In finance we are accustomed to talking about, and thinking about, and predicting, and measuring, returns. This is the first of a small set of articles that asks a very basic question: are returns a portable metric for comparing price changes?

What I mean is: does it make sense to say IBM went up 3.6% and CSCO went up 3.7% so IBM's performance was comparable to that of CSCO. Does taking a price change (IBM went up $3.04) and dividing it by the prior price (IBM's was $84.15/share) produce a quantity (3.61%) that it is meaningful to compare to that for CSCO (up $0.60 from $16.30/share implies a change of 3.68%) and to then conclude "they changed by the same amount."

The first reaction to this musing is: of course it makes sense --- you're telling me that making an equal investment in each company produced an equal return. But investment performance and price dynamics are not the same thing and this comparison is a little more subtle that it first appears. Tabulated below is the daily standard deviation for both stocks from approximately 800 business days.

Ticker St.Dev.
------ -------
CSCO    2.35 %
IBM     1.57 %

This shows that we should not expect both stocks to typically have the same scale of move on any given day. It casts our data (which is for today, 01/02/2009) into a different light. Statistically speaking, i.e. relative to the typical scale of daily moves, IBM moved more than CSCO (2.3 s.d. vs 1.6 s.d.).

Our comparison of the volatilities of the stocks has revealed a hidden assumption that exists when we compare returns: we assume that the volatility scales with the price of the stock in a linear fashion. i.e. We assume that because IBM costs over five times as much as CSCO per share that IBM's daily move should be of order five times as much as that of CSCO.

At this point, one could argue that all we've discovered is that volatility contains an idiosyncratic element and different stocks have different daily volatilities. i.e. That in the basic stochastic drift equation dS/S = m dt + k dX, all we need to say is that k is idiosyncratic. However, look at the equation: we've blythely included a scaling factor (1/S) to render our stochastic process model dimensionless. Ought we not ask, at least once, whether this assumption is empirically justified? Do we have any evidence that volatility scales with price level? After all, we know that volatility differs from stock to stock and we also know that volatility differs from day to day for the same stock. How certain are we that IBM's daily volatility at $84/share is twice what it was at $42/share?

Given that we know that although volatility varies, it varies somewhat slowly, let's take a readily available datum that scales with (the average of the) volatility for a given day, the absolute value of the daily price change, and see how this varies with price level.

This data, for recent history, is presented in
Scaling of Volatility with Price Level for U.S. Tech Stocks. This data set is a good candidate for analysis because it contains a substantial price excursion, and so is not confined solely to the current region of phase space. We see that not only do the regression lines differ substantially from a unit gradient, there is no consistency between the two stocks.

But, perhaps, this analysis is too idiosyncratic and too focused on recent history, so we should look at the behaviour of market aggregates over a substantial period of time (even though, for the operation of comparing returns to be useful, it is necessary that it be useful idiosyncratically). The second chart, Scaling of Volatility with Price Level for U.S. Stock Indices shows the same analysis for the DJIA and the S&P500 for daily data since 1928. In this chart we see a much smaller departure from unity, with an undeniable statistical and practical significance. Although the spectral index for the S&P500 is just 0.9 vs. 1.0, it is over 12 s.d. away based on the standard regression errors, and the effect of this scaling law vs. the null hypothesis, when aggregated daily for eighty years, will be quite substantial.

Again, one can poke holes in the methodology of (and very historical data for) both of these indices, I think that it's fair to conclude that this elementary conjecture is worthy of further study, which is what we shall do in Part II.