Bull and Bear Markets During the COVID-19 Pandemic
BBull and Bear Markets During the COVID-19 Pandemic
John M. Maheu ∗ Thomas H. McCurdy † Yong Song ‡ November 2020
Abstract
The COVID-19 pandemic has caused severe disruption to economic and financial ac-tivity worldwide. We assess what happened to the aggregate U.S. stock market duringthis period, including implications for both short and long-horizon investors. Using themodel of Maheu, McCurdy and Song (2012), we provide smoothed estimates and out-of-sample forecasts associated with stock market dynamics during the pandemic. Weidentify bull and bear market regimes including their bull correction and bear rally com-ponents, demonstrate the model’s performance in capturing periods of significant regimechange, and provide forecasts that improve risk management and investment decisions.The paper concludes with out-of-sample forecasts of market states one year ahead.Key Words: predictive density, long-horizon returns, Markov switching ∗ DeGroote School of Business, McMaster University, [email protected] † Rotman School of Management, University of Toronto, [email protected] ‡ Department of Economics, University of Melbourne, [email protected] a r X i v : . [ ec on . E M ] D ec Introduction
This paper dates and forecasts bull and bear markets for the COVID-19 pandemic periodbased on aggregate equity return data from 1885-2020. Using the model of Maheu et al.(2012) applied to weekly data from 1885-2020, we document where the market was and whathas happened to equity markets in 2020.There are several reasons for using the restricted 4-state Markov-Switching (MS) model ofMaheu et al. (2012). First, unlike ex post ex post dating methods are unable to distinguish be-tween bear rally and bull market states. Our approach does this, in part, due to probabilityestimates of risk differences across those states. We show the dramatic impact that theCOVID-19 pandemic has had on the return distribution and risk measures. The model pro-vides very accurate forecasts of turning points and these would have been available in real-timeto investors.Given the benefits of a full probability model of stock market phases, it is natural askwhether forecasts from this mixture-distribution model can improve investment and risk man-agement decisions. To this end, we define a pseudo Sharpe ratio to characterize in-sampleestimates of the state density parameters. We then extend this measure to an out-of-samplepredictive Sharpe ratio, derived from the predictive density of returns which is sensitive tothe forecasted market states. This measure can be useful in assessing the risk and return ofentering the market.Several market timing investments are explored. We show that simple timing rules di-2ecting when to exit and enter the market lead to improved investment decisions relative toa buy and hold strategy in 2020. These results are robust to different timing strategies andare a result of the precise turning points our model identifies and forecasts. Each of thesemarket timing strategies are in real time and would have been available to an investor usingthe model for forecasts.The paper concludes with long-horizon forecasts of state probabilities one year ahead. Ifthe effects of COVID-19 were to disappear today, then our model predicts months, and notweeks, till the stock market returns to normal times.Our paper is organized as follows. Sections 2 and 3 briefly review the structure andestimation of the Maheu et al. (2012) bull and bear market model; and Section 4 summarizesthe data. Section 5 reports the results for the pre and post COVID-19 periods. Notably,Section 5.3 provides forecasts based on one-week-ahead predictive densities and out-of-sampleforecasts of future states and the associated risk and return measures. Section 5.4 reportsmarket timing strategies that exploit those forecasts. Long-horizon forecasts are discussed inSection 5.5. Section 6 provides robustness results, comparing our model to competing modelsfrom several perspectives. Section 7 concludes and an Appendix provides additional resultsand model comparison details.
