key: cord-0056830-uu0wuopa authors: Hong, Harrison; Wang, Neng; Yang, Jinqiang title: Implications of Stochastic Transmission Rates for Managing Pandemic Risks date: 2021-02-09 journal: Rev Financ Stud DOI: 10.1093/rfs/hhaa132 sha: f6d6be52a52fba48e83e8848d0ed82a835337c74 doc_id: 56830 cord_uid: uu0wuopa We introduce aggregate transmission shocks to an epidemic model and link firm valuations to infections via an asset pricing framework with vaccines. Infections lower earnings growth but firms can mitigate damages. We estimate a large reproduction number [Formula: see text] and transmission volatility for COVID-19. Using these estimates, we quantify the bias of deterministic approximations based on [Formula: see text]. Our model generates predictions consistent with the data: unexpected infection resurgence, nonmonotonic mitigation policies, and higher price-to-earnings ratios during a pandemic. Valuations would be significantly lower absent mitigation and a high vaccine arrival rate. agents assume infection dynamics are deterministic (see, e.g., Atkeson 2020; Gourinchas 2020 ). Yet, both epidemiologists and economists recognize deterministic models are potentially crude approximations of stochastic epidemic dynamics. Aggregate transmission rate shocks due to environmental factors can play a large role in the evolution of infection dynamics (see Andersson and Britton 2012) . A case in point is the resurgence of COVID-19 in a number of countries during Summer 2020, including countries with prudent mitigation, such as South Korea. While epidemiologists recognize such stochasticity in fitting statistical models (see, e.g., Kucharski et al. 2020; Li et al. 2020) , it would be valuable to explicitly model how transmission volatility influence infection dynamics and optimal mitigation strategies. This is especially important when one considers financial damage of the sort mentioned by the Federal Reserve Board Financial Stability Report (2020): "Asset prices remain vulnerable to significant price declines should the pandemic take an unexpected course . . . ." Indeed, standard asset pricing theory suggests that aggregate transmission shocks ought to play a critical role in asset prices through a risk premium channel. Toward this end, we start with an extension of a widely used deterministic epidemic model of COVID-19 (Kermack and McKendrick 1927) featuring aggregate transmission rate shocks that are intended to capture that viral contagiousness is unpredictable due to environmental factors. 1 Epidemic models of COVID-19 have typically entertained multiple compartments in terms of tracking susceptible, infected, and resistant (including the recovered and dead). To transparently highlight the importance of transmission volatility and for tractability purposes, we focus on modeling just the infected population I t , via a susceptible-infectedsusceptible (SIS) as opposed to a susceptible-infected-recovered (SIR) setting. 2 For a number of economic and financial applications, the focus is typically on horizons of many years and the infected population is often the main state variable of interest since damages are likely to increase with infections. At this point, the scientific community has not reached consensus about whether recovery from COVID-19 confers decreased severity of reinfection or immunity. Following Gray et al. (2011) , who have used this approach for other viruses, we model the COVID-19 aggregate transmission shocks via a stochastic transmission rate, β t . This key input is modeled as a random variable with constant mean (predictable transmission captured by parameter β) and transmission shocks (mean zero but with volatility captured by parameter σ ). The exit rate from the infected state back into the susceptible state is assumed to be a constant γ . We further introduce a stochastic vaccine arrival into our epidemic model as a jump with a Poisson arrival rate λ. When the vaccine arrives, we assume the pandemic is over and infections go to zero. The resultant dynamics of the fraction of infected then follows a four-parameter nonlinear diffusion process. We then model the financial impact of infection forecasts through the lens of a dynamic asset pricing model. The unexpected arrival of COVID-19 has directly affected earnings through three channels. First, most industries, except for a few, such as technology (Landier and Thesmar 2020) , experience a significant negative jump in earnings. Second, earnings growth is also potentially adversely affected by higher COVID-19 infections since higher infection rates mean workers may be sick or caring for sick friends or family members and/or generally less productive. Third, stochastic transmission shocks also increase risk premiums. However, firms have access to a mitigation technology to reduce the drift of infections by paying both (flow) fixed and variable costs. Taking the stochastic discount factor used to price earnings as given, a representative firm optimally mitigates infections to reduce the damage on valuation via earnings, while taking into account the arrival of a vaccine. For instance, MarketWatch reported on May 2, 2020, that even a technology company like Amazon will spend $4 billion or more on COVID-19 mitigation responses, such as COVID-19 testing for its workers, potentially wiping out the company's Q2 profit. Reducing the spread of disease incurs costs and hence lowers earnings in the short term but increases the expected earnings in the future by sufficient amounts so that it is optimal for the firm even from a value-maximizing perspective. As a result, even absent health considerations, controlling COVID-19 comes with economic benefits. Our model is a partial-equilibrium one and ignores externalities associated with mitigation. Our approach is justified by a number of studies, including Andersen et al. (2020) and Farboodi, Jarosch and Shimer (2020) , that point to the importance of voluntary mitigation in social distancing by households and firms even before government-imposed lockdowns. 