dividends to retained earnings. Though this argument has been widely criticised and has not received strong empirical support, it has been supported by Gordon and Shapiro (1956), Lintner (1962) and Walter (1963). The main assumptions are:

that investors have imperfect information about the profitability of a firm;

that cash dividends are taxed at a higher rate than when capital gain is realized on the sale of a share; and

that dividends function as a signal of expected cash flows.

Signalling hypothesis

Though Miller and Modigliani (1961) assumed that investors and management have perfect knowledge about a firm, this has been countered by many researchers, as management who look after the firm tend to have more precise and timely information about the firm than outside investors. This, therefore, creates a gap between managers and investors; to bridge this gap, management use dividends as a tool to convey private information to shareholders (Al-Malkawi, 2007). Petit (1972) observed that the amount of dividends paid seems to carry great information about the prospects of a firm; this can be evidenced by the movement of share price. An increase in dividends may be interpreted as good news and brighter prospects, and vice versa. But Lintner (1956) observed that management are reluctant to reduce dividends even when there is a need to do so, and only increase dividends when it is believed that earnings have permanently increased.

Clientele effects of dividends theories.

Investors tend to prefer stocks of companies that satisfy a particular need. This is because investors face different tax treatments for dividends and capital gains and also face some transaction costs when they trade securities. Miller and Modigliani (1961) argued that for these costs to be minimised, investors tend towards firms that would give them those desired benefits. Likewise, firms would attract different clientele based on their dividend policies. Though they argued that even though clientele effect may change a firm's dividend policy, one clientele is as good as another, therefore dividend policy remains irrelevant. Al-Malkawi (2007) affirms that firms in their growth stage, which tend to pay lower dividends, would attract clientele that desire capital appreciation, while firms in their maturity stage, which pay higher dividends, attract clientele that require immediate income in the form of dividends. Al-Malkawi (2007) grouped the clientele effect into two groups, those that are driven by tax effects and those driven by transaction cost. He argued that investors in higher tax brackets would prefer firms that pay little or no dividends, to get reward in the form of share price appreciation, and vice versa. Transaction cost-induced clientele, on the other hand, arises when small investors depend on dividend payments for their needs; this clientele prefers companies who satisfy this need because they cannot afford the high transaction cost of selling securities.

Dividends form the hard core of stock values. As Justice Holmes remarked, "the commercial value of property consists in the expectation of income from it." (In Galveston, H. & S. A. Ry. Co. v. Texas, 210 U. S. 217, 226.) Black (1976) observed, 'The harder we look at the dividend picture, the more it seems like a puzzle, with pieces that just don't fit together'

Williams applied Fisher's work on stock valuation and developed the famous dividend discount model (DDM) (Fewings, 1979, p. 12). Williams defines ''the investment value of stock as the present worth of all the dividends to be paid upon it (Williams, 1956, p. 55).'' He further makes it clear that ''the investment value of a common stock is the present worth of its net dividend to perpetuity (Williams, 1956, p. 63).'' Amid this theoretical research, the academic world was divided and a fierce debate erupted concerning the irrelevance of dividend policy in the determination of the valuation of firms or their stocks. The inconsequence of dividend policy in the stock valuation contemplates the equivalence of the valuation using earning approach and the valuation using discounted dividend approach.

Fisher's inter-temporal investment and consumption model predicted that ''earnings which are reinvested at the going rate of capital instead of being released for consumption neither adds nor subtracts from the value of the overall stream of benefits (Fewings, 1979, p. 17).'' Thus, according to Fisher, dividend policy is irrelevant in the valuation of stocks. One must remember that Fisher's theory is applicable under perfect capital markets with certain futures. At the same time, Graham and Dodd (1934) developed their valuation methodologies based on the assumption that the firm's main objective is to pay dividends to shareholders.

Empirical evidence in the market suggested a positive correlation between stock prices and dividend payout (Harkavy, 1953), suggesting the relevance of dividends in the valuation of stocks. Gordon and Shapiro (1956), Walter (1956) and Solomon (1963) supported this hypothesis. In accordance with the relevance of the dividend policy on the valuation of stocks, Gordon extended Williams' model of stock valuation to include retained earnings. He further developed the model to include continuous equity financing. These dividend dependent models are called the ''bird-in-the-hand'' models by authors like Frankfurter et al. (2003), as they are based on the assumption that there are two opportunity rates - one for the firm and the other for the investor. The firm should retain 100 per cent of its earnings if the opportunity rate of a firm is greater than the opportunity rate of the investor.

