Price-Earnings Ratio. Note Quote.

The price-earnings ratio (P/E) is arguably the most popular price multiple. There are numerous definitions and variations of the price-earnings ratio. In its simplest form, the price-earnings ratio relates current share price to earnings per share.

The forward (or estimated) price-earnings ratio is based on the current stock price and the estimated earnings for future full scal years. Depending on how far out analysts are forecasting annual earnings (typically, for the current year and the next two fiscal years), a company can have multiple forward price-earnings ratios. The forward P/E will change as earnings estimates are revised when new information is released and quarterly earnings become available. Also, forward price-earnings ratios are calculated using estimated earnings based on the current fundamentals. A company’s fundamentals could change drastically over a short period of time and estimates may lag the changes as analysts digest the new facts and revise their outlooks.

The average price-earnings ratio attempts to smooth out the price-earnings ratio by reducing daily variation caused by stock price movements that may be the result of general volatility in the stock market. Different sources may calculate this figure differently. Average P/E is defined as the average of the high and low price-earnings ratios for a given year. The high P/E is calculated by dividing the high stock price for the year by the annual earnings per share fully diluted from continuing operations. The low P/E for the year is calculated using the low stock price for the year.

The relative price-earnings ratio helps to compare a company’s price-earnings ratio to the price-earnings ratio of the overall market, both currently and historically. Relative P/E is calculated by dividing the firm’s price-earnings ratio by the market’s price-earnings ratio.

The price-earnings ratio is used to gauge market expectation of future performance. Even when using historical earnings, the current price of a stock is a compilation of the market’s belief in future prospects. Broadly, a high price-earnings ratio means the market believes that that the company has strong future growth prospects. A low price-earnings ratio generally means the market has low earnings growth expectations for the firm or there is high risk or uncertainty of the firm actually achieving growth. However, looking at a price-earnings ratio alone may not be too illuminating. It will always be more useful to compare the price-earnings ratios of one company to those of other companies in the same industry and to the market in general. Furthermore, tracking a stock’s price-earnings ratio over time is useful in determining how the current valuation compares to historical trends.

Gordon growth model is a variant of the discounted cash flow model, is a method for valuing intrinsic value of a stock or business. Many researches on P/E ratios are based on this constant dividend growth model.

When investors purchase a stock, they expect two kinds of cash flows: dividend during holding shares and expected stock price at the end of shareholding. As the expected share price is decided by future dividend, then we can use the unlimited discount to value the current price of stocks.

A normal model for the intrinsic value of a stock:

V = D₁/(1+R)¹ + D₂/(1+R)² +…+ D_n/(1+R)ⁿ = ∑_t=1^∞ D_t/(1+R)^t (n→∞) —– (1)

In (1)

V: intrinsic value of the stock;

D_t: dividend for the t^th year

R: discount rate, namely required rate of return;

t: the year for dividend payment.

Assume the market is efficient, the share price should be equal to the intrinsic value of the stock, then equation (1) becomes:

P₀ = D₁/(1+R)¹ + D₂/(1+R)² +…+ D_n/(1+R)ⁿ = ∑_t=1^∞ D_t/(1+R)^t (n→∞) —– (2)

where P₀: purchase price of the stock;

D_t: dividend for the t^th year

R: discount rate, namely required rate of return;

t: the year for dividend payment.

Assume the dividend grows stably at the rate of g, we derive the constant dividend growth model.

That is Gordon constant dividend growth model:

P₀ = D₁/(1+R)¹ + D₂/(1+R)² +…+ D_n/(1+R)ⁿ = D₀(1+g)/(1+R)¹ + D₀(1+g)²/(1+R)² +….+ D₀(1+g)ⁿ/(1+R)ⁿ = ∑_t=1^∞ D₀(1+g)^t/(1+R)^t —– (3)

When g is a constant, and R>g at the same time, then equation (3) can be modified as the following:

P₀ = D₀(1+g)/(R-g) = D₁/(R-g) —– (4)

where, P₀: purchase price of the stock;

D₀: dividend at the purchase time;

D₁: dividend for the 1^st year;

R: discount rate, namely required rate of return;

g: the growth rate of dividend.

We suppose that the return on dividend b is fixed, then equation (4) divided by E₁ is:

P₀/E₁ = (D₁/E₁)/(R-g) = b/(R-g) —– (5)

where, P₀: purchase price of the stock;

D₁: dividend for the 1st year;

E₁: earnings per share (EPS) of the 1^st year after purchase;

b: return on dividend;

R: discount rate, namely required rate of return;

g: the growth rate of dividend.

Therefrom we derive the P/E ratio theoretical computation model, from which appear factors deciding P/E directly, namely return on dividend, required rate of return and the growth rate of dividend. The P/E ratio is related positively to the return on dividend and required rate of return, and negatively to the growth rate of dividend.

