# Temporal structure and gain/loss asymmetry for real and artificial stock indices

###### Abstract

We demonstrate that the gain/loss asymmetry observed for stock indices vanishes if the temporal dependence structure is destroyed by scrambling the time series. We also show that an artificial index constructed by a simple average of a number of individual stocks display gain/loss asymmetry — this allows us to explicitly analyze the dependence between the index constituents. We consider mutual information and correlation based measures and show that the stock returns indeed have a higher degree of dependence in times of market downturns than upturns.

###### keywords:

Temporal structure, gain/loss asymmetry, mutual information## Introduction

Inspired by research in the field of turbulence, Simonsen, Jensen, and Johansen investHorizon () considered “inverse statistics” of financial time series: what is the smallest time interval needed for an asset to cross a fixed return level ? Figure 1 shows the distribution of this random variable, the first passage time, for the Dow Jones Industrial Average index, for . As noted by Jensen, Johansen, and Simonsen investStatistics (), the most likely first passage time is shorter for than for , which they refer to as the gain/loss asymmetry.

In this paper, we show that the gain/loss asymmetry in the Dow Jones index vanishes if the time series is “scrambled” — that is, if one considers a new time series constructed by randomly permuting the returns. This basic fact, which seems to have gone unnoticed in the literature so far, has important implications: the gain/loss asymmetry is not due to properties of the unconditional index returns, like skewness, but is rather an expression of potentially complex temporal structure. This finding resonates with the results from Siven, Lins, and Lundbek Hansen multiscale (), where wavelet analysis is used to demonstrate that the gain/loss asymmetry is a long time scale phenomenon — it vanishes if enough low frequency content is removed from the index, that is, if the index is sufficiently “detrended.” Siven, Lins, and Lundbek Hansen multiscale () also present a generalization of the asymmetric synchronous market model from Donangelo, Jensen, Simonsen, and Sneppen fearModel () where prolonged periods of high correlation between the individual stocks during index downturns gives rise to a gain/loss asymmetry.

Whether the constituents of e.g. the Dow Jones index indeed tend to move with a greater degree of dependence during market downturns could in principle be tested empirically by analysis of the time series of the individual stocks. That is an awkward task, however, since the relative weights for different stocks in these indices have changed over time in complicated ways. To address this issue, we demonstrate that if one defines a new, artificial index by simply taking the average of a number of stocks, this index also displays gain/loss asymmetry. With the constituents readily available, we consider two measures based on correlation and mutual information, and show that there indeed is a higher degree of dependence between the stock returns during index downturns than upturns.

## Gain/loss asymmetry and temporal structure

For a given process , for instance daily closing prices of a stock index, the first passage time of the level is defined as

and is assumed to be independent of . The distribution of is estimated in a straightforward manner from a time series . Consider , and let be the smallest time point such that , if such a time point exists. In that case, is viewed as an observation of . (If , take instead such that .) Running from to gives a set of observations from which the distribution of is estimated as the empirical distribution. Given the empirical distribution, we follow Jensen, Johansen, and Simonsen investStatistics () and compute a fit of the density function for the generalized gamma distribution. This density is plotted as a solid line together with the empirical distribution in all figures, to guide the eye — we do not discuss the fitted parameters, nor claim that truly follows a generalized gamma distribution.

### Gain/loss asymmetry for the Dow Jones index

Figure 1 shows the estimated first passage time for the Dow Jones index. As discussed in the Introduction, there is a gain/loss asymmetry in that the most likely first passage time is shorter for than for . Next, we construct a scrambled version of the index by randomly re-arranging the log returns. Formally, if denotes the time series of daily closing prices of the Dow Jones index, let for and draw a random permutation of . We define

and let the scrambled index be given by and

Figure 1 shows that the scrambled index does not display a gain/loss asymmetry. This result is surprisingly strong: since the empirical return distributions are identical for an index and any of its scrambled versions, it shows that the gain/loss asymmetry is an expression of potentially complex temporal structure in the index. This fits nicely with the results from Siven, Lins, and Lundbek Hansen multiscale (), where a multiscale decomposition is used to demonstrate that the gain/loss asymmetry is a long rather than short scale phenomenon.

The gain/loss asymmetry in the asymmetric synchronous market model from Donangelo, Jensen, Simonsen, and Sneppen fearModel () does not disappear when the index returns are scrambled. This is to be expected, since the daily returns in that model are independent and identically distributed, so all statistical properties remain the same when the time series is scrambled. However, for the generalized model proposed in Siven, Lins, and Lundbek Hansen multiscale () the asymmetry does vanish, in perfect agreement with the Dow Jones index, see Figure 2.

### Gain/loss asymmetry for an artificial index

Consider stocks, and let denote the closing price of the th stock on day , for . We consider the artificial index constructed by averaging all the stocks,

The denominators give all stocks equal weight in the index at time .

We consider historical stock prices from January 1970 until December 2008 for the following 12 Dow Jones constituents: Boeing Co. (BA), Citigroup Inc. (C), El DuPont de Nemours & Co. (DD), General Electric Co. (GE), General Motors Corporation (GM), International business Machines Corp. (IBM), Johnson & Johnson (JNJ), JPMorgan Chase & Co. (JPM), The Coca-Cola Company (KO), McDonald’s Corp. (MCD), Procter & Gamble Co. (PG), and Alcoa Inc. (AA). These companies are chosen since long time series of stock returns are available, but our results are stable in the sense that adding or removing companies give very similar results.

