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The Shape of Financial Time


Ross M. Miller
Miller Risk Advisors
January 9, 2006

[This commentary appears in the January/February 2006 issue of Financial Engineering News. This is the raw, unedited version of that commentary.]

It is must be reassuring to those entering finance from more tangible disciplines to encounter a familiar concept amidst all the strange and seemingly arbitrary newness. That concept appears in the form of a variable represented by the letter t (or T) and is commonly known as "time."

It can take a while to discover that this t, financial time, is not the same thing as the more familiar t—the physical time that most of us inhabit and that our clocks measure. It is not that clock time does not matter in finance; it does, just not as much as one might imagine. Absolute times, such as the expiration date for an option, are (unless an exchange or the SEC happens to decides otherwise) usually remain absolute. The speed at which financial time passes is another story.

The field of economics has long recognized that it is unnecessary for financial time to pass at a uniform clip. In the Arrow-Debreu formalization of general equilibrium theory that underlies much of modern finance, the notion of financial time is only loosely linked to physical time. While physical time apparently exists so that everything does not happen all at once, financial time is necessary only as a window through which risk and uncertainty can climb into economic theory. In the Arrow-Debreu world, time flows however it wants as long as there is general agreement as to when certain points in time occur. In other words, financial time flies like a socially constructed Arrow.

While it was generally taken for granted that time in financial and economic models was identical to physical time, after a while doubts surfaced. In light of the mounting empirical evidence that asset returns did not look like they were drawn from independent and identical normal distributions as many popular financial models had assumed, something was wrong and the treatment of time was one place to look for answers. If the financial world truly obeyed physical law, it would make sense (via the central limit theorem) for asset returns to either themselves be normally distributed or else be derivable from a normal process. By changing the shape of financial time from a deterministic straight line to something more exotic—not necessarily continuous and certainly not deterministic—perhaps one could reunite the financial and physical realms.

When a market suffers from one its occasional fits of volatility, for example, it could simply be that the financial clock has begun to tick faster. What clocks out to be a single hour's worth of time in the physical world might very well translate into a day, a week, a month, or even a year of financial time. Indeed, if one tinkers sufficiently with the financial clock, one can generate almost any volatility pattern that one desires without having to distort the underlying normality of the process generating prices and returns.

While the idea that financial time might be only loosely related to physical time has yet to make waves in the financial mainstream, it has a perverse elegance and furthermore is laughably obvious to any trader. The time signature of the symphony played by the interaction of the various financial markets changes constantly. Some of these changes are predictable—markets often open to a scherzo before settling down to a midday waltz—others are not. Markets also watch the calendar, while physics knows no season.

If financial time and physical are meaningfully different, coming up with a way of translating between them might explain a lot. Perhaps by doing so, in one fell swoop many outstanding problems of finance could be solved or at least be made more tractable. Unfortunately, such a Rosetta stone for moving between financial time and physical time remains elusively.

In an obvious parallel to the Einsteinian physical world, financial time appears to be a local phenomenon—something this is permitted within the Arrow-Debreu framework, but is precluded by most financial models. In the absence of a single, universal financial clock, each market must keep its own time.

A common device for ticking off local financial time is to have trading volume serve as a metronome for the market. Hence, when a market is active, more financial time ticks away than when traders are napping or out to lunch. It would be convenient if the trades or trading volume were the key to financial time; however, a recent paper by Laszlo Gillemot, J. Doyne Farmer, and Fabrizio Lillo from the Santa Fe Institute entitled "There's More to Volatility than Volume" shows, things are not so simple.

Volatility, however, is not the only place that time matters in finance. Consider any of the several yield curves, which are expressed in functional form as r(t)-so that the interest rate, r, depends on time (presumably physical time), t.

The yield curve makes operational the clich้ that "time is money." Under the normal situation of a rising yield curve, we find that from the vantage point of the present a unit of time is worth more money the further out into the future that we go. In the case of the Treasury yield curve, however, time is not all that is money. Liquidity is money, too.

Time for Treasury securities is warped by the funding whims of the U.S. Treasury. On-the-run Treasury securities (those most recently issued at a given maturity) tend to trade at a somewhat lower yield than off-the-run (seasoned) Treasuries because they are more liquid. Such small discrepancies are consistent with Mom, apple pie, and efficient markets. During the early 1990s, however, a more significant warpage developed.

Back then, I happened to be doing some yield curve modeling for one of the parties responsible for denting the yield curve, a major insurance company. Taking the lead from the mortgage markets, the big insurers had gotten into some serious slicing and dicing of cash flows and the further out in time they were able to slice off cash flows, the better. Whenever a new 30-year Treasury bond was issued, it would be gobbled up and its price bid up, with the old 30-year Treasury cast off like last season's designer clothes.

For anyone trying to fit a model to Treasury yields, the 30-year Treasury bond was the sort of nuisance that were it not so important would have qualified as an outlier. It marred the otherwise beautifully Treasury yield curve that sloped smoothly upward. There was nothing special about 30 years, if the Treasury, to the delight of the insurance industry, had decided to issue a 40-year bond; it too would have commanded an unnaturally low yield. Physical time may lack a well-defined terminus, but for many big-money players, financial time ends with the last Treasury bond.

This anomaly, and the arbitrage opportunity it presented, did not last forever. Hedge funds capitalized on this opportunity until they had competed it away. Nonetheless, temporal anomalies of one kind or another continue to serve up feasts to those who hunt alpha for a living. In finance, it is nice to know what time it is, but it is even better to know what time is.

Copyright 2006 by Miller Risk Advisors and Financial Engineering News.