Contagion modeling between the nancial and insurance markets with time changed processes Donatien Hainaut
*
ISBA, Université Catholique de Louvain December 21, 2016
Abstract This study analyses the impact of contagion between nancial and non-life insurance markets on the asset-liability management policy of an insurance company. The indirect dependence between these markets is modeled by assuming that the assets return and non-life insurance claims are led respectively by time-changed Brownian and jump processes, for which stochastic clocks are integrals of mutually self-exciting processes. This model exhibits delayed co-movements between nancial and non-life insurance markets, caused by events like natural disasters, epidemics, or economic recessions. KEYWORDS: Self-exciting process, Cramer-Lundberg risk model, Stochastic optimal control.
1 Introduction Non life insurance claims, by nature, are not correlated to nancial markets, excepted in case of events like natural disasters, epidemics, or serious economic recession.
For exam-
ple, in 2003, the severe acute respiratory syndrome (SARS) spread across several countries and aected with a delay the insurance industry in dierent ways. Some areas of impacted insurance operations are clear event cancellations coverage, travel insurance and life and health policies. This epidemic also slowed down economic exchanges and indirectly caused turmoil in nancial markets. More recently, during the nancial crisis of 2008, the number of claims covered by credit insurances surged in US, as underlined in a recent report from the IMF (2016). As last example, we mention climate changes. It is already aecting and will over time signi cantly aect the incidence of natural conditions such as: tropical cyclones; winter storms; wild- res; hail storms; lightning strikes; droughts and oods. These events are expected to aect signi cantly property claims to non-life insurers. In parallel, climate change will have a huge economic and social impact and will lead to nancial instability. These observations motivate us to study the in uence of a potential contagion between the
* Postal address: ISBA, UCL,Voie du Roman Pays 20, B-1348 Louvain-la-Neuve (Belgique). E-mail to:
[email protected].
1
insurance and nancial markets on the asset-liability management policy of insurers. The literature about the modeling and management of non-life insurance company is vast. The starting point of research in this eld is the classical CramerLundberg (1903) risk model, in which the arrival of claims is modeled by a Poisson process. Since then, many extensions have been developed and proposed bounds on the insurer's ruin probability in various frameworks.
Later, Björk and Grandell (1988) and Embrechts et al.
(1993) introduced a Cox
process in the CramerLundberg model, for the modeling of claim arrivals. Albrecher and Asmussen (2006) studied a Cox process with shot noise intensity. Dassios and Zhao (2011 and 2012) analyzed the clustering phenomenon of claims, caused by a self-exciting process. Another strand of the literature focuses on the optimization of investment, reinsurance and dividend policies, in a Cramer-Lundberg approach. For example, Browne (1995) showed in a one-dimensional diusion model that the strategy maximizing the expected exponential utility of terminal wealth also minimizes the ruin probability. Asmussen and Taksar (1997) studied the optimal dividend policy for an insurer.
Hipp and Plum (2000) optimized the
investment policy of a non life insurer's surplus, in a Brownian setting. Schmidli (2002 and 2006) instead of maximizing the utility of the surplus or dividends, adapted the investment and reinsurance strategies to minimize the probability of ruin.
Kaluszka (2001, 2004) ex-
amined the optimal reinsurance problem under various meanvariance premium principles. Yuen et al. (2015) considered the optimal proportional reinsurance strategy in a risk model with multiple dependent classes of insurance business. And recently, Yin and Yuen (2015) studied the optimal dividend problems for a jump diusion model with capital injections and proportional transaction costs. Whereas Zheng et al. (2016) investigated the robust optimal portfolio and reinsurance problem for an ambiguity-averse insurer. This work studies the optimal proportional reinsurance, dividends and asset allocation for a non-life insurer, in presence of a contagion risk between nancial and insurance activities. Drawing on the theoretical and empirical background regarding the market time scale, as in Ané and Geman (2000) or Salmon and Tham (2007), we build time-changed dynamics with chronometers that are integrated positive Hawkes processes. This approach, inspired from Hainaut (2016(d)) introduces a non linear dependence between assets and liabilities. Hawkes processes developed by Hawkes (1971a) (1971b), Hawkes and Oakes (1974), are parsimonious self-exciting point processes for which the intensity jumps in response and reverts to a target level in the absence of event. clustering of shocks.
This dynamics is increasingly used in nance to model the
Empirical analysis conducted in Aït-Sahalia et al.
(2015) (2014) or
in Embrechts et al. (2011) emphasize the importance of this eect in stocks or CDS markets. They also underline that clustering is not characterized by a single jump but by the ampli cation of this movement that takes place over days. Recently, Hainaut (2016(a) and 2016 (b)) detects self-excitation in interest rate markets. And the paper of Hainaut 2016(c) analyzes the impact of the clustering of jumps on prices and risk of variable annuities. In this work, Hawkes processes determine the pace of market clocks. This research contributes to the literature in several directions.
Firstly, it is an elegant
method to introduce dependence between a geometric Brownian motion and a risk process. In this framework, we nd the main features of clocks:
2
means, variances and their joint
moment generating function (mgf ). Secondly, we show that the linear dependence between log-prices and claims is proportional to the risk premium of stocks and to the insurer's average pro t. In particular, when the insurer does not charge any fee above the pure premium, the correlation is null despite an evident dependence by construction.
When the insurer's margin is positive, the linear correlation
is induced by incomes from the insurance activity, reinvested in the nancial market. From an economic point of view, the linear dependence between between insurance and nancial markets in a time-changed model nd its origin in the existence of a risk premium in both segments. Thirdly, we prove that the insurer's ruin probability is still below the Cramer-Lundberg bound, if the surplus is not invested in the stocks market. Fourthly, we determine the optimal investment, reinsurance and dividend policies that maximize the exponential utility drawn from dividends and terminal surplus. Surprisingly, Optimal reinsurance and investment strategies are independent from markets clocks.
Whereas, the optimal dividend is a
linear function of the wealth and of intensities of chronometers. Finally, we compare with optimal strategies when the claim process is approached by a Brownian motion.
2 Stochastic clocks of nancial and insurance markets Papers of Ané and Geman (2000), Salmon and Tham (2007) provide pieces of evidence that the time scale of nancial markets is not chronological but rather driven by traded volumes. Starting from this observation, we respectively model nancial returns and insurance claims by a Brownian motion and a jump process, observed on distinct random scales of time. This approach allows us to replicate clustering of shocks observed in nancial and in insurance markets. It also introduces contagion and random correlation between assets and liabilities. The chronometers measuring the time scales of nancial and insurance markets are respecL S tively denoted by τt and τt . They are positive increasing processes de ned as the integrals of S L two processes λt and λt on a probability space Ω, endowed with a probability measure t t P and a natural ltration denoted by (Gt )t :
τtS
Z
t
λSs ds
:=
(1)
0
τtL
Z :=
t
λLs ds
0 S S By construction, the sample paths of random clocks are continuous and dτt = λt dt , L L S L dτt = λt dt . λt and λt are non-homogeneous processes that may be interpreted as the frequencies at which economic or actuarial information ows. Their dynamics is ruled by PNtS S PNtL L S L S two auxiliary jump processes Zt and Zt such that Zt = Jk and ZtL = k=1 Jk . k=1 t t S L S L Where Nt and Nt are point processes with random jumps Jk and Jk . The intensities t t S L of jump arrivals are assumed equal to the frequencies of information ows: λt and λt .