Define log-returns as r t , t = 1 , . . . , T and r t − = { r , . . . , r t − } . Consider the following 4-stateMarkov-Switching (MS4) model from Maheu et al. (2012) for returns, r t | s t ∼ N ( µ s t , σ s t ) p ij = p ( s t = j | s t − = i ) , i = 1 , ..., , j = 1 , ..., . in which s i , i = 1 , ..., , denote the latent states, parameterized as Normally distributed withmean µ s i and variance σ s i , and p i,j denote the state transition probabilities.The following restrictions and labels are imposed for identification purposes, µ < ,µ > ,µ < ,µ > . No restriction is imposed on { σ , . . . , σ } . Note that there are four states but we refer to twodistinct regimes by B t as B t = 1 if s t = 1 or 2 (bear regime) ,B t = 2 if s t = 3 or 4 (bull regime) . The transition matrix takes the following form. P = p p p p p p p p p p p p P we cancompute the vector of unconditional state probabilities: π = ( A (cid:48) A ) − A (cid:48) e (2.1)where A (cid:48) = [ P (cid:48) − I , ι ] and e (cid:48) = [0 , , , ,
1] and ι = [1 , , , (cid:48) . The long-run restrictions are E [ r t | bear regime , B t = 1] = π π + π µ + π π + π µ < E [ r t | bull regime , B t = 2] = π π + π µ + π π + π µ > . (2.3) We perform posterior simulation with Gibbs sampling steps which reject any draws that violatethe parameter restrictions, coupled with the simulation smoother of Chib (1996) to sample thelatent state vector. Estimation follows exactly from Maheu et al. (2012) with the same priorsemployed in our analyses. For the MCMC output simulation, consistent posterior momentsor predictive density quantities can be computed and are detailed in Maheu et al. (2012). Wecollect 30 ,
000 posterior draws for inference after dropping an initial 5 ,
000 draws for burn-in.
Daily equity capital gains for 1885 - 1927 are from Schwert (1990). Equity data from 1928 -2020 use the S&P500 index daily adjusted close reported by Yahoo Finance ( ˆGSPC symbol).Risk-free return data are from the U.S. Department of the Treasury. From these data, weeklycontinuously compounded returns, scaled by 100, are obtained for 1885 - 2020. Weekly returnsare computed using Wednesday data and Thursday if Wednesday data is missing. The lastobservation is November 25, 2020. In the following we refer to the equity index returns asthe S&P500 or the market. A matching weekly realized variance measure RV t , is computedas the sum of intra-week daily squared returns. Summary statistics for the weekly data arereported in Table 1. Full sample parameter estimates for the four states are found in Table 2. Those estimatesinclude posterior means and 0.95 probability density intervals for µ i and σ i associated witheach of the four states. Below those estimates, we report a pseudo Sharpe ratio, µ i /σ i , for eachstate i for i = 1 , , ,
4. This measures the expected return adjusted for risk assuming a zerorisk-free rate for each state. The parameter estimates are broadly similar to those in Maheuet al. (2012). The average return in the bear states, − .
94, is more negative than − .
11 in4he bull correction phases of the bull regimes. Analogously, the upward trend for returns inbull states (0 .
52) is stronger than 0 .
23 in the bear rally phase of the bear regime. Combiningthe return estimates with state volatilities, the Sharpe ratios also make sense, ranked fromhighest to lowest in the bull, bear rally, bull correction and bear states respectively.The posterior means of the state transition matrix P indicate persistence of bull and bearregimes in that states within the bear regime ( s t = 1 ,
2) and those in the bull regime ( s t = 3 , . . . . . . − . Figure 1 displays the cumulative log-return and realized volatility (top), the probability of abull regime P ( B t = 2 | r T ) (middle), and the probabilities of the 4 individual states (bottom)during 2019, the year before the COVID-19 pandemic. Very early in 2019, the market movedfrom a bear rally state into the bull state. Throughout 2019 the model decisively identifies abull regime with fluctuations between the bull state and bull corrections. Figure 2 reports the same information as Figure 1 but for the year 2020 during which COVID-19 erupted. The year began with strong evidence of a bull market, although with the proba-bility of a bull correction building. The week of January 29 began a sequence of large negativedrops in the market leading to increasing evidence of a transition from a bull market to a bearmarket. By February 26, the market had transitioned decisively from the bull correction stateto the bear market state. By April the probability of the bear market state declined until April22 revealed a transition to a bear rally state. At the end of our estimation window (November25, 2020), the probability of a bear rally state is still very high at 0 . .