3 Our model hence links risk management and valuation to epidemic data (infections, mitigation, reproduction numbers, transmission volatility, and vaccine arrival rate). To relate our model to the data, we first estimate our epidemic model by pooling COVID-19 case data from 16 countries (regions) that were at high risk during the period of January to February 2020. These countries had among the most air travel connections to Wuhan, China, and have been the basis of the modeling of the early dynamics of COVID-19 before the onset of mitigation strategies. Given the noisiness and brief time series of the data and our goal of demonstrating the influence of shocks, we fit one model for all 16 countries. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 4 1-42 The Review of Financial Studies / v 00 n 0 2020 Our estimate of β is 6.62 per month, which translates to an infected individual infecting one susceptible on average every five days (≈ 30/6.62.) Our estimate of monthly σ is 1.69, which translates to a standard deviation of plus or minus 1.69 infected individuals per month. The exit rate γ is equal to the inverse of the expected duration that an infected is sick and infective; it is typically not estimated based on aggregate data early in epidemics since there is a delay in when individuals leave the infected state. For our estimation of a population average, we simply use 14 days as the duration to infer the exit rate γ at 1/(14days), which is 2.17 per month. 4 These estimates then imply that our (basic) reproduction number R 0 using case data from January-February is 3.05 and the 90% confidence interval (CI) is (1.12, 6.52) based on the empirical distribution. Despite constraining one model for all countries (regions), our estimates are in line with leading studies of Along with these estimates that characterize the premitigation COVID-19 process, we choose the remaining parameters to target asset pricing moments and the magnitude of the decline in current earnings, which is nearly 40% for the typical firm (see Landier and Thesmar 2020) , and a postmitigation reproduction number of around one. We assume that the initial jump in earnings comes about equally from customers, who stop consuming when COVID-19 arrives and will not return until a vaccine arrives, and optimal mitigation on the firm. Our calibration assumes that the two channels equally contribute to reducing transmission rates. We set the vaccine arrival rate λ to one per year, so that the expected pandemic duration is around one year based on surveys of vaccine experts and scientists (for such a timeline, see McKinsey Report, 2020). Our calibration generates a number of new insights. First, how well a deterministic model's infection forecasts approximate those of our stochastic model depends on a vaccine's arrival. Deterministic model infection forecasts tend to overshoot our model's conditional forecasts. Mathematical epidemiologists reason that introducing noise into the transmission process will lead to a dampening of stationary distribution of infections (Andersson and Britton 2012; Brauer, Driessche, and Wu 2008) . Even if the reproduction number R 0 > 1, the disease might nonetheless become extinct because of the uncertainty of transmissions as opposed to in the deterministic setting. The further out the vaccine, the worse of an approximation is the deterministic model. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 5 1-42 Second, our stochastic model yields rich optimal strategies beyond those from purely deterministic considerations. Because our estimated reproduction number is high, the optimal strategy always involves incurring fixed costs over a large range of infections rates, starting at even a tiny fraction of the population being infected. However, transmission volatility generates an option value of waiting reflected by the optimal mitigation policy in the infection rate I t on the intensive margin. Optimal mitigation policy hence can be nonmonotonic in infection rates: initially increasing because of this option value of waiting and then declining because of standard congestion effects. Simply put, as infections rise, less of the population will be susceptible. These explanations match well with a standard epidemiological playbook that we have observed as we will discuss below. Third, however, the infection process cannot be perfectly controlled due to aggregate transmission shocks in contrast to deterministic models. Hence, mitigation at the intensive margin can then fluctuate with infection rates. But the optimally mitigated COVID-19 infection process will tend to be pulled toward a constant reproduction number over time. These features match well the experience of a number of countries, including the United States, that had to deal with an unexpected resurgence of COVID-19 in early March, followed by a period during which the reproduction number fell but then unexpectedly rose in Summer 2020. Fourth, we show that the price-to-earnings (net of mitigation costs) actually can be higher during a