Default and Systemic Risk in Equilibrium

# Default and Systemic Risk in Equilibrium

Agostino Capponi School of Industrial Engineering, Purdue University, West Lafayette, IN, 47906, Email:capponi@purdue.edu    Martin Larsson School of Operations Research, Cornell University, Ithaca, NY, 14853, Email: mol23@cornell.edu
July 21, 2019
###### Abstract

We develop a finite horizon continuous time market model, where risk averse investors maximize utility from terminal wealth by dynamically investing in a risk-free money market account, a stock written on a default-free dividend process, and a defaultable bond, whose prices are determined via equilibrium. We analyze the endogenous interaction arising between the stock and the defaultable bond via the interplay between equilibrium behavior of investors, risk preferences and cyclicality properties of the default intensity. We find that the equilibrium price of the stock experiences a jump at default, despite that the default event has no causal impact on the dividend process. We characterize the direction of the jump in terms of a relation between investor preferences and the cyclicality properties of the default intensity. We conduct a similar analysis for the market price of risk and for the investor wealth process, and determine how heterogeneity of preferences affects the exposure to default carried by different investors.

## 1 Introduction

The default of a systemically important entity can have an impact on the rest of the economy through a number of different mechanisms. For instance, firms that have exposures to the defaulted entity through market transactions, can experience a deterioration in fundamentals driving the value of their assets. Under adverse circumstances this can lead to a domino effect, where the default of one firm causes financial distress on entities with which the firm had business relations. This distress can propagate through the financial system causing a cascading failure, leading in the worst case to the collapse of a significant portion of the system (the recent credit crisis being a clear example). In the context of interbank lending, Giesecke and Weber (2006) propose a reduced form contagion model, while Amini et al. (2010) and Amini et al. (2011) use tools from random graph theory to analyze short term counterparty credit exposures. Dynamic contagion models are considered in Dai Pra et al. (2009), and more recently in  Cvitanić et al. (2010) and Giesecke et al. (2011).

Alternatively, there may be a purely informational effect, where the default of one firm triggers the market participants to update their perception of the state of the economy. For example, Collin-Dufresne et al. (2003) show that the unexpected default of an individual firm can lead to a market-wide increase in credit spreads, and demonstrate via calibration that the risk premium due to contagion risk may be considerable.

A third possibility is that the sudden shock associated with the default event leads to a re-allocation of wealth as the economy returns to equilibrium. This may in turn cause rapid price changes due to linkages that stem from the equality between supply and demand. The aim of the present paper is to study this mechanism in a continuous time financial model, including default risk, where prices are determined endogenously in equilibrium.

While models of economic equilibrium have been studied for a long time, it is only recently that fully dynamic stochastic models of equilibrium have received significant attention. Dumas (1988) considers a dynamic equilibrium model with two investors, and characterizes the equilibrium behavior of the wealth allocation and risk-free rate, assuming that the stock returns are specified exogenously. Chabakauri (2010) considers a similar economy, but allows for the possibility of portfolio constraints, and analyzes cyclicality properties of market price of risk and stock return volatilities. Bhamra and Uppal (2009) consider a continuous time economy populated by two power utility agents with heterogenous beliefs and preferences, and give closed form expressions for consumption policies, portfolio policies, and asset prices. The same model as in Bhamra and Uppal (2009) is considered by Cvitanic et al. (2011) and Cvitanic and Malamud (2011a), who extend the results by Bhamra and Uppal to the case of an arbitrary number of agents, including an asymptotic analysis for large time horizons.  Cvitanic and Malamud (2011b) provide decompositions into myopic and non-myopic components for market price of risk, stock volatility, and hedging strategies. In the same economic model, Wang (1996) studies how investor preferences affect the term structure of interest rates.

The literature on dynamic equilibrium models, including the papers mentioned above, has been concerned primarily with models where equilibrium prices have continuous paths. This means that dramatic and sudden changes, such as crisis events or major defaults, are absent—and indeed these papers have focused on other economic phenomena. An exception is Hasler (2011), which considers a Lucas economy with multiple defaults, where the default intensities are constant.