Figure 3 shows that the index constructed from these stocks display a gain/loss asymmetry, much like the Dow Jones index, and that the asymmetry vanishes if we scramble the time series.

In what follows, we will use this artificial index as a kind of proxy for a real stock index. This has the advantage that the individual index constituents are readily available for analysis. This is unlike the Dow Jones index for which the relative weights and indeed the set of constituents have changed over time.

## Dependence between constituents during periods of index upturns and downturns

Here, and in what follows, denotes the artificial index defined in the previous section.

Inspired by the generalized asymmetric synchronous market model from Siven, Lins, and Lundbek Hansen multiscale (), our general intuition is that the individual stocks tend to “move together” to a greater degree during index downturns than during upturns, resulting in more violent downturns than upturns. To quantify this, we first divide the price history of our artificial index into two parts, corresponding to upturns and downturns, respectively.

Fix a window length and consider the index return over the th window,

for , where denotes the largest integer smaller than or equal to . We define the set of indices for which the daily returns belong to a window over which the index went up,

respectively went down,

Note that the sets and are disjoint.

We will consider two measures of dependence between all the individual stocks and evaluate it for the returns corresponding to days , and compare that to the same measures evaluated for days . Before describing the first measure, the mean of mutual information, we establish some additional notation. Let the th index be defined by

The th index is simply the artificial index constructed by averaging all stocks except the th. Denote the log return at day in the th stock and index by and .

#### Mean mutual information

The mutual information of two discrete stochastic variables and is defined as

where denote the joint and and the marginal probability functions of and . Mutual information can be written as , where and are the marginal entropies, and is the joint entropy of and , and it is a measure of dependence in the sense that and are independent if and only if . Mutual information can estimated from a finite set of joint samples of in a number of different ways, see Paninski mutualInfo (). In the computations below we apply the most straightforward estimator, the so-called plug-in estimator.

Let and denote the mutual information of the returns of the th stock and index, estimated from the samples respectively . We average over to obtain the mean mutual information, which can be seen as a measure of the degree of dependence between all the stocks over periods of upturns, and, respectively, downturns of the index :

Figure 4 shows the mean mutual information for varying window length — there is clearly a higher degree of dependence between the stocks returns during index downturns. However, given the hypothesis that stocks tend to “move together” to a greater degree during index downturns, with the result that downturns are more dramatic than upturns, there is a potential problem with the measure: the mutual information between the th stock and index is large whenever there is a high degree of dependence, not only when they tend to move in the same direction. If some stocks tend to move up when the index moves down, this would moderate the downturns, contrary to our intuition, and yet result in high values for the mean mutual information. For this reason, we also consider a correlation based measure.

#### Mean correlation

Let and denote the correlation between the returns of the th stock and index, estimated from the samples respectively . We average over to obtain the mean correlation, which can be seen as a measure of the degree of dependence between all the stocks over periods of upturns respectively downturns of the index :

Figure 4 shows the mean correlation for varying window length — this measure of dependence between the stock returns also show markedly higher values during index downturns that during index upturns. Contrary to the mean mutual information, however, the presence of “defensive” stocks that move up during index downturns would give negative contributions.

## Conclusion

If the gain/loss asymmetry observed for stock indices were a property of the unconditional distribution of returns, then the phenomenon should remain invariant under random permutations of the returns — this is not the case, as we have demonstrated. We may begin to rely more confidently on expectations derived from the generalized asymmetric synchronous market model, which have previously demonstrated that differences in correlated movements in index constituents for down-moves and up-moves can give rise to the kind of temporal dependence structure that produces such asymmetry. However, there are practical difficulties in exploring the correlations between the time series of the individual constituents of real stock indices, since these are not readily available, so we have shown that the gain/loss asymmetry can also be reproduced in an artificial stock index constructed as a simple average of a number of individual stocks.

Considering two different measures of dependence, mean mutual information and mean correlation, we concluded that there indeed is a greater degree of dependence between the constituents of the artificial index during downturns than upturns. This part of our analysis can be seen as an attempt to overcome some of the general difficulties in formulating tractable ways of analyzing non-stationary dependence structure in multivariate stochastic processes. Future work in the direction of analyzing the dynamics of the changes in the level of dependence between asset prices would certainly be interesting — not least from the perspective of investors who seek diversification that does not break down at the worst possible time. For instance, is it possible to design a localized measure of the level of dependence between stock prices and zoom in even more on the points in time where it is changing?

## References

- [1] Ingve Simonsen, Mogens H. Jensen, and Anders Johansen (2002) ‘Optimal investment horizons’, European Physical Journal B, 27, 583–586.
- [2] Mogens H. Jensen, Anders Johansen, and Ingve Simonsen (2003)‘Inverse fractal statistics in turbulence and finance’, International Journal of Modern Physics B, 17, 4003–4012.
- [3] Johannes Siven, Jeffrey Lins, and Jonas Lundbek Hansen (2009), ‘A multiscale view on inverse statistics and gain/loss asymmetry in financial time series’, Journal of Statistical mechanics: Theory and Experiment, P02004.
- [4] Raul Donangelo, Mogens H. Jensen, Ingve Simonsen, and Kim Sneppen (2006) ‘Synchronization model for stock market asymmetry’, Journal of Statistical mechanics: Theory and Experiment, L11001.
- [5] Liam Paninski (2003), ‘Estimation of entropy and mutual information’, Neural Computation, 15, 1191–1253.