© 2000 John Petroff

There is no single valuation formula for stocks similar to that for bonds because no proceeds can be expected with any degree of certainty. There are three methods which depend on the confidence one can have predicting future dividends. As pointed out in Chapter 12 Section A-2, payment of dividends is never assured for common shareholders, but most companies follow policies of stable dividends. These stable dividend policies allow an approximation of the general value formula. In the case of preferred stocks, the calculation of value can be identical to that of bonds if preferred dividends are guaranteed and the preferred stock has a finite life (which some preferred stock do because they are considered temporary equity, as argued in Chapter 12 Section C). If the preferred stock has rights to convert into common shares, or to participate in earnings, then these extra benefits must be evaluated separately for what they represent, and added to the preferred stock valuation based on preferred dividends alone.

See review questions Q-3D.1 and Q-3D.2.

1)- Discounted dividends model with no growth

The discounted dividend model looks at a company through the eyes of shareholders who will only receive dividends. The value of a share is equated to the sum of all future dividends. The assumption necessary for this model is that a share be held at infinitum. At first, the assumption may seem unrealistic and contrary to definition of holding period stock rate of return (which is the combination of dividends and capital appreciation, as seen in Chapter 2 Section F). But in practice, this assumption is at lot easier to accept than many other estimations and approximations which are often required in finance. Moreover, this assumption makes practical sense for two reasons. First, because even if a moderate discount rate is used, present values of all benefits beyond 50 years is negligible, including any resale price no matter how large one can imagine it to become. Proof of this resides in actuarial tables where 50 year present value factors are less than 0.01 at 10% discount rate and too small be given for higher rates. Second, if some arbitrary date is set for sale of the stock, the purchasing price paid by a buyer will then be nothing else than the sum of present values of dividends beyond that date. So, summing present value of dividends before the sale and adding the sum of present values of dividends after the sale, is exactly the same as summing present values of dividends at infinitum. If there is no set date for sale of the stock, then any resale value can simply be ignored.

The stock value SV given by the discounted dividend model is written as

SV = D/(1 + k) + D/(1 + k)² + .... + D/(1 + k)ⁿ

where D = dividend currently paid annually

k = risk adjusted required rate of return for this type of stock

n = infinity

If an assumption of a constant dividend for the foreseeable future is acceptable, then the simplified formula with a constant annuity in perpetuity (as for consols discussed in Chapter 2 Section C-2) can be used. The result is a very elegant formulation of company stock value as

SV = D / k

where SV = stock value

D = current year expected annual dividend

k = risk adjusted required rate of return for that type of stock

As mentioned, such valuation is appropriate for preferred stocks, which normally offer a fixed dividend, and for all other securities which have a fixed periodic payment for an extended or indefinite length of time. (The consol calculation in a previous chapter falls in this class. So would calculation for corporate or government bonds with maturities beyond 50 years because the part stemming from the discounted principal is negligible.)

For example, the preferred stock of GM pays an annual dividend of $2.28. If the required rate of return for a preferred stock such that of GM is 8.5%: obtained with a risk-free rate of 6.5% (see Chapter 2 Section E) plus market-risk premium of 2% (which is indeed slightly higher than the market premium for corporate bonds), and a BETA estimate of 1 for GM in 1995, thus giving back the rate of 8.5%. The value of a share is SV of GM preferred stock = 2.28/0.085 = $26.82 On July 12, 1995, GM preferred G (with dividend of $2.28) closed at $27.75, and GM preferred Q (with dividend of $2.28 also) closed at $26.87. The formula does seem to offer rather good results. However, prices of these two stocks over the last 52 weeks ranged from $24.75 to $29.25. Thus, the actual price deviates from the estimated value a lot. One explanation are variations in the risk-free rate, but other reasons must be traced to investors' attitude about GM.

No Russian company offered a stock with a fixed dividend in the 1990's. In fact, a fixed dividend would not make sense in an inflationary context as the one experienced in Russia in the recent past. One could argue that if it were permissible to offer ruble equivalents of a dividend that is set at a constant amount in dollars, then such stock could have a strong investor appeal, at least from the sole fact that its estimated value would be straightforward.