94 from Table 1.
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 resultsfor which the model has been estimated at each point t using all past data 1 , . . . , t to producea forecast of the state one week ahead t + 1. 5igure 3 uses these model forecasts of the state probabilities one week ahead to generateout-of-sample regime forecasts. This figure compares the out-of-sample forecast of the marketregime probabilities with the full-sample (smoothed) probability estimates. It shows a rel-atively accurate week-by-week classification of regimes that was available in real-time to aninvestor using the model to forecast stock market cycles.One challenging period for the model forecasts was late August to early September. FromJuly to September the index displayed a strong positive trend, and the model forecasts allo-cated a nontrivial probability to a bull regime. In hindsight, this episode is precisely identifiedas a bear regime using the smoothed estimates.The breakdown of the regime forecasts into the constituent state forecasts one week aheadin seen in Figure 4. Given this additional disaggregation, there is more deviation between thestate forecasts and smoothed estimates, but overall there is a strong correspondence betweenthe two. Note again that, with the exception of the July to September period, the forecastsassign the highest probability to the bear rally state rather than a bull state.The COVID-19 period has had an important impact on several features of the returndistribution. For example, Figure 5 displays the one-week-ahead predictive density generatedby the model. From the middle of March 2020 there was a dramatic impact that flattened thereturn 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.Figure 6 illustrates the one-week-ahead predictive Sharpe ratios defined as E ( r t | r t − ) √ Var( r t | r t − ) .The flattened density of returns in March is accompanied by a sudden drop of the predictiveSharpe ratio. This ratio becomes positive in June and continues to improve over the summermonths until a decline in August. For the remainder of our sample, the predictive Sharperatio never does attain the values at the start of 2020 before COVID-19 struck.To illustrate another way, these changes in risk can be clearly seen from Value-at-Risk levelsestimated for our model and illustrated in Figure 7. The dashed Value-at-Risk levels obtainedfrom an assumption that returns are normally distributed would significantly understate riskas compared to those levels implied by our model that incorporates a mixture of four statedistributions. In this section, we consider some simple market timing strategies that exploit the forecastsfrom our model. All of the investment strategies are based on the out-of-sample predictionsof states and regimes; and the simple maxim to buy low and sell high. This can take severalforms 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 (bearrally). Holding the market during periods for which forecasts assign significant probability forthese states could be fruitful.In each case, the investor can buy the market and continue to hold the market; or sell andhold a risk-free asset. No short selling is allowed. Here are the market timing strategies weconsider.1. Strategy B: buy or continue to hold the market when P ( B t = 2 | r t − ) > τ B and otherwise sell.2. Strategy S: buy or continue to hold the market if P ( s t = 2 | r t − ) > τ S or P ( s t = 4 | r t − ) > τ S and otherwise sell.6he first strategy B only uses the aggregate regime information associated with the probabilityforecasts for B t . The second strategy exploits the positive expected return in both the bearrally ( s t = 2) and bull states ( s t = 4). We focus on using one cutoff value τ S for both of thosestates but this could be generalized and we present some evidence of this below.Table 4 shows the investment results for 2020. All values are annualized. The annualizedreturn is 13.1% with a Sharpe Ratio of 0.566 if the investor buys in the first week of 2020 andholds the position until the last week of our data sample (last Wednesday of November). Thiscompares with a hypothetical buy and hold return of 6.46% if an investor held the index forour entire sample from 1885.As reported in Table 4, using the market timing strategy B does not perform well in2020 even with a range of alternative cutoff values, τ B , for buying and selling. However,exploiting the additional information about states provided by the MS4 model (as in strategyS), yields positive results. For example, with τ S = 0 .