3
S For the sake of simplicity, jumps are exponential random variables with densities ν (z) = ρS e−ρS z 1{z≥0} and ν L (z) = ρL e−ρL z 1{z≥0} where ρS and ρL are positive and constant. The 1 1 average sizes of jumps are equal to µS = and µL = . Whereas the moment generating ρS ρL functions of jumps are respectively
φJS (ω) := E eωJ
S that the couple of intensities λt and
dλSt dλLt
S
φJL (ω) := E eωJ
and
L
. We assume
λLt obeys to the next dynamics:
dZtS ηSS ηSL αS θS − λSt . dt + dZtL ηLS ηLL αL θL − λLt {z } |
=
(2)
Ξ
αS
These processes revert respectively at speeds
ηLS , ηSS , ηSL , ηLL are constant and positive.
αL
and
toward
θS
or
θL .
The parameters
This last relation underlines the main features of
our approach: contagion, mutual and self-excitation. Indeed, when the clock of the nancial S L (resp. insurance) market speeds up due to a jump of Zt (resp. Zt ), the chronometer of the insurance (resp. nancial) market accelerates proportionally. This also raises the volatility as longer periods, measured on the market time scale, are observed on the same invariable chronological time scale. Another consequence of a jump is an instantaneous increase in the S L probability of observing a new nancial or actuarial shock as λt and λt are the intensities of L S point processes Zt and Zt . We check by direct dierentiation that intensities are the sum of a deterministic function and of two jump processes,
λSt
−αS t
= θS + e
λS0
t
Z
− θS + ηSS
e
−αS (t−s)
dZsS
= θL + e
−αL t
λL0
(3)
0 t
Z
e−αS (t−s) dZsL ,
+ ηSL
0
λLt
t
Z
− θL + ηLS
e
−αL (t−s)
dZsS
Z + ηLL
0
t
e−αL (t−s) dZsL .
0
These expressions are useful to nd closed form expressions of expected intensities, from which we will infer the conditions guaranteeing the stability of jump processes. Proposition 2.1.
mS (t) mS0 (t)
The expectations of λSt and λLt are equal to
:=
1 γ1
(eγ1 t − 1) 0 1 γ2 t (e − 1) 0 γ γt 2 S e1 0 λ0 +V V −1 0 eγ2 t λL0
E λSt |G0 = V E λLt |G0
V
−1
αS θS αL θL
(4)
where V is the following matrix V
=
ηSS ρS
− ηρSL L − αS − γ1
ηSS ρS
− ηρSL L − αS − γ2
.
If we note ∆ =
ηSS − αS ρS
−
4
ηLL − αL ρL
2 +4
ηLS ηSL ρS ρL
(5)
then γ1 , γ2 are constant and de ned by the following relations 1 ηSS ηLL = + − (αS + αL ) + 2 ρS ρL 1 ηSS ηLL = + − (αS + αL ) − 2 ρS ρL
γ1 γ2
Proof.
Consider the functions
g S = λSt
and
1√ ∆, 2 1√ ∆. 2
(6)
g L = λLt .
According to equations (3) and if the S L in nitesimal generators of these functions are denoted by Ag and Ag , their expectations satisfy the relation
S
E(Ag ) = αS (θS − E = αS θS − E E(Ag L ) = αL θL − E = αL θL − E The rst moments
mS (t)
Z Z +∞ +∞ S L ) + E λt ηSS z dνS (z) + E λt ηSL z dνL (z) −∞ −∞ λSt + E λSt ηSS µS + E λLt ηSL µL Z Z +∞ +∞ L S L λt + E λt ηLS z dνS (z) + E λt ηLL z dνL (z) −∞ −∞ λLt + E λSt ηLS µS + E λLt ηLL µL
λSt
and
mL (t)
are then solutions of a system of ordinary dierential
equations (ODE's) with respect to time:
∂ ∂t
mS mL
=
αS θS αL θL
+
(ηSS µS − αS ) ηSL µL ηLS µS (ηLL µL − αL )
Finding a solution requires to determine eigenvalues
γ
mS mL
.
(7)
(v1 , v2 ) of the matrix
and eigenvectors
present in the right term of this system:
(ηSS µS − αS ) ηSL µL ηLS µS (ηLL µL − αL )
v1 v2
=γ
v1 v2
.
Eigenvalues cancel the determinant of the following matrix:
(ηSS µS − αS ) − γ ηSL µL ηLS µS (ηLL µL − αL ) − γ
det
=0
and are solutions of the second order equation:
γ 2 − γ ((ηSS µS − αS ) + (ηLL µL − αL )) + (ηSS µS − αS )(ηLL µL − αL ) − ηSL ηLS µL µS = 0 which has for discriminant:
∆ = ((ηSS µS − αS ) − (ηLL µL − αL ))2 + 4 ηSL ηLS µL µS . Then
γ1 , γ2
are constants de ned by the following relations
γ1 γ2
1 ηSS ηLL = + − (αS + αL ) + 2 ρS ρL 1 ηSS ηLL = + − (αS + αL ) − 2 ρS ρL 5
1√ ∆, 2 1√ ∆. 2
(8)
An eigenvector is orthogonal to each rows of the matrix:
(ηSS µS − αS ) − γ ηSL µL ηLS µS (ηLL µL − αL ) − γ
v1 v2
= 0,
then necessary,
Let
D = diag(γ1 , γ2 ).
V
=
−ηSL µL (ηSS µS − αS ) − γi
f or i = 1, 2.
The matrix in the right term of equation (7) admits the representation:
where
v1i v2i
(ηSS µS − αS ) ηSL µL ηLS µS (ηLL µL − αL )
= V DV −1 ,
is the matrix of eigenvectors, as de ned in equation (5). If two new variables are
de ned as follows:
uS uL
= V
−1
mS mL
The system (7) is decoupled into two independent ODE's:
∂ ∂t
uS uL
= V
−1
αS θS αL θL
+
γ1 0 0 γ2
u1 u2
.