See review questions Q-3D1.1 through Q-3D1.6.

See research assignment R-3.8 and R-3.9.

2)- Discounted dividend model with constant growth of dividends

The more general formula used for common stock assumes that dividends will grow at a constant growth rate g. Then the formula for stock value SV becomes

SV = D / (k-g)

where SV = stock value

D = current year expected annual dividend

k = risk adjusted required rate of return for that type of stock

g = rate of growth of dividends

For companies that do not distribute dividends, the formula obviously does not make any sense. This is the case of Silicon Graphics offered (discussed in an example Chapter 2 Section E), as it is also very often the case for all growth stock which need to reinvest their earnings into the business rather than distribute them as dividends (see Chapter 12 Section 12A-2e). For most companies that do distribute dividends, this formula still does not give satisfactory estimates of common share prices. The difficulty is primarily in estimating a rate of growth of dividends. Another source of difficulty already mentioned comes from bullish or bearish attitudes and trends in stock markets. Furthermore, many stocks are affected by market trader's attitudes that typically overbid the price of nationally traded stocks and underbid the price of new, small or poorly capitalized companies. For some companies, especially those that have a very steady market and the rate of growth of dividends is moderate and predictable, the formula is useful. For instance, utilities generally grow at a rate that correlates with population growth and growth of the regional economy they serve.

Let us take the case of ConEdison of New York common stock quoted at a closing price of $29.25 on July 12, 1995. The dividend is $2.04, the required rate of return is 9.1% (that is, 6.5% riskless rate from Table T-2.2 , a risk premium of 5.3% , or the excess of 11.8% from Table 2-1 over 6.5%, and a BETA for electrical utility of 0.5 from Table T-2.4, combining to 6.5 + 5.3(0.5) = 9.1%), and the growth the New York economy is estimated at 2.0%. The stock value of ConEdison common share is SV = 2.04/(0.091 - 0.02) = 2.04/0.071 = $28.73  The value of $28.73 obtained is reasonably close to the quoted price of $29.95. But realistically, buy or sell decision on the basis of such calculations would be misguided because there is clearly other elements in investors' pricing of $29.95.

The estimated value in the example above appears realistic. It rests, however, on a BETA and a growth rate which are gross approximations.

In numerous textbooks one can find models that attempt to incorporate elements of product life cycle (discussed in Chapter 14 Section B), by summing dividends over periods of high, moderate and no growth phases. Unless the pattern of sales growth can realistically be predicted over periods of several years (which is very questionable, as argued in Chapter 9 Section G-4), the approach seems theoretically appealing but highly speculative and suspicious in practice.

For Russian stocks, the discounted dividend model with the growth of dividends presents in practice even more difficulty than the fixed dividend version because the growth of dividends in real terms cannot be predicted with any accuracy.

See review questions Q-3D2.1 through Q-3D2.3.

See research assignment R-3.10.

3)- Earnings multiple or P/E ratio

Another approach which is used as a short-cut by a large number of investors, is the earnings multiple. It is sometimes referred to as earnings multiplier, and it is most commonly known as price-to-earnings or P/E ratio. In many instances, the approach, rather than being an oversimplification, can be an improvement over the previous format. In its most common presentation, the idea is that the price P of a share should be a multiple m of its earnings per share E (which is defined and discussed in Chapter 13 Section B-2). The multiple m is an industry average because it is assumed that all companies in an industry face similar marketing, technological and resource challenges, and thus, should have similar organizational and production patterns.

P = E . m

where P = estimated price of company stock

E = earnings per share

m = industry average price to earnings multiplier

One may note that the multiple is the inverse of a rate of return. Choosing an earnings multiple is very much like estimating a required rate of return. The earnings multiple is an older approach which existed before modern portfolio theory, but is still widely used.

Table T-3.5 below lists average P/E ratios of 18 industries calculated on the basis of a selected major companies within each industry. (Note that because the companies are large, the P/E ratios are likely to be higher than if smaller firms had been used. Another distortion comes from inclusion of newly acquired company revenues in operating income pushing the sales growth rate average of 0.09 to twice the annual growth rate of GDP for that period.)