Realistically speaking, most investors relate high P/E ratios to corporations with fast growth of future profits. However, the risk closely linked the speedy growth is also very important. They can counterbalance each other. For instance, when other elements are equal, the higher the risk of a stock, the lower is its P/E ratio, but high growth rate can counterbalance the high risk, thus lead to a high P/E ratio. P/E ratio reflects the rational investors’ expectation on the companies’ growth potential and risk in the future. The growth rate of dividend (g) and required rate of return (R) in the equation also response growth opportunity and risk factors.

Financial indices such as Dividend Payout Ratio, Liability-Assets (L/A) Ratio and indices that reflecting growth and profitability are employed in this paper as direct influence factors that have impact on companies’ P/E ratios.

Derived from (5), the dividend payout ratio has a direct positive effect on P/E ratio. When there is a high dividend payout ratio, the returns and stock value investors expected will also rise, which lead to a high P/E ratio. Conversely, the P/E ratio will be correspondingly lower.

Earnings per share (EPS) is another direct factor, while its impact on P/E ratio is negative. It reflects the relation between capital size and profit level of the company. When the profit level is the same, the larger the capital size, the lower the EPS will be, then the higher the P/E ratio will be. When the liability-assets ratio is high, which represents that the proportion of the equity capital is lower than debt capital, then the EPS will be high and finally the P/E ratio will led to be low. Therefore, the companies’ L/A ratio also negatively correlate to P/E ratio.

Some other financial indices including growth rate of EPS, ROE, growth rate of ROE, growth rate of net assets, growth rate of main business income and growth rate of main business profit should theoretically positively correlate to P/E ratios, because if the companies’ growth and profitability are both great, then investors’ expectation will be high, and then the stock prices and P/E ratios will be correspondingly high. Conversely, they will be low.

In the Gordon growth model, the growth of dividend is calculated based on the return on retained earnings reinvestment, r, therefore:

g = r (1-b) = retention ratio return on retained earnings.

As a result,

P₀/E₁ = b/(R-g) = b/(R-r(1-b)) —– (6)

Especially, when the expected return on retained earnings equal to the required rate of return (i.e. r = R) or when the retained earnings is zero (i.e. b=1),

There is:

P₀/E₁ = 1/R —– (7)

Obviously, in (7) the theoretical value of P/E ratio is the reciprocal of the required rate of return. According to the Capital Asset Pricing Model (CAPM), the average yields of the stock market should be equal to risk-free yield plus total risk premium. When there not exists any risk, then the required rate of return will equal to the market interest rate. Thus, the P/E ratio here turns into the reciprocal of the market interest rate.

As an important influence factor, the annual interest rate affect on both market average and companies’ individual P/E ratios. On the side of market average P/E ratio, when interest rate declines, funds will move to security markets, funds supply volume increasing will lead to the rise of share prices, and then rise in P/E ratios. In contrast, when interest rate rises, revulsion of capitals will reflow into banks, funds supply will be critical, share prices decline as well as P/E ratios. On the other side on the companies’ P/E ratio, the raise on interest rate will be albatross of companies, all other conditions remain, earnings will reduce, then equity will lessen, large deviation between operation performance and expected returns appears, can not support a high level of P/E ratio, so stock prices will decline. As a result, both market average and companies’ individual P/E ratios will be influenced by the annual interest rate.

It is also suitable to estimate the market average P/E ratio, and only when all the above assumptions are satisfied, that the practical P/E ratio amount to the theoretical value. However, different from the securities market, the interest rate is relatively rigid, especially to the strict control of interest rate countries; the interest rate adjustment is not so frequent, so that it is not synchronous with macroeconomic fundamentals. Reversely, the stock market reflects the macroeconomic fundamentals; high expectation of investors can raise up the stock prices, sequent the growth of the aggregate value of the whole market. Other market behaviors can also lead to changes of average P/E ratios. Therefore, it is impossible that the average P/E ratio is identical with the theoretical one. Variance exits inevitably, the key is to measure a rational range for this variance.

For the market average P/E ratio, P should be the aggregate value of listed stocks, and E is the total level of capital gains. To the maturity market, the reasonable average P/E ratio should be the reciprocal of the average yields of the market; usually the bank annual interest is used to represent the average yields of the market.

The return on retained earnings is an expected value in theory, while it is always hard to forecast, so the return on equity (ROE) is used to estimate the value.

(6) can then evolve as,

P₀/E₁ = b/(R-g) = b/(R-r(1-b)) = b/(R-ROE(1-b)) —– (8)

From (8) we can know, ROE is one of the influence factors to P/E ratio, which measures the value companies created for shareholders. It is positively correlated to the P/E ratio. The usefulness of any price-earnings ratio is limited to firms that have positive actual and expected earnings. Depending on the data source you use, companies with negative earnings will have a “null” value for a P/E while other sources will report a P/E of zero. In addition, earnings are subject to management assumptions and manipulation more than other income statement items such as sales, making it hard to get a true sense of value.