5, the market timing strategy generatesa 22% annualized return with a Sharpe Ratio of 1.203. This investment strategy performssignificantly better than the buy and hold strategy.Figure 8 displays returns for strategy S as a function of different values of τ S . For mostvalues of τ S ∈ (0 . , .
9) a positive return is achieved with the best performance for values lessthan 0 . τ S associated with both bear rally and bull states. This figure fixes τ S at 0 . P ( s t = 2 | r t − ) > .
5; while allowing τ S to vary for the bull state, that is, buy if P ( s t = 4 | r t − ) > τ S and sell otherwise. Theresults show that it is possible to achieve even larger gains by separating the thresholds instrategy S. This is further evidence that the added value associated with using informationinherent in the state probabilities and predictive state densities for investment strategies isquite robust. We have focused on one-week-ahead predictions from the model and how they might be usedfor investment decisions. However, the model estimates have implications for the long-runbehaviour of stock market returns. Being stationary, our MS4 model implies that any long-horizon predictions converge to the implied stationary distribution. Figure 10 and 11 reportone-year-ahead probability forecasts for states and associated regimes looking forward fromour most recent observations at the end of November 2020. What is notable is that thetransition back to normal times in the form of the long-run values of the states is slow. Evenif the effects of COVID-19 were to disappear today, our model predicts months, and not weeks,until the stock market returns to normal times.
There are a number of alternative models and comparisons we have conducted. The detailsare collected in the Appendix. Here we highlight a few of these results. Table 5 reports log-predictive likelihood values for the 2020 data for several alternative models. Included is ourproposed 4-state model (MS4), an unrestricted version of MS4, a MS4 model with Student-tinnovations, a GARCH model and a MS2 model. None of these specifications improve onour proposed MS4 model. In the Appendix, we show how these alternative models date theturning points in 2020 and generally see a close correspondence with the MS4 model. One7otable exception is a 2-state Markov switching model (MS2). The MS2 classifies the summermonths as a bull state while our MS4 identifies this period as a bear market rally. This couldbe considered as a drawback of the 2-state model in that it does not allow for intra-regimedynamics.
This paper estimates and forecasts bull and bear markets during the COVID-19 pandemic.Using the model of Maheu et al. (2012), we document where the market was and what hashappened to U.S. equity markets in 2020. We find that the market moved from a bull state atthe start of 2020 to a bull correction and quickly to a bear market on February 26. This bearmarket dominated until April 22 when the market transitioned to a bear rally, remaining inthis phase until the current date. We show the dramatic impact of the COVID-19 pandemic onthe forecasts of the return distribution and how effective market timing strategies can exploitmodel forecasts. Long-horizon forecasts from the model predict months, and not weeks, untilthe stock market returns to normal times.
Figure 12 shows the smoothed standard deviations from the MS4 versus a GARCH(1,1) model.The MS4 model picks up the volatility surge quickly at the beginning of 2020 and doesnot have the long tapering-off period associated with the GARCH model. The implicationsfor Value-at-Risk are illustrated in Figure 13. The quick adjustment of the Value-at-Riskestimates generated by the MS4 model match the high potential losses after March 2020.This adjustment is substantially delayed and less flexible using a GARCH(1,1) model. Thelatter is due to the single exponential decay parameter in the GARCH model which invokeshigher persistence and less flexible adjustment to shocks.
Figure 14 shows the probability of a bull market regime inferred from our 4-state model versusa simple 2-state model for the period 2019-2020. There is one striking difference between thesetwo models in the period from April to September 2020. While the MS4 model classifies thisperiod as a bear regime (in a bear rally state), the MS2 model signals a bull regime. Thisdifference is due to the 2-state model not having the structure to incorporate intra-regimedynamics. The September and October evidence from the 2-state model had the bull marketprobability plunge to the bottom with small humps, while the 4-state model always estimatedthat the bull regime had not been confirmed yet.