The seminal paper of Miller and Modigliani (1961) argued the irrelevance of dividend policy and the equivalence of the valuation of stocks using four approaches, namely the discounted cash flow (DCF) approach, the current earnings plus future investment opportunities approach, the discounted dividend approach and the stream of earnings approach. This equivalence was proven under assumptions of perfect capital markets, rational behaviour and perfect certainty. In addition, Miller and Modigliani (1961) point out that the dividend policy may be relevant when a revision in the dividend policy points to some information that the investors do not know. This information content of dividends argument led to the development of dividend signaling models. The irrelevance of dividends is not resolved. Academics are still divided into two, if not more, schools of thought on the subject.

## Price Earnings Ratio

A firm's profitability, risk, quality of management, and many other factors are reflected in its stock and security prices. Hence, market value ratios indicate the market's assessment of the value of the firm's securities. The price/earnings (P/E) ratio is simply the market price of the firm's common stock divided by its annual earnings per share. Sometimes called the earnings multiple, the P/E ratio shows how much investors are willing to pay for each dollar of the firm's earnings per share. Earnings per share comes from the income statement, so it is sensitive to the many factors that affect the construction of an income statement, from the choice of GAAP to management decisions regarding the use of debt to finance assets. The price/earnings ratio is stated as:

Stock prices are determined from the actions of informed buyers and sellers in an impersonal market. Stock prices reflect much of the known information about a company and are fairly good indicators of a company's true value. Although earnings per share cannot reflect the value of patents or assets, the quality of the firm's management, or its risk, stock prices can and do reflect all of these factors. Comparing a firm's P/E to that of the stock market as a whole, or with the firm's competitors, indicates the market's perception of the true value of the company. While the P/E ratio measures the market's valuation of the firm relative to the income statement value for per-share earnings, the price-to-book value ratio measures the market's valuation relative to balance sheet equity. The book value of equity is simply the difference between the book values of assets and liabilities appearing on the balance sheet. The price-to-book-value ratio is the market price per share divided by the book value of equity per share. A higher ratio suggests that investors are more optimistic about the market value of a firm's assets, its intangible assets, and the ability of its managers. The price-to-book value ratio is stated as:

Market value indicators reflect the market's perception of the true worth of a firm's future prospects. As such, market perceptions of a firm's value are important to the financial analyst. However, the market may not be perfect; investors may become overly optimistic or pessimistic about a firm. The fact that a firm presently has a higher P/E or price-to-book-value ratio than its competition does not automatically imply that the firm is better managed or really deserves its higher valuation. Some firms may have low market value ratios because they truly deserve them; other firms may suffer from extreme and undeserved pessimism on the part of the market. High market value ratios can be similarly deceptive. The analyst must determine whether a firm deserves its market value ratios or not.

## Empirical Studies on the P/E model

Ball and Brown (1968) are amongst the first pioneers who provided evidence that accounting earnings are potentially useful to investors for the valuation of equity. Furthermore, Beaver, Clarke and Wright (1979) also concluded that earnings act as a major determinant for equity valuation. Despite many researchers were inspired by the work of Ball and Brown prior study on price earnings ratio may trace back to 1934 when Graham and Dame considered that the major factors affecting price earnings ratio are factors coming from investors and companies. Internal scholars pay more attention to price earnings ratio status and qualitative or quantitative studies using cross sectional data model or time serials model are made in detail when stock market is established. As an important index measuring stock investment value and reflecting stock market development status, price earnings ratio is not only useful for department of banking custody to make sound regulation measures but helpful for investors to distinguish stock investing risk and select advisable invest strategy.

Shroff (1995) cites that earnings of firms with high P/E ratio and high return on equity exhibit higher explanatory powers for stock returns. According to Barth et al. (1998) income statement plays fundamental role for equity valuation. Burgstahler & Dichev (1997) found that book value and earnings being interrelated, act as component of equity value. Therefore it implies that the value of the firm can be expressed as a function of both earnings and book value of equity. Consequently, the higher is the earnings to book value ratio, the more relevant earnings will be as a determinant of equity value. While a lower earnings to book value ratio will imply book value being more important determinants of equity value.

According to the work of Jan & Ou (1995) firms which are reporting net losses, their earnings explain very little of equity price, but on the other hand book value of equity is an important determinant of stock price. Penman (1998) finds that book value provides greater relevance than earnings in equity valuation for firms with an extreme earnings to book ratio. Collins et al. (1997) further report that the value-relevance of earnings and book value of equity moves inversely to each other. Ou and Penman (1989) note that P/E ratios are good predictors of future earnings while changes in share price are poor predictors of future earnings.

