Pandemic portfolio choice

1. Introduction

COVID-19 has shown that a pandemic can significantly increase biometric risk and at the same time trigger crashes of the stock market. Consequently, the threat of (future) pandemics should have a significant effect on financial decisions including the consumption-portfolio choice of a household. Taking this threat into account, the consumption-portfolio problem of an agent becomes a complex stochastic control problem. To capture the different states of the economy (normal vs. pandemic) and the health of the agent, our model involves a continuous-time Markov chain. The stock dynamics are modeled by a jump-diffusion process with state-dependent coefficients. We allow for a stock market crash when there is a transition into the pandemic state. Apart from these financial aspects, there are also effects on the personal situation of the agent: During the pandemic, she can be infected by the pandemic disease and recover or die. In other words, financial risk and health/mortality risk are intertwined, which distinguishes a pandemic from other crises.

We thus have to solve a dynamic stochastic control problem that contains a continuous-time Markov chain, different types of jump processes, as well as regime-dependent financial and non-financial parameters. The particular link between financial and health risks in the context of portfolio choice is new and leads to an involved dynamic optimization problem where the first-order conditions are mathematically challenging compared to pure regime-switching frameworks. Likewise, proving existence and uniqueness of the relevant solutions must follow a non-standard routine. First, we prove a verification theorem and show that the solution to the problem is characterized by a system of Hamilton-Jacobi-Bellman (HJB) equations. This system is coupled with optimality conditions, which complicates the analysis. Second, we show that there exists a unique solution to this coupled system. It is given by the unique solution to a system of non-linear ordinary differential equations. Another contribution of our paper is that we analyze our dynamic portfolio problem also for recursive utility in the sense of Duffie & Epstein (1992), which is a generalization of continuous-time-additive utility.

Our numerical results show that pandemics have a sizable effect on the portfolio and consumption strategy of an agent. In general, the mere threat of a pandemic significantly reduces her stock investment outside of a pandemic (normal times). The possibility of a pandemic crash links the portfolio decision to the optimal consumption policy so that a potential reduction of consumption during a pandemic amplifies the reduction of the stock investment when there is no pandemic. More precisely, we show that a higher probability of a pandemic significantly reduces the stock share in normal times. Furthermore, higher additional costs during a pandemic also lead to a reduction of the stock share outside of a pandemic. Finally, the expected duration of a pandemic reduces the stock share during a pandemic, whereas the effect is moderate in normal times. Since we also consider recursive utility, we can disentangle the effects of risk aversion and elasticity of substitution (EIS) on the agent’s decision. We find that her risk aversion affects both portfolio and consumption decisions, whereas the EIS mainly affects her consumption decision.

Our paper contributes to a strand of the literature that studies an individual’s consumption-portfolio decision which is formulated as a continuous optimization problem. Continuous-time portfolio theory was established by Merton, 1969, Merton, 1971. Kolm, Tütüncü, & Fabozzi (2014) provide a comprehensive survey over discrete-time portfolio theory.1 There are extensions in multiple directions: Korn & Kraft (2001) study a dynamic portfolio problem with stochastic interest rates. Liu, Longstaff, & Pan (2003) consider such a problem with event risk that does not contain a Markov chain. Ang & Bekaert (2002) and Guidolin & Timmermann (2008) study international portfolio choice with a regime-switching model in discrete-time settings. Honda (2003) considers a portfolio problem with unobservable state. Sass & Haussmann (2004) and Bäuerle & Rieder (2007) study portfolio problems with a hidden Markov chain. Ait-Sahalia, Cacho-Diaz, & Hurd (2009) derive an explicit solution to a problem with jumps. Ait-Sahalia & Hurd (2016) study a consumption-portfolio problem in a class of multidimensional jump-diffusion models in which asset prices are subject to mutually exciting jump processes. Closely related are papers studying regime-switching models such as Sotomayor & Cadenillas (2009), Fu, Wei, & Yang (2014), and Liu & Chen (2018). Sotomayor & Cadenillas (2009) analyze a portfolio problem for a regime-switching model without jumps and derive a verification theorem in this setting. Fu et al. (2014) consider a regime-switching model without jumps and add derivatives to their framework. Liu & Chen (2018) study a dynamic portfolio problem and regime switching with time-consistent risk measures. All these papers do not contain a framework as in our paper where the first-order conditions and HJB equations are coupled. Bäuerle & Rieder (2009) analyze a financial market with jumps only. Furthermore, there are papers that focus on applications in credit risk such as Kraft & Steffensen (2009). In these models, however, the Markov chains are typically directed, since the number of entities is shrinking over time. This is not the case in our framework. Finally, none of the above papers studies agents with recursive utility.

