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Financ Res Lett. 2021 Oct; 42: 102091.
Published online 2021 Apr 27. doi:10.1016/j.frl.2021.102091
PMCID: PMC8450758
PMID: 34566535
John M. Maheu,a Thomas H. McCurdy,⁎,b and Yong Songc
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Abstract
The COVID-19 pandemic has caused severe disruption worldwide. We analyze the aggregate U.S. stock market during this period, including implications for both short and long-horizon investors. We identify bull and bear market regimes including their bull correction and bear rally components, demonstrate our model’s performance in capturing periods of significant regime change, and provide weekly forecasts that improve risk management and investment decisions. An investment strategy that uses out-of-sample forecasts for market states outperforms a buy and hold strategy during the pandemic by a wide margin, both in terms of annualized returns and Sharpe ratios.
Keywords: COVID-19, Equity returns, Markov switching, Predictive density, Investment strategies
1. Introduction
This paper dates and forecasts bull and bear markets for the COVID-19 pandemic period based on aggregate equity return data from 1885 to 2020. We document market phases prior to the pandemic; and then use both smoothed and out-of-sample forecasts to analyze what has happened to equity markets during 2020. Appying those forecasts to risk management and investment strategies during the pandemic period demonstrates the value of using a 4-state Markov-Switching (MS) model to direct the dynamics of the distribution of stock market returns.
There are several reasons for using a restricted 4-state Markov-Switching (MS) model, as in Maheuetal.(2012). First, unlike ex post dating methods (Pagan, Sossounov, 2003, Lunde, Timmermann, 2004), the Markov-Switching model treats the market states as latent and provides probability estimates for future states and regimes. This probability framework provides a full specification of the data generating process. As such, it generates probability statements about the market dynamics which are essential for investment and risk management decisions.
Secondly, in contrast to simple specifications which focus on two states of the market, our model allows for four states including bull corrections and bear rallies. Conventional methods of partitioning regimes do not identify intra-regime dynamics that can be very important for forecasts and investment decisions. For example, if a rally (positive sub-trend) starts during a bear regime, what is the probability that it will continue and transition to a bull market regime as opposed to falling back to the bear market state? Will a bull market correction (negative sub-trend) continue to a bear regime or recover to stay in the bull market regime?
Thirdly, higher-order moments of the state-specific distributions also provide useful information – for example, risk assessments associated with different states. They combine to provide the aggregate mixture distribution that governs the market dynamics and the associated predictive mean and density forecasts that are key for decisions.
Finally, the parameters and states are identified with economic restrictions that are consistent with investors attitudes to market phases. The data supports these economically motivated restrictions compared to an unrestricted 4-state model.
We find that the market moved from a bull state at the start of 2020 to a bull correction and quickly to a bear market on February 26. This bear market dominated until June 3 when the market transitioned to a bear rally and has remained in this phase. This is in contrast to the dating methods used in the popular press (“New bull market in stocks could last three years and may produce another 30% in gains, veteran strategist says”, Aug 31, 2020, https://www.marketwatch.com/). Those methods focus solely on price trends and ignore risk. Our model classifies the recent market into the bear rally due to the larger variance in returns. This elevated risk is not consistent with past bull market phases that display lower variability.
Traditional 2-regime models using ex post dating methods are unable to distinguish between bear rally and bull market states. Our approach does this, in part, due to probability estimates of risk differences across those states. We show the dramatic impact that the COVID-19 pandemic has had on the return distribution and risk measures. The model provides very accurate forecasts of turning points and these would have been available in real-time to investors.
Given the benefits of a full probability model of stock market phases, it is natural to ask whether forecasts from this mixture-distribution model can improve investment and risk management decisions. To this end, we define a pseudo Sharpe ratio to characterize in-sample estimates of the state density parameters. We then extend this measure to an out-of-sample predictive Sharpe ratio, derived from the predictive density of returns which is sensitive to the forecasted market states. This measure can be useful in assessing the risk and return of entering the market.
