US Equity Risk Premiums during the COVID-19 Pandemic
aa r X i v : . [ q -f i n . GN ] A p r US Equity Risk Premiums during the COVID-19 Pandemic
Alan L. Lewis ∗ April 30, 2020
Abstract
We study equity risk premiums in the United States during the COVID-19 pandemic.
COVID-19 is shorthand for the novel corona virus disease with origins in Wuhan, China in thefall of 2019. It spread in 2020 to become a global pandemic. As of this writing (late April,2020), there have been approximately 3 million identified cases worldwide, and 200,000 deaths.Of those, the US totals are approximately 946,000 cases and 53,000 deaths . Fig. 15 shows thedaily US development to date.Worldwide, governments have urged or mandated shelter-in-place policies, and mandatedshutdowns of most ‘non-essential’ business. In the US this approach has achieved the immediategoal of buying time for hospitals to prepare for future COVID-19 patients and not be over-whelmed by current ones.But, broad lock-downs are unsustainable for more than a few months. Indeed, many USstates are preparing to carefully open up in mid-May and beyond. The lock-downs have resultedin enormous economic stress. In the US: 27 million lost jobs and counting at this juncture.These health and economic issues have been a key driver of recent extreme volatility in financialmarkets. Fig. 16 shows the S&P500 index levels and returns during 2020. For perspective, notethat in normal times, a ±
3% move would merit a mention and explanation on the nightly news.Here, we study an interesting and important financial question: what new return expecta-tions have accompanied all this increased volatility? At first glance, expectations might seemimpossible to discern. Indeed, getting an answer is rather subtle and requires both the optionsmarket and estimates of risk aversion.Risk averse investors hold equities only if they feel they will be fairly compensated for theperceived risks. Fair compensation, in the aggregate, is called the “equity risk premium” (ERP).More carefully, the ERP is the market’s forward-looking, expected rate of return – after sub-tracting an available riskless rate (say a US Treasury rate). Because of the subtraction, it’s calledan excess return. Thus, the ERP can be thought of as the “required (excess) rate of return”,conditioned on what the market knows, to keep all stocks held. Another way to say it: whatexcess return is needed to clear the equity markets?Expectations both require a horizon and change with the passage of time: the market learnsnew things. Thus, each day t , we have ERP t,T where the T are various time horizons. Ourhorizons range from one day ahead to just under 3 years. Fixing t , a graph of ERP t,T vs. T is an ERP term structure plot. Unlike a familiar interest rate term structure (a yield curve),the ERP term structure is not directly visible and needs to be estimated. Like a yield curve,regardless of the time to the horizon, we always quote ERP’s as annual percentage rates . ∗ Newport Beach, California, USA; email: alewis@financepress.com source:
1e study the effect of the pandemic events on the ERP term structures in the United States,from late January through mid-April 2020. Estimates are found using the methods recentlydeveloped in (Lewis, 2019). We take the S&P500 Index as a broad equity market proxy. Then,in brief, daily S&P 500 index option quotes are combined with estimates of a risk-aversionparameter κ to develop the ERP term structures. We briefly review how that works in Sec. 2below. Further computational details may be found in Appendix 2 here and the Lewis article. What have I learned?
In Sec. 3, one finds the detailed results. My approach is to presentbrief key-event timelines, show corresponding ERP’s, and supply some brief commentary. ERPplots come with a central estimate (dotted) surrounded by an uncertainty interval in gray.Unsurprisingly, there is a strong general association between volatility (VIX levels, for exam-ple, as seen in Fig. 17) and the ERP’s. It is well-known that when the market gets very stressedby something, VIX rises and the whole VIX term structure ‘inverts’. In other words, short-term(risk-neutral) volatility expectations rise above long-term ones. Correspondingly, the ERP termstructure also strongly inverts. During the pandemic, this ERP inversion first happened circaFeb 24, 2020; it remained inverted through the end of the study data on Apr 15, 2020.Qualitatively, based upon my earlier experience with the model in (Lewis, 2019), I expectedto see these strong inversions. But, I was surprised, quantitatively, by the extraordinary heightsreached by the short-term ERP’s during mid-March 2020. For example, the March 12 termstructure (Fig. 10) shows the short-dated (one day horizon) ERP reaching a mid-point estimateof approximately 540% per year! For comparison, long-run ERP estimates typically lie in the3-6% per year range. Indeed, 3-6% characterized the pandemic-period US market through mid-February. ERP levels as high as the March 12 ones may be record-setting. To say for sure requiresapplying the current methodology to option quotes during the 2008-2009 Financial Crisis – thishas not yet been done.Finally, some related analysis and updates will be provided in follow-up research to be postedonline. See the Outlook at the end for how to locate that.