Notice that our paper uses a primal approach. Recently, there have also been several papers using dual approaches. For instance, Bick, Kraft, & Munk (2013) introduce a method delivering explicit close-to-optimal solutions in incomplete markets. This approach is generalized by Kamma & Pelsser (2021). Chen & Vellekoop (2017) use a duality approach to study a portfolio problem where terminal wealth is allowed to be negative. By contrast, Kraft & Steffensen (2013) analyze portfolio problems with additional constraints.

The remainder of our paper is structured as follows: Section 2 introduces the model setup. Section 3 describes the agent’s stochastic control problem. Section 4 derives the system of HJB equations and the optimality conditions. Section 5 provides economic intuition behind our results. In Section 6, we prove a verification theorem for our setup and show that a unique solution to the system of HJB equations exists. This solution determines the optimal controls of the problem. Section 7 extends our results to continuous-time recursive utility in the sense of Duffie & Epstein (1992). Section 8 provides a detailed calibration of our model. Section 9 reports numerical results. Section 10 concludes. An online appendix contains additional material.

2. Model

We study the consumption-portfolio problem of an investor (agent) who faces the risk that a pandemic starts to spread. To capture this, we introduce a four-state Markov chain $Z$ with state space $\mathcal{J}=\{0,1,2,3\}$. Here $Z_t=0$ means that there is currently no pandemic and $Z_t=1$ means that there is one. The probabilities of switching back and forth between these two states are modeled by the state-dependent transition intensity ${\kappa }^j$ where ${\kappa }^0$ is the intensity for jumping from state 0 to state 1, i.e., for being hit by a pandemic. On the other hand, ${\kappa }^1$ is the intensity for jumping back to the non-pandemic state. This could for instance happen if a vaccine has been developed and herd immunity is reached.

Regardless of whether there is currently a pandemic, the agent can die from an ordinary cause. The corresponding intensity is denoted by $\lambda \left(t\right)$, which is assumed to be a continuous function of time $t$.2 Notice that time can be interpreted as the age of the investor, i.e., time-dependent is equivalent to age-dependent. If she dies from an ordinary cause, then the Markov chain jumps to state 2. During a pandemic there is also some excess mortality to die from the pandemic disease. If this happens, then the Markov chain jumps from state 1 to state 3. The corresponding intensity is denoted by $\Delta \lambda \left(t\right)$ and is assumed to be a continuous function of time. To summarize, state 0 and 1 characterize the pandemic situation, whereas state 2 and 3 identify the investor’s cause of death. Formally, state 0 and 1 are transient, whereas state 2 and 3 are absorbing.

Motivated by the experience during COVID-19, pandemics do not only affect the mortality risk of the agent, but also have an impact on the financial market. Therefore, the asset dynamics can depend on the pandemic situation, i.e., on whether the Markov chain is in state 0 or 1. Notice that our model ends with the death of the investor. Therefore, we do not specify dynamics for state 2 or 3. Fig. 1 depicts the Markov chain that characterizes our setup.3