In the present paper we study a finite horizon continuous time model, where rational investors maximize utility from terminal wealth. Three securities are liquidly and dynamically traded: a money market represented by a locally risk-free security, i.e. investors can borrow from or lend to each other without default, a stock representing shares of the aggregate endowment, and a defaultable bond which represents the corporate bond index (for example, the Dow Jones corporate bond index). We assume a constant recovery model, in which case the default of the bond index is interpreted as the default of one (or more) of the index bonds, which reduces the total payment of the index. The intensity of the defaultable bond may, but need not, depend on the dividend process.

As we demonstrate in the present paper, introducing a defaultable security in the economy leads to new insights regarding the behavior of securities prices, market price of risk, and wealth allocation. For instance, we find that the equilibrium price of the stock typically jumps when default occurs, despite the fact that the underlying dividend process is entirely unaffected by the default event. Moreover, the direction of the jump (up or down) depends in a non-trivial way on the interplay between investor preferences and the cyclicality properties of the default intensity. In particular, we show that upward jumps in the stock price are possible if, roughly speaking, the default intensity is sufficiently counter-cyclical. The precise statement is given in Theorem 2. We also show that a similar analysis, with similar conclusions, can be carried out for the wealth processes of individual investors, see Section 5. In this connection, we investigate how heterogeneity of preferences affects the exposure to the default carried by the different investors.

Due to the possibility of default, there are two sources of risk in our model: diffusion risk and jump risk. Using techniques from the theory of filtration expansions, which has a long and successful history in credit risk modeling, we are able to guarantee market completeness, even in the presence of jumps, see also Bielecki et al. (2006a) and Bielecki et al. (2006b) for a detailed analysis of market completeness and replication strategies in reduced form models of credit risk. This allows us to identify a unique market price of risk process, corresponding to diffusion risk, and default risk premium process, corresponding to jump risk. It turns out that the two quantities are intimately linked, see Proposition 2. By means of a quite delicate mathematical analysis, these quantities are studied in the case of constant interest rate and default intensity.

The most natural interpretation of the phenomena we study is as a form of systemic risk, arising in an economy consisting of securities carrying both market and default risk. While systemic risk effects generated from equilibrium models have been studied, for instance in Allen and Gale (2000) and Freixas et al. (2000), these papers use static discrete time models, exclusive of default, where the focus is on characterizing optimal risk sharing across banks with different credit profiles, or belonging to different geographical sectors. Differently from most research efforts, our model exhibits an endogenous interaction between the stock and the defaultable bond, which arises via the interplay between equilibrium behavior of the investors and their risk preferences.

The rest of the paper is organized as follows. Section 2 introduces the economic model. Section 3 analyzes the market price of risk in equilibrium along with its behavior at the default event. Section 4 characterizes the behavior of the equilibrium stock price at default via a relation between cyclicality properties of short rate and default intensity, and investor preferences. Section 5 performs a similar analysis for the wealth process of a risk-averse agent, and, in the case of a power utility investor, provides monotonicity relations between the size of the jump and the level of risk aversion. Section 6 concludes the paper. The proofs of the necessary lemmas are deferred to the appendix.

## 2 The Model

### 2.1 The Probabilistic Model

Let be a complete probability space, supporting a standard Brownian motion . Let be the augmented filtration generated by , which satisfies the usual hypotheses of completeness and right continuity. We use a standard construction (also called Cox construction) of the default time , based on doubly stochastic point processes, using a given nonnegative  adapted intensity process . To this end, we assume the existence of an exponentially distributed random variable defined on the probability space , independent of the process . The default time is then defined as

 τ=inf{t≥0:∫t0λsds≥χ}.

The market filtration , which describes the information available to investors, is given by

 Gt=⋂u>tFu∨σ(τ∧u).

That is, it contains all information in , together with the knowledge of whether has occurred or not, and has been made right-continuous. It is a well-known result (see e.g. Bielecki and Rutkowski (2001), Section 6.5 for details) that the process

 Mt=1{τ≤t}−∫t∧τ0λsds

is a -martingale under . In other words, is the default intensity (or hazard rate) of .