(9)
And introducing the following notations
V
−1
αS θ S αL θL
=
1 2
,
leads to the solutions for the system (9):
1 γ1 t e − 1 + d1 eγ1 t γ1 2 γ2 t uL (t) = e − 1 + d2 eγ2 t γ2 uS (t) =
d = (d1 , d2 )0 is such that d = V −1 (λS0 , λL0 ). Or in in matrix form, γt S 1 γ1 t (e − 1) 0 uS αS θS e1 0 λ0 −1 −1 γ1 = V + V 1 γ2 t γ2 t (e − 1) αL θL 0 e λL0 uL 0 γ2
where
S According to this last result, intensities are stable, in the sense that the limits of λt and L λt exist when t → +∞ if and only if γ1 and γ2 are negative. In this case, the asymptotic expectations are equal to
lim
t→∞
1 0 − γ1 E λSt |G0 αS θS −1 V . = V 0 − γ12 αL θL E λLt |G0
(10)
γ1 > 0 or γ2 > 0, the frequency of claim arrivals and the volatility of stocks are not bounded when t → +∞. For this reason, we assume in the remainder of this work that γ1 ≤ 0 or γ2 ≤ 0. The next corollary links the expected values of chronometers to the chronological If
time. And it may be proven by direct integration of expressions (4).
6
Corollary 2.2.
The expected values of τtS and τtL are given by 1 γ12
E τtS |G0 = V E τtS |G0
+V
t γ1
(eγ1 t − 1) − 0
1 γ1
(eγ1 t − 1) 0
1 γ2
! 0 αS θS −1 V 1 αL θL (eγ2 t − 1) − γt2 γ22 S 0 λ0 V −1 γ2 t λL0 (e − 1)
(11)
The variances of intensities do not admit any closed form expression. However, we can compute them numerically by solving a system of ordinary dierential equations:
2 2 λSt − E λSt 2 2 .Their covariance is VSL (t) = E λSt λLt − E λSt E λLt . − E λLt and VL (t) = E λLt VS , VL and VSL are solutions of the following system of ODE's: η2 2 2 2 ηSL 2 SS ρ2 VS ρ L ∂ η 2 2S mS (t) 2 2 η VL = + (12) LS ρ2 LL ρ2 S L mL (t) ∂t 2 2 VSL ηSS ηLS ρ2 ηLL ηSL ρ2 L S 0 2ηSL ρ1L 2 ηSS ρ1S − αS VS VL , 0 2 ηLL ρ1L − αL 2ηLS ρ1S VSL ηSS ρ1S + ηLL ρ1L − αS − αL ηLS ρ1S ηSL ρ1L
Proposition 2.3.
Let us denote the variances of λS and λL by VS (t) = E
and satisfy the initial conditions Vi (0) = 0
Proof.
Let us introduce the notations:
f or i = S, L, SL. 2 2 gS = λSt , gL = λLt and gSL = λSt λLt .
In nitesimal
generators of these functions are:
AgS =
2λSt αS θS
−2
2 λSt
αS +
λSt
+∞
Z
2λSt ηSS z + (ηSS z)2 dνS (z)
−∞
+λLt
Z
+∞
2λSt ηSL z + (ηSL z)2 dνL (z) ,
−∞
AgL =
2λLt αL θL
−2
2 λLt
αL +
λLt
Z
+∞
2λLt ηLL z + (ηLL z)2 dνL (z)
−∞
+λSt
Z
+∞
2λLt ηLS z + (ηLS z)2 dνS (z) ,
−∞
AgSL = αS (θS − λSt )λLt + αL (θL − λLt )λSt Z +∞ S +λt λSt + ηSS z λLt + ηLS z − λSt λLt dνS (z) −∞ Z +∞ +λLt λSt + ηSL z λLt + ηLL z − λSt λLt dνL (z) . −∞ 7
we denote and
νSL
vS = E
2 , λSt
vL = E
2 , λLt
vSL = E
λSt λLt , and
ξS = E
J
S 2
νS , νL
.
are solutions of a system of ODE's:
∂ vS = 2mS (t)αS θS − 2vS (t)αS + 2vL (t)ηSS µS + ∂t 2 2 2 2 + 2vSL (t)ηSL µL + mL (t)ηSL E JL mS (t)ηSS E JS ∂ vL = 2mL (t)αL θL − 2vL (t)αL + 2vL (t)ηLL µL + ∂t 2 2 2 2 + 2v3 (t)ηLS µS + m1 (t)ηLS E JS mL (t)ηLL E JL
(13)
∂ vSL = mL (t)αS θS − αS vSL (t) + mS (t)αL θL − αL vSL (t) ∂t 2 +vS (t)ηLS µS + vSL (t)ηSS µS + mS (t)ηSS ηLS E J S 2 +vL (t)ηSL µL + vSL (t)δLL µL + mL (t)ηLL ηSL E J L As centered second moments
Vi (t), are linked to non centered ones, vi
by the next dierential
equations
(
∂ ∂ ∂ V = ∂t vi − 2mi ∂t mi ∂t i ∂ ∂ ∂ V = ∂t vSL − mS ∂t mL ∂t SL
i = S, L ∂ − mL ∂t mS
Combining equations (7) and (13) allows us to conclude. The variances and correlation of intensities are then computable by an Euler's method applied to ODE's (12). The next proposition allows us to calculate numerically the joint L S moments of τt and τt . This result is used later to evaluate covariances and correlations. Proposition 2.4.
intensities
S
The generating function of τtS and τtL is an exponential a ne function of L
E eωS τs +ωL τs | Gt
S
L
= eωS τt +ωL τt × exp A(t, s) +
BS (t, s) BL (t, s)
>
where A(.), B(.) and C(.) satis es the system of ODE's ∂ A = −αS θS BS − αL θL BL ∂t ∂ BS = −ωS + αS BS − φJS (BS ηSS + BL ηLS ) − 1 ∂t ∂ BL = −ωL + αL BL − φJL (BS ηSL + BL ηLL ) − 1 ∂t
with the terminal conditions BS (s, s) = BL (s, s) = 0 and A(s, s) = 0.
8
λSt λLt
! ,
Proof.