The agent can invest in a Treasury bond with constant interest $r$ and a stock index (short: stock) that is exposed to pandemic risk.4 For $Z_t\in \{0,1\}$, the asset dynamics are5

where ${\mu }^j$ is the state-dependent drift and ${\sigma }^j$ is the state-dependent volatility of the stock index, $j=1,2$. However, the drift and the volatility are constant over time in each state. The stock dynamics involve a Brownian motion $W$ and point processes $N^j$, $N_R^j$, and $N_P^j$. The meaning of the point processes is as follows: $N^j$ models a stock market crash in state $j$. This happens with intensity ${\chi }^j$ and the relative jump size is ${\ell }^j$. $N_R^j$ models a stock market recovery in state $j$. This happens with intensity ${\zeta }^j$ and the relative jump size is $p^j$. Conditioning on being in state $j$, both $N^j$ and $N_R^j$ are Poisson processes. Besides, $N_P^0$ counts the number of jumps into a pandemic, whereas $N_P^1$ counts the number of jumps out of a pandemic, i.e.,

where the intensity of $N_P^j$ is ${\kappa }^j$. These jumps also trigger pandemic crashes or recoveries of the stock market and the corresponding jump size is denoted by $L^j$. A pandemic crash could be an event such as the stock market crash on March 9, 2020. A pandemic recovery can be thought of as a pandemic-induced stock rebound, for instance if stock market participants realize that a majority of the population can be vaccinated and herd immunity will be reached. Therefore, $L^0$ is positive number leading to a loss due to the minus in front of $d{N}_{Pt}^{Z_{t-}}$ in (2.1), whereas $L^1$ is negative number leading to a gain. Since $N_P^0$ cannot jump in state 1 and $N_P^1$ cannot jump in state 0, the stock dynamics (2.1) involve only one of these processes depending on $Z$.

Additionally, we assume that the investor can get seriously sick in state 0 or 1. This happens with (state-indpendent) intensity $\nu$. Such a sickness is costly for the agent (e.g., co-payments) and for simplicity we assume that a fraction of wealth is lost, which is captured by the relative loss $k$.6 In the pandemic state, $Z_t=1$, it is also possible to be infected by the pandemic disease and survive. This happens with intensity $\Delta \nu$ and the relative costs are $k^P$. During the pandemic, the agent can get infected more than once. By introducing an additional state of the Markov chain, the number of infections could be restricted to be at most one, but it turns out to be less relevant in our numerical examples.

Now, let be ${\eta }^Z\equiv {\mu }^Z-r$ be the expected excess return of the stock index and $\pi$ the portfolio strategy (proportion invested in stock). Our model ends at the stopping time $\tau$ defined as

where ${\tau }_P$ is the time of death if she dies from the pandemic disease and ${\tau }_o$ is the time of death if she dies from an ordinary cause. Formally, ${\tau }_o\equiv \inf \{t\ge 0:Z_t=2\}$ and ${\tau }_P\equiv \inf \{t\ge 0:Z_t=3\}$. The constant $T$ is a time point where it is extremely unlikely that the agent is still alive and mortality tables typically end, e.g. an age of 120.7 For $t\in [0,\tau )$, the investor’s wealth dynamics thus read

where $b^1$ captures additional relative expenses during the pandemic and we set $b^0\equiv 0$. The meaning of the counting processes is as follows: $N_H$ counts the number of ordinary non-deadly health shocks and $N_{HP}^Z$ counts the number of pandemic non-deadly health shocks. Notice that the intensity of $N_{HP}^Z$ is zero in state 0 and $\Delta \nu$ in state 1, i.e., the intensity can be written as $Z_t\Delta \nu$ for $Z_t\in \{0,1\}$. Furthermore, $N_D$ jumps if the agent dies from an ordinary cause and $N_{DP}^Z$ jumps if the agent dies from the pandemic disease. The intensity of $N_{DP}^Z$ can be written as $Z_t\Delta \lambda$ for $Z_t\in \{0,1\}$.

The investor consumes at a rate of $c$. In our benchmark case, she optimizes time-additive utility of consumption and bequest with respect to the utility function

where $\gamma >1$ is her risk-aversion coefficient.8 Her time preferences are captured by the time-preference rate $\delta$. For some strategy $(c,\pi )$ and $t\in [0,\tau ]$, we can thus define the corresponding utility index by

where $j=0,1$ and $\epsilon$ is her weight of bequest. We assume

In Section 8, we calibrate our model. Table 1 summarizes the corresponding parameters.