An important consequence of the previous construction is that Hypothesis (H) holds, i.e. every  martingale remains a  martingale, see Bielecki and Rutkowski (2001). It then follows from a result by Kusuoka (Theorem 5 in Appendix A) that every square integrable  martingale may be represented as a stochastic integral with respect to and .

### 2.2 The market model

We consider a market model, which is an extension of the standard setting in Cvitanić and Malamud (2010). We assume that there is an underlying dividend process with dynamics

 dDtDt=μD(Dt)dt+σD(Dt)dBt,D0>0. (1)

It is assumed that and are such that a strictly positive, strong solution exists. We also assume that and are infinitely differentiable on , and that .

There are two risky assets in the economy, a stock which carries market risk, and a defaultable bond which carries default risk. At terminal time , the stock pays a terminal dividend , while the defaultable bond pays a terminal dividend . The latter is given by

 PT=1{τ>T}+ε1{τ≤T}.

Here is a constant recovery value paid at time in case default happens at or before . We assume that is deterministic, although many calculations would still be valid as long as is -measurable. Neither the stock, nor the defaultable bond generates any intermediate dividends. We also assume the existence of a locally risk free money-market account with interest rate . Finally, we assume that the default intensity and interest rate are of the form

 λt=λ(Dt)andrt=r(Dt)

for deterministic functions and . The same assumption has also been used by Cvitanić and Malamud (2010) for the interest rate.

In our model, both the stock and the defaultable bond are positive net supply assets. In contrast, the zero money-market account is assumed to be available in zero net supply.

The market price at time of the stock is denoted by , and that of the defaultable bond by . These processes are determined in equilibrium, and their dynamics is of the form

 dStSt− =μStdt+σStdBt+ρStdMt dPtPt− =μPtdt+σPtdBt+ρPtdMt.

The existence of such representations follows from Theorem 5 together with the fact that in equilibrium, both and are semimartingales with absolutely continuous finite variation parts. Furthermore, we conjecture that the matrix

 (σStρStσPtρPt)

will be invertible in equilibrium. This immediately implies that the market is complete, via application of Theorem 5. It is then well known, see e.g. Cvitanić and Zapatero (2004), that there exists a unique state-price density process

 ξ=(ξt)0≤t≤T.

The time price of a payoff received at time  is given by .

### 2.3 The investors

There are a finite number of investors, indexed by , who optimize expected utility from final consumption. They are all assumed to have identical beliefs given by the historical probability , but can have different utility functions . These are assumed to be twice continuously differentiable, strictly concave, and satisfy Inada conditions at zero and infinity:

 limx↓0U′k(x)=∞andlimx→∞U′k(x)=0.

Two important measures of risk aversion, which will be used extensively in this paper, are the coefficients of absolute and relative risk aversion, both defined in Pratt (1964). The coefficient of absolute risk aversion is defined as

 ℓU(x)=−dlogU′(x)dx=−U′′(x)U′(x). (2)

Pratt related this measure to the agent’s risk behavior by showing that an agent with utility is more risk averse than an agent with utility if and only if for all . The coefficient of relative risk aversion is defined as

 LU(x)=−dlogU′(x)dlogx=−xU′′(x)U′(x). (3)

The :th investor chooses a dynamic portfolio strategy , a  predictable and -integrable process, where is the proportion of wealth invested in the stock at time , and is the proportion of wealth invested in the defaultable bond. The remaining wealth is invested in the money market account to make the strategy self-financing. The investor must choose his strategy so that the corresponding wealth process, given by

 dWktWkt−=rtdt+πSkt(dStSt−−rtdt)+πPkt(dPtPt−−rtdt), (4)

stays strictly positive for . The portfolio strategy is chosen to maximize the expected utility

 E[Uk(WkT)].

Market completeness allows one to use standard duality methods (see Cvitanić and Malamud (2010)) to show that the optimal final wealth in equilibrium is given by

 WkT=Ik(ykξT), (5)

where the number is the solution to the budget constraint equation,

 E[Ik(ykξT)ξT]=Wk0.