The generating function of
τsS
and
τsL
for
s≥t
is denoted by
f (t, τtS , τtL , λSt , λLt ).
and
is solution of an Itô SDE:
0 = ft + fτ S λSt + αS (θS − λSt ) fλS + fτ L λLt + αL (θL − λLt ) fλL Z +∞ S f t, τtS , τtL , λSt + ηSS z, λLt + ηLS z − f ν S (dz) +λt Z0 +∞ +λLt f t, τtS , τtL , λSt + ηSL z, λLt + ηLL z − f ν L (dz)
(14)
0
τtS , τtL and λSt , λLt : > S > S ! BS (t, s) λt CS (t, s) τt f = exp A(t, s) + + , L BL (t, s) λt CL (t, s) τtL
Let us assume that
f
is an exponential a ne function of
where coe cients are functions of time. Under this assumption, the partial derivatives of
f
are given by:
∂ A+ ∂t
ft =
fτ S = CS f
∂ B ∂t S ∂ B ∂t L
>
λSt λLt
fτ L = CL f
+
∂ C ∂t S ∂ C ∂t L
fλS = BS f
>
τtS τtL
! × f,
fλL = BL f
And integrands in equation (14) are rewritten as follows:
f t, τtS , τtL , λSt + ηSS z, λLt + ηLS z − f = f [exp ((BS ηSS + BL ηLS ) z) − 1] f t, τtS , τtL , λSt + ηSL z, λLt + ηLL z − f = f [exp ((BS ηSL + BL ηLL ) z) − 1] If
φJS (z) =
ρS and ρS −z
φJL (z) =
ρL are the moment generating functions of jumps, we obtain ρL −z
that
∂ > S ∂ > S ∂ B λ C τt S S t 0 = A + ∂t + ∂t + CS λSt + αS (θS − λSt ) BS + ∂ ∂ L L λ τ B C ∂t t t ∂t L ∂t L L L S JS CL λt + αL (θL − λt ) BL + λt φ (BS ηSS + BL ηLS ) − 1 + λLt φJL (BS ηSL + BL ηLL ) − 1 Then necessary
CS = ωS
and
CL = ωL .
The Itô equation becomes then:
> S ∂ ∂ B λt S + ωS λSt − αS BS λSt + 0 = A + αS θS BS + αL θL BL + ∂t ∂ L λ B ∂t t ∂t L L L S JS ωL λt − αL BL λt + λt φ (BS ηSS + BL ηLS ) − 1 + λLt φJL (BS ηSL + BL ηLL ) − 1
and regrouping terms allows us to conclude that
∂ A = −αS θS BS − αL θL BL ∂t ∂ BS = −ωS + αS BS − φJS (BS ηSS + BL ηLS ) − 1 ∂t ∂ BL = −ωL + αL BL − φJL (BS ηSL + BL ηLL ) − 1 . ∂t
9
From this last proposition, we infer that the rst cross moment of chronometers is equal to
E
τtS τtL
| G0
∂ ∂ ωS τsS +ωL τsL | Gt E e ∂ωS ∂ωL ωS =ωL =0 > S ! ∂ ∂ BS (t, s) λt A(t, s) + L λ B (t, s) ∂ωS ∂ωL L t
= =
ωS =ωL =0
that unfortunately does not admit any closed form expression. However, this cross-moment is computable numerically by a nite dierence method.
To understand the in uence of
mutual and self excitation between intensities on standard deviations and correlations, ve numerical tests are conducted. The sets of parameters considered for this exercise and results are reported in table 1.
Parameters
Test 1
Test 2
Test 3
Test 4
Test 5
ε1 αS , αL θS , θL ρS , ρL ε2 E(τ1S )=E(τ1L ) std(τ1S )=std(τ1L ) corr(τ1S , τ1L )
0.00
0.04
0.08
0.12
0.16
19.91
19.91
19.91
19.91
19.91
0.48
0.48
0.48
0.48
0.48
0.49
0.49
0.49
0.49
0.49
5.72
5.72
5.72
5.72
5.72
1.07
1.13
1.18
1.24
1.30
4.12
4.26
4.46
4.70
4.98
0.00
0.03
0.07
0.10
0.12
Table 1: This table reports the one year expectations, standard deviations of
τtS , τtL
and
their correlation, for ve sets of parameters.
Speeds and levels of reversion for
λ1t
and
λ2t
are assumed equal. Whereas the matrix
Ξ
of
mutual excitation weights, is parametrized as follows:
Ξ= is null,
ηSL
= 1
cos(2 ) sin(2 ) sin(2 ) cos(2 )
.
are null and stochastic clocks are independent. Increasing 2 S L raises the correlation between τ1 and τ1 from 0% to 12%. It also speeds up on average the clocks and standard deviations slightly move up. This con rms that a large range of positive
When
2
ηSS ηSL ηLS ηLL
and
ηLS
correlations may be achieved by introducing mutual excitation in the dynamics of intensities. S Notice also that with this parametrization of Ξ, expectations and standard deviations of τt L and τt are equal and independent from 2 . The left plot of gure 1 shows the term structure S L of standard deviations of τt and τt , for dierent initial values of intensities. It emphasizes S L S L that just after a jump of λt or λt , the standard deviations of τt and τt step up to a higher level. At longer term, the growth of these standard deviations becomes nearly linear. The
10
S L right plot of the same gure presents the curve of correlations between τt and τt for three S L S L dierent sets of initial values of λt and λt . When λ0 and λ0 take their lowest values, that are θS and θL , the term structure of correlation is an increasing function of time that reverts L S at medium term to a constant level. When λ0 and λ0 are signi cantly higher than θS and θL , the curve of correlations is a humped function of time, that also reverts to a constant level.
Correlation between τS and τLt t
Standard deviation of τS , τLt t 18
0.08 S
L
S
L
std(τt )=std(τt ), λ0 =λ0=θS std(τS )=std(τLt), [λS λL]’=[θS θL]’+ Ξ×[2/ρS t 0 0 S S L std(τ ), [λ λ ]’=[θ θ ]’+ Ξ×[3/ρ 0] t 0 0 S L S std(τLt), [λS λL]’=[θS θL]’+ Ξ×[3/ρS 0] 0 0
16
2/ρL] 0.07
14 0.06
12 0.05 10 0.04 8
λS =λL0=θS 0 [λS λL]’=[θS θL]’+ Ξ×[2/ρS 2/ρL] 0 0
0.03
[λS λL]’=[θS θL]’+ Ξ×[3/ρS 0] 0 0
6
0.02 4
0.01
2
0
0
0.5
1 t
1.5
0
2
0
0.5
1 t
1.5
2
L S evolution of variances of τt , τt with respect to time. Right plot: S L correlation between τt , τt with respect to time. Parameters used are these reported in S L table 1, in the column labeled Test 3. Three dierent sets of values for λ0 and λ0 are S L S L > considered : 1) λ0 = λ0 = θS = θL 2) (λ0 , λ0 ) = (θS , θL )> + Ξ( ρ2S , ρ2L )> 3) (λS0 , λL0 )> = (θS , θL )> + Ξ( ρ3S , 0)> . Figure 1:
Left plot:
3 Financial assets and the insurance claims process We develop now the dynamics of markets in which the insurance company operates.
