The author has declared that no competing interests exist.

Market timing is an investment technique that tries to continuously switch investment into assets forecast to have better returns. What is the likelihood of having a successful market timing strategy? With an emphasis on modeling simplicity, I calculate the feasible set of market timing portfolios using index mutual fund data for perfectly timed (by hindsight) all or nothing quarterly switching between two asset classes, US stocks and bonds over the time period 1993–2017. The historical optimal timing path of switches is shown to be indistinguishable from a random sequence. The key result is that the probability distribution function of market timing returns is asymmetric, that the highest probability outcome for market timing is a below median return. Put another way, simple math says market timing is more likely to lose than to win—even before accounting for costs. The median of the market timing return probability distribution can be directly calculated as a weighted average of the returns of the model assets with the weights given by the fraction of time each asset has a higher return than the other. For the time period of the data the median return was close to, but not identical with, the return of a static 60:40 stock:bond portfolio. These results are illustrated through Monte Carlo sampling of timing paths within the feasible set and by the observed return paths of several market timing mutual funds.

Market timing is an investment technique whereby an investment manager (professional or individual) attempts to anticipate the price movement of asset classes of securities, such as stocks and bonds, and to switch investment money away from assets with lower anticipated returns into assets with higher anticipated returns. Market timing managers use economic or other data to calculate propitious times to switch. Market timing seems a popular approach to investment management, with Morningstar listing several hundred funds in its tactical asset allocation (TAA) category—TAA being an industry name for market timing—and mainstream fund managers advertising their ability to switch to defensive assets when stock markets seem poised for a downturn. The antithesis of market timing, and another broadly popular investing approach, is buy-and-hold, whereby investment managers allocate static fractions of their monies to the available asset classes and then ignore market price gyrations.

Is market timing likely to be successful relative to investing in a static allocation to the available asset classes? The literature in this area is focused on developing sophisticated statistical tools that can detect and measure the market timing ability of professional fund managers [

My goal here is both different and simpler than statistical tests to detect market timing. I want to create a simple model to ask the question, what is the likelihood of successful market timing? Or more precisely, what is the return probability distribution function (PDF) for market timing? Is the PDF of market timing returns symmetric? If it is hard to obtain above average returns by market timing, is it also hard to obtain below average returns? What is the most basic mathematics of market timing?

I try in this paper to evoke a similar spirit to Sharpe’s “The Arithmetic of Active Management” [

In the rest of the paper my approach will be to calculate the boundaries of the feasible set of market timing portfolios using fund data for perfectly timed (by hindsight) switching between two asset classes, stocks and bonds. From this analysis I also obtain the historically optimal timing path of switches, which the NIST (National Institute of Standards and Technology, U.S. Department of Commerce,

The data consists of time series of quarterly returns for three index funds starting in 1993, the advent of the youngest of the three funds, and ending in Q3 2017. The series covers 24 years, and there are

Quarterly return time series for stock and bond total market index funds, 1993–2017. Returns are in multiplicative form.

Since the way to calculate total return is to multiply the sub-period returns together, I trivially transform the original data to multiplicative form, e.g. a +3% return becomes 1.03 and a −3% return becomes 0.97. The differences between multiplicative and additive random processes will be important in the subsequent analysis.

Here I define the simple two asset market timing model with all or nothing quarterly switches, emphasizing the deliberate choice to assume a simple model in order to gain insight into the fundamental mathematics. Using perfect hindsight, it is easy to identify the best and worst possible market timing portfolios, which form the boundaries of the feasible return paths for all market timing portfolios, i.e. all possible market timing portfolios lie between the boundaries of the feasible set. (Technically it is all market timing portfolios that conform to the assumptions of the model; however, in the discussion section we will see that real, non-conforming market timing funds fall within the feasible set.) I reveal the optimal (highest possible return) timing sequence and test it for randomness. A later section focuses on deriving the return PDF for the model.

The model consists of quarterly all or nothing switches between stocks and bonds. In the _{i} the return of stocks is denoted _{si} and the return of bonds is denoted _{bi}. A _{i} that is
_{i}. The

With perfect hindsight the best and worst performing return paths are easily found. In the notation of Matlab code _{b} and worst _{w} possible return paths
_{b} is given by _{b} = (stocks > bonds); similarly for _{w}. _{b} and _{w}, while _{b}, and _{w}. There are no surprises: partitioning returns by _{b}, while excluding the negative return, left tail. The reverse happens to _{w}.

Two asset, all or nothing market timing model switches to whichever of the two assets classes will have the better return that quarter. (a) Quarterly returns of the best and worst market timing portfolios as a function of time in multiplicative form. (b) Histograms of returns for the indicated data sets. (c) Feasibility envelope plotted on semi-log axes. Thick red lines are the best and worst possible return paths over this time period. Blue lines are the three data sets: stocks (

The best and worst possible return paths demark the feasible set of return paths for the two asset model. All possible return paths (all possible market timing paths _{i}) fall inside the envelope made by _{b} and _{w}. As the model has all or nothing switches, the number of possible paths of length ^{N}. As the data set has ^{99} ∼ 10^{29}, which is large.

_{b} that produces the highest possible return path _{b} over the time period. Black regions have _{i} = 1 (stock return > bond return). White regions have _{i} = 0 (bond return > stock return). It will be convenient to define _{b} and dividing by _{b} ≈ 0.64: over this time period approximately 2/3 of the time stocks returned more than bonds. While the optimal timing path _{b} is not random like a coin flip (_{b} ≠ 1/2), _{b} random?

Optimal timing path _{b} that would have produced the highest possible return path _{b} over the time period. Black regions have _{i} = 1 (stocks > bonds). White regions have _{i} = 0 (bonds > stocks).

