Download Equilibrium Analysis of Expected Shortfall

Equilibrium Analysis of Expected Shortfall
Pengyu Wei∗
April 23, 2017
ABSTRACT
This article studies optimal, dynamic portfolio and wealth/consumption policies
of expected utility maximizing investors who must also manage market-risk exposure
which is measured by Expected Shortfall (ES). We find that ES managers can incur
larger losses when losses occur, compared to both VaR and benchmark managers. A
general-equilibrium analysis reveals that the presence of ES managers increases the
market volatility during periods of significant financial market stress, in both pureexchange and production economies.
Key Words: portfolio selection; risk measure; value-at-risk; expected shortfall; general
equilibrium; asset pricing
∗
Mathematical Institute and Oxford-Man Institute of Quantitative Finance, The University of Oxford,
Oxford OX2 6GG, UK. E-mail: [email protected]. Preliminary Version, Comments Welcome
1
Introduction
This article analyzes the impact of market-risk regulation on portfolio choice and assets
prices. We study the impact of Expected Shortfall (ES), its partial equilibrium incentives, and the general equilibrium asset-pricing implications. This is motivated by the
recent advancement in risk measurement.
Value-at-Risk (VaR) has long been an industry standard by choice or by regulation Jorion (1997); SEC (1997); Dowd (1998); Saunders (2000); Jorion (2002); BCBS
(2011). However, ever since its introduction around 1994, VaR has been criticized in
both academia and industry, for its weaknesses as the benchmark. VaR fails to capture
“tail risk” and it is not subadditive, defying the notion of diversification.
Recognizing the shortcomings of VaR, there is an advocacy, especially recently, to
replace VaR with ES, both in academia Artzner et al. (1999); Rockafellar and Uryasev
(2000); Acerbi and Tasche (2002); Rockafellar and Uryasev (2002); Embrechts et al.
(2014) and in industry BCBS (2012, 2013, 2016). For example, as stated in BCBS
(2016), one of the key enhancements in the revised market risk framework is a shift
from Value-at-Risk (VaR) to an Expected Shortfall (ES) measure of risk under stress.
Use of ES will help to ensure a more prudent capture of ”tail risk” and capital adequacy during periods of significant financial market stress. Nevertheless, there is limited understanding of the potential impact of the ES regulation. This paper contributes
towards filling this gap.
We first study dynamic portfolio selection of expected utility maximizing investors
who must also manage market-risk exposure measured by ES. In the partial equilibrium,
investors with the ES constraint are induced to take on a larger risk exposure than in the
unconstrained setting during the adverse states of the market, thereby incurring larger
losses. We then develop a production and a pure-exchange general equilibrium model
featuring ES risk managers respectively. We find that ES managers optimally choose
larger risk exposure and thus increase the market volatility during periods of significant
financial market stress in both economies. These findings are similar to those of VaR
in Basak and Shapiro (2001). One common criticism on VaR is that it fails to take the
magnitude of losses into account. Although ES takes the sizes of losses into account, it
2
does not suffice to alleviate risk-taking behaviors.
Our paper is closely related to Basak and Shapiro (2001); Leippold et al. (2006).
Basak and Shapiro (2001) study the effects of VaR on optimal wealth and consumption
policies of risk managers. They find that VaR risk managers may take on more risk than
non-risk managers and thus increase the stock market volatility during market downturns. Leippold et al. (2006) study the asset-pricing implications of VaR regulation in
incomplete continuous-time economies with stochastic opportunity set and heterogeneous attitudes to risk, where the VaR constraint is applied dynamically. In contrast,
we analyze the asset pricing implications of ES, in both pure-exchange and production
economies.
Finally, let us comment on the setting of our model. Following Basak and Shapiro
(2001); Basak et al. (2006), we measure the investment risk by applying a risk measure
on the terminal wealth. The risk is evaluated at the beginning of the investment and the
agent needs to commit himself to comply with the constraint in all future dates. This is
due to the static nature of ES. There are papers, e.g., Yiu (2004); Cuoco and Liu (2006);
Leippold et al. (2006); Cuoco et al. (2008), that apply VaR dynamically. We do not
attempt this approach in the current paper for the following reasons. Firstly, investors
may not monitor the risk of an investment continuously as in these papers. Regulators
typically require investors to report their risks over a certain time period. For example,
it is stated in SEC (1997) that companies must disclose quantitative information at the
end of a fiscal year, whereas in BCBS (2011) banks are required to compute VaR on
a daily basis. The time horizon T in our paper can be chosen to match the industry
practices. Secondly, it is shown in Cuoco et al. (2008) that dynamic risk constraints can
be translated into position limits, which reduce investors’ risk exposure. In particular,
when VaR limits are dynamically updated, no unappealing incentives arise. However,
it is well understood that VaR regulation increases risk exposures, making such models questionable. Moreover, in such models, constraints under most risk measures are
qualitatively the same and there is no point in understanding impacts of different risk
measures. Thirdly, when risk constraints are applied dynamically, closed-form solutions
are unavailable and one must resort to numerical solutions or asymptotic solutions. In
3
contrast, our setting allows us to calculate relevant quantities in the general equilibrium
models explicitly.
The rest of the paper is organized as follow: In Section 2, we formulate the ESbased risk management (ES-RM) problem, which is then analyzed in Section 3. Section
4 carries out general equilibrium analysis. Section 5 concludes. All remaining proofs
are placed in Appendix A.
2
Model
In this section, we formulate our ES-based risk management (ES-RM) problem.
2.1
Market
Let T > 0 be a given terminal time and (Ω, F, (Ft )0≤t≤T , P) be a filtered probability
space on which is defined a standard (Ft )t∈[0,T ] -adapted n-dimensional Brownian motion W (t) ≡ (W 1 (t), , W n (t))> with W (0) = 0, and hence the probability space is
atomless. It is assumed that Ft = σ{W (s) : 0 ≤ s ≤ t}, augmented by all the P-null
sets in F. Here and henceforth, A> denotes the transpose of a matrix A. We define a
continuous-time financial market following Karatzas and Shreve (1998). In the market
there are n + 1 assets being traded continuously. One of the assets is a bank account