pandemic than in a normal period, consistent with data assuming there is optimal mitigation. The median annual price-to-earnings ratio in the stock market has risen from around 19 before the pandemic to 24 during the pandemic. With both optimal mitigation and an expected vaccine arrival rate around one per year, mitigation while costly is temporary, earnings reduction is thus also temporary and moreover earnings is expected to discretely jump upward (because of both the elimination of mitigation costs and customers' return once the pandemic is over). For these reasons, prices, being the present value of all future earnings, fall much less than earnings do. Fifth, we calculate a counterfactual for what would happen to stock prices absent any mitigation. The market would be down 15% relative to the optimal mitigation scenario. Our counterfactual can provide an account of asset price dynamics around COVID-19, namely, a dramatic plunge in late February 2020 and an equally dramatic rebound in March 2020. For instance, investors might have thought that society failed to control the virus initially (which corresponds to our counterfactual of no mitigation leading to lower stock market values) but subsequently learned that they were going to (and hence the market rebounded to the equilibrium prices under optimal mitigation). Finally, we consider comparative statics exercises that further speak to observed asset price dynamics. Asset valuations are highly sensitive to the vaccine arrival rate, consistent at least with anecdotes of stock market response to news on vaccine developments. Our paper contributes to several literatures. In epidemiology, aggregate transmission shocks are used in epidemiological forecasting models to capture deviations of infections from deterministic projections (see Dureau et al. 2013 for how parameter perturbation is employed). The analytical treatment of transmission volatility is fully articulated in Gray et al. (2011) , who characterize the stationary distribution of our nonlinear diffusion process absent a vaccine. Zhao and Jiang (2014) extend the baseline model setup in Gray et al. (2011) by allowing for a third compartment for the vaccinated. In their paper, a fraction of the population can become vaccinated. They explore stationary distributions in this setting. Given that a goal of our analysis is to analyze the impact of an effective vaccine arrival on both the spread of disease and valuation, we model a vaccine via a Poisson jump process and explore the impact on conditional distributions, particularly how the deterministic model approximations depend on this Poisson arrival rate, and also valuation. Hence, our model contributes to the epidemiology literature by simultaneously accounting for a stochastic vaccine arrival, studying conditional distributions through Kolmogorov equations far from steady state, and deriving the optimal mitigation strategy. Our optimally mitigated stochastic SIS process is new to the epidemiology literature as work on stochastic control in epidemics is limited. Other approaches to stochastic epidemics using Markov chains are available. For instance, Allen and Burgin (2000) work with Markov chain models with a discrete state space and one absorbing state that guarantee that the disease is eventually driven to extinction. That is, the stationary distribution is degenerate. Our model (with no vaccine) has a nondegenerate stationary distribution, which is better suited to explore implications of stochastic transmission shocks. Also, our model features a vaccine, whereas theirs does not. Clancy (2005) also considers a Markov chain SIS model setup with no diffusion shocks where indirect transmissions (e.g., environmental bacteria and zoonotic diseases, where harmful germs travel from animals to human) are possible. While this channel is relevant for some types of diseases (e.g., Zika, a mosquito-borne flavivirus), it does not seem important for COVID-19. In economics, recent theoretical models on controlling epidemics have used deterministic SIR models. The work closest to ours is Kruse and Strack (2020) where they show using a deterministic SIR model that the optimal solution is typically to act early unless herd immunity is within reach. 6 Our contribution is to show how aggregate transmission shocks significantly influence optimal mitigation strategies in an SIS setting. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 7 1-42 For tractability, we work with an SIS setup rather than an SIR model as doing so yields an ordinary differential equation (ODE) rather than a PDE for the price-to-earnings ratio. R 0 is no longer a sufficient statistic in richer SIR, SEIR, and other even richer models with multiple compartments. 7 Our SIS model nonetheless captures first-order insights and mechanisms. As the mitigated I process tends to be low (near zero) most of the time in our calculations, the recovered fraction (in an SIR model) also would be close to zero. As a result, our approximation (which is effectively ignoring the recovered population) is likely to be sensible and hence our insights regarding R 0 and deterministic approximations also have implications for SIR analysis. In finance, our contribution is to link asset prices to underlying epidemiological data. Our model is consistent with Gormsen and Koijen (2020) , who use a fundamentals-based asset pricing model along with dividend futures to isolate a large impact of COVID-19 via the earnings growth channel. Hong, Kubik, Wang, Xiao and Yang (2020) combine our model with analyst forecasts to infer market expectations regarding the arrival rate of an effective vaccine that returns earnings to normal, direct effect of infections on growth rates of earnings, and the damage to earnings due to mitigation. In this section, for pedagogical purposes, we construct our stochastic model by starting with the classic Kermack and McKendrick (1927) model. Time is continuous, and the horizon is infinite. We normalize the total population size to one, and births and deaths do not occur in the population. A key motivation is the design of a tractable and parsimonious model with which to conduct risk management applications; thus, we model two compartments (groups): infected and infectious (I) and susceptible (S) (or equivalently uninfected). 