3. Optimization problem of the agent

As usual, we focus on strategies that are said to be admissible. To formalize this idea, let $(\Omega ,\mathcal{F},{\left\{\mathcal{F}\right\}}_{t\in [0,T]},P)$ be a probability space where ${\left\{\mathcal{F}\right\}}_{t\in [0,T]}$ is the canonical filtration that is generated by the Brownian motion $W$, the Markov Chain $Z$, and the jump processes introduced in the previous section.9 Then we can define:

Definition 3.1 Admissible Strategy —

An ${\mathbb{R}}^2$-valued predictable process $(c,\pi )$ is admissible if

• $\left(a\right)$ for any initial value $X_t=x>0$, $t\in [0,\tau )$, the stochastic differential Eq. (2.2) has a unique solution,

• $\left(b\right)$ the utility index is finite,10 i.e., $J^j(t,x;c,\pi )>-\infty$, and

• $\left(c\right)$ the following integrability conditions are satisfied

  $${\text{E}}^{t,x,j}\left[{\int }_t^{\tau }{\left(X_s^{1-\gamma }{\pi }_s\right)}^{2}ds\right]<\infty ,{\text{E}}^{t,x,j}\left[{\int }_t^{\tau }{X}_s^{1-\gamma }{(1-{\ell }^{Z_s}{\pi }_s)}^{1-\gamma }ds\right]<\infty ,$$

  $${\text{E}}^{t,x,j}\left[{\int }_t^{\tau }{X}_s^{1-\gamma }{(1+p^{Z_s}{\pi }_s)}^{1-\gamma }ds\right]<\infty ,{\text{E}}^{t,x,j}\left[{\int }_t^{\tau }{X}_s^{1-\gamma }{(1-L^{Z_s}{\pi }_s)}^{1-\gamma }ds\right]<\infty .$$

The set of admissible strategies is denoted by $\mathcal{A}(t,x)$.

Remarks. (a) The integrability conditions make sure that the Brownian and Poissonian integrals are true martingales, since the integrands are ${\mathcal{L}}^2$- or ${\mathcal{L}}^1$-integrable, respectively. This excludes true local martingales and thus policies that are clearly inferior to an optimal strategy.

(b) A portfolio strategy $\pi$ is a stochastic process that can depend on time $t$, wealth $X$, and the state of the economy $Z$. If the economy is in a particular state $j$, it is sometimes convenient to denote the policy for that state by ${\pi }^j$. From this perspective, the optimal $\pi$ of our problem is characterized by the policies ${\pi }^0$ and ${\pi }^1$ in state 0 and 1.

We will show below that the optimal strategy satisfies all these requirements. Notice that conditions (i) and (ii) are reasonable and weak requirements: Condition (i) makes sure that the wealth equation is well-defined. Condition (ii) excludes strategies with extremely negative utility that cannot be optimal anyway. We now define the maximum size of the potential crashes and recoveries:

The following proposition shows that all strategies which do not lead to poor outcomes if a jump occurs and which involve consumption rates being at most linear in wealth are admissible.

Proposition 3.2 Tame Strategies Are Admissible —

Let$(c,\pi )$be an${\mathbb{R}}^2$-valued predictable process satisfying the following conditions:

• (i)

  For state 0, there exist a lower bound${\underset{\_}{\pi }}^0>-1/p^0$and an upper bound${\overline{\pi }}^0<1/{\overline{\ell }}^0$such that${\underset{\_}{\pi }}^0\le {\pi }^0\le {\overline{\pi }}^0$.

• (ii)

  For state 1, there exist a lower bound${\underset{\_}{\pi }}^1>-1/{\underset{\_}{p}}^1$and an upper bound${\overline{\pi }}^1<1/{\ell }^1$such that${\underset{\_}{\pi }}^1\le \pi \le {\overline{\pi }}^1$.

• (iii)

  Consumption is at most linear in wealth, i.e., there is a constant$k>0$such that$c\le kX$.

Then$(c,\pi )$is admissible. We refer to admissible strategies satisfying (i)-(iii) as tame. Analogously, the portfolio strategy$\pi$of an admissible policy$(c,\pi )$is called tame if it satisfies (i) and (ii).