We
assume that the nancial market is composed of two assets: cash and stocks. The risk free rate earned by the cash account is denoted by
r
and is constant. The stock price, noted
St ,
is a time-changed geometric Brownian motion:
dSt = r dt + (µ − r)dτtS + σdWτSS t St 11
(15)
where
µ
and
σ
are positive constants and
WtS
is a Brownian motion. In this approach, only
the risk premium depends upon the market time scale. The Brownian motion being a stable process, this relation is equivalent to the following one which emphasizes that the drift and the the volatility of stock prices are random and proportional to the intensity of the nancial clock:
dSt = St
r + (µ −
r)λSt
q dt + σ λSt dWtS .
In a certain way, the dynamics of stock prices is comparable to these of stochastic volatility models, with a random drift. In our approach, the volatility is indeed mean reverting like in the Heston's model, excepted that it is driven by a bivariate self-exciting jump process. Using the Itô's lemma leads to the following relation for the log-price
d ln St =
q σ2 λSt dt + σ λSt dWtS r+ µ−r− 2
and from which we infer the stock price
St
Z t Z tp σ2 S S S λs ds + σ λs dWs , = S0 exp r t + µ − r − 2 0 0 σ2 S S τt + σWτ S = S0 exp r t + µ − r − t 2
(16)
On the side of liabilities, we consider a time-changed version of a risk process that is the dierence accumulated premiums and aggregated claims:
N
Lt = c
τtL
−
L
τt X
Ji
i=1 before reinsurance.
c
is the premium rate,
Ji
are the random claims and
Nt
is a Poisson
process with a constant intensity λ. The statistical distribution of J is not speci ed but J ωJ we assume that its moment generating function, denoted by φ (ω) = E e , exists. Its J probability density function is written ν (.). The premium rate is assumed strictly bigger than the expected claims,
c > λE (J).
If this condition is not ful lled, the ruin occurs with
certainty. Notice also that the dynamics of liabilities is equivalent to the following one:
dLt = c dτtL − JdNτtL
(17)
= cλLt dt − JdNtb where
Ntb
R t J and an intensity equal to λ 0 λLs ds . k claims till (chronological) time t is equal
is a point process with marks of size
Indeed by de nition, the probability of observing to
R
P NτtL = k
τtL 0
=
k λds
k! R
=
t 0
= P 12
λλLs ds
k
k! =k .
Ntb
L
e−λτt e−
Rt 0
λλL s ds
The natural ltration associated to tion carrying all the information is
St and Lt is denoted by (Ht )t and the augmented ltraFt := Ht ∨ Gt (remember that Gt is the natural ltration
of chronometers). In the speci cation of our model, the claims process is not directly correlated to the nancial market. However a dependence appears through the correlation that exists between stochastic clocks driving insurance and nancial markets. This dependence is measured by the next proposition that has also an interesting economic interpretation. Proposition 3.1.
is given by
The covariance between the logarithm of stocks prices and the risk process
(E (ln (St ) Lt |F0 ) − E (ln (St ) |F0 ) E (Lt ) |F0 ) = 1 2 E τtS τtL |G0 − E τtS |G0 E τtL |G0 (c − λE (J)) µ − r − σ 2
Proof.
To prove this result, it is su cient to remember the expressions of log prices and of
the risk process:
ln St
σ2 = ln S0 + r t + µ − r − τtS + σWτSt 2 N
Lt = L0 + c τtL −
L
τt X
Ji
i=1 A direct calculation leads to the following expressions for the cross expectation and product of expectations (we momentarily forget the ltration with respect to which we calculate these quantities to lighten notations):
L
E (ln (St ) Lt ) = ln S0 L0 + ln S0 c E τt
N L τt X − ln S0 E Ji i=1
N L τt X 1 2 L +r L0 t + r t c E τt − r t E Ji + L0 µ − r − σ E τtS 2 i=1 NL τt 2 X σ 1 2 S L S +c µ − r − σ E τt τt − µ − r − E τ t Ji 2 2 i=1 and
E (ln St ) E (Lt ) = ln S0 E τtS + c ln S0 E τtL − ln S0 E τtS
N
L
τt X
Ji
i=1
N L τt X 1 2 L +r L0 t + r t c E τt − r t E Ji + L0 µ − r − σ E τtS 2 i=1 N L τt 2 2 X σ σ S L S +c µ − r − E τt E τt − µ − r − E τt E Ji 2 2 i=1 13
The covariance is then
(E (ln (St ) Lt ) − E (ln (St )) E (Lt )) = 1 2 c µ−r− σ E τtS τtL |G0 − E τtS |G0 E τtL |G0 2 i σ2 h S S L − µ−r− E (J) E τt NτtL − λE (J) E τt |G0 E τt |G0 . 2 Finally, nesting conditional expectations leads to
E τtS NτtL |F0 = E E τtS NτtL |F0 ∨ Gt |F0 = λE τtS τtL |F0
This result appeals several comments.
Firstly, the correlation between log-prices and
claims is proportional to the one of stochastic clocks. Secondly, it depends on the product 1 2 of the average risk premium of stocks, (µ − r − σ ), and of the insurer's average pro t, 2 (c − λE (J)), that may also be interpreted as an insurance risk premium. If c is equal to the pure premium rate,
c = λE (J),
the expected surplus and the covariance are both null.
In this case, we do not observe any linear dependence between assets and liabilities. And this despite the fact that the asset price and risk process are clearly not independent by construction. The covariance diers from zero only if the insurance business is pro table on average. The linear dependence is in this case induced by insurer's incomes that are next reinvested in the nancial market. From an economic point of view, the correlation between insurance and nancial markets comes then exclusively from the existence of a risk premium in both segments. The next proposition introduces the joint mgf of log-prices and liabilities. Proposition 3.2.
The joint moment generating function of ln St and Lt is given by
E eωS ln Ss +ωL Ls | Ft
= StωS eωS r (s−t) eωL Lt E exp ωSb (τsL − τtL ) + ωLb (τsL − τtL ) | Gt (18)
where ωSb ωLb
1 σ2 = µ−r− ωS + ωS2 σ 2 2 2 J = ωL c + λ φ (−ωL ) − 1
and E exp ωSb (τsL − τtL ) + ωLb (τsL − τtL ) | Gt > S ! BS (t, s) λt = exp A(t, s) + , BL (t, s) λLt
with A, BS and BL de ned in proposition 2.4. 14
Proof.