Figure 16 shows the bull regime probability (middle) and the probability of each individualstate (bottom) associated with assuming a Student-t distribution for each state. There is noqualitative difference from our proposed MS4 model in Figure 2.8 .4 Predictive Likelihood Comparisons
Figure 17 illustrates differences in the cumulative predictive likelihood associated with alter-native models for year 2020, using the GARCH(1,1) model as the benchmark value. Thevalues at time t are log predictive Bayes factor as log (cid:104) p ( r t | r before 2020 , M) p ( r t | r before 2020 , GARCH(1,1)) (cid:105) , where M indicates various alternative models including the MS4. According to Kass & Raftery(1995), a value p ( r | M ) p ( r | M ) that is larger than 5 indicates strong data evidence to support model M . The 2020 data clearly favours the MS4 model against the GARCH(1,1). In addition,having a Student-t distribution for each state does not provide any additional value. P Economic intuition suggests a zero restriction on p , p , p and p . Without that restriction,the 4-state model can still identify key dates of regime change in 2020 as shown in Figure 18.However, a closer look at Figure 19, showing the bull regime and 4-state probabilities fromboth MS4 and MS4 with unrestricted P , reveals that our proposed MS4 model, which has arestricted transition probability structure, has less uncertainty between a bull market state anda bear rally state. From the figure, during February and July 2020 the MS4 with unrestricted P has higher bull state probability than the MS4, because the unrestricted transition matrixhas no structure to prevent transition from the bull state to a bear rally state. As a result,at the aggregate level, (top panel of Figure 19), the MS4 has sharper identification of regimesthan without the restriction on P . Further evidence can be seen from Figure 17, whichreveals that the out-of-sample performance of the unrestricted MS4 model is dominated byour proposed MS4 model.Interestingly, the MS4 model can be interpreted as a two-level hierarchical hidden Markovmodel (Fine et al. (1998)), in which each level has two states with simple restrictions. Thefirst restriction is on the vertical down probability to incorporate the zero restrictions onthe transition matrix. The second restriction is the mean restriction for state-identificationpurposes. These restrictions add value to bull and bear regime identification, and proves theusefulness of a priori restrictions motivated by economic intuition.9 eferences Chib, S. (1996), ‘Calculating posterior distributions and modal estimates in markov mixturemodels’,
Journal of Econometrics , 79–97.Fine, S., Singer, Y. & Tishby, N. (1998), ‘The hierarchical hidden markov model: Analysisand applications’, Machine learning (1), 41–62.Kass, R. E. & Raftery, A. E. (1995), ‘Bayes factors’, Journal of the American StatisticalAssociation (420), 773–795.Lunde, A. & Timmermann, A. G. (2004), ‘Duration dependence in stock prices: An analysisof bull and bear markets’, Journal of Business & Economic Statistics (3), 253–273.Maheu, J. M., McCurdy, T. H. & Song, Y. (2012), ‘Components of bull and bear markets: Bullcorrections and bear rallies’, Journal of Business & Economic Statistics (3), 391–403.Pagan, A. R. & Sossounov, K. A. (2003), ‘A simple framework for analysing bull and bearmarkets’, Journal of Applied Econometrics (1), 23–46.Schwert, G. W. (1990), ‘Indexes of u.s. stock prices from 1802 to 1987’, Journal of Business (3), 399–426. 10able 1: Weekly Return StatisticsN Mean Mean( RV . ) Skewness Excess Kurtosis7064 0.125 1.938 -0.565 8.007 Table 2: Posterior Estimates mean 95% DIbear µ -0.94 (-1.09, -0.79)bear rally µ µ -0.11 (-0.21, -0.02)bull µ σ σ σ σ µ /σ -0.17 (-0.20, -0.14) µ /σ µ /σ -0.06 (-0.12, -0.01) µ /σ P = .
906 0 .
092 0 0 . .
013 0 .
968 0 0 . .
013 0 0 .
891 0 . .