whose price process S0 (t) is subject to the following equation:
dS0 (t) = r(t)S0 (t)dt, t ∈ [0, T ]; S0 (0) = s0 > 0
(1)
where the interest rate r(·) is a uniformly bounded, (Ft )t∈[0,T ] -progressively measurable, scalar-valued stochastic process. The other n assets are stocks whose price processes Si (t), i = 1, · · · , n, satisfy the following stochastic differential equation (SDE):
dSi (t) = Si (t)[µi (t)dt +
n
X
σij (t)dW j (t)], t ∈ [0, T ]; Si (0) = si > 0
j=1
4
(2)
where µi (·) and σij (·), the appreciation and volatility rates respectively, are scalarvalued, (Ft )t∈[0,T ] -progressively measurable stochastic processes with
Z
0
T
n
n X
n
X
X
[
|µi (t)| +
|σij (t)|2 ]ds < ∞, a.s..
i=1
i=1 j=1
Define
µ(t) := (µ1 (t), · · · , µn (t))> ,
σ(t) := (σij (t))n×n ,
B(t) := (µ1 (t) − r(t), · · · , µn (t) − r(t))> .
Basic assumptions imposed on the market parameters are summarized as follows:
Assumption 2.1. There exists an unique, (Ft )t∈[0,T ] -progressively measurable, Rn 1
valued process θ(t) with Ee 2
RT
0
|θ(t)|2 dt
< ∞ such that
σ(t)θ(t) = B(t), a.s., a.e. t ∈ [0, T ].
Consequently, we have a complete model of a securities market. Consider an economic agent, with an initial endowment X(0) > 0 and an investment horizon [0, T ],
whose total wealth at time t ≥ 0 is denoted by X(t). Assume that the trading of shares
takes place continuously in a self-financing fashion and there are no transaction costs.
Then X(·) satisfies
dX(t) = X(t)[r(t)+B > (t)π(t)]dt+X(t)π > (t)σ(t)dW (t), t ∈ [0, T ]; X(0) = x, (3)
where πi (t) denotes the fractions of the agents wealth in stock i at time t.
The process π(·) ≡ (π1 (·), · · · , πm (·)) is called an admissible portfolio if it is Ft progressively measurable with
Z
T
>
2
Z
|X(t)σ (t)π(t)| dt < ∞ and
0
T
|X(t)B > (t)π(t)|dt < ∞, a.s.,
0
and is tame, i.e., the corresponding wealth process X(·) is almost surely bounded from
below- although the bound may depend on π(·). It is standard in the continuous-time
5
portfolio choice literature that a portfolio is required to be tame so as to, among other
things, exclude the doubling strategy.
With the complete market assumption, we can define the pricing kernel or state price
density process
Z t
Z t
1
>
2
θ(s) dW (s) .
ξ(t) := exp − [r(s) + |θ(s)| ]ds −
2
0
0
Denote ξ := ξ(T ). It is clear that under 2.1 and the uniformly boundedness of r(·), 0 <
ξ < +∞ a.s. and 0 < Eξ < +∞. Then, in view of the standard martingale approach,
Pliska (1986); Cox and Huang (1989); Karatzas and Shreve (1998), finding the optimal
portfolio in this economy is equivalent to finding the optimal terminal wealth.
2.2
Benchmark Agent
A Benchmark agent is a standard utility maximizer with the following problem,
max E[u(X(T ))]
X(T )
(4)
subject to E[ξX(T )] ≤ X(0).
u is the utility function with the following assumptions as in most literatures:
Assumption 2.2. u : R+ → R is strictly increasing and continuously differentiable.
0
0
Furthermore, u (·) is strictly decreasing and satisfies the Inada condition, i.e., u (0+) =
0
+∞ and u (+∞) = 0.
2.3
Value-at-Risk-based Risk Management
Recognizing that risk management is typically not an economic agent’s primary objective, Basak and Shapiro (2001) focus on portfolio choice within the familiar (continuous
time) complete markets setting, with the assumption that agents need to limit their risks,
measured by the VaR, while maximizing expected utility.
6
The Value-at-Risk-based risk management (VaR-RM) model is given by
max E[u(X(T ))]
X(T )
subject to E[ξX(T )] ≤ X(0),
(5)
P(X(T ) ≥ X) ≥ 1 − α.
where P(X(T ) ≥ x) ≥ 1 − α is the VaR constraint, X is the ’floor’ terminal wealth
specified exogenously. VaR describes the loss that can occur over a given period, at a
given confidence level. It has long been a industry standard by choice or by regulation
SEC (1997); Jorion (1997); Dowd (1998); Saunders (2000); BCBS (2011).
2.4
Expected-Shortfall-based Risk Management
Since its introduction around 1994, VaR has been criticized both in academia and industry, for its weaknesses as the benchmark. Recognizing the shortcomings of VaR,
there is an advocacy to replace VaR with Expected Shortfall (ES), also known as Tail
Conditional Expectation (TCE) or Conditional Value-at-Risk (CVaR), both in academia
Artzner et al. (1999); Rockafellar and Uryasev (2000); Acerbi and Tasche (2002); Rockafellar and Uryasev (2002); Embrechts et al. (2014) and in industry BCBS (2012, 2013,
2016). ES measures the riskiness of a position by considering both the size and the
likelihood of losses above a certain confidence level. ES at level α is defined as
1
ESα (X) = −
α
Z
α
GX (z)dz,
0
where GX (z) is the quantile function of X, that is, the right inverse of X’s cumulative
distribution function. If α = 0, then ES0 (X) = −GX (0).
We follow Basak and Shapiro (2001) to embed ES-RM into standard utility maximization: an economic agent is maximizing his expected utility, while maintaining his
ES-measured market-risk exposure below a prescribed threshold. The ES-RM model is
given by
7
max E[u(X(T ))]
X(T )
subject to E[ξX(T )] ≤ X(0),
(6)
ESα (X(T )) ≤ −X.
If α = 0, then the agent is a portfolio insurer; if α = 1, then the optimal solution
does not exist, as shown in Wei (2016).
3
Partial Equilibrium
In this section, we perform partial equilbriulum analysis for the benchmark agent problem (4), the VaR agent problem (5) and the ES agent problem (6). We follow the general
scheme developed in Wei (2016) to solve the problem. To this end, we impose the following assumption on ξ,
Assumption 3.1. ξ is atomless.
Denote Fξ (·) the cumulative distribution function of ξ, and Fξ−1 (·) the quantile function of ξ, which is strictly increasing since ξ is atomless. Let us introduce the following
assumption,
Assumption 3.2. ess inf ξ = 0, ess sup ξ = +∞, i.e., Fξ (0) = 0 and Fξ (1) = +∞.
This assumption stipulates that, for any given (positive) value, there exists a state
of nature in which the market offers a return that is greater (less) than that value. In
particular, this assumption is valid when the investment opportunity set, i.e., the triplet
(r(·), b(·), σ(·)), is deterministic and ξ is lognormally distributed (which is the case with
a Black-Scholes market).
We will not discuss feasibility and well-posedness of the problem, as these issues
have been examined thoroughly in Wei (2016) for general risk measures. We assume
all parameters are within reasonable ranges so that optimal solutions exist.
3.1
Optimal Solution
We impose the following integrability assumption throughout the paper.
8
0
Assumption 3.3. E[(u )−1 (λξ)] < ∞ for all λ > 0.
This integrability assumption is standard in expected utility maximization problems.
Proposition 3.1. We have the following assertions:
1. The optimal time-T wealth of the Benchmark agent (4) is
XB (T ) = I(λB ξ),
0
where I(·) = (u )−1 (·) and λB solves E[ξXB (T )] = X(0).
2. The optimal time-T wealth of the VaR agent (5) is