Several market-timing investment strategies are explored. We show that simple timing rules directing when to exit and enter the market lead to improved investment decisions relative to a buy and hold strategy in 2020. These results are robust to different timing strategies and are a result of the precise turning points our model identifies and forecasts. Each of these market-timing strategies are in real time and would have been available to an investor using the model for forecasts.
We have implemented a number of model comparisons to evaluate the robustness of our results, including log-predictive likelihood values for the alternative models, none of which improve on our proposed model. Details and further robustness tests are reported in our online Appendix. That Appendix also includes additional results, including long-horizon forecasts of state probabilities four weeks ahead throughout 2020 and one year ahead from the final week of our sample.
Our paper is organized as follows. Section2 briefly reviews the structure and estimation of the restricted 4-state Markov-Switching bull and bear market model. Section3 summarizes the data and reports the results for the pre and post COVID-19 periods. Notably, Section3.3 provides forecasts based on one-week-ahead predictive densities including out-of-sample forecasts of future states and the associated risk and return measures. Section3.4 reports market-timing strategies that exploit those forecasts. Section4 reports comparisons of our model to competing models. Section5 concludes.
2. Model and estimation
Define log-returns as and . Consider the following 4-state Markov-Switching (MS4) model, based on Maheuetal.(2012), for returns:
in which denote the latent states, parameterized as Normally distributed with mean and variance and denote the state transition probabilities.
The following restrictions and labels are imposed for identification purposes,
No restriction is imposed on . Note that there are four states but we refer to two distinct regimes by as
The transition matrix takes the following form.
This specification implies that bear states and bear rally states cannot move to a bull correction and bull states and bull correction states cannot move to a bear rally state. This is done to avoid confounding these states that display common mean trends up or down in prices. For instance, it would be difficult to separate states 1 and 3 which both display negative average growth. The restricted transition matrix also means that the start of a new regime must be from a bear state or a bull state.
To further enforce economic restrictions on this model, the following long-run trends are imposed on the model. Solving for the stationary distribution associated with we can compute the vector of unconditional state probabilities:
(2.1)
where and and . The long-run restrictions are
(2.2)
(2.3)
We perform posterior simulation with Gibbs sampling steps which reject any draws that violate the parameter restrictions, coupled with the simulation smoother of Chib(1996) to sample the latent state vector. Estimation follows from Maheuetal.(2012) with the same priors employed in our analyses. For the MCMC output simulation, consistent posterior moments or predictive density quantities can be computed and are detailed in Maheuetal.(2012). We collect 30,000 posterior draws for inference after dropping an initial 5000 draws for burn-in.
3. Data and results
Daily equity capital gains for 1885–1927 are from Schwert(1990). Equity data from 1928 to 2020 use the S&P500 index daily adjusted close reported by Yahoo Finance (ĜSPC symbol). Risk-free return data are from the U.S. Department of the Treasury. From these data, weekly continuously compounded returns, scaled by 100, are obtained for 1885–2020. Weekly returns are computed using Wednesday data and Thursday if Wednesday data are missing. The last observation is November 25, 2020. In the following we refer to the equity index returns as the S&P500 or the market. A matching weekly realized variance measure is computed as the sum of intra-week daily squared returns. Summary statistics for the weekly data are reported in Table1.
Table 1
Weekly return statistics.
N | Mean | Mean () | Skewness | Excess kurtosis | |
---|---|---|---|---|---|
7064 | 0.125 | 1.938 | −0.565 | 8.007 |
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Although, in what follows, our model has been applied to weekly returns associated with a broad equity market index, it could also be applied to other asset classes at the individual security or portfolio level. For example, the latter could include balanced equity - fixed income portfolios and risk-parity portfolios.
Full sample parameter estimates for the four states are found in Table2. Those estimates include posterior means and 0.95 probability density intervals for and associated with each of the four states. Below those estimates, we report a pseudo Sharpe ratio, for each state for . This measures the expected return adjusted for risk assuming a zero risk-free rate for each state. The parameter estimates are broadly similar to those in Maheuetal.(2012). The average return in the bear states, is more negative than in the bull correction phases of the bull regimes. Analogously, the upward trend for returns in bull states (0.52) is stronger than 0.23 in the bear rally phase of the bear regime. Combining the return estimates with state volatilities, the Sharpe ratios are consistent with economic intuition, ranked from highest to lowest in the bull, bear rally, bull correction and bear states respectively. Note that the 0.95 probability density intervals do not overlap for these estimates. For example, the 0.95 probability density interval (0.06,0.13) associated with the pseudo Sharpe ratio for the bear rally (state 2), is very different than that (0.35,0.65) for the bull market (state 4). These estimates highlight the fact that a distinction between a bear rally and a bull state is worth modeling.
