Volatility for the Delta Challenged

Nobody really knows how to calculate the “volatility” of a stock price or, more commonly, the volatility of stock price returns. We might even go further and ask, What is it supposed to mean, anyway? What usable insight does it provide about future stock prices or returns that we might bet on? The answer is none and if we are betting on the stock market then the only expectation that we are entitled to have is that we will lose our money (please see our previous Letter, Run Rabbit! Run, June 2012).

The issue is easily researched (see, for example, Volatility (Finance) – Wikipedia) but one goes away from that research with a queasy feeling about the supposed nature of stock prices or stock price returns which the authors hypothesize as being some type of “random walk” (a Wiener Process, perhaps) or an even more unfortunate Lévy Distribution, neither of which is provable nor, one wonders, even plausible.

If we buy a stock today for P1 (dollars) then our expectation is that at some time in the future it will be worth P2>P1 and that P2 is the sum of earned dividends and capital appreciation that we expect (or hope) will at least exceed the rate of inflation so that we can expect to buy the same things then as now.

In effect, in the manner of bonds in which the time and capital is guaranteed (for most practical purposes), we have bought P2 now at a discount and the discount is calculated as P2-P1 = (1-P1/P2)×P2. For example, if we buy the $100 T-bill for $95 today, we are almost certain that we will receive $100 in ninety days (for example) and our estimate of inflation is no more than 5% during that time (else, why would we buy the T-bill if we expect to have only the same, or less, cash then as now?).

Our “price of risk” is then exactly P1 and given the way in which P1 is determined in the case of T-bills, by sealed but frequent public auction, P1 is the price of risk and the “risk” is that inflation will be no more than 5% during the next several months.

The case of equities is only superficially different. There will be a wide range of opinion for what and when P2 might obtain but the fact that there is a P1 at which we can buy or sell the stock means that P1 is the price of risk for P2 now – it just might not (unlike the bond) be the price of risk tomorrow or even a moment from now because neither the amount nor the time of P2 is guaranteed.

In other words, the price of a stock, P1, now, has no predictive value at all – we have no idea from P1 when or what P2 might be or what other investors have in mind when P2 (whatever that might be or could be) can be bought for P1.

However, we can return to terra firma by thinking about the bond (T-bill) of our example. We would “like” N*=$100 in three months time but we are willing to pay – based on our estimation of inflation and the certainty of our return – no more than V=$95 for it today nor can we get it for less than $95 because of the competitive auction (although there are surely lots of people who will sell it to us for more than $95, including the government which would like to get more than $95, but can’t).

In other words, the “price of risk” (SF) can be described as

SF=[V/N*]×N*=[1-(N*-V)/N*]×N*

or, for the case at hand,

SF=[95/100]×100=[1-5/100]×100

which looks strangely familiar (with P1=V and P2=N*) and says exactly that we are “happy” to pay P1=V=$95 today for P2=N*=$100 in ninety days.

In the case of “bonds”, we know what N* is – it is the “reward” that we are sure to obtain, regardless of what its future “worth” might be – and V is to be determined by an open bidding on a common presumed rate of inflation – one unknown or degree of freedom.

In the case of equities, however, we know what V is (the price of the stock, quoted every minute) but we have no idea of what N* is if all we know is V every minute, hour, day, week, month, et cetera.

Economists, of course, have no idea what N* might be and therefore hypothesize about V – that it is some kind of “random” variable determined by efficient and frictionless markets somehow labouring towards an “equilibrium” and, therefore, that the current price, V, is just an error.

“Value investors”, on the other hand, believe in “earnings per share” (EPS) available at “low multiples” (PE) and their “price of risk” can be described as

SF= [(V/Share)/(N*/Share]×N*/Share=[PE]×EPS

where V is the “market value” of the firm, “Share” is the Common Shares Outstanding (typically) and N* are the recent or projected “earnings” (of some sort, possibly “consensus”) of the company. When we put it that way, then one notices that the “price of risk”, so defined, is hyperbolic and, based on the demonstrated behaviour of the markets, no “price of risk” is either too low or too high for these folks. And to put it yet another way, it ought to be clear that “if you don’t know the price of risk, then you don’t know anything about the stock price.” (Goetze 2009)

John Maynard Keynes, the economist and mathematician, was both wildly successful and, at times, wildly unsuccessful in stock market investing and provides us with the sobering thoughts:

 “ All sorts of considerations enter into market valuation which are in no way relevant to the prospective yield.” – J. Maynard Keynes, The General Theory of Employment, Interest and Money, New York, 1936.

“It is interesting that the stability of the system and its sensitiveness to changes in the quantity of money should be so dependent on the existence of a variety of opinion about what is uncertain. Best of all that we should know the future. But if not, then, if we are to control the activity of the economic system by changing the quantity of money, it is important that opinions should differ.” – ibid. J. Maynard Keynes, 1936.

Disclaimer

Investing in the bond and stock markets has become a highly regulated and litigious industry but despite that, there remains only one effective rule and that is caveat emptor or “buyer beware”.

Nothing that we say should be construed by any person as advice or a recommendation to buy, sell, hold or avoid the common stock or bonds of any public company at any time for any purpose. That is the law and we fully support and respect that law and regulation in every jurisdiction without exception and without qualification to the best of our knowledge and ability.

We can only tell you what we do and why we do it or have done it and we know nothing at all about the future or the future of stock prices of any company nor why they are what they are, now.

The author retains all copyrights to his works in this blog and on this website. The Perpetual Bond®™ is a registered trademark and patented technology of The RiskWerk Company and RiskWerk Limited (“Company”) . The Canada Pension Bond®™ and The Medina Bond®™ are registered trademarks or trademarks of the Company as are the words and phrases “Alpha-smart”, “100% Capital Safety”, “100% Liquidity”,  “price of risk”, “risk price”, and the symbols “(B)” , “(N)” and N*.