I(λV aR ξ) ξ < ξ V aR ,


XV aR (T ) =
X
ξ V aR ≤ ξ < ξ V aR ,



 I(λ
V aR ξ) ξ V aR ≤ ξ,
0
where ξ V aR ≡ u (X)/λV aR , ξ V aR is such that P(ξ > ξ V aR ) ≡ α, and λV aR ≥ 0
solves E[ξXV aR (T )] = X(0). The VaR constraint in (5) is binding if, and only
if, ξ V aR < ξ V aR .
3. The optimal time-T wealth of the ES agent (6) is



I(λES ξ)


XES (T ) =
I(λES ξ ES )



 I(λ ξ − 1 λ
ES
α
ξ < ξ ES ,
ξ ES ≤ ξ < ξ ES ,
ES µES )
ξ ES ≤ ξ,
where ξ ES > 0 solves hµES (1 − Fξ (ξ ES )) = 0, ξ ES = ξ ES + α1 µES , and λES >
0, µES ≥ 0 solve
E[ξXES (T )] = X(0)
ESα (XES (T )) ≤ −X.
Here, hµ (·) is define by (17). The ES constraint is binding, if and only if ESα (XB (T )) =
−X (or µES > 0). Moreover, if the ES constraint is binding, P(ξ > ξ ES ) >
α, P(ξ > ξ ES ) < α.
9
If X ≤ I(λB Fξ−1 (1 − α)), then the VaR agent becomes the Benchmark agent.
Rα
Moreover, since ESα (XB (T )) = − α1 0 I(λB Fξ−1 (1 − z))dz > −I(λB Fξ−1 (1 − α)),
when the VaR constraint is binding, the ES constraint is also binding. To compare
optimal wealth of different agents, we assume X > I(λB Fξ−1 (1 − α)).
Proposition 3.2. We have the following assertions:
1. I(λES ξ ES ) > X;
2. λES > λV aR > λB ;
3. 0 < ξ ES < ξ V aR < ξ V aR < ξ ES ;
4. XV aR (T ) < XB (T ) when ξ ≥ ξ V aR ;
5. XES (T ) < XB (T ) when ξ > ξ 1 :=
λES µES
;
α(λES −λB )
6. XES (T ) < XV aR (T ) when ξ > ξ 2 :=
λES µES
.
α(λES −λV aR )
Moreover, ξ 2 > ξ 1 .
Figure 1 depicts the optimal terminal wealth of a benchmark agent, a VaR agent
and an ES agent, with identical X and 0 < α < 1. Here, X V aR := I(λV aR ξ V aR ) and
X ES := I(λES ξ ES ).
Similar to the VaR agent, the ES agent endogenously classify the scenarios of the
future into three states, but his economic behaviours in these states are quite different. In
the good states [ξ < ξ ES ], the ES agent behaves like the Benchmark and the VaR agent
but he is willing to accept a lower wealth level than the Benchmark and the VaR agent,
i.e. XES (T ) < XV aR (T ) < XB (T ). In return, he fully insures against the intermediate
states [ξ ES ≤ ξ < ξ ES ]. Moreover, both the scope and the wealth level of the insured
states are larger than that of the VaR agent, i.e. ξ ES < ξ V aR < ξ V aR < ξ ES and
X ES > X. He chooses to maintain a loss in the bad states [ξ ≥ ξ ES ], because these are
the most expensive states to insure against. However, in case of a significant financial
market stress [ξ ≥ ξ 1 ], the ES agent will incur larger losses than the Benchmark agent,
and even larger than the VaR agent in the worst states [ξ ≥ ξ 2 ].
This highlights an disencouraging feature of the ES-RM: ES only helps to insure
against losses in the intermediate states, at the expense of larger extreme losses. Though
10
X(T)
B
VaR
ES
X ES
X
X V aR
0
ξ ES
ξ V aR
ξ V aR
ξ ES
ξ1
ξ2
ξ
Figure 1: Optimal horizon wealth
The figure plots the optimal terminal wealth of a Benchmark agent, a VaR agent and an
ES agent, with same X and 0 < α < 1, as functions of the horizon state price density
ξ. The black line is for the unconstrained Benchmark agent, the red line is for the VaR
agent and the blue line is for the ES agent.
11
the scope of the bad states is smaller, the wealth in the extreme bad states is below the
benchmark and even the VaR wealth, i.e. the ES-RM reduces the probability of the
loss but increases the magnitude of extreme losses. Use of ES is believed to be able to
help to ensure a more prudent capture of ”tail risk” and capital adequacy during periods
of significant financial market stress BCBS (2016), but it is shown that the losses are
more severe than without the ES-RM, defeating the very purpose of ES. One common
criticism on VaR is that it fails to take the magnitude of losses into account. Our results
reveal that even though ES takes the sizes of losses into account, it is far from enough.
We also note that the ES agent chooses both ξ ES and ξ ES endogenously, which
depend on the agent’s preferences and endowment, the market ξ and the ES constraint
α and X. In contrast, ξ V aR solely depends on α and the market. Moreover, the terminal
wealth of the VaR agent has a discontinuity at ξ V aR , whereas the wealth of the ES agent
is continuous across all states.
Of course, VaR and ES at the same level α are not comparable. For example, as
stated in BCBS (2016), in calculating the ES, a 97.5th percentile, one-tailed confidence
level is to be used, while 99th percentile is used in calculating the VaR. For this reason,
we will from now on compare the ES agent only to the Benchmark agent.
3.2
Properties
In this section we study trading behaviors under the ES-RM. For analytical tractability,
we specialize the setting to CRRA preferences,
u(X) =


