In January of this year, I gave a talk in Paris, at University of Paris X – Nanterre, to a group of economists on “Value and price.” The paper was published online in April in “La gazette permanente du MAUSS” (1). It is in French – as suits speaking in Paris. There is however an abstract in English that reads like this:

“What is being referred to as the “value” of a share in a company’s stock is in truth only one of two possible ways for establishing its price: what Adam Smith called its natural price by contrast to its market price. The market price is being generated through the spontaneous interaction of a stock exchange’s traders while the value or natural price is computed, reflecting an “additive” concept of price. The underlying principle is that it is feasible to decompose the price of any commodity as the set of prices of each of its components and that the price for the whole (the “price”) needs to equate the sum of the components’ prices (the “value”), forcing the prior to align with the latter.
With a share, a synthetic product of its underlying components can’t be realized. It is therefore so to say conventionally that the publication of a company’s quarterly figures – which allow in a stylized way a computation of the price of its components (its future dividends) results in an alignment of its market price with an additive interpretation of its price, that is, with its value.”

The reason I’m mentioning this here is because I’ve gone back to the paper by Refet S. Gurkaynak that I mentioned in “Now that the housing bubble has burst, who’s to blame?” where he states that “We are still unable to distinguish bubbles from time-varying or regime switching fundamentals” (2). It occurred to me that there might be a solution to the question he raises in the paper I gave in Paris.

It is an empirical fact that there are at times two distinct prices: the fair price (Adam Smith’s natural price) and the speculative price (Adam Smith’s market price). The popular distinction between “value” and “growth” stocks reflects this: the investor in value stocks looks for companies whose stock is underpriced relative to its fundamentals or fair price, while the investor in growth is focused on companies whose stock price has a tendency to take off compared to its fundamentals, i.e. is speculative.

When Gurkaynak wants to assess the existence or not of bubbles, his approach is, to say the least, simplistic. His starting point is the classical model for a stock’s price: the present value of future dividends – in other words, a discounted cash flow model – he then adds a bubble factor that he gingerly calls “B” for bubble but could as well be “B” for “black-box” even if he links its value to the cost of carry. His conclusion that econometric approaches are unable to establish the existence of bubbles is then only to be expected: “bubble tests do not do a good job of differentiating between misspecified fundamentals and bubbles” (p. 27), indeed, the fundamentals can be redefined so as to make “B” tend to zero and reproduce therefore whatever market price is observed. His other conclusion that “the bubble is a catch-all for stock price movements not explained by the model” (28), is correct but only to the extent that this is precisely the way he has characterized his “B” factor: as a catch-all.

Benjamin Graham introduced the by now widely accepted concept that the non-speculative natural price of a stock can be calculated in an “additive” way as the sum of all discounted future dividends, to which should be added the present value of the company’s equity per share, should the company fold some time in the future.

The dynamics of the market price is of an entirely different nature and its evolution is best described as a discrete dynamical system where the most recent settlement price is a function of past prices.

It can be represented as

MaP t = F(MaP t-1, MaP t-2, …);

with MaP t standing for Market Price at time t.

A market price is clearly dynamic as its value changes with time; it is also discrete as each transaction generates a settlement price that applies to the specific volume of shares exchanged between seller and buyer, and it is a function of past states as all agents base at any point in time their decisions to buy or sell on an analysis of past prices – be it crude or sophisticated. When news kick in – such as quarterly earnings figures – realignment on the natural price may take place. At other times, the dynamics that is purely driven by past prices re-establishes itself.

The speculative value (SpV) of the stock is simply Market Price minus Natural Price.

SpV t = MaP t – NaP t.

A bubble arises when SpV keeps growing.

Should one wish to prevent the market price from ever departing from the natural price, the best way would be to generate an arbitrage opportunity by trading a stock synthetic ©®™ (3) replicating the cash flows of the stock’s fundamentals. This can be achieved by creating an instrument that pays an amount equal to the expected dividends at the same date as the share does, plus an amount equivalent to the company’s equity per share, should it fold. To calculate the price of the synthetic stock, the amounts representing future dividends would be discounted at spot (“zero-coupon”) rates. Dividends’ distributions recur potentially an infinite number of times but should for practical purposes be limited to the years when their present value exceeds 1 cent. The discounted equity per share for any particular year can be calculated as the present value of the company’s equity per share, times the probability the company goes belly up in that particular year. Here again, the number of years should be determined as the maximum that still returns at least one cent. A value for that probability can be derived from a CDS (Credit Default Swap) quote used as a proxy.

With such synthetic stocks being traded, any speculative value (SpV t > $0.00) generates an arbitrage opportunity that eager traders would erase in no time. The bubble question would be solved once and for all – at least as far as the stock market is concerned.

(1) “Le prix et la « valeur » d’une action boursière“, Revue permanente du MAUSS, 2007.

(2) Refet S. Gurkaynak, “Econometric Tests of Asset Price Bubbles : Taking Stock”, Finance and Economics Discussion Series, Divisions of Research & Statistics and Monetary Affairs, Federal Reserve Board, Washington, D.C.2005-04.

(3) I seem to have exhausted all symbols suggesting intellectual priority.