Remark. Conditions (i) and (ii) are weak requirements. Notice that empirically jump sizes for the S&P500 are less then 20%. For such a calibration, condition (ii) implies that the portfolio strategy must be between $-500\%$ and $+500\%$. Strategies with more leverage that do not satisfy this condition are extremely unrealistic.

Proof of Proposition 3.2..

Conditions (i) and (ii) guarantee that the SDE (2.2) has a unique solution with ${\text{E}}^{t,x}\left[X^{1-\gamma }\right]<\infty$. Therefore, we also get (b) of Definition 3.1. Since ${\text{E}}^{t,x}\left[X^{2(1-\gamma )}\right]<\infty$ as well and all terms involving $\pi$ in the integrability conditions of (c) in Definition 3.1 are bounded, we obtain (c) and thus admissibility of $(c,\pi )$. □

We now define the value function of our consumption-portfolio problem:

for the initial values $X_t=x\in (0,\infty )$ and $Z_t=j\in \{0,1\}$, $t\in [0,\tau )$.

4. Hamilton-Jacobi-Bellman equations

As we will formally show in Section 6, the solution of the problem can be determined by studying two-coupled HJB equations for state 0 and 1 that are given by

where $\tilde{\lambda }\equiv \lambda +\Delta \lambda$ is the aggregated intensity to die from an ordinary case or due to the pandemic(“ord. or pan. death”). For both HJB equations, the terminal condition is

In both states, optimal consumption reads $c^{j*}={\left(G_x^j\right)}^{-\frac{1}{\gamma }}$, $j=0,1$. We conjecture

and obtain

The HJB equations simplify to

with terminal conditions $g^j\left(T\right)=\epsilon$, $j=0,1$, where

The optimal strategy ${\pi }^0$ in state 0 satisfies the first-order condition

whereas the optimal strategy ${\pi }^1$ in state 1 satisfies the first-order condition

5. Economic interpretation of optimal policies

The optimal portfolio strategy ${\pi }^{0*}$ in the normal state can be written as11

Compared to the classical Merton result, the optimal portfolio strategy has three additional terms. The first additional term captures the positive effect of a stock market recovery. The last two terms are crash terms that start with a ratio: expected jump size, ${\chi }^0{\ell }^0$ or ${\kappa }^0{L}^0$, divided by diffusive variance ${\left({\sigma }^0\right)}^2$ scaled by the risk aversion $\gamma$ of the agent. Both terms have a negative sign and thus reduce the stock proportion. However, the pandemic crash term also involves the ratio $g^1/g^0$. To understand its effect, consider the first-order conditions for consumption

where $\Delta c\equiv (c^0-c^1)/c^0$ is the relative consumption loss upon entering the pandemic state. The ratio $g^1/g^0$ thus measures the severeness of this loss. Consequently, the pandemic crash term is the product of an ordinary crash term and an adjustment for this loss:

The adjustment term increases in the consumption loss and reduces the optimal stock proportion further compared to ordinary stock crashes. Therefore, pandemic crashes lead to a novel link between consumption and investments that does not exist in frameworks with ordinary crashes only.

Notice that effects of a pandemic-induced stock crash and consumption reduction are not simply additive. To see this, we focus on the last two factors in (5.2) and apply a first-order approximation

The last term, ${\gamma }^2{\pi }^{0*}{L}^0\Delta c^{*}$ is a cross term showing that there is an interaction between stock crash and consumption reduction.

6. Verification, existence, and uniqueness

This section is concerned with proving two theorems: First, we show that if there is a solution to the HJB Eqs. (4.1) and (4.2), then it is indeed the solution to the consumption-portfolio problem. This result is summarized in Theorem 6.1. Second, we prove that there exists a unique solution to the HJB equations. This is summarized in Theorem 6.2.