Table 2
Posterior estimates.
Mean | 95% DI | |
---|---|---|
bear | −0.94 | (−1.09, −0.79) |
bear rally | 0.23 | (0.14, 0.32) |
bull correction | −0.11 | (−0.21, −0.02) |
bull | 0.52 | (0.42, 0.64) |
5.60 | (5.21, 6.03) | |
2.44 | (2.27, 2.61) | |
1.85 | (1.69, 2.04) | |
1.09 | (0.97, 1.21) | |
−0.17 | (−0.20, −0.14) | |
0.10 | (0.06, 0.13) | |
−0.06 | (−0.12, −0.01) | |
0.49 | (0.35, 0.65) |
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The posterior means of the state transition matrix indicate persistence of bull and bear regimes in that states within the bear regime () and those in the bull regime () are likely to cluster together so that moves between regimes will be infrequent. Interestingly, a bull correction state is much more likely (0.097) to transition back to the bull state than to move to a bear state (0.013); whereas, a bear rally state is more likely (0.019) to transition to the bull state than it is to fall back to a bear state (0.013). Furthermore, regime changes are almost always transitions from a bear rally or from a bull correction. Directly jumping from a bull state to a bear state or vice versa is very rare.
Table3 reports the unconditional probabilities associated with the four states from which one can compute the unconditional probabilities for the regimes. For example, the bull market regime has a long-run probability of 0.672. Using the formulae in Eq.(2.2) and Eq.(2.3), one can compute that the associated long-run mean of weekly returns in bull regimes as 0.186; whereas, that for bear regimes is −0.069.
Table 3
Unconditional state probabilities.
Mean | |
---|---|
bear | 0.084 |
bear rally | 0.245 |
bull correction | 0.356 |
bull | 0.316 |
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3.1. Before COVID-19
Fig.1 displays the cumulative log-return and realized volatility (top), the probability of a bull regime (middle), and the probabilities of the 4 individual states (bottom) during 2019, the year before the COVID-19 pandemic. Recall that the bull regime combines the probabilities associated with the bull market and bull correction states. Very early in 2019, the market moved from a bear rally state into the bull state. Throughout 2019 the model decisively identifies a bull regime with fluctuations between the bull state and bull corrections.
Fig. 1
Estimates for 2019.
3.2. After COVID-19
Fig.2 reports the same information as Fig.1 but for the year 2020 during which COVID-19 erupted. The year began with strong evidence of a bull market, although with the probability of a bull correction building. The week of January 29 began a sequence of large negative drops in the market leading to increasing evidence of a transition from a bull market to a bear market. By February 26, the market had transitioned decisively from the bull correction state to the bear market state. By April the probability of the bear market state declined until April 22 revealed a transition to a bear rally state. At the end of our estimation window (November 25, 2020), the probability of a bear rally state is still very high at 0.824. This is because, even with a visible upward trend in the index since April, the market volatility is still too high and variable to be consistent with a typical bull regime. This conclusion is supported, in part, by the observation that the weekly realized volatility (red line in the top panel) is variable, with spikes (for example, during September and October 2020), and higher on average in comparison to its historical average value 1.94 from Table1.
Fig. 2
Estimates for 2020.
3.3. Forecasts
Although one can date stock market cycles after the fact with smoothed probability estimates, it is much more difficult to forecast changes out-of-sample. In this section, we report results for which the model has been estimated at each point using all past data to produce a forecast of the state one week ahead .