Ou & Sepe (2002) find that the larger the spread between analysts' forecasts of a firm's future earnings and reported current earnings, the less value-relevant current earnings and the more the market relies on book value for equity valuation.

Researches undertaken by Nicholson (1960), McWilliams (1966), Latane et al. (1969), Dowen and Bauman (1986), Keim (1990), and Fama and French (1992) provide evidence that stock returns are linked to P/E ratios.

Penman (1996) notes that the P/E ratio acts not as a predictor of share price or returns but of future earnings levels. Allen et al. (1998) conclude similarly as their results indicate that firms with high E/P stocks have relatively low earnings growth while companies with low E/P shares experience high earnings growth. Furthermore, Fuller et al. (1992) conclude that low P/E ratio stocks generate low future earnings growth while high P/E ratio shares result in high earnings growth.

Another line of research (e.g., Beaver, 1989; Mande, 1994) provides strong evidence that earnings aids investors in evaluating a firm's dividend paying ability. As Larcker (1989) notes, share price is determined in the market through capitalisation (i.e., discounting) of the future cash flows or dividends expected to accrue to stockholders. Since earnings provide an information signal about future cash flows, stock price is affected by expectations concerning earnings. Because P/E ratios act as predictors of future earnings, these ratios are also linked to share price or returns.

Moreover Nelson and Kim (1993) and Campbell and Shiller (1988) have documented that dividend yield predicts stock returns with some success, as it capture expectations about dividend growth as well as expected returns. While Lamont (1998) argues that the P/E ratio has independent predictive power for excess returns in addition to the dividend price ratio. Ang and Bekaert (2003) detect a strong role for the P/E ratio as a predictive instrument for future dividend growth. Since the P/E ratio is a function of expected growth in earnings, obviously expected growth in earnings are eminent in the valuation of a stock.

## Limitations of P/E

The P/E is a fairly simple tool for assessing company value. But it has been argued thatÂ the P/E ratio is not always reliable. There are plenty of reasons to be wary of P/E based stock valuations.

The P/E ratio is supposed to enumerate how many years' worth of current earnings a company will need to produce in order to arrive at its current market share value. Naturally, investors want to be able to buy more earnings for every dollar they pay, so the lower the P/E ratio, the less expensive the stock.

The calculation of the ratio sounds simple enough, but here are some of the dangers associated with taking P/E ratios at face value. The first part of the P/E equation, price, is straightforward. The market price is easily available from the stock exchange market. On the other hand, coming up with an appropriate earnings number can be tricky. You have to make a lot of decisions how to define earnings. Earnings are not always clear cut. Earnings can be affected by unusual gains or losses which sometimes obscure the true nature of the earnings metric. What's more, reported earnings can be manipulated by company management to meet earnings expectations, while creative accounting choices, shifting depreciation policies or adding or subtracting non-recurring gains and expenses, can make bottom line earnings numbers bigger and, in turn, P/E ratios, smaller and the stock appear less expensive. Investors need to be wary of how companies arrive at their reported EPS numbers. Appropriate adjustments often have to be done in order to obtain a more accurate measure of earnings than what is reported on the balance sheet.

Then there is the matter of whether to use trailing earnings or forward earnings figures. Located right in the company's latest published income statement, historic earnings are easy to find. Unfortunately, they are not much use for investors, since they say very little about what earnings are in store for the year and years ahead. It's the company's future earnings that investors are interested in most since as they reflect a stock's future prospects.

The biggest limitation of the P/E ratio is that it tells investors next to nothing about the company's EPS growth prospects. If the company is growing quickly, you will be comfortable buying it even it had a high P/E ratio, knowing that growth in EPS will bring the P/E back down to a lower level. If it is not growing quickly, you might shop around for a stock with a lower P/E ratio. It is often difficult to tell if a high P/E multiple is the result of expected growth or if the stock is simply overvalued.

A P/E ratio, even one calculated using a forward earnings estimate, does not always tell whether or not the P/E is appropriate to the company's forecasted growth rate.

Finally, there's the tricky issue of a company's debt load. The P/E ratio does nothing to factor in the amount of debt that a company carries on its balance sheet. Debt levels have an impact on financial performance and valuation, yet the P/E does not allow investors to make comparisons between debt-free firms and those bogged down with outstanding loans and liabilities.

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