In this section we give a brief, technical explanation of how the ERP’s are computed. If you’renot interested in these details, feel free to skip ahead to the results in Sec. 3.With E t denoting a (real-world) expectation conditional on date- t information I t – broadlyspeaking: the “state of the world” – we define:ERP t,T = E t (cid:2) R et,T (cid:3) − R ft,T = E t h R et,T − R ft,T i , where at time t : (1) • R et,T is a future random total return on the equity market from t to T , and • R ft,T is a time- t observable risk-free return (using US Treasury instruments).Returns in (1) are simple total returns: R et,T = ( ¯ S T − ¯ S t ) / ¯ S t , where ¯ S is a total-return indexincorporating reinvested dividends. (Without a bar, S t is the price series without dividends).Call R et,T − R ft,T the excess total return . Like interest rates, we’ll always give estimated ERP’s onan annualized percentage basis . For those, we multiply the ERP calculated from (1) by 100 × f ann ,where the annualization factor f ann = 1 / ( T − t ), with time measured in years.With logarithmic variables ¯ X T = log ¯ S T / ¯ S t , and corresponding probability density p ¯ X T ( x ),(1) is equivalent to ERP t,T = Z e x p ¯ X T ( x ) dx − (1 + R ft,T ) . (2)2ow do we find p ¯ X T ( x )? It turns out that, from the options market (specifically, options on theSPX index), we can estimate a closely related probability density q ¯ X T ( x ), the so-called “risk-neutral” density. A simple transformation between them exists under the additional assumptionthat investors in the aggregate can be characterized as having a constant measure of risk-aversion,which we write as κ . Pronounced “kappa”, it’s a single number, which I have estimated fromhistorical SPX returns as κ = 3 ± .
5. More carefully, it’s called the Coefficient of Relative RiskAversion and has been heavily studied. Indeed, to get from q to p , one just applies a simpleexponential transformation p ¯ X T ( x ) = e κx q ¯ X T ( x ) R e κx q ¯ X T ( x ) dx . (3)OK – so how do we find q ¯ X T ( x )? My approach estimates q by parameterizing it as a Gaussianmixture model . Specifically, I take q X T ( x ) = N X i =1 w i e − ( x − µ i τ ) / (2 σ i τ ) p πσ i τ , (4)where τ = T − t , and N is a small integer (5 in my fits). The fitted parameters are N positiveweights, { w i } , and 2 N drifts and volatilities, { µ i , σ i } . After a normalization and martingalecondition, this leaves 3 N − each ( t, T ) pair associated to a trade date andan option expiration. Free parameters are adjusted to fit option quotes: minimizing an objectivefunction given in (Lewis, 2019). Finally, after algebra, now find – on an annualized percent basis:ERP ( ann %) t,T ( κ ) = 100 T − t × ( e δ t,T τ N X i =1 ˜ w i e α i +( κ + ) v i ! − e r t,T τ ) , using τ = T − t, α i = µ i τ, v i = σ i τ,γ i = κ α i + κ v i , and ˜ w i = w i e γ i / N X i =1 w i e γ i . (5)New parameters which have just appeared are r t,T and δ t,T : the cost-of-carry parameters. Theycorrespond to the continuously compounded riskless rate and dividend yield associated to optionexpiration T . For example, if C t,T and P t,T denote call and put prices with strike K , we havethe model-independent, put-call parity relation: C t,T − P t,T = S t e − δ t,T τ − K e − r t,T τ = e − r t,T τ ( F t,T − K ) , (6)where F t,T is an (option-implied) forward price.Once all the parameters are estimated, we evaluate (5) with κ = 3 to get the mid-pointestimates (dotted lines) for all the ERP charts in Sec. 3. Similarly, we use κ = 2 . κ = 3 . differ from the discussion in(Lewis, 2019) or would be unresolved if you simply turned to that reference. My estimates are in line with one of the most classical of these studies (Friend & Blume, 1975). See (Lewis,2019) for further commentary. Timelines with Equity Risk Premium Term Structures
Fig. 1 shows some early events in the development of the pandemic. Figure 1:
Timeline 1
Wuhan placed under quarantineFirst case outside China reported in ThailandFirst death in China recordedIdentification of new virus: SARS - CoV - -
09 Dec -
23 Jan -