001 0 0 .
122 0 . Table 3: Unconditional State Probabilities meanbear π π π π Return Sharpe RatioStrategy B a : τ B = 0 . b : τ S = 0 . a Buy if P ( B t = 2 | r t − ) > τ B and sell otherwise. b Buy if P ( s t = 2 | r t − ) > τ S or P ( s t = 4 | r t − ) > τ S . Sell otherwise.. Table 5: Log-Predictive Likelihood in 2020
MS4 MS4 Unrestricted MS4t GARCH MS2-127.6 -128.7 -127.9 -146.3 -132.8
Cumulative Log Returnsqrt of RV
Prob of Bull bearbear rallycorrectionbull
Cumulative Log Returnsqrt of RV
Prob of Bull bearbear rallycorrectionbullFeb 26 April 22 Bear Regime (smoothed)Bear Regime (forecast)
Bull Regime (smoothed)Bull Regime (forecast)
Figure 3: Out-of-sample: One-week-ahead Regime Probability Forecasts15 bear (smoothed)bear (forecast) bear rally (smoothed)bear rally (forecast) bull correction (smoothed)bull correction (forecast) bull (smoothed)bull (forecast)
Figure 4: Out-of-sample: One-week-ahead State Probability Forecasts16igure 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 probabilitydensity function values.
Figure 6: One-week-ahead Predictive Sharpe Ratios
Cumulative Log ReturnPredictive Sharpe Ratio
The Predictive Sharpe ratio is defined as the ratio of predictive mean and standard deviation, E ( r t | r t − ) √ Var( r t | r t − ) . data1%5%10%Normal 1%Normal 5%Normal 10% Figure 7: Out-of-sample: One-week-ahead Value-at-Risk Forecasts18igure 8: Market Timing Returns as a Function of Signal Threshold
Bull and Bear Rally State Threshold -0.0500.050.10.150.20.25 A nnua li s ed R e t u r n The blue line is the return in 2020 till the end of the sample period as a function of τ S for theinvestment strategy S: buy or continue to hold the market if P ( s t = 2 | r t − ) > τ S or P ( s t =4 | r t − ) > τ S and otherwise sell. Bull State Threshold: s A nnua li s ed R e t u r n The blue line is the return in 2020 till the end of the sample period as a function of τ S for investmentstrategy S: buy or continue to hold the market if P ( s t = 2 | r t − ) > . P ( s t = 4 | r t − ) > τ S andotherwise sell. Bear StateForecast
Bear RallyForecast
Bull CorrectionForecast
Bull StateForecast
Figure 11: Regime Estimates for 2020 and One-Year Forecasts
Bull RegimeForecast GARCH(1,1)MS-4
Figure 12: Standard Deviations from GARCH(1,1) and MS4 in 2020. data1%5%10%GARCH 1%GARCH 5%GARCH 10%
Figure 13: Out-of-sample: One-week-ahead Value-at-Risk Forecasts22
MS2 BullMS4 Bull
Figure 14: Bull Regime Probability: MS2 vs MS4 Model
MS2 BullMS4 BearMS4 Bear RallyMS4 Bull CorrectionMS4 Bull
Figure 15: Bull Regime Probability: MS2 & State Probabilities: MS423igure 16: State Probability Estimates: MS4t -40-30-20-1001020 0510152025
Cumulative Log Returnsqrt of RV
Bull Regime Probability bearbear rallycorrectionbull
MS4 v.s. GARCH(1,1)MS4t v.s. GARCH(1,1)MS4 Unres v.s. GARCH(1,1)MS2 v.s. GARCH(1,1) P Cumulative Log Return
Prob of Bull bearbear rallycorrectionbullFeb 26 April 22 P Restrictions
Probability of Bull
MS4MS4 unrestricted P
MS4 bearbear rallycorrectionbull
MS4 unrestricted P bearbear rallycorrectionbullbearbear rallycorrectionbull