7. Recursive utility

We now study an investor whose preferences are captured by continuous-time recursive utility (syn. stochastic differential utility), which was introduced by Duffie & Epstein (1992). In contrast to time-additive utility, recursive utility allows us to disentangle the agent’s risk aversion from her elasticity of intertemporal substitution (EIS). This is important since these concepts capture different phenomena: Risk aversion determines how much an agent invests in risky assets, whereas EIS is decisive for how the agent distributes her consumption over time. The utility index for some strategy $(c,\pi )$ and $t\in [0,\tau )$ is defined by the recursion

where $\epsilon >0$ is the weight on the terminal condition. The aggregator $f$ is given by

where $\delta$ is the time-preference rate, $\gamma$ the relative risk aversion, and $\psi \equiv 1/\varphi$ the elasticity of intertemporal substitution (EIS).

Notice that $\gamma >\varphi$ implies preferences for early resolution of uncertainty, i.e., the agent prefers to obtain any piece of information rather earlier than later.14 For $\gamma =\varphi$, one obtains time-additive utility as in the previous sections, which implies neutrality towards the resolution of uncertainty. The value function is defined by

Duffie & Epstein (1992) show that the solution to problems with stochastic differential utility is also characterized by a system of coupled HJB equations where the aggregator $f$ replaces the term $u\left(c\right)-\delta G^j$.15 The corresponding HJB equations can be found in Online Appendix C. A similar conjecture as for CRRA utility,

leads to the following first-order conditions for consumption

Notice that optimal consumption (7.4) is different from the time-additive case (4.5) since the exponent is now $-\psi /\theta$ (compared to $-1/\gamma$). Consumption thus depends on the agent’s risk aversion and her EIS. The functions $g^j$ are determined by a system of simplified HJB equations:

for the non-pandemic state (state 0) and

for the pandemic state (state 1).

The terminal condition (4.3) and the conjecture (7.3) lead to the terminal conditions

Notice that in contrast to consumption, the first-order conditions for the portfolio policy do not change compared to time-additive utility and are again (4.8) and (4.9). Quantitatively, however, the results do change, since $g^0$ and $g^1$ will be different compared to the time-additive case. This is a consequence of the pandemic related jumps $L_0$ and $L_1$. For ordinary portfolio problems where the stock price follows a jump-diffusion process as in Merton (1971), this is not the case and the strategies are the same for time-additive and recursive utility.

We now prove that the above HJB equations have a unique solution if the EIS is below unity and the risk aversion is above two. For our benchmark choice of $\gamma =5$, we can thus freely vary the EIS between zero and unity and still be sure that a unique solution exists. Notice that for $\gamma =5$, values of $\psi$ below $1/\gamma$ are less relevant, since they imply preferences for late resolution of uncertainty. In any case, we have enough degrees of freedom to analyze the effect of the EIS on consumption and portfolio choice in the context of pandemics.

Proposition 7.1 Existence and Uniqueness for Recursive Utility —

If$\psi \in (0,1)$and$\gamma +\psi >2$, then the results from Theorem6.2also hold for(7.5)and(7.6).

Proof

Under the conditions of this proposition, we have $\theta >0$ and $1-\frac{\psi }{\theta }\in (0,1)$. Therefore, all arguments of Theorem 6.2 go through since the non-linear terms are qualitatively of the same form. □

Remark. The case $\gamma +\psi =2$ is of minor importance, since this restriction implies preferences for late resolution of uncertainty. See Kraft, Seifried, & Steffensen (2013) for a discussion in a different setting.

8. Calibration

In our benchmark case, we consider an agent with time-additive utility and a risk aversion of $\gamma =5$. The weight on bequest is $\epsilon =1$.16 We use the mortality table provided by the US social security administration to calculate the mortality intensity $\lambda \left(t\right)$ of a female agent.17 To estimate the excess mortality of COVID-19, we use US data that is available on the website of the Center for Disease Control Prevention (CDC).18

The average percentage of COVID-19 related deaths compared to all deaths is about 15% for females above 55 years. Since the agent of our benchmark case is female and has an initial age of 60 years, we thus set

To calibrate ${\kappa }^0$ and ${\kappa }^1$, which are the intensities to enter or leave the pandemic state, we use significant historic pandemics since 1950 that had a worldwide impact.19