To prove this result, we use again nested conditional expectations and the fact that
Gs , log prices and liabilities are independent E eωS ln Ss +ωL Ls | Ft = E E eωS ln Ss +ωL Ls | Ft ∨ Gs | Ft E E eωS ln Ss | Ft ∨ Gs E eωL Ls | Ft ∨ Gs | Ft .
conditionally to ltration
If we remind the expression (16) for
Ss ,
the rst term in this last product is equal to
E (SsωS | Ft ∨ Gs ) = E eωS ln Ss | Ft ∨ Gs σ2 1 2 2 ωS ωS r (s−t) S S = St e exp µ−r− ωS + ωS σ (τs − τt ) . 2 2 On the other hand, the moment generating function of the claims process, in absence of any time change, is equal to
ω (c s−
E e
P Ns
i=1
Ji )
ω(c s)
= e
P s −ω ( N i=1 Ji )
E e = exp ω c + λ φJ (−ω) − 1 s
Then
E eωL Ls | Ft ∨ Gs
= eωL Lt exp
ωL c + λ φJ (−ωL ) − 1 (τsL − τtL )
and we can conclude with the proposition 2.4.
The generating function of cross-moments is derived numerically to calculate the correlation between
St
and
Lt
reported in table 2. We consider the ve sets of parameters of table
1, that were used to analyze the impact of mutual excitation on the correlation between stochastic clocks. In our example, the one year correlation between assets and liabilities is comparable to the correlation between stochastic clocks. For example, a correlation of 7% S L between τ1 and τ1 induces a correlation of 8% between S1 and L1 . The correlation also depends upon the time horizon that is considered. This point is illustrated in gure 2 that S shows the term structure of correlations between ln St and Lt , for dierent values of λ0 and L λ0 . The correlation between ln St and Lt is clearly negligible at short term and next reverts at medium term to a constant level.
This means that a delay is induced between the oc-
currence of an event in one market and the reaction of the other market. In other words, there is well contagion between the insurance and nancial markets but the dependence is not instantaneous. Some time is needed to assimilate the information ow from a market and to eventually cause a shock in the other market.
15
Correlation between ln St and Lt 0.12
0.1
0.08
0.06
λS =λL0=θS 0 [λS λL]’=[θS θL]’+ Ξ×[2/ρS 2/ρL] 0 0
0.04
[λS λL]’=[θS θL]’+ Ξ×[3/ρS 0] 0 0
0.02
0
0
0.2
0.4
0.6
0.8
1 t
Figure 2: Evolution of the correlation between
1.2
St
1.4
and
Lt
1.6
1.8
2
with respect to time. Parameters
used are these reported in table 1, in the column labeled Test 3. Three dierent sets of values 2 > L > S L S L S , 2 )> for λ0 and λ0 are considered : 1) λ0 = λ0 = θS = θL 2) (λ0 , λ0 ) = (θS , θL ) + Ξ( ρS ρL 3 S L > > 3) (λ0 , λ0 ) = (θS , θL ) + Ξ( , 0)> . ρS
corr(τ1S , τ1L ) E (ln S1 ) E (L1 ) std(ln S1 ) std(L1 ) corr(S1 , L1 )
Test 1
Test 2
Test 3
Test 4
Test 5
0.00
0.03
0.07
0.10
0.12
0.04
0.04
0.05
0.05
0.05
22.43
23.50
24.63
25.82
27.06
0.21
0.22
0.23
0.23
0.24
42.23
43.46
45.16
47.33
49.97
0.00
0.04
0.08
0.12
0.15
Table 2: This table reports the 1 year expectations, standard deviations and correlation of S L S L assets and liabilities. λ0 = λ0 = θS = θL . Parameters de ning λt and λt , are these presented in table 1. Others parameters de ning the asset and risk processes are: µ = 5%, r = 0%,
σ = 20%, λ = 200. Claims λ premium rate is c = 1.10. ρ
are exponential random variables with a parameter
ρ = 1.
The
In practice, the calibration of this model is a challenging exercise given that assets and liabilities depend on two hidden state variables. This point is detailed in a study of Hainaut (2016 (d)) that proposes an approached method to calibrate similar Lévy process, timechanged by an integrated self-excited subordinator. Here, we only summarize how to adapt this procedure to our ALM framework and refer to the original article for details. The method is based on the premise that intensities remains constant and equal to their asymptotic
16
averages
λS∞
and
λL∞
:
as de ned by equation (10). cesses.
λS∞ λL∞
:=
lim
t→∞
E λSt |G0 E λLt |G0
Under this assumption,
ln St
and
(19)
Lt
become stationary pro-
The probability density function (pdf ) for a given set of parameters
Θ
may then
be computed by inverting numerically the moment generating function (18) with a two dimensions Discrete Fourier Transform (2D DFT). The set of parameters
Θ
is next tted by
maximization of the log-likelihood calculated with the numerical pdf. The 2D DFT algorithm is provided in Hainaut (2016 (d)). This study also reveals that despite the bias introduced S L by the hypothesis λt , λt = λS∞ , λL∞ , the t is of good quality. An alternative approach S L consists rstly to develop a particle lter to retrieve the sample path of λt and λt (again we refer to Hainaut (2016 (d)) for a presentation of this lter). And secondly to combine it with a Particle Markov Chain Monte-Carlo method, as explained in Andrieu et al. (2010). Before investigating the problem of optimal asset allocation, we propose a bound on the ruin probability when insurer's earnings are not invested in nancial markets. We look for ∗ ∗ an upper bound of P (τ ≤ ∞), where τ is the rst time such that Lt < 0. To infer this bound, that is a common indicator of risk in the actuarial literature, we rst determine the conditions under which an exponential combination of processes of the form
Mt (gS , gL , gR , ξ) := exp gS λSt + gL λLt + gR Lt − ξt
(20)
is a local martingale.
If for any ξ ∈ R+ there exists a suitable triplet (gS , gL , gR ) solution of the system of equations Proposition 3.3.
0 = −ξ + gS αS θS + gL αL θL 0 = −gS αS + φJS (gS ηSS + gL ηLS ) − 1 0 = −gL αL + φJL (gS ηSL + gL ηLL ) − 1 +gR c + λ φJ (−gR ) − 1
(21)
then Mt is a local martingale. Proof. Mt
is a local martingale if and only if its in nitesimal generator
AMt = −ξM + gS αS (θS − λSt )M + gL αL (θL − λLt )M + gR c λLt M Z Z S S (gS ηSS +gL ηLS )z L +λt M e − 1 ν (dz) + λt M e(gS ηSL +gL ηLL )z − 1 ν L (dz) Z L +λλt M e−gR z − 1 ν J (dz) is null. Regrouping terms leads to the system (21).
17
Notice that at this stage, we made assumptions on the distribution of on the claims size
J.
JS
and
JL
but not
The only constraint is that its mgf exists. We nally have the following
expression for the asymptotic probability of ruin: Proposition 3.4.
If ξ ≥ 0 and gR (ξ) ≤ 0 then
P (τ ∗ ≤ ∞) =
Proof.