X 1−γ
1−γ
 ln X
γ > 0 and γ 6= 1,
(7)
γ = 1,
and to lognormal state prices with constant interest and market price or risk.
The following proposition presents explicit expressions for the Benchmark’s and the
ES agent’s optimal wealth and portfolio strategies before the horizon.
Proposition 3.3. Assume u is given by (7), and r, µ, and σ are constant.
For the Benchmark agent:
12
1. The optimal time-t wealth is
XB (t) =
where Γ(t) :=
1−γ
(r
γ
+
kθk2
)(T
2γ
eΓ(t)
,
1
(λB ξ(t)) γ
− t) and λB is as in Proposition 3.1.
2. The fraction of wealth invested in stocks is
πB (t) =
1 0 −1
(σ ) θ,
γ
(8)
For the ES agent:
1. The optimal time-t wealth is
XES (t) =
eΓ(t)
eΓ(t)
−
1
(λES ξ(t)) γ
1
(λES ξ(t)) γ
N (−d1 (ξ ES ))
+ X ES e−r(T −t) [N (−d2 (ξ ES )) − N (−d2 (ξ ES ))]
+ e−r(T −t) G(ξ ES , γ).
where ξ ES , λES and µES are as in Proposition 3.1, and
2
x
ln ξ(t)
+ (r − kθk
)(T − t)
2
√
d2 (x) :=
,
kθk T − t
√
1
d1 (x) := d2 (x) + kθk T − t,
γ
1
ξ ES := ξ ES + µES ,
α
Z
+∞
G(x, γ) :=
φ(z)
d2 (x)
(λES ξ(t)e
1
√
kθk2
kθk T −tz−(r− 2 )(T −t)
1
dz,
− α1 λES µES ) γ
N (·) is the standard normal distribution function and φ(·) is the standard normal
probability density function.
2. The fraction of wealth invested in stocks is
πES (t) = qES (t)πB (t),
13
where
qES (t) := 1 −
X ES e−r(T −t) (N (−d2 (ξ ES )) − N (−d2 (ξ ES )))
XES (t)
1
e−r(T −t)
γ
+ λES µES
G(ξ ES ,
).
α
XES (t)
1+γ
(9)
3. The exposure to risky assets relative to the benchmark is bounded below: qES (t) ≥
0, and
lim qES (t) = lim qES (t) = 1.
ξ(t)→0
ξ(t)→∞
4. When the ES constraint is binding, qES (t) > 1 if ξ(t) > ξES (t) for some deterministic ξES (t).
We find that when the state price density process is sufficiently low, the ES agent
has to invest more in stocks to meet the ES constraint. The ES constraint increases
the risk exposure (represented by the fraction of wealth invested in risky assets) in the
extremely bad states, which is consistent with our previous findings.
Figure 2 depicts the optimal time-t wealth of a benchmark agent and an ES agent.
The time-t wealth of the ES agent is similar to its time-T behaviors. The ES agent
behaves like a Benchmark agent in the good and bad states, whereas in the intermediate
states, his wealth flattens as time approaches the horizon. In the good and extremely
bad states, his wealth is below the the Benchmark agent’s wealth.
Figure 3 depicts the ES agent’s time-t relative risk exposure qES (t), which is Sshaped. We may classify it into five states. In the two extremes of ξ(t), the B behavior
dominates. In the good states, his relative risk exposure is low and increase as the
market transit into bad states. His risk exposure is high in the bad states but decreases
as the market continues to deteriorate.
14
X(t)
B
ES
1.6
1.2
0.8
0.4
0
2
4
6
8
10
12
ξ(t)
Figure 2: Optimal time-t wealth
The figure plots the optimal time-t wealth of a Benchmark agent and an ES agent, as
functions of the time-t state price density ξ(t). We assume CRRA preferences and
lognormal state price density. The parameters are T = 1, t = 0.5, r = 0.05, θ =
0.4, X(0) = 1, X = 0.8, α = 0.05, γ = 1.
4
General Equilibrium Analysis
4.1
Equilibrium in a Production Economy
In Section 3.1, we illustrated that, the ES agent optimally shifts wealth from good and
bad states to intermediate states to meet the requirement on ES. This motivates us to
consider a production economy in which aggregate consumption/wealth can be postponed or shifted and aggregate nonzero holdings in a riskless investment are allowed.
We consider a continuous-time, finite horizon [0, T ] production economy similar
to that in Basak (2002); Basak and Shapiro (2005), which is a variation on the Cox
et al. (1985) economy. In this economy, the investment opportunities available are
constant-returns-to-scale production technologies, using the single consumption good
as their only input and producing the consumption good as output. The production
technologies have perfectly elastic supplies, and net returns given by (1) and (2), where
the (exogenously specified) parameters r, µ, and σ are assumed constant. The first
15
q ES(t)
B
ES
1.2
1
0.8
0.6
0.4
0.2
0
4
8
12
16
ξ(t)
Figure 3: Time-t realtive risk exposure
The figure plots the ES agent’s time-t exposure to risky assets relative to the Benchmark
agent, as a function of the time-t state price density ξ(t). We assume CRRA preferences
and lognormal state price density. The parameters are T = 1, t = 0.5, r = 0.05, θ =
0.4, X(0) = 1, X = 0.8, α = 0.05, γ = 1.
production technology is riskless and the others risky. The economy is populated by two
types of agents whose initial wealth is exogenously specified as units of consumption
good:
1. The Benchmark agent, who solves
max E[u(XB (T ))]
X(T )
subject to E[ξXB (T )] ≤ XB (0);
2. the ES agent, who solves
max E[u(XES (T ))]
XES (T )
subject to E[ξXES (T )] ≤ XES (0),
ESα (XES (T )) ≤ −X,
where u is given by (7) and X is a given positive constant.
16
We refer to this economy in which the first agent is the Benchmark agent and the
second agent is the ES agent as the ES economy, and the economy in which both agents
are Benchmark agents as the Benchmark economy.
Equilibrium in this production economy requires both agents to act optimally, and
for all wealth to be invested in the production technologies. The optimal trading strategies in the partial equilibrium setting (Section 3.2) persist. Our goal is to compare
equilibrium in the presence of the ES constraint with equilibrium in the benchmark
economy without the constraint. In particular, we are interested in the impact of the ES
constraint on the market value dynamics. The price of the market portfolio, XM , is
defined as the aggregate wealth invested in the production technologies, which equals
both agents’ net worth:
XM (t) := XB (t) + XES (t).
The equilibrium market-price dynamics can be represented by
dXM (t) = XM (t)[µM (t)dt +
n
X
σM,j (t)dW j (t)],
j=1
where µM (t) is the market drift and kσM k :=
qP
n
2
j=1 (σM,j (t))
