Research Article: The mathematics of market timing

Date Published: July 18, 2018

Publisher: Public Library of Science

Author(s): Guy Metcalfe, Alejandro Raul Hernandez Montoya.


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.

Partial Text

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.

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 N = 99 data points per series. The funds, all from Vanguard, are Total Stock Market, Total Bond Market, and Balanced Index, the last a static portfolio of 60% Total Stock and 40% Total Bond. Other information on these funds is in S1 Appendix. Fig 1 shows the quarterly return time series for stocks and bonds. Because the data are from live funds, calculated return paths are net of management and trading costs; however, tax consequences are ignored. For quarterly switching taxes would likely be substantial, but the effect would only dampen the spread of net returns and change only the quantitative, not the qualitative results of the model. Note that because fund data are the basic building blocks of the model, all return paths calculated could have been obtained by an investor during the time period.

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.

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 f = 0.6 balanced index fund.

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 f sequences) is not representative of actual market timing. There are two answers to this. One is that the feasible set is well-defined and that it is simply a fact that all market timing paths, no matter how they are generated, are contained in the feasible set. As such, any sampling of the feasible set generates valid timing paths. The second answer is in Fig 6, which reproduces Fig 4 with the addition of the return paths (yellow lines) of two funds that Morningstar classifies as TAA funds and for which there are publicly available returns data from 1994, almost as long as for the index funds data series. S4 Appendix has details about these two funds, which are rated by Morningstar as above average. While these market timing funds were neither limited to two asset classes, nor did they make all or nothing switches, yet their return paths are, as expected, contained inside the feasible set. The conclusion is that real-life market timers are correctly characterized—except for costs—by the PDF within the feasible set, and that random sampling of the PDF does properly characterize the return distribution expected from market timing schemes.

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.