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 +…+ Dn/(1+R)n = ∑t=1 Dt/(1+R)t (n→∞) —– (1)

In (1)

V: intrinsic value of the stock;

Dt: dividend for the tth 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 +…+ Dn/(1+R)n = ∑t=1 Dt/(1+R)t (n→∞) —– (2)

where P0: purchase price of the stock;

Dt: dividend for the tth 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 +…+ Dn/(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 1st 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 1st 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.

Arbitrage, or Tensors thereof…


What is an arbitrage? Basically it means ”to get something from nothing” and a free lunch after all. More strict definition states the arbitrage as an operational opportunity to make a risk-free profit with a rate of return higher than the risk-free interest rate accured on deposit.

The arbitrage appears in the theory when we consider a curvature of the connection. A rate of excess return for an elementary arbitrage operation (a difference between rate of return for the operation and the risk-free interest rate) is an element of curvature tensor calculated from the connection. It can be understood keeping in mind that a curvature tensor elements are related to a difference between two results of infinitesimal parallel transports performed in different order. In financial terms it means that the curvature tensor elements measure a difference in gains accured from two financial operations with the same initial and final points or, in other words, a gain from an arbitrage operation.

In a certain sense, the rate of excess return for an elementary arbitrage operation is an analogue of the electromagnetic field. In an absence of any uncertanty (or, in other words, in an absense of walks of prices, exchange and interest rates) the only state is realised is the state of zero arbitrage. However, if we place the uncertenty in the game, prices and the rates move and some virtual arbitrage possibilities to get more than less appear. Therefore we can say that the uncertanty play the same role in the developing theory as the quantization did for the quantum gauge theory.

What of “matter” fields then, which interact through the connection. The “matter” fields are money flows fields, which have to be gauged by the connection. Dilatations of money units (which do not change a real wealth) play a role of gauge transformation which eliminates the effect of the dilatation by a proper tune of the connection (interest rate, exchange rates, prices and so on) exactly as the Fisher formula does for the real interest rate in the case of an inflation. The symmetry of the real wealth to a local dilatation of money units (security splits and the like) is the gauge symmetry of the theory.

A theory may contain several types of the “matter” fields which may differ, for example, by a sign of the connection term as it is for positive and negative charges in the electrodynamics. In the financial stage it means different preferances of investors. Investor’s strategy is not always optimal. It is due to partially incomplete information in hands, choice procedure, partially, because of investors’ (or manager’s) internal objectives. Physics of Finance



High Frequency Markets and Leverage


Leverage effect is a well-known stylized fact of financial data. It refers to the negative correlation between price returns and volatility increments: when the price of an asset is increasing, its volatility drops, while when it decreases, the volatility tends to become larger. The name “leverage” comes from the following interpretation of this phenomenon: When an asset price declines, the associated company becomes automatically more leveraged since the ratio of its debt with respect to the equity value becomes larger. Hence the risk of the asset, namely its volatility, should become more important. Another economic interpretation of the leverage effect, inverting causality, is that the forecast of an increase of the volatility should be compensated by a higher rate of return, which can only be obtained through a decrease in the asset value.

Some statistical methods enabling us to use high frequency data have been built to measure volatility. In financial engineering, it has become clear in the late eighties that it is necessary to introduce leverage effect in derivatives pricing frameworks in order to accurately reproduce the behavior of the implied volatility surface. This led to the rise of famous stochastic volatility models, where the Brownian motion driving the volatility is (negatively) correlated with that driving the price for stochastic volatility models.

Traditional explanations for leverage effect are based on “macroscopic” arguments from financial economics. Could microscopic interactions between agents naturally lead to leverage effect at larger time scales? We would like to know whether part of the foundations for leverage effect could be microstructural. To do so, our idea is to consider a very simple agent-based model, encoding well-documented and understood behaviors of market participants at the microscopic scale. Then we aim at showing that in the long run, this model leads to a price dynamic exhibiting leverage effect. This would demonstrate that typical strategies of market participants at the high frequency level naturally induce leverage effect.