is the market volatility.
The following proposition presents the equilibrium market price, volatility, and risk
premium and contrasts those with the Benchmark economy.
Proposition 4.1. The equilibrium market price, volatility, and risk premium in the
Benchmark economy are given by
B
XM
(t) =
B
XM
(0)
1
,
(ξ(t)) γ
1
B
kσM
(t)k = kθk,
γ
1
2
µB
M (t) − r = kθk .
γ
The equilibrium market price, volatility, and risk premium in the ES economy are
17
given by
ES
XM
(t) =
eΓ(t)
1
(λB ξ(t)) γ
+
eΓ(t)
1
(λES ξ(t)) γ
−
eΓ(t)
1
(λES ξ(t)) γ
N (−d1 (ξ ES ))
+ X ES e−r(T −t) [N (−d2 (ξ ES )) − N (−d2 (ξ ES ))],
1
XES (t) ES
kσM
(t)k = 1 − (1 − qES (t)) ES
kθk,
γ
XM (t)
1
XES (t) 2
kθk .
µES
1 − (1 − qES (t)) ES
M (t) − r =
γ
XM (t)
ES
B
B
Thus, kσM
(t)k > kσM
(t)k and µES
M (t) − r > µM (t) − r, if ξ(t) > ξES (t).
Proposition 4.1 reveals that the equilibrium market volatility and risk premium are
increased by the presence of the ES constraint, in the bad states of the world (ξ(t) >
ξES (t)). This is because, as reflected in Proposition 3.3, in the bad states, the ES agent
has a higher demand for risky investment opportunities than the Benchmark agent. In
these scenarios, the ES economy’s aggregate investment in the risky technologies is
higher than that in the Benchmark economy, leading to higher market volatility and risk
premium. This may be a source of concern for policy-makers: ES increases rather than
decreases the market volatility in the most adverse states of the world.
4.2
Equilibrium in a Pure-Exchange Economy
In the production economy with exogenous investment technologies, we show the market volatility and risk premium is increased by the presence of ES. We re-evaluate this
in an economy in which all quantities except the aggregate consumption process are determined endogenously. We follow Basak (1995); Basak and Shapiro (2001) to develop
a pure-exchange general equilibrium model featuring ES risk managers.
We assume that the economy is populated by two types of agents, the Benchmark
and the ES agent, who derive utility from intertemporal (continuous) consumption over
0
their lifetime [0, T ]. As opposed to the Benchmark agent, the ES agent is subject to the
0
additional ES constraint as in (6) over time-T wealth, where T < T . We refer to the
economy in which the first agent is the Benchmark agent and the second agent is the ES
agent as the ES economy, and the economy in which both agents are Benchmark agents
18
as the Benchmark economy. For simplicity, we assume there is a single consumption
good serving as the numeraire and n = 1, i.e., all the uncertainty is represented by the
one-dimensional Brownian motion W .
There are two investment opportunities: one instantaneously riskless and the other
risky.The riskless investment is a bond in zero net supply; the risky investment is a stock
in constant net supply of 1 and paying out a dividend stream at rate δ. We assume the
(exogenously given) dividend process to follow a geometric Brownian motion:
dδ(t) = δ(t)[µδ dt + σδ dW (t)],
with constant µδ , σδ and δ(0) > 0.
In light of Basak (1995); Basak and Shapiro (2001), we anticipate that the constraint
applied at the ES horizon T may results in jumps in the equilibrium security and state
prices. The price processes of the riskless asset S0 (t) and the risky asset S(t) are subject
to the following equations:
dS0 (t) = S0 (t)[r(t)dt + ηdA(t)],
dS(t) + δ(t)dt = S(t)[µ(t)dt + σ(t)dW (t) + ηdA(t)],
where r, b and σ are endogenous and are determined in equilibrium, ηdA(t) is the
changes of security prices at time T . A(t) is a right-continuous step function defined
by A(t) := 1t≤T , and the jump size η is an {FT }-measurable random variable. To prevent arbitrage opportunities, the jump size q in all security prices must equal, see Basak
(1995); Basak and Shapiro (2001) for a detailed discussion. The state price density
process ξ is given by
Z t
Z t
1
2
>
ξ(t) := exp − [r(s) + kθ(s)k ]ds −
θ(s) dW (s) − ηA(t) .
2
0
0
(10)
where θ is the unique bounded, {Ft }-progressively measurable market price of risk,
given by θ(t) :=
µ(t)−r(t)
.
σ(t)
At time 0, each agent is endowed with ei , i = B, ES, units (exogenous) of the risky
asset, that is, an initial wealth of Xi (0) = S(0)ei and we assume eB +eES = 1. Assume
19
that the trading of shares takes place continuously in a self-financing fashion and there
are no transaction costs. Then the wealth process of agent n, Xi (·) satisfies
dXi (t) = Xi (t)[r(t)dt+ηdA(t)]−ci (t)dt+Xi (t)[µ(t)−r(t)]πi (t)dt+Xi (t)πi σ(t)dW (t),
where πi (t) and ci (t) denote the fractions of the agents wealth in the risky asset and
consumption at time t, respectively. We require πi (t) and ci (t) to be Ft -progressively
measurable with
Z
T
|Xi (t)σπi (t)|2 dt < ∞, a.s.,
0
Z
T
|Xi (t)[µ(t) − r(t)]πi (t)|2 dt < ∞, a.s.,
0
Z
T
|ci (t)|2 dt < ∞, a.s.,
0
0
and Xi (T ) ≥ 0, a.s..
Given the dynamics of the state price density (10), it is well known
Lemma 4.1.
Z T0
1
0
Xi (t) =
ξ(s)ci (s)ds|Ft ], t ∈ [0, T ].
E[
ξ(t)
t
(11)
For tractability, we assume that all agents have logarithmic utility. The Benchmark
agent solves the following problem:
Z
T
0
max E[
cB
ln(cB (t))dt]
0
Z
T
(12)
0
ξ(t)cB (t)dt] ≤ XB (0).
subject to E[
0
20
The ES agent solves the following problem:
Z
max
cES ,XES (T −)
T
0
ln(cES (t))dt]
E[
0
Z
T
ξ(t)cES (t)dt + ξ(T −)XES (T −)] ≤ XES (0),
subject to E[
(13)
0
Z
T
0
ξ(t)cES (t)dt] ≤ ξ(T −)XES (T −), a.s.,
E[
T
ESα (XES (T −)) ≤ −X.
The following proposition characterizes each agent’s optimal policy, under Assumption 3.1 and 3.2 1 . We will verify later that in the equilibrium the state price density
process indeed satisfies these assumptions.
Proposition 4.2. We have the following assertions:
1. The optimal consumption policies and time-T wealth of the Benchmark agent
(12) is
1
0
, t ∈ [0, T ],
λB ξ(t)
0
T −T
XB (T −) =
.
λB ξ(T −)
cB (T ) =
0
where λB =
T
.
XB (0)
2. The optimal consumption policies and time-T wealth of the ES agent (13) is
cES (t) =
XES (T −) =