8 Within each group, the population is homogeneous and well mixed. Let I t and S t denote the mass of the infected population and the susceptible at time t, respectively. As I t +S t = 1 at all t, we only need to keep track of the evolution for I t , which is the single state variable in our model. How does disease transmit from an infectious individual to a susceptible individual? The probability that an infectious individual meets a susceptible individual is proportional to the product of their population mass, I t (1−I t ), with an effective transmission rate, 7 A model with N compartments naturally calls for N −1 state variables as the only restriction across the N compartments is that the population sums to one (as a normalization). 8 The epidemiology literature features various generalized formulations of these compartmental models. Widely used ones include SIR (susceptible, infected, recovered) and SEIR (susceptible, exposed, infected, and recovered) models. See Andersson and Britton (2012) and Brauer, Driessche, and Wu (2008) for textbook treatments. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 8 1-42 The Review of Financial Studies / v 00 n 0 2020 which we denote by β. Thus, over the interval [t,t +dt), the total number of new infections is βI t S t dt = βI t (1−I t )dt . The infectious individual recovers and becomes the susceptible individual in our model. Let γ >0 denote the rate at which an infectious individual recovers. Hence, 1/γ is the duration for an infected individual to be infective. Subtracting the mass for the recovered γ I t dt over the interval [t,t +dt) from the newly infected individual βI t (1−I t )dt, we obtain the following process for dI t , the net change of I t : (1) The solution to (1) satisfies the following logistic function: 9 (2) 2.1.2 Basic reproduction number R 0 . The basic reproduction number, R 0 is defined as the expected number of secondary infections generated by a single (representative) infected individual in a completely susceptible population: If R 0 ≤ 1 (when β ≤ γ ), the disease is eventually driven to extinction, as (2) implies lim t→∞ I t =0. If R 0 > 1, the infected population I t reaches the maximum level, I ∞ =1− R −1 0 > 0 as t →∞, provided that I 0 =1−R −1 0 . We will use the terms basic reproduction number and reproduction number interchangeably. The literature sometimes refers to the effective reproduction number at time t, which is the basic reproduction number multiplied by the susceptible mass. The effective reproduction number R 0 (1−I t ) is time varying in classic deterministic models. Figure 1 plots the infected mass I t at t in panel A and the net change of the infected mass dI t /dt in panel B with the initial value of I 0 =66/(3.28×10 8 )=2×10 −7 (as there were 66 infective individuals on March 1, 2020, in the United States and the U.S. population as of 2019 is 328 million.) The solid-blue lines represent the solution for our deterministic case using our estimate of the transmission rate for COVID-19 that we discuss in Section 4.2. By reducing β by half from 6.616 to 3.308 per month, such as using economywide lockdowns, we lower the basic reproduction 9 If β = γ , by applying L'Hopital's rule to (2), we obtain I t = βt + 1 number R 0 by half from 3.045 to 1.522 (unlike the three structural parameters, R 0 is invariant to the time horizon we choose). As a result, the eventual infected fraction, I ∞ , decreases by half from 67.1% to 34.3% of the entire population. Panel B captures the widely discussed flattening the curve argument (see, e.g., Atkeson 2020; Gourinchas 2020). Here, the curve refers to the net change of the infected population, dI t /dt, as a function of time t. If the society successfully reduces β by half via social distancing and other interventions, this deterministic evolution curve is indeed significantly flattened and postponed. Specifically, this curve peaks at a bit over 1 year (t =12.657 months) if β =3.308 rather than at a bit over one quarter (t =3.384 months). The curve of the net change, dI t /dt, is substantially flattened. Note the very sharp increase of I t at the very early stage. This is because early on I t is close to zero, and we can thus effectively drop the (1−I t ) terms and approximate I t as an exponential process: dI t ≈ (β −γ )I t dt with the approximate solution: I t ≈ I 0 e (β−γ )t . Obviously, exponential growth at a large rate β −γ is incompatible with convergence of I t to I ∞ =1−R −1 0 as t →∞. This is due to the dampening effect of I t on its own growth. As the fraction of the infected increases, fewer are susceptible, which lowers dI t /I t . That is, the higher the level of I , the lower the infection growth rate dI t /I t . where both β and σ are constant parameters and t is a mean-zero standard normal random variable. 