06 Jan - Wuhan, China was placed under quarantine on Jan 23, 2020, with rail and services suspended.Two days earlier, on Jan 21, the first US case was identified in Washington state – a man in his30’s who had returned from a trip to Wuhan. Fig 2 shows the estimated ERP term structure.It’s within typical long-run ERP estimates of 3-6% per year: the US equity market was notconcerned. Figure 2:
US ERP term structure ( yrs ) ( Ann Percent ) Jan 22, 2020 Timeline events are drawn from the World Economic Forum ( ), the Wall StreetJournal of Mar 21-22 2020, Zack’s Equity Research (Stock market news), and misc. online news sources. .2 First death in Europe Fig. 3 shows some next events in the development of the pandemic.Figure 3:
Timeline 2
First
European COVID -
10 death announced in France3600 pasengers are quarantined on Diamond Princess cruise shipFirst death outside China ( Philippines ) US restricts entry by foreign nationals with recent China travelWHO declares a Public Health EmergencyUS 1st confirmed case, WA stateJan -
27 Feb -
03 Feb -
10 Feb - Feb 14, 2020 marks the end of this segment with the announcement of the first European COVID-19 death, in France. Fig 4 shows the estimated ERP term structure. I think it’s fair to say theUS equity market remained in “business as usual” mode.Figure 4:
US ERP term structure ( yrs ) ( Ann Percent ) Feb
14, 2020 .3 Italy starts lockdowns Fig. 5 shows some next events in the development of the pandemic.Figure 5:
Timeline 3
Italy sees a major surge of cases and many towns are locked downA church in South Korea is linked to a surge of casesIran announces two COVID -
19 casesFeb -
19 Feb -
20 Feb -
21 Feb -
22 Feb - Feb 23, 2020 marks the end of this segment with start of lock-downs in Italy. As seen in Fig. 6,the ERP chart for the next day, the market is now definitely paying attention. The S&P500 hasfallen 4.7% from its record peak of 3386.15 on Feb 19, and the CBOE’s VIX index has risen to25.03. Fig 6 shows the estimated ERP term structure. It’s become strongly inverted with theshort-term ERP (the “required return”) rising to about 40%. Note the long end of the curve,representing the Dec 21, 2022 maturity – about 2.8 years away. It’s around 8%, slightly higherbut not too far from the longer-term value in the previous plots.Figure 6:
US ERP term structure ( yrs ) ( Ann Percent ) Feb
24, 2020 .4 Early March – the Fed acts Fig. 7 shows some events from late February and early March.Figure 7:
Timeline 4
Fed cuts rates, but market falls % S&P 500 rallies 4.6 % on anticipation of Fed interest rate cutFirst US death. Travel restrictions are announcedFirst case in sub - Saharan Africa ( Nigeria ) Feb -
28 Feb -
29 Mar -
01 Mar -
02 Mar -
03 Mar - Early March begins a Federal Reserve monetary policy response. On Mar 2, 2020 the marketrallies by almost 5% – likely anticipating an interest rate cut. The VIX index (which closedat 40.1 on Feb 28), correspondingly eased to 33.4 on Mar 2. Indeed, the Fed announces a ratecut on Mar 3, the first unscheduled, emergency rate cut since the 2008-2009 Financial Crisis.However, the market fell and the VIX climbed back to 36.8.In general, VIX’s above 40 are a sign of a very high level of systematic market stress (seeFig. 17). The associated ERP term structure (Fig. 8) significantly steepens from our last plot,showing an estimated short-term required return of 100-130 percent per annum.Figure 8:
US ERP term structure ( yrs ) E(cid:0)(cid:1) ( Ann Percent ) Mar .5 Mid March I – time to panic Fig. 9 shows some events from mid-March.Figure 9:
Timeline 5
US announces a 30 - day ban on some travel from EuropeItaly q(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) US e(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20) trigger circuit b(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) 1(cid:29)(cid:30)(cid:31) T ! hits record l" of 0.5 % O$% falls ( &’() / R*+,-. argue ) Mar -
06 Mar -
08 Mar -
10 Mar - The week of Mar 9-13 is very ugly on many fronts. COVID-19 cases are rising exponentially inEurope and the US. In the absence of mitigation, there are predictions of millions of deaths tooccur in the US before so-called ‘herd immunity’ is achieved. A vaccine is predicted to be atleast 18 months away, and not certain even then.On Thurs Mar 12, 2020, the VIX index closes at 75.5% and the short-term (1-day) annualizedERP reaches 500-600% (see Fig. 8). Although hard to discern in the chart, the ERP over thelongest term (2 years) has risen to approximately 17% annualized. Credit markets are highlystressed. Figure 10: US ERP term structure ( yrs ) ( Ann Percent ) Mar /02 .6 Mid March II – lockdowns, the IHME becomes influential Fig. 11 shows some events from mid-March.Figure 11:
Timeline 6
CA governor
Newsom declares statewide shutdownTrump backs massive stimulus plan US stocks have worst day since Oct '87 crashSunday: Fed cuts benchmark rate to near 0Mar -
15 Mar -
16 Mar -
17 Mar -
18 Mar -
19 Mar -
20 Mar - By the end of Mar 16-20 week, stresses begin to ease somewhat: see Fig. 12. The short-termERP estimate ended the week at 200-250%, the lowest of the week. VIX ended at 66.What prompted the ease? Prospective stimulus likely helped. Also, the U. Washington’sInstitute for Health Metrics and Evaluation (IHME) was gaining influence. Their earliest pre-dictions (Mar 25), based upon curve fitting to the Wuhan experience, suggested cumulative USdeaths to total 38,000-162,000 through Aug 2020 – premised on lock-downs. These estimateswere significantly lower than the previous ‘millions’ of others (without mitigation), and not ter-ribly disproportionate to the annual mortality from the flu. Indeed, as time passed, the IHMEUS death estimates were tightened and lowered: 48,000-123,000 at this writing (late April 2020),with 54,000 actual deaths to date.Figure 12:
US ERP term structure ( yrs ) ( A78 P9:;<=> ) Mar ?@B See https://covid19.healthdata.org/united-states-of-america .7 Late March to mid-April – signs of optimism Fig. 13 shows some events through April 15, 2020.Figure 13:
Timeline 7
CDFGHI starts to ease
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Johnson enters ‘ac for dfghi ( jkmno rpstuvwx ) yz{|}~(cid:127)(cid:128)(cid:129) cases reach 1 (cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138) case (cid:139)(cid:140)(cid:141)(cid:142)(cid:143) reaches (cid:144)(cid:145)(cid:146)(cid:147)(cid:148)(cid:149)(cid:150) (cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156)(cid:157) (cid:158)(cid:159)(cid:160)¡¢£⁄¥ƒ§¤' “«‹›fifl(cid:176)–† ‡·(cid:181)¶• ‚„”»… ‰(cid:190)¿(cid:192)`´ˆ˜¯ ˘˙¨(cid:201)˚ ¸(cid:204)˝˛ˇ —(cid:209)(cid:210)(cid:211) Mar - (cid:212)(cid:213) Mar - (cid:214)(cid:215) (cid:216)(cid:217)(cid:218) - (cid:219)(cid:220)(cid:221) - (cid:222)(cid:223) April 15, 2020 marks the end of our ERP study. The last timeline events reflect increasedoptimism that (at least the initial phase of) the pandemic has plateaued or peaked in manyareas of the world. VIX ended at 40.8, and the short-term ERP has fallen to the 50-60% range.The long end of the ERP curve remains quite elevated at 16.7%.Figure 14:
US ERP term structure ( yrs ) (cid:224)Æ(cid:226)ª (cid:228)(cid:229)(cid:230) ( (cid:231)ŁØ Percent ) Œº(cid:236)
15, 2020 Outlook and Future Work
The outlook for the course of the pandemic and the economy at this writing is encouraging. Forexample, in New York state, the hardest hit US state, hospitalizations are down significantlyfrom a month ago. Encouraged by that, Governor Andrew Cuomo, says he will extend thePAUSE regulations in many parts of the state, but some less-affected regions can reopen on May15. I hope to see the same in my state, California. It’s clear that reopening will be done carefullyeverywhere, with social distancing and protective measures an ongoing recommended part of lifefor many months to come.There are many to-be-answered questions: what exactly is the mortality rate, how many havebeen infected, are infected but asymptomatic or recovered people now immune, etc? Severalrecent studies suggest that the mortality rate is much lower than many original estimates.In terms of my financial analysis here, there are also some unanswered questions. For ex-ample, what exactly is the risk-return trade-off here? This can be answered by computing the(annualized) variance rate σ t,T associated to the inferred real-world p -distributions, and plottingERP t,T vs. σ t,T . Another project on my “to-do” list is to organize the ERP’s for standardized maturities, say 3 days, 1 week, 1 month, and so on. Finally, I would like to continue to updatethe results as time progresses. As I work through these projects, I’ll update this preprint. References
Friend, I., & Blume, M. E. (1975). The Demand for Risky Assets.