1
λES,1 ξ(t)
t ∈ [0, T ),
1
λES,2 ξ(t)
t ∈ [T, T ],
0
0
T −T
λES,1 ξ(T −)
ξ(T −) < ξ ES ,
0
T −T
λES,1 ξ ES
ξ ES ≤ ξ(T −) < ξ ES ,
0
T −T
1
λES,1 ξ(T −)− α
λES,1 µES
ξ ES ≤ ξ(T −),
where ξ ES > 0 solves hµES (1 − Fξ (ξ ES )) = 0, ξ ES = ξ ES + α1 µES , λES,2 =
1
From now on we denote by Fξ the quantile function of ξ(T −)
21
0
T −T
,
ξ(T −)XES (T −)
and λES,1 > 0, µES ≥ 0 solve
T
λES,1
+ E[ξ(T −)XES (T −)] = XB (0),
ESα (XES (T −)) = −X.
hµ (·) is define by (17). Here, Fξ is the quantile function of ξ(T −).
Note that when µES = 0 we recover the Benchmark agent. We next define and then
characterize the equilibrium in our setting.
Definition 4.1. An equilibrium is a collection of (r, b, σ) and optimal (cB , cES , πB , πES ),
0
such that the good, stock, and bond markets clear, that is, ∀t ∈ [0, T ],
cB (t) + cES (t) = δ(t),
πB (t)XB (t) + πES (t)XES (t) = S(t),
XB (t) + XES (t) = S(t).
The following proposition characterize the equilibrium state price density, interest
rate and market price of risk explicitly.
Proposition 4.3. We have the following assertions:
1. The equilibrium state price density is given by