10 Mapping (4) into our continuous-time formulation, we obtain β t dt = βdt +σ dZ t , where Z t is a standard Brownian motion. By using β dt given in (5) to replace βdt in (1) and then combining drift and diffusion terms, we obtain the following stochastic differential equation (SDE) for I t : The drift term is the same as in the deterministic SIS model, while the diffusion term captures the uncertainty of the epidemiological evolution process. When no one is infected (I t = 0), the disease is driven to extinction: dI t = 0 as both drift and volatility terms in (6) are zero. If the entire population is infected (I t = 1), the volatility has to be zero and the drift has to be negative so that the model is well posed. 11 Unlike I t =0, I t = 1 is not an absorbing state as γ >0. Note that both the drift and volatility of the growth rate for the infected population, dI/I , depend on (1−I ), the population of the susceptible. Specifically, the higher the level of I , the lower the drift (i.e., the expected infection growth rate) of dI t /I t . As the fraction of the infected increases, fewer are susceptible, which dampens the drift of dI/I . To complete the description of our compartmental model, below we report the dynamics for the susceptible population S t implied by Equation (6): 2.2.2 Permanence of initial transmission shocks. The process for I t given in (6) is not a geometric Brownian motion (GBM) process widely used in economics and finance. But at a very early stage, I t is close to zero; therefore, we can effectively drop the (1−I t ) terms in both drift and volatility functions and approximate I t via a GBM process: dI t ≈ (β −γ )I t dt +σ I t dZ t . That is, in the early stage, I t evolves as Unlike the exponential growth approximation for I t in the deterministic model, in our stochastic model, I t is not only driven by R 0 but also driven by the (exponential) martingale, the second exponential term in (8). This second term is equally important in driving the dynamics of I t as the first (exponential) term involving R 0 . Because very few individuals are infective early on, the change in I t is highly idiosyncratic as the diffusion term dominates the drift term. A few negative shocks early on have outsized permanent effects on the evolution of I t . On the other hand, if few shocks occur early on, then the total number of the infected population stays low for an extended period of time. That is, in the very early stage, the sequence of realized values of β, not the expected transmission rate β used in the deterministic compartmental epidemic models, drives the speed at which disease spreads. We now have a threeparameter (β,γ , and σ ) nonlinear diffusion process. By applying Ito's lemma to (6), we obtain where the drift for lnI t is a quadratic function in I t : Equations (9) and (10) are convenient to work with when we analyze the stationary distribution. Gray et al. (2011) show that I t persists in the long run if and only if R 0 > 1, where Whereas R 0 = β/γ > 1 determines whether the disease breaks out in a deterministic model, the analogous outbreak condition is R 0 > 1. Therefore, even at a reproduction number R 0 above one, a sufficiently large value of σ 2 can cause R 0 < 1, which, in turn, implies that the disease is extinct in the limit. Next, we turn to the stochastic steady state (SS) and stationary distribution to gain some intuition for why R 0 is an insufficient statistic for managing COVID-19 risks. The long-run distributional properties of the infected fraction I depend on all three parameters in a nonlinear way. Simply relying on R 0 , which is ratio between the expected transmission rate β and exit rate γ can be quite misleading. Unlike in the deterministic model, which generates a single number for I t at any t, in order to fully capture the dynamics of disease transmission, we next characterize the time 0 conditional distribution of I t for all t. Let f (I t ,t;I 0 ) denote the time 0 conditional density function for I t , the infected mass at t given the initial infected mass I 0 . The density function, f (I,t), satisfies the following Kolmogorov forward equation: The first term is the time effect on f (I,t), the second term is the drift effect on f (I,t), and the last term is the volatility effect on f (I,t). In Section 4.3, we show how uncertainty substantially alters the transmission dynamics. Next, we incorporate the impact of stochastic vaccine arrival. We assume that COVID-19 will disappear following a successful vaccine development. 12 Specifically, we use the following SDE to model the evolution of I t : We capture this vaccine arrival effect on I t via the third term, where J t is a (pure) jump process with a constant arrival rate, which we denote by λ. When a vaccine is successfully developed, that is, dJ t = 1, the disease is driven to extinction and the pandemic ends. We can generalize our model to allow for a multiple-stage vaccine development process with a gradual reduction of the infected population without losing much analytical tractability. 13 In this section, we develop a parsimonious yet operational model to capture the impact of pandemic shocks on fundamentals-based valuation. We show how COVID-19 parameters β (equivalently R 0 ) and σ together with asset pricing specifications impact valuation. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 13 1-42 In Section 3.1, we propose a valuation model before unanticipated pandemic arrival. In Section 3.2, we consider optimal mitigation after an unanticipated pandemic arrival in an asset pricing framework. We explicitly allow for stochastic vaccine arrival in our analysis. To ease our exposition and set up the basic apparatus in which we later incorporate COVID-19 shocks, we first introduce a simple asset pricing model with no pandemic shocks, that is, under a normal business-as-usual environment or when I t =0. We start by specifying the following process for the widely used simple stochastic discount factor (SDF), M t , in the normal regime: where B t is the standard Brownian motion for the aggregate shock. Here, r is the risk-free rate and η B is the market price of risk for the aggregate shock. 14 For simplicity, let r and η B be constant. Equation (14) implies a one-factor model where the factor can be the aggregate consumption growth shock as in a representative-agent general-equilibrium model of Lucas (1978) , or the market portfolio return in the capital asset pricing model (CAPM) of Sharpe (1964) , the option pricing model of Black and Scholes (1973) , or the portfolio choice problem of Merton (1971) . Here, η B is positive as a positive shock dB t to the aggregate consumption growth or market return is good news which lowers the investor's marginal utility or equivalently M t . We assume that the earnings process in the normal regime, Y t , follows: where B t is the aggregate shock introduced in (14) and W t is the standard Brownian motion driving earnings idiosyncratic risk. By construction, B t and W t are orthogonal. In (15), g is the expected earnings growth (drift) and φ is the volatility of earnings growth, which includes the aggregate component ρφ and the idiosyncratic component 1−ρ 2 φ. That is, ρ is the correlation coefficient between the aggregate shock B t and the asset's earnings process. For simplicity, we let g, φ, and ρ all be constant. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 14 1-42 The Review of Financial Studies / v 00 n 0 2020 3.1.2 Pricing formula. Under the assumption that investors price earnings without expecting the possibility of a pandemic arrival, the firm's value in the normal regime (pre-and postpandemic) satisfies the following standard asset pricing equation (Duffie 2001): In Appendix C.3, using (14) and (15) and solving (16), we show that the firm's value is proportional to its earnings, P t = p Y t , where the price-to-earnings ratio, p, is a constant: Equation (17) is the well-known Gordon growth model where r +ρφη B is the firm's constant cost of capital (discount rate) and g is the earnings growth rate. This firm earns a risk premium of ρφη B , which is given by the the product of the market price of risk η B and ρφ, the systematic volatility component of φ and consistent with the one implied by the widely used CAPM. Next, we incorporate pandemic shocks into our valuation model and consider the effect of mitigation responses by both customers (consumers) and firms. After COVID-19 unexpectedly arrived in the United States in late February 2020, a fraction of consumers voluntarily engaged in social distancing and took various other precautionary measures. These voluntary actions by consumers (customers) substantially cut the transmission rate but also lower corporate earnings. In addition to customer mitigation which reduces earnings and transmission rates, we will model the optimal mitigation strategy of firms, which take as given the effects of customer mitigation. Let Y t denote the earnings process during the pandemic regime. To capture the impact of customer mitigation, we assume that logarithmic earnings drop at the moment of COVID-19 arrival time t 0 by a stochastic fraction n(I t 0 ): where Y t 0 − is the prepandemic arrival earnings and Y t 0 is the postpandemic arrival earnings. Changes in peoples' behaviors, for example, quarantining, working from home, and social distancing, reduce the speed at which disease spreads, and, thereby, the transmission rate is also lower. Let where β t 0 − is the transmission rate of disease β (when disease is not contained and in the absence of any behavioral response.) The parameter ψ measures the fraction of β reduction due to customers' voluntary behavioral adjustments. Next, we discuss the earnings process in the pandemic regime after t 0 , which will then depend on the optimal mitigation strategy of the firm. 3.2.1 Earnings process. Before the vaccine arrival at τ , we assume that Y t is given by COVID-19 influences Y t as follows. First, the infection shock dZ t directly causes additional earnings growth volatility, v(I ). Second, the expected earnings growth rate (absent vaccine arrival) is changed to g(I ) from g. Third, the stochastic arrival of vaccine (dJ t =1) at t = τ causes an instantaneous (logarithmic) change of earnings from the prejump level, Y t− , to the postjump level Y t at t = τ . For expositional simplicity, we assume that the percentage of earnings upward jump at the moment of vaccine arrival τ is equal to the percentage of earnings downward decrease at the moment of pandemic arrival is the same as the n(·) function in Equation (18). We set n(0) = 0 so that earnings re continuous Now consider a counterfactual case that helps us understand the mechanism of the model. Suppose τ − = t 0 , which occurs if λ →∞. For this case, earnings are not affected by the two jumps (an unexpected pandemic and the arrival of a vaccine) as the disease is driven to extinction in no time: Additionally, earnings are still subject to the business-as-usual aggregate shock dB t and idiosyncratic shock dW t with volatility ρφ and 1−ρ 2 φ, respectively. All shocks are orthogonal to each other. 