Amer. Econ. Review , 900-922.Lewis, A. L. (2019). Option-based Equity Risk Premiums. arXiv:1910.14522 [q-fin.CP].11
Appendix 1 – Basic reference charts
Figure 15:
US COVID-19 development through late April 2020
Jan -
27 Feb -
10 Feb - (cid:237)(cid:238) Mar -
09 Mar - (cid:239)(cid:240) æ(cid:242)(cid:243) - (cid:244)ı(cid:246) - (cid:247)ł øœ ß(cid:252) (cid:253)(cid:254)(cid:255)C(cid:0) - d(cid:1)(cid:2)(cid:3)(cid:4) n(cid:5)(cid:6) cases Jan -
27 Feb -
10 Feb - Mar -
09 Mar - (cid:8)(cid:9) A(cid:10)(cid:11) - (cid:12)(cid:13)(cid:14) - - (cid:27)(cid:28)(cid:29)(cid:30)(cid:31) deaths S&P500 Index: levels and percent returns
Jan Feb Mar !" ’()*+,-./015 S&P500
I6789: D;< =>? - @BE FGH / JK / LM N / O / PQ - - RSTVW returns
XYZ : [\] ^_‘ - abc efg Figure 17:
VIX Index: longer run and latest one year hijk lm VIX: Jun 3, 2004 - Apr 23, 2020
May Jul Sep Nov Jan Mar020406080
VIX: Apr 29, 2019 - Apr 23, 2020 Appendix 2 – More computational details
Here, I discuss some differences from my previous study: (Lewis, 2019).
Data.
As in my previous study, option quotes were sourced from the CBOEs LiveVol service:“End-of-Day Option Quotes with Calcs”. These quotes are recorded at 15:45 New York time, 15minutes from the close of the regular session. The CBOE advertises them as a “more accuratesnapshot of market liquidity than the end of day market”.With my previous study data, every non-zero bid option was accompanied by a larger non-zero ask. In my data for this study, which was sampled almost completely for all SPX tradedates and quotes from Jan 2, 2020 through April 15, 2020, there were a few exceptions to thisrule. For example, on March 16, 2020 there were some quotes for the Dec 16, 2022 expirationshowing a bid > Dual expirations.
In my previous study, I included both AM and PM options on the Fridayswhere these both occurred. These were treated as distinct expirations because times were mea-sured to 15 min accuracy. Here, for simplicity, for such dual expirations, only the PM optionswere included. However, certainly the AM options were included here for any expiration whenthose were the sole options expiring. Also for simplicity, all times T here (in years), were simplymeasured as T=(days)/365, where days was the integer number of days from the trade date tothe expiration date. Cost-of-carry methodology.
I adopted the put-call parity regression method of the previousstudy. One change was that I only included 50% of the put-call pairs: those closest to themoney. For very short-dated expirations, while the regression method produces very plausibleforward prices (which are key), the inferred interest rate and dividend yield are rather erratic,often negative. This erratic effect was lessoned by the 50% inclusion (the previous study using100%). The previous study showed that two different cost-of-carry methods will produce almostidentical ERP’s as long as the inferred forward prices are similar.
Simplified rules.
In my previous study, besides my nominal objective function, I also adopteda secondary objective related to certain
OutStats , which are explained there. I sought to achievemy secondary objective by switching the number of Gaussian components from N=4 to N=5, ormaking other adjustments such as introducing a minimum bid on put quotes higher than the 0 . OutStats here were not as good as in my previous study, but generally I found myERP’s to be quite robust to my choices for various optimizer parameters. Optimizer pa-rameters consisted of the PrecisionGoal (PG=4), the maximum number of optimizer steps(MAXSTEPS=600), and a scaling parameter ( sigMULT=1.2 ). This last one fixed the upper limitof the allowed fitted volatility to be sigMULT × IVMAX , where the second term was the highestimplied volatility observed at that expiration. On a few expirations, if the