T , we have that exp gS λST + gL λLT + gR LT − ξT
(22)
is a local ∗ martingale if conditions (21) are ful lled. On the other hand, the minimum between τ and
T
For any given time horizon
M0 (gS , gL , gR , ξ) limT →∞ E (Mτ ∗ (gS , gL , gR , ξ) |F0 , τ ∗ ≤ T )
is a stopping time and according to the optional stopping theorem, we infer that
E (Mτ ∗ ∧T |F0 ) = P (τ ∗ > T ) E (MT |F0 , τ ∗ > T ) + P (τ ∗ ≤ T ) E (Mτ ∗ |F0 , τ ∗ ≤ T ) = M0 . If
gR < 0
and
ξ≥0
and as processes
λST
and
λLT
do not explode, then the rst term of the
previous equation converges to zero:
lim E (MT |F0 , τ ∗ > T )
T →∞
= lim exp (−ξT ) E exp gS λST + gL λLT + gR LT |F0 , τ ∗ > T T →∞
= 0. and
M0
is then equal to
M0 = P (τ ∗ ≤ ∞) lim E (Mτ ∗ |F0 , τ ∗ ≤ T ) . T →∞
On the other hand
lim E (Mτ ∗ |F0 , τ ∗ ≤ T )
T →∞
= lim E exp gS λSτ∗ + gL λLτ∗ + gR Lτ ∗ − ξτ ∗ |F0 , τ ∗ ≤ T T →∞
and we can conclude. Finally, we infer the following upper bound on the asymptotic probability of ruin: Corollary 3.5.
If gR de ned as the solution of the non linear equation 0 = gR c + λ φJ (−gR ) − 1
is negative, the asymptotic probability of ruin admits the following bound: P (τ ∗ ≤ ∞) ≤ M0 (0, 0, gR , 0) = exp (gR L0 )
18
(23)
Proof.
ξ ≥ 0, the asymptotic probability of ruin admits indeed the representation (22). And if ξ = 0, gL (ξ) = 0 and gS (ξ) = 0 satisfy the rst equation of the system of equations (21). Mt is a martingale if gR is solution of the equation (23). As Lτ ∗ < 0, it follows that If
E (Mτ ∗ (gS , gL , gR , ξ) |F0 , τ ∗ ≤ T ) = E (exp (gR Lτ ∗ ) |F0 , τ ∗ ≤ T ) > 1
are exponential random variables of parameter ρ, then the equation (23) gR = λc − ρ . As the premium rate is assumed strictly bigger than 1 the pure premium, c > λ , gR < 0 and the asymptotic probability of ruin is bounded by ρ λ P (τ ∗ ≤ ∞) ≤ e( c −ρ)L0 . This upper bound is identical to the bound on the asymptotic ruin If claims sizes
J
admits the solution:
probability, in a Cramer-Lundberg model.
Safety
Simulated
Cramer
Numerical
Margin
Ruin
Lundberg
Approximation of
Probabilities (%)
Bound (%)
equation (22) (%)
2.5%
83.0
88.5
85.2
5.0%
70.6
78.8
73.3
7.5%
60.1
70.6
63.6
10%
52.1
63.5
55.3
12.5%
45.8
57.4
48.8
15.0%
40.5
52.1
43.3
17.5%
36.5
47.5
38.7
20.0%
32.2
43.5
34.0
Table 3: Comparison of simulated ruin probabilities with the Cramer-Lundberg upper bound and the ruin probabilities computed with equation (22).
So as to evaluate the accuracy of this Cramer-Lundberg bound and to validate numerically the equation (22), simulations are performed. Parameters used to simulate stochastic clocks are those reported in the third column of table 1. Claims are exponential random variables with
ρ = 1 whereas the frequency of claims is set to λ = 200.
The premium includes a safety λ margin, from 2.5% to 20% and the premium rate is such that c = (1+saf ety margin). The ρ initial provision L0 , is equal to 5. We use an Euler discretized version of equations (2) and (17) to simulate 5000 sample paths of
Lt
over a period of 100 years, with time step of 0.005.
The results reported into table 3 emphasizes that the gap between the real ruin probabilities and the upper bound varies between 5.52% for a margin of 2.5% to 11.60% for a margin of 12.5%. According to equation (22), the asymptotic probability of ruin whenξ and
gS (ξ) = 0
is given by
P (τ ∗ ≤ ∞) =
exp (gR L0 ) . E (exp (gR Lτ ∗ ) |F0 , τ ∗ ≤ T ) 19
= 0, gL (ξ) = 0
To check this relation, we evaluate numerically the expectation present in the denominator of this quotient. The probabilities of ruin computed by this way are reported in the third column of table 3 and relatively close to real ones. This con rms the validity of equation (22). The spread between these probabilities of default varies from 1.74% to 3.49%, depending upon the level of safety margin. This spread is due to the limited number of simulations, to the time horizon and to numerical errors generated by the Euler discretization.
4 Optimal asset allocation, reinsurance and dividends This section focuses on the asset-liability management (ALM) policy of an insurance company investing in nancial markets the incomes from the insurance activity. We denote by
πt
the percentage of the total asset managed by the insurer that is invested in stocks. Fur-
thermore, we assume that the claims process is proportionally reinsured. A such reinsurance treaty foresees the transfer of a fraction,
1 − qt
of collected premiums to the reinsurer, in
exchange of the covering of the same fraction of claims. Finally, the insurer also distributes to shareholders a continuous dividend that is noted
dt .
In this framework, the economic value of the insurance company also called surplus and denoted by
Xt ,
is the dierence between the assets and liabilities.
It obeys to the next
dynamics:
dSt − dt St +dLt − (1 − qt ) c dτt + (1 − qt )JdNτt
dXt = (1 − πt )Xt r dt + πt Xt
The rst line is related to nancial operations whereas the second line is the income from dSt insurance activities. If we replace in this last equation and dLt by their expressions (15) St and (17), we obtain the following SDE for the surplus:
dXt =
where
Ntb
rXt + πt (µ − r)λSt Xt − dt + c qt λLt dt q +πt Xt σ λSt dWtS − qt JdNtb .
is a point process with an intensity equal to
ment is noted
R t λ 0 λLs ds .
If the horizon of manage-
T , the insurer optimizes the investment, reinsurance and dividend policies.
The
criteria of optimization are the discounted utility of dividends distributed over this period and the discounted utility of the terminal surplus. If these utilities are respectively denoted by
U1 (.)
and
U2 (.),
and if the discount rate is
β,
the value function of the insurer is de ned
by the following relation:
V
(t, Xt , λSt , λLt , τtS , τtL )
Z = max E πt ,dt qt
T −β(s−t)
e t
20
−β(T −t)
U1 (ds )ds + e
U2 (XT ) |Ft .