Moreover, the wealth at times is given by

 Wkt=E[ξTWkT∣Gt]ξt. (6)

### 2.4 The equilibrium

We employ the usual notion of equilibrium:

###### Definition 1

The market is said to be in equilibrium if each investor behaves optimally and all the securities markets clear.

Again by market completeness, standard equilibrium theory, see  Constantinides (1982), shows that security prices coincide with those in an artificial economy populated by a single, representative investor. We denote the corresponding utility function by , and assume that is twice continuously differentiable, strictly concave, and satisfies Inada conditions at zero and infinity. The state-price density is then given by

 ξT=U′(DT+PT). (7)

Furthermore,

 ξt=e−∫t0rsdsZt,

where is the Radon-Nikodym density process corresponding to the (unique) risk-neutral measure ,

Using Equation (7), the definition of and , and Lemma 4 in Appendix A, we can separate the state price density into a pre- and post-default component. More precisely, we have

 ξt=1{τ>t}ξpret+1{τ≤t}ξpostt,

where

 ξpostt =E[e∫TtrsdsU′(DT+ε)∣∣∣Ft] ξpret =E[(1−e−∫Ttλsds)e∫TtrsdsU′(DT+ε) +e−∫Ttλsdse∫TtrsdsU′(DT+1)∣∣∣Ft]. (8)

Remark. Assume that the intensity is deterministic and, for simplicity, that . We then have

 ξpret=P(τ≤T|τ>t)E[U′(DT+ε)∣Ft]+P(τ>T|τ>t)E[U′(DT+1)∣Ft],

indicating that the pre-default state price density is the weighted average of the state price density in an economy where default will surely happen, and the state price density in a default-free economy. The weights are, respectively, the probability that default will, or will not, take place before , given that it has not occurred up to time .

The equilibrium market price processes are computed using the state price density . They are given by

 St =E[ξTDT∣Gt]ξt=EQ[e−∫TtruduDT∣Gt] (9) Pt =E[ξTPT∣Gt]ξt=EQ[e−∫TtruduPT∣Gt].

Therefore, again relying on Lemma 4 in Appendix A, we obtain

 St=1{τ>t}Spret+1{τ≤t}Spostt, (10)

where

 Spostt = 1ξposttE[DTU′(DT+ϵ)∣∣∣Ft] Spret = 1ξpretE[(1−e−∫Ttλsds)DTU′(DT+ϵ)+e−∫TtλsdsDTU′(DT+1)∣∣∣Ft].

## 3 Equilibrium market price of risk

In this section we derive expressions for the market price of (diffusion and default) risk, as well as the risk premium of the stock. The risk premium is defined as the excess growth rate of the asset above the risk-free rate, namely .

By Theorem 5 the density process associated with the risk-neutral measure has the representation

 dZtZt−=−θtdBt+κtdMt

for some  predictable processes and . An application of Girsanov’s theorem shows that

 BQt=Bt+∫t0θsdsandMQt=Mt−∫t∧τ0κsλsds

are  local martingales, and in particular is  Brownian motion. Note that we can write

 MQt=1{τ≤t}−∫t∧τ0λs(1+κs)ds,

so that the risk-neutral default intensity is given by . The quantity is called the default risk premium, and is called the market price of risk. We fix this notation from now on.

###### Proposition 1

The market price of risk is given by

 θt=θpret1{τ≥t}+θpostt1{τ

where is the volatility of , and is the volatility of . The default risk premium is given by

 κt=ξposttξpret−1.

The risk premium associated with the stock, or the equity risk premium, is given by

 μSt−rt=σStθt−⎛⎝SposttSpret−1⎞⎠⎛⎝ξposttξpret−1⎞⎠λt1{τ≥t}.

Proof. The assertions concerning and follow from Lemma 5 and the definition of and , since . Let us establish the expression for the risk premium. The relations between and , respectively and , together with the -dynamics of the stock price yield

 dStSt−=[μSt−σStθt+ρStκtλt1{τ≥t}]dt+σStdBQt+ρStdMQt.