15 To highlight the role of stochastic transmission shocks on both earnings and valuation in a simple way, we assume that parameters for the business-as-usual aggregate variables and idiosyncratic risks, do not change with the unexpected pandemic arrival. We can of course also allow the business-as-usual parameters to also change as well without technical difficulties, but leave these extensions out for brevity. As COVID-19 is clearly an aggregate shock, it changes the equilibrium SDF. We model the SDF M t in the pandemic regime as follows: As a positive pandemic shock dZ t (which increases I ) is bad news for the aggregate economy, investors' marginal utility (the SDF M t ) should increase with I t , which means η Z < 0, in contrast to a positive η B for the businessas-usual aggregate shock dB t . The last term captures the effect of stochastic vaccine arrival on the SDF M t and this jump term is a martingale under The Review of Financial Studies / v 00 n 0 2020 the physical measure (to be consistent with no arbitrage. 16 Upon successful vaccine development at t = τ , that is, dJ t = 1, the SDF immediately changes discretely from M τ − by M τ = e κ M τ − . As a vaccine arrival is good news for the aggregate economy, investors' marginal utility (SDF) should decrease after vaccine arrival, which implies that the market price of vaccine arrival risk is negative, that is, κ <0. Let {X t ; t 0 t 0 , the firm further lowers the transmission rate from β t 0 to β t 0 −h t , where h t ≥ 0. This additional reduction obtained by mitigation X t captures the effects of corporate actions. Let where h(x) is increasing and concave in x. The motivation for the homogeneity (in earnings Y t ) assumption underpinning Equation (22) is that to cut the transmission rate by the same magnitude h(x t ), one firm whose earnings is twice the size of another needs to spend twice as much to achieve the same levels of reduction of the transmission rate. This assumption is reasonable as the benefit scales up with earnings and also makes our analysis tractable. The evolution of I t given in (13) with mitigation is Recall that β t 0 is the constant transmission rate after customers respond to the pandemic by taking precautionary measures. That is, absent mitigation but with customers' response, the transmission rate is lowered from β t 0 − to β t 0 . The basic reproduction number is then at time t after the pandemic arrival but before the vaccine arrival. Additionally, we assume that mitigation is costly and lowers earnings by more than the level of mitigation X. For simplicity we assume that mitigation incurs a (flow) fixed cost that is proportional to Y , that is, πY t , where π >0 is a constant measuring the size of flow fixed costs. Let Y * t denote the firm's earnings net of both fixed and variable costs. With the above assumptions, the net earnings is where 1 X t >0 is an indicator function that equals one if mitigation is strictly positive (X t > 0) and zero otherwise. [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 17 1-42 Notice that this equation implies that right after the pandemic arrives at t 0 and earnings has jumped due to customer mitigation from prepandemic levels at t 0 −, firm mitigation entails a further jump in earnings at t 0 + and onward as long as a vaccine has not arrived. These costs have to be paid until a vaccine arrives, at which point earnings will then jump back up for two reasons: customers return and firms can stop paying these mitigation costs. Next, we state the optimization problem. The firm chooses mitigation X to maximize the following risk-adjusted present value in the pandemic regime: where τ is the stochastic vaccine arrival time. Inside the expectation operator in Equation (26), two terms contribute to the discounted stochastic value of earnings: the first term is the value before the arrival of vaccine at τ , and the second term is the value after τ . Note that there is no need to spend on mitigation after τ , but anticipation of a vaccine's arrival at τ fundamentally affects the agent's optimal mitigation before τ . Let C t denote the corresponding present value of mitigation costs: Because the earnings process features geometric growth, the firm's value is proportional to its earnings Y t : where p(I t ) is the equilibrium price-to-earnings ratio. As mitigating disease involves a fixed flow cost πY t , the firm may choose not to mitigate when the benefit of doing so is sufficiently low. Conditional on choosing X >0, the optimal scaled mitigation, x = X/Y , satisfies the following first-order condition (FOC): The level of mitigation X t > 0 at the margin lowers the infected population mass satisfies Equation (29).In Appendix C.4, we obtain the following ODE for p(I ): where is the risk-adjusted disease transmission rate (i.e., under the risk-neutral measure Q) and λ Q is the risk-adjusted vaccine arrival rate (under the risk-neutral measure Q): We expect the risk-adjusted duration of the pandemic (under the risk-neutral measure Q), 1/λ Q , to be longer than 1/λ, which is the expected duration of the pandemic (under the physical measure P). This is because vaccine arrival is good news for the aggregate economy, which means κ <0, as we discussed early. Next, we turn to boundary conditions. First, as the no-infection state is absorbing, the price-to-earnings ratio at I = 0 is equal to the value in the normal regime: Second, consider the extreme (and counterfactual) case