One could argue that transactions take place at the finest frequencies and prices are revealed through order book type mechanisms. Therefore, it is an obvious fact that leverage effect arises from high frequency properties. However, under certain market conditions, typical high frequency behaviors, having probably no connection with the financial economics concepts, may give rise to some leverage effect at the low frequency scales. It is important to emphasize that leverage effect should be fully explained by high frequency features.

Another important stylized fact of financial data is the rough nature of the volatility process. Indeed, for a very wide range of assets, historical volatility time-series exhibit a behavior which is much rougher than that of a Brownian motion. More precisely, the dynamics of the log-volatility are typically very well modeled by a fractional Brownian motion with Hurst parameter around 0.1, that is a process with Hölder regularity of order 0.1. Furthermore, using a fractional Brownian motion with small Hurst index also enables to reproduce very accurately the features of the volatility surface.


The fact that for basically all reasonably liquid assets, volatility is rough, with the same order of magnitude for the roughness parameter, is of course very intriguing. Tick-by-tick price model is based on a bi-dimensional Hawkes process, which is a bivariate point process (Nt+, Nt)t≥0 taking values in (R+)2 and with intensity (λ+t, λt) of the form


Here μ+ and μ are positive constants and the functions (φi)i=1,…4 are non-negative with associated matrix called kernel matrix. Hawkes processes are said to be self-exciting, in the sense that the instantaneous jump probability depends on the location of the past events. Hawkes processes are nowadays of standard use in finance, not only in the field of microstructure but also in risk management or contagion modeling. The Hawkes process generates behavior that mimics financial data in a pretty impressive way. And back-fitting, yields coorespndingly good results.  Some key problems remain the same whether you use a simple Brownian motion model or this marvelous technical apparatus.

In short, back-fitting only goes so far.

  • The essentially random nature of living systems can lead to entirely different outcomes if said randomness had occurred at some other point in time or magnitude. Due to randomness, entirely different groups would likely succeed and fail every time the “clock” was turned back to time zero, and the system allowed to unfold all over again. Goldman Sachs would not be the “vampire squid”. The London whale would never have been. This will boggle the mind if you let it.

  • Extraction of unvarying physical laws governing a living system from data is in many cases is NP-hard. There are far many varieties of actors and variety of interactions for the exercise to be tractable.

  • Given the possibility of their extraction, the nature of the components of a living system are not fixed and subject to unvarying physical laws – not even probability laws.

  • The conscious behavior of some actors in a financial market can change the rules of the game, some of those rules some of the time, or complete rewire the system form the bottom-up. This is really just an extension of the former point.

  • Natural mutations over time lead to markets reworking their laws over time through an evolutionary process, with never a thought of doing so.


Thus, in this approach, Nt+ corresponds to the number of upward jumps of the asset in the time interval [0,t] and Nt to the number of downward jumps. Hence, the instantaneous probability to get an upward (downward) jump depends on the arrival times of the past upward and downward jumps. Furthermore, by construction, the price process lives on a discrete grid, which is obviously a crucial feature of high frequency prices in practice.

This simple tick-by-tick price model enables to encode very easily the following important stylized facts of modern electronic markets in the context of high frequency trading:

  1. Markets are highly endogenous, meaning that most of the orders have no real economic motivation but are rather sent by algorithms in reaction to other orders.
  2. Mechanisms preventing statistical arbitrages take place on high frequency markets. Indeed, at the high frequency scale, building strategies which are on average profitable is hardly possible.
  3. There is some asymmetry in the liquidity on the bid and ask sides of the order book. This simply means that buying and selling are not symmetric actions. Indeed, consider for example a market maker, with an inventory which is typically positive. She is likely to raise the price by less following a buy order than to lower the price following the same size sell order. This is because its inventory becomes smaller after a buy order, which is a good thing for her, whereas it increases after a sell order.
  4. A significant proportion of transactions is due to large orders, called metaorders, which are not executed at once but split in time by trading algorithms.

    In a Hawkes process framework, the first of these properties corresponds to the case of so-called nearly unstable Hawkes processes, that is Hawkes processes for which the stability condition is almost saturated. This means the spectral radius of the kernel matrix integral is smaller than but close to unity. The second and third ones impose a specific structure on the kernel matrix and the fourth one leads to functions φi with heavy tails.