The drift term equals since the discounted stock price is a martingale under . The proof follows by substituting the expressions for and into the above equation (the latter follows from Lemma 5.)

Remark. The risk premium can alternatively be expressed in terms of the risk-neutral default intensity , using that . The result is

 μSt−rt=σtθt−⎛⎝SposttSpret−1⎞⎠⎛⎝1−ξpretξ% postt⎞⎠λQt1{τ≥t}.

It is clear from the definition of and that we always have . The contribution to the equity risk premium coming from default risk therefore has the same sign as . This quantity is minus the size of the jump in the stock price, were default to happen at time . In particular, if the stock price jumps down at default, then the investors require a premium for holding the stock, as they want to be compensated for the loss incurred upon default. On the other hand, if the stock jumps up at default, then it becomes an attractive security to hold, and therefore the investors are willing to pay a premium for holding it. We will study the sign of the jump in more detail in Section 4; suffice it to say here that positive price jumps, while atypical, are indeed possible.

There is an interesting relationship between the sensitivity of with respect to changes in the level of the dividend process, and the market price of diffusion risk. To state the result, first observe that the Markovian structure allows us to write

 κt=κ(t,Dt)

for some measurable function . We now have

###### Proposition 2

The function is differentiable, and the derivative is given by

 κx(t,Dt)=−1DtσD(Dt)ξposttξpret(θpostt−θ% pret)

Proof. As for , the Markovian structure allows us to write for and measurable functions . As in the proof of Theorem 1 below, we may apply Theorem 6.1 in Janson and Tysk (2006) to obtain the smoothness of , and hence of since by Proposition 1. Differentiating this relation yields

 κx=ξpostξpre(ξpostxξpost−ξprexξpre).

Now, the volatility of a positive  adapted semimartingale of the form is given by , as can be seen from Itô’s formula. By Proposition 1, is equal to minus the volatility of , which yields the result.

Observe that is the size of the jump in , if default were to occur at time . Proposition 2 shows in particular that if this quantity is positive, the default risk premium moves in the opposite direction to the dividend: an increase in the dividend process is accompanied by a decrease in the default risk premium, and vice versa. This appears to suggest that, upon default, a risk averse investor who sees an upward jump in the market price of risk, prefers to shift wealth from the risky stock to a default-free bond, giving a sure payoff of at maturity. If, on the other hand, is negative, the default risk premium moves in the same direction as the dividend.

We proceed to study how the market price of risk behaves at default. As we have just seen, this also provides information about the sensitivity of the default risk premium to changes in . The following result unfortunately requires us to assume constant interest rate and constant default intensity—already in this case the analysis is non-trivial (in particular it is much more delicate than for the jump in the stock price.) Extending it to more general and is an interesting problem that we leave for future research.

###### Theorem 1

Assume that the interest rate and default intensity are constant. If the representative investor has a strictly decreasing absolute risk aversion, then the market price of risk has a nonnegative jump at .

The rest of this section is devoted to proving Theorem 1. First, let us introduce some notation. For each , define the function

 uα(t,x)=E[U′(DT+α)∣Dt=x].

Using, for instance, Theorem 6.1 in Janson and Tysk (2006), we deduce that satisfies the PDE

 uαt+12x2σD(x)2uαxx+xμD(x)uαx=0,uα(T,x)=U′(x+α),

where the subscripts denote partial derivatives. Standard results then imply that has the same degree of smoothness as and on , see e.g. Theorem 10 in Chapter 3 of Friedman (2008). Since we assume that and are infinitely differentiable, the same holds for .

Proof of Theorem 1. Due to Lemma 1 below, the theorem will be proved once we establish that the quantity

 −∂∂xloguα(t,x)

is decreasing in . This is done in two stages: Lemma 2 gives the result when is bounded, and Lemma 3 then extends this to unbounded .

###### Lemma 1

Assume that the interest rate and default intensity are constant. If

 −uεx(t,x)uε(t,x)>−u1x(t,x)u1(t,x)

for all , then on .