Fig.3 uses these model forecasts of the state probabilities one week ahead to generate out-of-sample regime forecasts. This figure compares the out-of-sample forecast of the market regime probabilities with the full-sample (smoothed) probability estimates. It shows a relatively accurate week-by-week classification of regimes that was available in real-time to an investor using the model to forecast stock market cycles.
Fig. 3
Out-of-sample: one-week-ahead regime probability forecasts.
One challenging period for the model forecasts was late August to early September. From July to September the index displayed a strong positive trend, and the model forecasts allocated a nontrivial probability to a bull regime. In hindsight, this episode is precisely identified as a bear regime using the smoothed estimates.
The breakdown of the regime forecasts into the constituent state forecasts one week ahead in seen in Fig.4. Given this additional disaggregation, there is more deviation between the state forecasts and smoothed estimates, but overall there is a strong correspondence between the two. Note again that, with the exception of the July to September period, the forecasts assign the highest probability to the bear rally state rather than a bull state.
Fig. 4
Out-of-sample: one-week-ahead state probability forecasts.
The COVID-19 period has had an important impact on several features of the return distribution. For example, Fig.5displays the one-week-ahead predictive density generated by the model. From the middle of March 2020 there was a dramatic impact that flattened the return distribution for the rest of the year, along with more subtle changes in the location. This implied a sharp increase in risk associated with holding the S&P500 portfolio.
Fig. 5
Out-of-sample: one-week-ahead predictive densities. The X-axis is a grid of possible return values. The Y-axix is the time. The Z-axis is the probability density function values.
Fig.6 illustrates the one-week-ahead predictive Sharpe ratios defined as . The flattened density of returns in March is accompanied by a sudden drop of the predictive Sharpe ratio. This ratio becomes positive in June and continues to improve over the summer months until a decline in August. For the remainder of our sample, the predictive Sharpe ratio never does attain the values at the start of 2020 before COVID-19 struck.
Fig. 6
One-week-ahead predictive sharpe ratios. The predictive Sharpe ratio is defined as the ratio of predictive mean and standard deviation, .
To illustrate another way, these changes in risk can be clearly seen from Value-at-Risk levels estimated for our model and illustrated in Fig.7. The dashed Value-at-Risk levels obtained from an assumption that returns are normally distributed would significantly understate risk as compared to those levels implied by our model that incorporates a mixture of four state distributions.
Fig. 7
Out-of-sample: one-week-ahead value-at-risk forecasts.
3.4. Market-timing investment strategies
In this section, we consider some simple market-timing strategies that exploit the forecasts from our model. All of the investment strategies are based on the out-of-sample predictions of states and regimes; and the simple maxim to buy low and sell high. This can take several forms such as selling at the end of a bull market and buying at the end of a bear market. However, our MS4 model provides much more detailed and useful information. For example, the states that identify increasing prices are state 4 (bull state) and the riskier state 2 (bear rally). Holding the market during periods for which forecasts assign significant probability for these states could be fruitful.
In each case, the investor can buy the market and continue to hold the market; or sell and hold a risk-free asset. No short selling is allowed. Here are the market-timing strategies we consider, in which and are threshold probabilities associated with investment decisions for the regime versus the state-directed strategies.
1.
Strategy B: buy or continue to hold the market when
and otherwise sell.
2.
Strategy S: buy or continue to hold the market if
or and otherwise sell.
The first strategy B only uses the aggregate regime information associated with the probability forecasts for . The second strategy exploits the positive expected return in both the bear rally () and bull states (). We focus on using one cutoff value for both of those states but this could be generalized and we present some evidence of this below.
Table4 shows the results for the out-of-sample investment period from January 2, 2020 to November 25, 2020. The total number of transactions required to implement the various strategies for that period are reported in the last column. Since at most 3 or 4 transactions would be required, and transaction costs are very low (for example, 0.3 cents per share on NYSE Arca for the SPDR security that tracks the S&P500 index), the reported returns exclude transaction costs which would be negligible.
Table 4
Investment returns: 2020.
Return | Sharpe ratio | Number of transactions | |
---|---|---|---|
StrategyBa: | −0.009 | −0.048 | 3 |
StrategySb: | 0.220 | 1.203 | 4 |
Buy-and-hold | 0.131 | 0.566 | 0 |
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The returns are annualized. Transaction costs are 0.3 cents per share.
aBuy if and sell otherwise.
bBuy if or . Sell otherwise.