 (1 +
λB
ξ(t) =
 (1 +
λB
1
) 1
λES,1 δ(t)
1
) 1
λES,2 δ(t)
t ∈ [0, T ),
0
(14)
t ∈ [T, T ],
where λB , λES,1 , and λES,2 are given in Proposition 4.2 with (14) substituted in.
(14) satisfies Assumption 3.1 and 3.2.
2. The equilibrium interest rate and market price of risk are constants, at all t ∈
0
[0, T ], given by
r = µδ − kσδ k2 ,
θ = kσδ k.
22
3. The jump size is
η = ln((
1
1
1
1
+
)) − ln((
+
)).
λB λES,2
λB λES,1
The price of the equity market portfolio, XM (t), is defined as the aggregate optimally invested wealth in the risky asset, i.e.,
XM (t) := πB (t)XB (t) + πES (t)XES (t) = XB (t) + XES (t) = S(t).
The equity market value is always equal to the stock price since there is only one stock
in the market 2 .
We represent the equilibrium market dynamics as
dXM (t) + δ(t)dt = XM (t)[µM (t)dt + σM (t)dW (t)],
where µM is the equity market drift, |σM | is the equity market volatility and µM − r is
the equity market risk premium.
The following proposition presents these quantities in equilibrium and contrasts
them with the Benchmark economy.
Proposition 4.4. We have the following assertions:
1. The equilibrium market price, volatility, and risk premium in the Benchmark
economy are given by
0
B
XM
(t) = (T − t)δ(t),
B
|σM
(t)| = |σδ |,
(15)
2
µB
M (t) − r = |σδ | .
2. Before the ES horizon, the equilibrium market price, volatility, and risk premium
2
If there are multiple stocks in the market, the equity market value will be the sum of values of all
stocks.
23
in the ES economy are given by
0
ES
XM
(t)
λB (T − T )
=(T − t)δ(t) −
δ(t)N (−dˆ1 (δ))
λB + λES,1
0
λB (T − T ) −(µδ −|σδ |2 )(T −t)
+
δe
[N (−dˆ2 (δ)) − N (−dˆ2 (δ))]
λB + λES,1
0
0
+ (T − T )e−(µδ −|σδ |
2 )(T −t)
Ĝ(δ, 1),
(16)
ES
|σM
(t)| =q̂(t)|σδ |,
2
µES
M (t) − r =q̂(t)|σδ | ,
where
δ :=
δ :=
1
λB
+
1
λES,1
ξ ES
1
λB
+
1
λES,1
ξ ES
,
,
ln δ(t)
+ (µδ − 12 |σδ |2 )(T − t)
x
ˆ
√
d1 (x) :=
,
|σδ | T − t
√
dˆ2 (x) := dˆ1 (x) − |σδ | T − t,
Z +∞
1
√
φ(z) λB +λES,1
Ĝ(x, y) :=
dz,
3
2
|σ
|
T
−tz−(µ
−
δ 2 |σδ | )(T −t)
− α1 λES,1 µES )y
( λB δ(t) e δ
dˆ2 (x)
0
λB (T − T )
2
δe−(µδ −|σδ | )(T −t) [N (−dˆ2 (δ)) − N (−dˆ2 (δ))]
q̂(t) := 1 −
ES
(λB + λES,1 )XM (t)
0
1
(T − T )e−(µδ −|σδ |
+ λES,1 µES
ES
α
XM
(t)
2 )(T −t)
Ĝ(δ, 2).
After the ES horizon, market prices, volatility, and risk premiums in both economies
are identical.
ES
B
3. For t ∈ [0, T ), XM
(t) > XM
(t) and
ES (t)
XM
δ(t)
>
B (t)(t)
XM
.
δ(t)
ES
B
B
∗
4. For t ∈ [0, T ), |σM
(t)| > |σM
(t)| and µES
M (t) > µM (t) if δ(t) < δ (t), for some
deterministic δ ∗ (t).
The ES economy is qualitatively the same as the VaR economy in Basak and Shapiro
(2001). We find that the prehorizon market price in the ES economy is always higher
than in the Benchmark economy. This is because the ES agent values the dividend at
24
T more than the prehorizon consumption to meet the ES constraint. The equity market
value is then pushed up as it is the claim against future dividends. The price-dividend
ratio is thus increased. When the market state is bad, that is, the output δ(t) is low and
the state price density ξ(t) is high, the market volatility in the ES economy is amplified
compared to the Benchmark economy. The interest rate and the market price of risk are
constant in the equilibrium. When the output is low, the ES agent will need to invest
more in the risky asset. The market volatility is then increased and the market risk
premium must also increase accordingly to keep the market price of risk constant. This
may be a source of concern for policy-makers: ES increases the market volatility during
periods of significant financial market stress.
Figure 4 depicts the time-t market prices of the Benchmark economy and the ES
economy. We may classify it into three states. In the two extremes of δ(t), the market
price of the ES economy evolves like the B economy, whereas in the intermediate states
it is significantly higher.
Figure 5 depicts the ES economy’s time-t market volatility and risk premium relative
to the Benchmark economy, given by q̂ES (t) =
ES (t)|
|σM
B (t)|
|σM
=
µES
M (t)−r
.
µB
M (t)−r
In the two extremes
of δ(t), the ES economy is similar to the Benchmark economy. As the output increases,
the ES market volatility and risk premium relative to the Benchmark economy first increase from one, then decrease, and finally increase to one. When the output is high,
these quantities in the ES economy is significantly lower than in the Benchmark economy. However, the ES market is more volatile and the risk premium is higher during
periods of market stress, that is, when the output is low.
5
Conclusion
We study the effects of Expected Shortfall on portfolio choice of expected utility maximizers, who derive utility from wealth at some horizon and must comply with a ES constraint imposed at that horizon. In the partial equilibrium analysis, we find ES agents
will insure against only intermediate states and incur larger losses than both VaR and
Benchmark agents. We then study general equilibrium asset pricing models featuring
ES agents. It is shown that the market volatility and risk premium in the ES economy
25
XM (t)
B
ES
4
3
2
1
0
0
0.5
1
1.5
2
2.5
δ(t)
Figure 4: Time-t market price
The figure plots the time-t market prices of the Benchmark economy and the ES economy, as functions of the time-t dividend δ(t). The parameters are µδ = 0.1, σδ =
0.2, λB = 1, λES,1 = 1.07, µES = 0.02.
26
B
ES
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0
0.5
1
1.5
2
2.5
(t)
Figure 5: Time-t realtive risk exposure
The figure plots the ES economy’s time-t market volatility and risk premium relative to
the Benchmark economy, as a function of the time-t dividend δ(t). The parameters are
0
T = 2, T = 1, t = 0.5, µδ = 0.1, σδ = 0.2, λB = 1, λES,1 = 1.07, µES = 0.02.
27
are larger than in the Benchmark economy when the output is low, in both production
and pure-exchange models. This provides a critique on the (potentially) wide use of
ES. Finally, we admit that real world managers and policy-makers may not use ES as
our model does, and therefore, it is of great interests to study the impact of ES under
different industry practices.
A
Proofs
A.1
Proof of Proposition 3.1
The first claim is standard. The second can be found in Basak and Shapiro (2001). We
now follow Wei (2016) to solve the ES agent problem. Define
 R
 − 1 F −1 (1 − s)ds + y α−z
ξ
α
z
ϕy (z) =
R
1
 − F −1 (1 − s)ds
z
ξ
z ∈ [0, α],
z ∈ (α, 1].
We now try to find δy (·), the concave envelope of ϕy (·). First, we derive AES :=
0
{y : y > 0, δy (z+) > 0, z ∈ [0, 1)}.
Lemma A.1.
y
AES = {y : y > 0 and ϕy (1 − Fξ ( )) < 0}
α
0
Proof of Lemma A.1. First, if y ≤ αFξ−1 (1 − α), then ϕy (z+) > 0, z ∈ [0, 1) and
consequently ϕy (z) < 0, z ∈ [0, 1).
0
0
Next, for any y > αFξ−1 (1 − α), ϕy (z) > 0, z ∈ [0, 1 − Fξ ( αy )), ϕy (1 − Fξ ( αy )) = 0,
0
0
ϕy (z) < 0, z ∈ (1 − Fξ ( αy ), α), and ϕy (z) > 0, z ∈ (α, 1). Therefore, ϕy (z) < 0, z ∈
[0, 1) if and only if ϕy (1 − Fξ ( αy )) < 0. The rest of the proof follows from Lemma B.3
in Wei (2016).
For α < t ≤ t := 1 − Fξ ((Fξ−1 (1 − α) − αy ) ∨ 0), define
sy (t) = 1 − Fξ (Fξ−1 (1 − t) +
28
y
),
α
0
0
and we have ϕy (sy (t)) = ϕy (t), 0 < sy (t) < α < t ≤ t. Define
0
hy (t) = ϕy (t) − ϕy (sy (t)) − ϕy (t)(t − sy (t)).
(17)
The following lemma is crucial in deriving δy (·).
Lemma A.2. For y ∈ AES ,there exist α < t∗ < 1 such that hy (t) < 0 for α < t < t∗ ,
hy (t∗ ) = 0, and hy (t) > 0, t∗ < t < 1.
Proof of Lemma A.2. Note
0
hy (t) = ϕy (t) − ϕy (sy (t)) − ϕy (t)(t − sy (t))
Z t
α − sy (t)
Fξ−1 (1 − z)dz − y
=
− Fξ−1 (1 − t)(t − sy (t)),
α
sy (t)
we have
hy (t) > Fξ−1 (1 − t)(t − sy (t)) − y
= −y
α − sy (t)
− Fξ−1 (1 − t)(t − sy (t))
α
α − sy (t)
,
α
and
hy (t) < Fξ−1 (1 − sy (t))(t − sy (t)) − y
=
α − sy (t)
y
− (Fξ−1 (1 − sy (t)) − )(t − sy (t))
α
α
y
(t − α).
α
Thus, hy (α+) < 0. If αFξ−1 (1 − α) ≥ y, t = 1 − Fξ (Fξ−1 (1 − α) − αy ), sy (t) = α, and
we have hy (t) > 0. If αFξ−1 (1 − α) < y, t = 1, sy (t) = 1 − Fξ ( αy ). By Lemma (A.1),
we have
y
y
0
hy (1) = ϕy (1) − ϕy (1 − Fξ ( )) − ϕy (1)(1 − 1 + Fξ ( ))
α
α
y
= 0 − ϕy (1 − Fξ ( ))
α
> 0.
Thus, hy (t) > 0. Since h(·) is continuous, there exists at least one t such that hy (t) = 0.
29
Next, for α < t1 < t2 ≤ t,
hy (t1 ) − hy (t2 )
=[ϕy (t1 ) − ϕy (t2 )] − [ϕy (sy (t1 )) − ϕy (sy (t2 ))]
0
0
0
0
− [ϕy (t1 )t1 − ϕy (t2 )t2 ] + [ϕy (sy (t1 ))sy (t1 ) − ϕy (sy (t2 ))sy (t2 )]
Z t1
Z t1
Z sy (t1 )
Z t1
0
0
0
0
zdϕy (z)]
ϕy (z)dz +
ϕy (z)dz − [
ϕy (z)dz −
=
sy (t1 )
Z
Z
0
sy (t1 )
t1
Z
0
zdϕy (z) −
=
0
zdϕy (z)
t2
sy (t2 )
t1
Z
Z
0
t1
sy (z)dϕy (sy (z)) −
=
0
zdϕy (z)]
sy (t2 )
sy (t2 )
Z
sy (t1 )
ϕy (z)dz +
+[
t2
t2
sy (t2 )
t2
0
zdϕy (z)
t2
t2
Z
t2
=−
0
[sy (z) − z]dϕy (z)
t1
<0.
Thus, h(·) is strictly increasing. This completes the proof.
Proposition A.1. For y ∈ AES ,