Proof. It follows from (8) and the assumption of constant and that

 ξpostt=er(T−t)uε(t,Dt)

and

 ξpret=er(T−t)((1−e−λ(T−t))uε(t,Dt)+e−λ(T−t)u1(t,Dt)).

The volatility of a positive  adapted semimartingale of the form is given by , as can be seen from Itô’s formula. By Proposition 1 and the above expressions for and it then follows that

 θpostt=−uεx(t,Dt)uε(t,Dt)DtσD(Dt)

and

 θpret=−(1−e−λ(T−t))uεx(t,Dt)+e−λ(T−t)u1x(t,Dt)(1−e−λ(T−t))uε(t,Dt)+e−λ(T−t)u1(t,Dt)DtσD(Dt).

A calculation using that and are strictly positive reveals that if and only if

 −uεx(t,Dt)uε(t,Dt)>−u1x(t,Dt)u1(t,Dt).

The result now follows.

###### Lemma 2

Assume that the conditions of Theorem 1 are satisfied. Assume also that there is a constant such that and for all . Then

 −∂∂xloguα(t,x)

is strictly decreasing in .

Proof. Define . It can be readily verified that satisfies the terminal value problem

 ˜uαt+12x2σD(x)2˜uαxx+xμD(x)˜uαx+12x2σD(x)2(˜uαx)2 =0, ˜uα(T,x) =logU′(x+α).

Now define , and differentiate the above equation with respect to to see that satisfies the nonlinear PDE

 vαt+12x2σD(x)2vαxx +(xμD(x)+12[x2σD(x)2]x)vαx +[xμD(x)]xvα−12[x2σD(x)2(vα)2]x=0,

with terminal condition

 vα(T,x)=−U′′(x+α)U′(x+α)=ℓU(x+α).

Let us pick , and define . We want to prove that . The function satisfies the terminal value problem

 wt+12a(x)wxx+b(t,x)wx+c(t,x)w =0 (12) w(T,x) =ℓU(x+β)−ℓU(x+α),

where

 a(x) =x2σD(x)2 b(t,x) =xμD(x)+12[x2σD(x)2]x−12x2σD(x)2(vα(t,x)+vβ(t,x)) c(t,x) =[xμD(x)−12x2σD(x)2(vα(t,x)+vβ(t,x))]x.

Notice that , as we are assuming that the coefficient of absolute risk aversion is strictly decreasing. Moreover, the coefficients and are smooth due to the smoothness of , , and . The latter functions are smooth since they are the derivatives of the logarithm of the infinitely differentiable functions and .

Now, let be the solution to the SDE

 dXt=√a(Xt)dBt+b(t,Xt)dt,X0=D0.

The smoothness of and implies that a unique strong solution exists up to an explosion time, but since and for all , we have and there, so no explosion can occur. Indeed, holds for , almost surely.

Next, define a process by

 Yt=e∫t0c(s,Xs)dsw(t,Xt).

Itô’s formula and the fact that satisfies (12) show that

 dYt=e∫t0c(s,Xs)dswx(t,Xt)√a(Xt)dBt,

and since remains in a compact set and , and are continuous, the integrand in front of is bounded. Therefore is a martingale, and its final value is due to the boundary condition of . We deduce that for every almost surely, and hence that , as desired.

###### Lemma 3

Assume that the conditions of Theorem 1 are satisfied. Then

 −∂∂xloguα(t,x)

is nonincreasing in .

Proof. Fix . The goal is to show that . For each , let and be infinitely differentiable and coincide with , respectively , on , while being zero outside the interval . Denote by the solution to

 dDntDnt=μn(Dnt)dt+σn(Dnt)dBt,Dn0=D0,

and define . An application of Lemma 2 shows that

 −uα,nxuα,n>−uβ,nxuβ,n

for each . It thus suffices to prove that and pointwise. The latter follows from the former using interior Schauder estimates, for instance by applying the corollary of Theorem 15 in Chapter 3 of Friedman (2008) on each subdomain , (using the PDE representation of , and noticing that on each subdomain the coefficients of the parabolic operator associated to are Hölder continuous, and is bounded away from zero for all sufficiently large .)