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Quantifier – Ontological Commitment: The Case for an Agnostic. Note Quote.

What about the mathematical objects that, according to the platonist, exist independently of any description one may offer of them in terms of comprehension principles? Do these objects exist on the fictionalist view? Now, the fictionalist is not committed to the existence of such mathematical objects, although this doesn’t mean that the fictionalist is committed to the non-existence of these objects. The fictionalist is ultimately agnostic about the issue. Here is why.

There are two types of commitment: quantifier commitment and ontological commitment. We incur quantifier commitment to the objects that are in the range of our quantifiers. We incur ontological commitment when we are committed to the existence of certain objects. However, despite Quine’s view, quantifier commitment doesn’t entail ontological commitment. Fictional discourse (e.g. in literature) and mathematical discourse illustrate that. Suppose that there’s no way of making sense of our practice with fiction but to quantify over fictional objects. Still, people would strongly resist the claim that they are therefore committed to the existence of these objects. The same point applies to mathematical objects.

This move can also be made by invoking a distinction between partial quantifiers and the existence predicate. The idea here is to resist reading the existential quantifier as carrying any ontological commitment. Rather, the existential quantifier only indicates that the objects that fall under a concept (or have certain properties) are less than the whole domain of discourse. To indicate that the whole domain is invoked (e.g. that every object in the domain have a certain property), we use a universal quantifier. So, two different functions are clumped together in the traditional, Quinean reading of the existential quantifier: (i) to assert the existence of something, on the one hand, and (ii) to indicate that not the whole domain of quantification is considered, on the other. These functions are best kept apart. We should use a partial quantifier (that is, an existential quantifier free of ontological commitment) to convey that only some of the objects in the domain are referred to, and introduce an existence predicate in the language in order to express existence claims.

By distinguishing these two roles of the quantifier, we also gain expressive resources. Consider, for instance, the sentence:

(∗) Some fictional detectives don’t exist.

Can this expression be translated in the usual formalism of classical first-order logic with the Quinean interpretation of the existential quantifier? Prima facie, that doesn’t seem to be possible. The sentence would be contradictory! It would state that ∃ fictional detectives who don’t exist. The obvious consistent translation here would be: ¬∃x Fx, where F is the predicate is a fictional detective. But this states that fictional detectives don’t exist. Clearly, this is a different claim from the one expressed in (∗). By declaring that some fictional detectives don’t exist, (∗) is still compatible with the existence of some fictional detectives. The regimented sentence denies this possibility.

However, it’s perfectly straightforward to express (∗) using the resources of partial quantification and the existence predicate. Suppose that “∃” stands for the partial quantifier and “E” stands for the existence predicate. In this case, we have: ∃x (Fx ∧¬Ex), which expresses precisely what we need to state.

Now, under what conditions is the fictionalist entitled to conclude that certain objects exist? In order to avoid begging the question against the platonist, the fictionalist cannot insist that only objects that we can causally interact with exist. So, the fictionalist only offers sufficient conditions for us to be entitled to conclude that certain objects exist. Conditions such as the following seem to be uncontroversial. Suppose we have access to certain objects that is such that (i) it’s robust (e.g. we blink, we move away, and the objects are still there); (ii) the access to these objects can be refined (e.g. we can get closer for a better look); (iii) the access allows us to track the objects in space and time; and (iv) the access is such that if the objects weren’t there, we wouldn’t believe that they were. In this case, having this form of access to these objects gives us good grounds to claim that these objects exist. In fact, it’s in virtue of conditions of this sort that we believe that tables, chairs, and so many observable entities exist.

But recall that these are only sufficient, and not necessary, conditions. Thus, the resulting view turns out to be agnostic about the existence of the mathematical entities the platonist takes to exist – independently of any description. The fact that mathematical objects fail to satisfy some of these conditions doesn’t entail that these objects don’t exist. Perhaps these entities do exist after all; perhaps they don’t. What matters for the fictionalist is that it’s possible to make sense of significant features of mathematics without settling this issue.

Now what would happen if the agnostic fictionalist used the partial quantifier in the context of comprehension principles? Suppose that a vector space is introduced via suitable principles, and that we establish that there are vectors satisfying certain conditions. Would this entail that we are now committed to the existence of these vectors? It would if the vectors in question satisfied the existence predicate. Otherwise, the issue would remain open, given that the existence predicate only provides sufficient, but not necessary, conditions for us to believe that the vectors in question exist. As a result, the fictionalist would then remain agnostic about the existence of even the objects introduced via comprehension principles!