ϕ (z)

 y
0
δy (z) =
ϕy (sy (t∗y )) + ϕy (sy (t∗y ))(z − sy (t∗y ))



 ϕ (z)
z ∈ [0, sy (t∗y )),
z ∈ [sy (t∗y ), t∗y ],
z ∈ (t∗y , 1],
y
and



F −1 (1 − z) −

 ξ
0
δy (z) =
Fξ−1 (1 − t∗y )



 F −1 (1 − z)
y
α
z ∈ [0, sy (t∗y )),
z ∈ [sy (t∗y ), t∗y ],
z ∈ (t∗y , 1],
ξ
where α < t∗y < 1 is the unique root of hy (t) = 0, and sy (t∗y ) = 1 − Fξ (Fξ−1 (1 − t∗y ) +
y
)
α
∈ (0, α).
Proof of Proposition A.1. δy (·) is obviously concave. Note δy (z) = ϕy (z), z ∈ [0, sy (t∗y )]∪
0
0
0
0
[t∗y , 1], δy (z) > ϕy (z), z ∈ (sy (t∗y ), α), and δy (z) < ϕy (z), z ∈ (α, t∗y ), we have
0
δy (z) > ϕy (z), z ∈ (sy (t∗y ), t∗y ). Moreover, since δy (·) is constant on (sy (t∗ ), t∗ ), we
30
conclude δy is the concave envelope of ϕy .
Proof of Proposition 3.1. The claim follows from Theorem 4.2 in Wei (2016).
A.2
Proof of Proposition 3.2
Proof of Proposition 3.2. First, since P(ξ > ξ ES ) > α, P(ξ > ξ ES ) < α and P(ξ >
ξ V aR ) = α, we have ξ ES < ξ V aR < ξ ES . Next, if I(λES ξ ES ) ≤ X, then the ES
constraint cannot be satisfied as XES (T ) < I(λES ξ ES ) ≤ X when ξ > ξ ES . Thus,
I(λES ξ ES ) > X.
λV aR > λB is due to Basak and Shapiro (2001) or Wei (2016). If λES ≤ λV aR ,
then ξ ES > ξ V aR , due to the fact that I(λES ξ ES ) > X = I(λV aR ξ V aR ). Moreover,
since I(λES ξ) ≥ I(λV aR ξ), I(λES ξ − α1 λES µES ) > I(λV aR ξ), we have XES (T ) ≥
XV aR (T ), a.s., and the inequality is strict when ξ > ξ V aR , violating the budget constraint. Therefore, λES > λV aR .
Similarly, since I(λV aR ξ ES ) > I(λES ξ ES ) > X = I(λV aR ξ V aR ), we must have
ξ ES < ξ V aR . The rest of the proof follows from direct comparison.
A.3
Proof of Proposition 3.3
Proof of Proposition 3.3. We only show the ES agent part.
1. It is well-known that ξ(t)XES (t) is a martingale:
ξ(t)XES (t) = E[ξ(T )XES (T )|Ft ].
When r, σ and θ are constant, conditional on Ft , ln ξ(T ) is normally distributed
with mean ln ξ(t) − (r +
kθk2
)(T
2
− t) and variance kθk2 (T − t). Evaluating the
conditional expectation gives XES (t).
2. Applying Ito’s lemma to the expression of XES (t) and comparing the coefficient
of the dW (t) term with that of (3), we obtain the expression of πES (t). Dividing
31
it by πB (t) given by (8) yields
qES (t) =
1
eΓ(t)
[
XES (t) (λES ξ(t))
1
γ
eΓ(t)
−
(λES ξ(t))
1
γ
γ
)].
1+γ
(18)
N (−d1 (ξ ES )) + e−r(T −t) G(ξ ES ,
Rearranging (18) gives (9).
3. (18) reveals that it is non-negative. The limits are straightforward to verify.
4. Define
F (ξ(t)) := −X ES (N (−d2 (ξ ES )) − N (−d2 (ξ ES ))) +
1
γ
λES µES G(ξ ES ,
),
α
1+γ
0
it suffices to show F (ξ(t)) < 0 for ξ(t) large enough. We have
1
√
kθk T − tξ(t)
φ(d2 (ξ ES )))
1
√
+ λES (ξ ES − ξ ES )
1+ γ1 kθk T − tξ(t)
(λES ξ ES )
Z +∞
− λES (ξ ES − ξ ES )
φ(z)
0
F (ξ(t)) = − X ES (φ(d2 (ξ ES )) − φ(d2 (ξ ES )))
(19)
d2 (ξ ES )
√
kθk2
kθk T −tz−(r− 2 )(T −t)
·
=
(1 + γ1 )λES e
√
(λES ξ(t)ekθk
T −tz−(r−
kθk2
)(T −t)
2
1
− α1 λES µES )2+ γ
dz
φ(d2 (ξ ES ))
g(d2 (ξ ES )),
√
1
(λES ξ ES ) γ kθk T − tξ(t)
where
(ln a)2
−
2kθk2 (T −t)
g(x) :=a − e
Z
−
φ(x)
e
φ(z)
x
√
1
(1 + γ1 )(λES ξ ES )2+ γ kθk T − ta(a − 1)
√
kθk T −t(z−x)
+∞
·
a :=
a
x
√
kθk T −t
(λES ξ ES
√
ekθk T −t(z−x)
1
− α1 λES µES )2+ γ
dz,
ξ ES
,
ξ ES
and φ(·) is the standard normal probability density function. It is a simple exercise
to show limx→−∞ g(x) = −∞, thereby completing the proof.
32
A.4
Proof of Proposition 4.1
Proof of Proposition 4.1. Summing over the two Benchmark agents’ time-t wealth in
Proposition 3.3, and expressing the Lagrange multipliers in terms of the initial wealth
gives the Benchmark economy’s time-t market value. Summing over the Benchmark
agent’s and the ES agent’s time-t wealth gives the ES economy’s time-t market value. In
virtue of (3) and Proposition 3.3, we obtain the expressions for the equilibrium market
volatility and risk premiums in both economies. The last claim is due to the property of
qES (t).
A.5
Proof of Proposition 4.3
Proof of Proposition 4.3. The Benchmark agent’s optimal consumption and time-T wealth
are standard in the literature. For the ES agent, we first consider the following problem:
Z
max
cES (t),t∈[T,T 0 ]
T
0
ln(cES (t))dt|FT ]
E[
T
Z
T
0
ξ(t)cES (t)dt|FT ] ≤ ξ(T −)XES (T −), a.s..
subject to E[
T
The optimal consumption is
cES (t) =
1
0
λES,2 ξ(t)
where
, t ∈ [T, T ],
0
λES,2
T −T
=
,
ξ(T −)XES (T −)
and the optimal value is
Z
(T − T ) ln(XES (T −)) + E[
0
T
33
T
0
ln(
ξ(T −)
)dt|FT ].
(T − T )ξ(t)
0
Consequently, we can consider the following problem:
Z
max
cES (t),t∈[0,T ],XES (T −)
T
0
ln(cES (t))dt + (T − T ) ln(XES (T −))]
E[
0
Z
T
ξ(t)cES (t)dt + ξ(T −)XES (T −)] ≤ XES (0),
subject to E[
0
ESα (XES (T −)) ≤ −X.
The rest of the proof is a straightforward extension of Proposition 3.1
A.6
Proof of Proposition 4.3
Proof of Proposition 4.3. Clearing the consumption good market gives (14). The proof
that clearing the good market implies all other markets are cleared appears in Basak
(1995). Morevoer, it is straightforward to verify (14) satisfies Assumption 3.1 and 3.2.
r and θ are determined by applying Ito’s lemma to (14).
A.7
Proof of Proposition 4.4
Proof of Proposition 4.4.
1. We only prove the ES part, since the Benchmark economy is a special case of the
ES economy when the ES constraint is not binding, i.e., µES = 0.
2. By Lemma 4.1, we have
XM (t) =XB (t) + XES (t)
Z T
1
=
E[
ξ(s)(cB (s) + cES (s))ds|Ft ]+
ξ(t)
t
1
E[ξ(T −)(XB (T −) + XES (T −))|Ft ], t ∈ [0, T ).
ξ(t)
By Proposition 3.3, 4.2 and 4.3, we arrive at (16). Applying Ito’s lemma to XM (t)
ES
yields the expression for kσM
(t)k and µES
M (t).
3. Note that when XES (T −) ≥
and when XES (T −) <
0
0
T −T
λES,1 ξ ES
T −T
λES,1 ξ ES
then XB (T −) + XES (T −) = δ(T −),
then XB (T −) + XES (T −) > δ(T −). Hence,
34
ES
B
XM
(T −) ≥ XM
(T −) and the inequality is strict with a non-zero probability,
proving the result.
4. The proof is as of Proposition 3.3.
References
Acerbi, C. and Tasche, D. (2002). On the coherence of expected shortfall. Journal of
Banking & Finance, 26(7):1487–1503.
Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999). Coherent measures of risk.
Mathematical Finance, 9:203–228.
Basak, S. (1995). A general equilibrium model of portfolio insurance. Review of Financial Studies, 8(4):1059–1090.
Basak, S. (2002). A comparative study of portfolio insurance. Journal of Economic
Dynamics and Control, 26(7):1217–1241.
Basak, S. and Shapiro, A. (2001). Value-at-risk-based risk management: optimal policies and asset prices. Review of Financial Studies, 14(2):371–405.
Basak, S. and Shapiro, A. (2005). A model of credit risk, optimal policies, and asset
prices. The Journal of Business, 78(4):1215–1266.
Basak, S., Shapiro, A., and Teplá, L. (2006). Risk management with benchmarking.
Management Science, 52(4):542–557.
BCBS (2011). Revisions to the Basel II market risk framework. Basel Committee on
Banking Supervision, Bank for International Settlements: Basel, Switzerland.
BCBS (2012). Consultative Document May 2012. Fundamental Review of the Trading
Book. Basel Committee on Banking Supervision, Bank for International Settlements:
Basel, Switzerland.
35
BCBS (2013). Consultative Document October 2013. Fundamental Review of the Trading Book: A Revised Market Risk Framework. Basel Committee on Banking Supervision, Bank for International Settlements: Basel, Switzerland.
BCBS (2016). Standards January 2016. Minimum capital requirements for market risk.
Basel Committee on Banking Supervision, Bank for International Settlements: Basel,
Switzerland.
Cox, J. C. and Huang, C.-f. (1989). Optimal consumption and portfolio policies when
asset prices follow a diffusion process. Journal of Economic Theory, 49(1):33–83.
Cox, J. C., Ingersoll Jr, J. E., and Ross, S. A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica: Journal of the Econometric Society,
pages 363–384.
Cuoco, D., He, H., and Isaenko, S. (2008). Optimal dynamic trading strategies with
risk limits. Operations Research, 56(2):358–368.
Cuoco, D. and Liu, H. (2006). An analysis of var-based capital requirements. Journal
of Financial Intermediation, 15(3):362–394.
Dowd, K. (1998). Beyond value at risk: the new science of risk management.
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R., and Beleraj, A. (2014). An
academic response to basel 3.5. Risks, 2(1):25–48.
Grossman, S. J. and Zhou, Z. (1996). Equilibrium analysis of portfolio insurance. Journal of Finance, 51(4):1379–1403.
Jorion, P. (1997). Value at risk: the new benchmark for controlling market risk. Irwin
Professional Pub.
Jorion, P. (2002). How informative are value-at-risk disclosures?
The Accounting
Review, 77(4):911–931.
Karatzas, I. and Shreve, S. E. (1998). Methods of mathematical finance, volume 39.
Springer Science & Business Media.
36
Leippold, M., Trojani, F., and Vanini, P. (2006). Equilibrium impact of value-at-risk
regulation. Journal of Economic Dynamics and Control, 30(8):1277–1313.
Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Mathematics of Operations Research, 11(2):371–382.
Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk.
Journal of Risk, 2:21–42.
Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss
distributions. Journal of Banking & Finance, 26(7):1443–1471.
Saunders, A. (2000).
Financial institutions management: a modern perspective.
McGraw-Hill College.
SEC (1997). Disclosure of accounting policies for derivative financial instruments and
derivative commodity instruments and disclosure of quantitative and qualitative information about market risk inherent in derivative financial instruments, other financial
instruments, and derivative commodity instruments. Other Financial Instruments,
and Derivative Commodity Instruments, SEC, Washington, DC.
Wei, P. (2016). Risk management with weighted var. Working Paper, University of
Oxford.
Yiu, K.-F. C. (2004). Optimal portfolios under a value-at-risk constraint. Journal of
Economic Dynamics and Control, 28(7):1317–1334.
37