To prove that , first note that and by the Markov property. Since is bounded, the desired convergence follows from the Bounded Convergence Theorem if almost surely, with . But this is clear: pathwise uniqueness and the construction of and imply that and coincide on the event

 An={n−1≤Ds≤n for\ all\ 0≤s≤T−t},

so . Since , almost surely, and the proof is finished.

## 4 Equilibrium stock price

In this section we are interested in how the market price of the stock changes when default occurs. If , there may be a jump in the stock price at . Under certain cyclicality assumptions on the default intensity and the interest rate, it turns out that the sign of the jump must be negative. On the other hand, in specific circumstances it can happen that the jump is positive. The following results gives the precise conditions. The proofs rely on a number of lemmas, which are stated and proved in Appendix B.

###### Theorem 2

Assume that the interest rate is counter-cyclical, and that the representative investor has strictly decreasing absolute risk aversion, as well as relative risk aversion bounded by one. Define

 g(t,x)=E[e−∫Ttλudu∣DT=x]andϕ(x)=1−U′(x+1)U′(x+ε). (13)

Then the following hold.

• If is strictly increasing in for every , the stock price has a strictly positive jump at .

• If is strictly decreasing in for every , the stock price has a strictly negative jump at .

Proof. Equations (8), (10) and (LABEL:eq:Sprepost2) show that the jump in the stock price is given by

 ΔSτ=atct−at−btct−dt∣∣∣t=τon{0<τ≤T}, (14)

where

 at =E[DTU′(DT+ε)∣Ft] bt =E[e−∫TtλuduDT(U′(DT+ε)−U′(DT+1))∣Ft] ct =E[e∫TtruduU′(DT+ε)∣Ft] dt =E[e∫Tt(ru−λu)du(U′(DT+ε)−U′(DT+1))∣Ft).

Using that and , elementary manipulations yields

 ΔSτ =1ξpret[Covt(e−∫Ttλuduϕ(DT),DTU′(DT+ε)) −SposttCovt(e−∫Ttλuduϕ(DT),e∫TtruduU′(DT+ε))]t=τ

on , where denotes -conditional covariance, and is defined in (13). It suffices to analyze the two covariances, since both and are strictly positive. Let us fix . By the Markov property of (and using that and ), we may without loss of generality assume that (and think of as ), as long as the starting point is allowed to be arbitrary.

By conditioning on , we find

 Cov(e−∫T0λuduϕ(DT),DTU′(DT+ε))=Cov(g(DT)ϕ(DT),DTU′(DT+ε))

and

 Cov(e−∫T0λuduϕ(DT),e∫T0ruduU′(DT+ε))=Cov(g(DT)ϕ(DT),f(DT)U′(DT+ε)),

where , and is given in (13). Since is counter-cyclical, is decreasing by Lemma 6, and hence is also decreasing. Moreover, the function has a derivative , which is strictly greater than zero if and only if

 1>−xU′′(x+ε)U′(x+ε)=−xx+ε(x+ε)U′′(x+ε)U′(x+ε)=xx+εLU(x+ε).

This is indeed the case since the relative risk aversion is less than or equal to one. Thus is strictly increasing.

Under the assumption of , is strictly increasing, so the first covariance is strictly positive, while the second is strictly negative. This uses the fact that for positive, strictly increasing functions and , and any non-constant random variable , , while if is strictly decreasing, .

Under the assumption of that is strictly decreasing, the situation reverses and the jump becomes strictly negative.

We also provide the following result, which shows that the stock price jump will be negative under more general conditions than those of Theorem 2.

###### Theorem 3

Assume that the interest rate is counter-cyclical and the default intensity pro-cyclical. If the representative agent has strictly decreasing absolute risk aversion, then the stock price has a strictly negative jump at .

Proof. Let , , and be as in the proof of Theorem 2. From Equation (14) we see that a sufficient condition for having a strictly negative jump is that for all . As in the proof of Theorem 2 it suffices to consider .

By Lemma 8 and the cyclicality of and , we have

 E[e∫T0(ru−λu)du∣DT]≥E[e∫T0rudu∣DT]E[e−∫T0λudu∣