The annualized return is 13.1% with a Sharpe Ratio of 0.566 if the investor buys in the first week of 2020 and holds the position until the last week of our data sample (last Wednesday of November). This compares with a hypothetical buy and hold return of 6.46% if an investor held the index for our entire sample from 1885.
As reported in Table4, using the market-timing strategy B does not perform well in 2020 even with a range of alternative cutoff values, for buying and selling. However, exploiting the additional information about states provided by the MS4 model (as in strategy S), yields very positive results. For example, with the market-timing strategy generates a 22% annualized return with a Sharpe Ratio of 1.203. This investment strategy performs significantly better than the buy and hold strategy.
Fig.8 displays returns for strategy S as a function of different values of . For most values of a positive return is achieved with the best performance for values less than 0.65. Given the relatively short sample associated with the first eleven months of 2020, it is clear that being in the market early (values of closer to 0.5) to capture bear market rallies is fruitful. As increases, the length of time in the market decreases so the annualized return decreases and is more variable given the effect of the heightened market volatility during the bear market rally.
Fig. 8
Market-timing returns as a function of signal threshold.
The line is the return in 2020 till the end of the sample period as a function of for the investment strategy S: buy or continue to hold the market if or and otherwise sell.
One can achieve even larger gains by separating the thresholds in strategy S, that is, relaxing the constraint of a common associated with both bear rally and bull states. For example, fixing at 0.5 for the bear rally state (buying or continuing to hold the market if ); while allowing to vary for the bull state (buy if and sell otherwise), generates annualized returns as high as 28%. This is further evidence that the added value associated with using information inherent in the state probabilities and predictive state densities for investment strategies is quite robust.
4. Robustness results
We have implemented a number of model comparisons to evaluate the robustness of our results. Table5reports log-predictive likelihood values for the 2020 data for several alternative models. Included is our proposed 4-state model (MS4), the MS4 model with Student-t innovations, the MS4 model with an unrestricted transition probability matrix, a MS2 model and a GARCH model. None of these specifications improve on our proposed MS4 model. We have reported more details about these model comparisons and further robustness tests in our online Appendix.
Table 5
Log-predictive likelihood: 2020.
MS4 | MS4t | MS4 unrestricted | MS2 | GARCH | |
---|---|---|---|---|---|
−127.6 | −127.9 | −128.7 | −132.8 | −146.3 |
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5. Conclusion
This paper estimates and forecasts bull and bear markets and their component states during the COVID-19 pandemic. We document market phases prior to the pandemic period and analyze what has happened to the U.S. equity market in 2020. We find that the market moved from a bull state at the start of 2020 to a bull correction and quickly to a bear market on February 26. This bear market dominated until April 22 when the market transitioned to a bear rally, remaining in this phase until the current date (end of November, 2020).
We show the dramatic impact of the COVID-19 pandemic on the forecasts of the return distribution and how effective market-timing strategies can exploit model forecasts. Investment strategies which use the conventional bear and bull market regime classifications to forecast turning points do not outperform a buy and hold strategy during 2020. However, using the more detailed information inherent in our 4-state model, particularly the risk differences across states, contributes significantly to market-timing strategies. Using forecasts for market states to guide an investment strategy outperforms buy and hold by a wide margin, both in terms of annualized returns and Sharpe ratios.
Our online Appendix includes additional results, including long-horizon forecasts, investment returns associated with alternative thresholds for state probability forecasts, and additional details concerning a number of model comparisons which show our results to be robust.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Footnotes
☆We thank Marco Salerno, Xiaofei Zhao, Judy Mao and workshop participants at the Rotman School of Management, Deakin University, Shanghai University of Finance and Economics, Kobe University and the NBER-NSF SBIES conference for comments. Maheu and McCurdy thank SSHRC for financial support.
Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2021.102091
Appendix A. Supplementary materials
Supplementary Data S1:
Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/
Click here to view.(1008K, pdf)Supplementary Data S1
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