It is worth distinguishing random and unpredictable. The historically optimal timing path is not a random bit sequence because ones occur about two-thirds of the time. Nonetheless, the important question is can I predict the next element in the sequence, given knowledge of the previous elements of the sequence? How can a sequence be not random but at the same time unpredictable? Consider a 6-sided die, of which four sides have a one and two sides have a zero. For each fair roll of the die there is a two-thirds probability of a one and a one-third probability of a zero, i.e. the chance of a 1 or 0 is similar to that observed in the data. Since each fair roll of the die is independent of all rolls that have come before, there is no way to predict from the past sequence of rolls what the next roll of the die will produce. Although _{b} is not known _{b} could be different over different time periods or markets—the unpredictable die analogy holds for all _{b} by changing the number of sides to the die.

Leaving the details to _{b} is close to the suggested minimum length. Again, leaving details to _{b} is random (unpredictable) at the 99% confidence level.

While the historically optimal timing sequence _{b} is clearly special in some sense—the probability of that particular sequence to occur is 2^{−99}—the question is what, if anything, distinguishes _{b} from any other random timing paths? If we look at _{b} and randomly generated timing paths _{b} from the masses of possible timing paths? If _{b} is random, as the NIST tests say it is, there is nothing to tell why it is special, which says that it is not special, that just by a 2^{−99} random chance, it was special for this time period and that, in itself, _{b} is unpredictable, i.e. it contains

As the optimal timing path is indistinguishable from a random sequence, I review elementary properties of random multiplicative processes, from which it follows that the highest probability outcome of market timing is a return less than the median of the PDF of market timing returns. The return PDF is estimated by Monte Carlo sampling of random timing paths. The median of the return PDF can be directly calculated as the weighted average of the returns of the assets with the weights given by the fraction of time each asset has a higher return than the other. For the time period covered by the data the median return was close to the

The distribution of typical returns of the model can be estimated by Monte Carlo methods. Generate _{b}. This is done by using Matlab’s _{b} or to zero if _{b}. ^{5} return paths as thin gray lines in a semi-log plot similar to _{b} and _{w}, while the thick black line is the data for the

Return paths (gray) for ^{5} randomly generated timing paths. Red lines are the best and worst market timing return paths. The black line is the observed

A

Nonetheless, what can be done is to take the log of the geometric mean of ^{μ} is the median and ^{μ − σ2} is the mode of the log-normal PDF: the mode, which is the most probable outcome, is less than the median of the log-normal PDF. Thus from elementary considerations the most probable outcome from market timing is a return that is less than the median of the return PDF.

To illustrate, ^{5} grossly under samples the order 10^{29} distinct paths in the feasible set, convergence to a Gaussian PDF is evident, as predicted by the form of _{w} and _{b}. The orange bar marks the median log-return and the log -return for the

Probability distribution function of (a) log-return and (b) return estimated from ^{5} trials with _{b} ≈ 0.64. Green and purple vertical bars are respectively the worst and best timing portfolios. The orange bar is the median of the PDF and the observed return of the

The expectation value operator _{b} is the observed fraction of time periods that the stock return exceeds the bond return. The median of the distribution of log returns is given by the log of the weighted average of the two assets with the weights given by the fraction of time periods ^{μ}.

Note that because over the time period of the data _{b} ≈ 0.64, that using the log return for the _{b}, which, of course, cannot be known

It is important to note that

Several critiques could be leveled at the analysis in this paper. For example, adherents of market timing would claim that their timing systems are not random, therefore they would be able to choose timing paths to have returns far out on the right tail of the PDF, i.e. that the strategy to generate random paths (random

Reprise of

A more subtle criticism is that I have not disproved market timing. This is because of the possibility of hidden variables. Hidden variables represent information, such as earnings, book value, anything, that a market timer could put into a function that produces a timing path. While the observed optimal timing path _{b} is random to the extent that it passes the NIST tests, it is possible that there was a set of hidden variables that could have been combined in a function that would have produced the optimal timing path _{b}. Good pseudo-random number generators also pass the NIST tests but are produced by deterministic systems. Taking into account the fund data of

I have examined a two asset, all or nothing market timing model with 24 years of data from US stock and bond total market index funds from 1993–2017. The model is deliberately kept simple in order to see the basic mathematics of market timing at work answering the question, what is the likelihood of successful market timing? The boundaries of the feasible set of market timing paths, within which all market timing return paths must lie, is easy in hindsight to calculate by always choosing the higher or lower returning asset each quarter. The historical optimal timing path is, however, indistinguishable from a random sequence; it is unpredictable and encodes no information about the future optimal timing path.

The key observation is that return is a multiplicative process and so its PDF is log-normal. The implication is the mathematical fact that the most probable outcome from market timing is a below median return—even before accounting for costs. This stems from an elementary property of the logarithm. Put another way, simple math says the most likely outcome of market timing is under performance. Exactly what this under performance is can be ascertained because the median of the market timing return PDF can be directly calculated as a weighted average of the returns of the model assets with weights given by the fraction of time periods each asset has a higher return than the other. For the time period of the data the median return was close to the return of the static 60:40 stock:bond balanced index; althrough, the value of _{b} need not be fixed for all time.

For simplicity of analysis and clarity of results the model in this paper has only two asset classes; however, it is clear that the methodology could be extended to any number of asset classes.

Matlab code with all data for calculations and figures.

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Fund data scrapped from Yahoo Finance on 2 November 2017; applicable terms of service were complied with.

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The NIST test suite [

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Fund data and objectives summaries scrapped from Yahoo Finance 6 November 2017; applicable terms of service were complied with.

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