Table T-3.5 makes it very clear that statistics bear out the proposition that companies in the same industry should have the same earning multiple. Earning multiples naturally vary over time and across different countries. For instance, with NYSE bullish market in early 2000, P/E's have risen with prices: NYNEX P/E has doubled to over 20 (from what appears in Table T-3.5), while those of ALLTEL and BellSouth are in the 15 range. Thus, only P/E's of the same period should be used. Comparing countries, Japan has had historically much higher P/E ratios than the U.S. This seems to be attributable to higher bank leverage customary in Japan. P/E ratios of Japanese stocks slumped during Asian crisis and Japanese recession of the late 1990's, but they have gone back up since. Conversely, European firms have lower P/E ratios (around 7 or 8 historically) than American firms (whose P/E ratios have averaged historically around 10, as observed in Table T-3.5), because European firms must rely on greater internal financing.

Because P/E ratios appear in most financial publication (such as the NY Times, Wall Street Journal, International Herald Tribune and Financial Times), it is just as easy to use the ratios as such without even calculating any stock value. One could indeed verify that P/E ratios are obtained as

P/E = market price / earnings per share

Such P/E ratio can be used to compare similar company stocks and determine if one stock is overvalued or undervalued, and possibly why. The improvement in this presentation (over the discounted dividends model) comes from the fact that current earnings can be distributed in the form of dividends or retained to make future dividend growth possible. Thus, rather than guessing at some rate of growth g (as in the previous approach), why not allow for reinvestment of earnings for future dividend growth (as it is implied in P/E calculation).

Some authors do explain the price-to-earnings ratio with the help of the parameters used above. Starting with the constant growth discounted dividend model

P = D / (k-g)

and replacing D by the proportion of earnings that is pay out at a payout rate p

D = EPS . p

But since what is not paid out is retained, and the sum of payout rate p and retention rate r is one

D = EPS (1 - r)

Thus

P = EPS(1-r) / (k-g)

which transforms into

P/E = (1-r) / (k-g)

where P/E = price to earnings ratio

EPS = earnings per share

r = retention rate, that is, the proportion of earnings which is not distributed as dividends

p = 1- r = payout rate (i.e. p + r = 1)

EPS(1-r) = dividends distributed (equal to D in discounted dividend model)

k = required rate of return

g = growth rate of dividends

Looking at the components of P/E formula allows an understanding of the causes for differences between P/E's in Table T-3.5 above. The industries are ranked by P/E in Table T-3.5a below.

The two industries which have the highest P/E ratios, are computers (with 49) and internet (with 43). This can be explained by the fact that these industries were expected to have the highest growth rate in sales and earnings.

Another interesting use of this approach is to investigate how common stock equivalents affect common stock price. As discussed in Chapter 13 Section B-2, there are indeed two earnings-per-share reported in most corporate annual financial statements. One is basic earnings-per-share which is simply net profit after tax divided by number of shares outstanding. The second is diluted earnings-per-share, in which the number of shares outstanding is increased by the number of shares that would have to be issued to all those who have rights to convert bonds or preferred stocks into common shares. If the gap between the two is large, it suggests that the company must rely on a substantial sweetners to raise the capital it needs, as covered in Chapter 13 Section 13D-3.

Here is an example of how the P/E ratio can be used for ConEdison stock studied previously. The historic industry earnings multiplier is 10 (as shown in Table T-3.5), and ConEdison earnings per share can be estimated as dividends of $2.04 divided by 70% payout rate (see Table T-3.5), or $2.91. This gives an estimated price of ConEdison P = 2.91 . 10 = $29.10 Comparing this estimate to the actual price of $29.25 in July 1995, one would conclude that the market slightly overpriced the stock. However, remember that the market in the Summer 1995 is in a bullish trend with P/E ratios rising for the majority of stock. One would question then, why the price of ConEdison stock has not gone up more.

This second example of ConEdison calculation illustrates that the use of P/E ratio for stock valuation is not inferior to the discounted dividends model.

See review questions Q-3D3.1 through Q-3D3.7.

See research assignments R-3.11 and R-3.12.

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