in which everyone is infected, I = 1. In this case, naturally there is no need to spend on mitigation, x = 0. The ODE (30) is then simplified as This boundary condition ties p (I ) with p(I ) at I =1. Next, we return to the optimal mitigation choice x. As the effective transmission rate depends on the product of I and S =1−I , mitigation is more valuable in an interior region of I , ceteris paribus. Given the fixed cost πY t of mitigation, it is optimal for the firm to mitigate only when I is neither too high nor too low. The preceding reasoning implies that the solution features three mutually exclusive regions: two inaction regions and one mitigation region. I and I denote the endogenously determined cutoff levels of the infected population mass for the three regions. In the regions 0 ≤ I 0. The ODE for p(x) in the mitigation region, that is, I 0 and 0 <ζ 2 < 1. Without historical data to nail down these parameters, for our baseline we choose ζ 1 = 3 and ζ 2 = 0.25 so that g(0.1) = −2/3× g and g(1) = −2× g. These are moderate long-run declines in growth rates absent mitigation. We set v(I ) = 0 for simplicity. That is, infections will only affect the drift, but not the volatility, of earnings. Finally, we specify the n(I ) function in the earnings process that appear at t 0 and τ as n(I )=α 1 I α 2 , where α 1 > 0 and α 2 > 0. Recall that I = 0 is an absorbing state and n(0)=0. We obtain α 1 = 18% and α 2 = ln(2/3) ln(I t 0 ) by targeting n(I t 0 ) = 12%, where I t 0 =2×10 −7 , and n(1)=1.5×n(I t 0 ) = 18%. We set the annual risk-free rate r at 4%, the annual stock market volatility σ m at 20%, and the annual stock-market risk premium (r m −r) at 6%. The implied annual Sharpe ratio of the stock market η B =(r m −r)/σ m , which is also the annual market price of business-as-usual risk in our CAPM model for the normal regime, is equal to 6%/20% = 30%. The implied asset's beta is (ρφ/σ m )×(r m −r)=ρη B φ =1×30%×20%=6% and the cost of capital for this asset is equal to 4%+1×6% = 10%. As we set g = 5%, the price-to-earnings ratio in normal times, p, is then equal to 1/(10%−5%) = 20, given by the Gordon growth model in Equation (17). 18 Next, we set the pandemic asset pricing parameters. We set the market price of pandemic risk η Z at −0.4 and the market price of vaccine arrival timing risk κ at −1. As v(I ) = 0, the only effect of η Z is that for pricing purposes we need to use the risk-adjusted β Q , which is different from β under the physical measure. Equation (31) implies the risk-adjusted transmission rate is larger than the real transmission rate for the pandemic (controlling for customers' voluntary precautionary response): We set the vaccine arrival rate, λ, at 1.1 per annum, consistent with optimistic assessment by scientists in the media. With κ = −1, the risk-adjusted vaccine arrival rate is λ Q /λ =1.1×e κ =0.404. That is, the risk-adjusted expected duration of the pandemic regime (under the risk-neutral measure Q), 1/λ Q , is about 2.5 years, which is much longer than the expected duration of the pandemic regime under the physical measure, 1/λ =0.9 years. We take the parameter values for COVID-19 dynamics absent mitigation from our estimates. For the period t , satisfies d =(1−I t ) 2 σ 2 dt . Therefore, we have Discretizing the preceding equation, we obtain the following estimate of σ 2 : By setting to one, we obtain (46). [16:34 20/12/2020 RFS-OP-REVF200145.tex] Page: 40 1-42 The Review of Financial Studies / v 00 n 0 2020 Appendix C. Derivation Details for Valuation In the normal regime no arbitrage implies that the drift under the physical measure for M t Y t dt + d( M t P t ) is zero (Duffie (2001) ). By applying Ito's lemma to this martingale, we obtain The above equation implies P ( Y ) satisfies the following pricing equation: By solving the above ODE, we obtain P ( Y )= p Y , where p is given by Equation (17). In the pandemic regime, no arbitrage implies that the drift under the physical measure for M t− Y t− −(πY t− +X t− )1 X t− >0 dt +d (M t P t ) is zero. The pricing function depends on both Y t and I t . Hence, we write P t = P (Y t ,I t ). By applying Ito's lemma to this martingale, we obtain By using the homogeneity property, P (Y,I )=p(I )Y , and substituting this expression into (C3), we obtain the ODE (30) for p(I ). Using the first-order condition (FOC) for the HJB Equation (C3), we obtain Equation (29) for the optimal mitigation x(I ). Comparison of deterministic and stochastic SIS and SIR models in discrete time A simple planning problem for COVID-19 lockdown Implications of Stochastic Transmission Rates for Managing Pandemic Risks Pandemic, shutdown and consumer spending: Lessons from Scandinavian responses to COVID-19. arXiv, preprint Stochastic epidemic models and their statistical analysis What will be the economic impact of COVID-19 in the US? Rough estimates of disease scenarios The pricing of options and corporate liabilities Lecture notes in mathematical epidemiology A stochastic SIS infection model incorporating indirect transmission Asset pricing, revised edition Dynamic asset pricing theory Capturing the time-varying drivers of an epidemic using stochastic dynamical systems The macroeconomics of epidemics Can the COVID bailouts save the economy? 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We haveThe second derivative of I t isLet t * denote the time at which the peak of the net change dI t /dt is reached, that is, when d 2 I t /dt 2 = 0. It is immediate to conclude that the curve dI t /dt peaks at t * , where