If we refer to the theory of stochastic optimal control, the value function of this optimization problem is the solution of a Hamilton Jacobi Bellman equation (HJB):
rX + π(µ − r)λS X − d + c q λL VX + U1 (d) π,d,q Z 1 2 2 2 S S L L J + π X σ λ VXX + Vτ S λ + Vτ L λ + λλt V (X − qz ) − V ν (dz) 2 Z S S S S V (λSt + ηSS z, λL + ηLS z) − V ν S (dz) +α θ − λ VλS + λ Z L L L S L L L V (λt + ηSL z, λ + ηLL z) − V ν (dz) +α θ − λ VλL + λ
0 = Vt − βV + max
where
Vt , VX , VXX , VλS
and
VλL
(24)
are the partial derivatives of the value function with respect
to driving stochastic processes. In the previous equation, we momentarily forget the index S L S L related to time so as to lighten notations. The terminal condition is V (T, XT , λT , λT , τT , τT )= U2 (XT ). If we derive the HJB equation with respect to π , we infer that the optimal investment policy is:
π∗ = −
(µ − r) VX . σ 2 X VXX
(25)
Using the same approach allows us to infer the optimal dividend policy: 0−1
d∗ = U1 (VX ) .
(26)
On the other hand, the optimal reinsurance satis es the following relation
Z 0 = c VX − λ
VX (X − q ∗ z ) z ν J (dz)
(27)
and if we insert these results into the relation (24), the HJB equation may be rewritten as follows:
! 1 (µ − r)2 VX S 0−1 0−1 λ + c q λL − U1 (VX ) VX + U1 (U1 (VX )) (28) 0 = Vt − βV + rX − 2 2 σ VXX Z S L L J +Vτ S λ + Vτ L λ + λλ V (X − qz ) − V ν (dz) Z S S S S +α θ − λ VλS + λ V (λSt + ηSS z, λL + ηLS z) − V ν S (dz) Z L L L L V (λSt + ηSL z, λL + ηLL z) − V ν L (dz) +α θ − λ VλL + λ Utility functions are assumed exponential:
U1 (y) = − γ11 e−γ1 y
and
U2 (y) = − γ12 e−γ2 y .
In this
particular case, it is possible to infer a semi-closed form expression for the value function: Proposition 4.1. The value function solving the HJB equation (24) is an exponential a ne function of the wealth and of chronometers intensities:
V (T, XT , λST , λLT ) = −
1 exp A(t, T ) + γ2 21
BS (t, T ) BL (t, T )
>
λSt λLt
! + C(t, T )Xt ,
(29)
where A(.), BS (.), BL (.) and C(.) are deterministic functions of time, solutions of the next ODE's: ∂ 1 1 A(t, T ) = β − ln − C(t, T ) + A(t, T ) − 1 C(t, T ) (30) ∂t γ1 γ2 −αS θS BS (t, T ) − αL θL BL (t, T ) ∂ 1 C(t, T ) = −rC(t, T ) − C(t, T )2 ∂t γ1 1 (µ − r)2 ∂ 1 S BS (t, T ) = − C(t, T ) − α BS (t, T ) + ∂t γ1 2 σ2 − φS (ηSS BS (t, T ) + ηLS BL (t, T )) − 1 ∂ 1 L BL (t, T ) = − C(t, T ) − α BL (t, T ) − c C(t, T ) qt∗ − λ φJ (−C(t, T ) qt∗ ) − 1 ∂t γ1 − φL (ηSL BS (t, T ) + ηLL BL (t, T )) − 1
with the terminal conditions A(T, T ) = 0, C(T, T ) = −γ2 , BS (T, T ) = BL (T, T ) = 0. qt∗ is here the optimal reinsurance solution of the next relation: +∞
Z
z e−C q z ν J (dz) =
0
Proof.
c . λ
(31)
To prove this, we use a veri cation argument. Under the assumption that the value
function has an exponential form of the type (29), the partial derivatives of
V
with respect
to risk factors and time are given by
Vt = (At + Ct X + BS t λS + BL t λL )V VX = C V VXX = C 2 V VλS = BS V VλL = BL V V (λSt + ηSS z, λLt + ηLS z) − V = V e(ηSS BS +ηLS BL ) z − 1 V (λSt + ηSL z, λLt + ηLL z) − V = V e(ηSL BS +ηLL BL ) z − 1 V (X − qz ) − V = V e−Cqz − 1 . On the other hand,
0
U1 (y) = e−γ1 y
and
0
U1−1 (y) = − γ11 ln (y)
then
1 1 1 S L U1 (VX ) − ln (C V ) = − ln − C + A + C X + BS λ + BL λ γ1 γ1 γ2 1 1 0−1 U1 (U1 (VX )) = − eln(VX ) = − C V γ1 γ1 0 −1
22
If we insert these intermediate results in the HJB equation, we obtain that
1 (µ − r)2 S 0 = (At + Ct X + BS t λS + BL t λL ) − β + rCX − λ + cC q ∗ λL 2 σ2 1 1 2 S L + C ln − C + AC + C X + BS Cλ + BL Cλ γ1 γ2 1 − C + λL λ φJ (−Cq) − 1 γ1 +αS θS − λS BS + λS φS (ηSS BS + ηLS BL ) − 1 +αL θL − λL BL + λL φL (ηSL BS + ηLL BL ) − 1 and the optimal reinsurance
q∗
! (32)
is solution of the next equation:
Z c−λ
ze
−Cq ∗ z
J
ν (dz)
= 0.
The relation (32) is satis ed whatever the value of risk factors if and only if the relations (30) hold.
By direct dierentiation, we can check that expression:
C(t, T )
admits the following closed form
−1 1 −r(T −t) 1 1 −r(T −t) 1−e − e C(t, T ) = − . γ1 r γ2
We will see that this function plays a crucial role in the determination of the optimal reinsurance and investment policy. The optimal ratio of reinsurance is indeed solution of a non linear equation that does not admit any closed form expression. However, if we approach the exponential in equation (31) by a rst order Taylor's development, we infer that the optimal reinsurance rate satis es approximately the next relation:
Z
+∞
ze
−C q z
J
Z
ν (dz) ≈
0
+∞
z − C q z 2 ν J (dz)
0
E(J) − CqE(J 2 ) =
c , λ
from which we obtain nally that
qt∗
1 c ≈ E(J) − , C(t, T )E(J 2 ) λ 1 1 1 −r(T −t) 1 c −r(T −t) = 1−e + e − E(J) . γ1 r γ2 E(J 2 ) λ
(33)
C(t, T ) is strictly negative, the optimal reinsurance rate is proportional to the safety marc gin on claims size ( − E(J)), embedded in the premium rate. Notice that the reinsurance λ As
ratio exclusively depends on frequency of claims measured on the insurance market scale and not on the chronological time scale. From this last relation, we also infer that no claim is
23
re-insured if
c
is equal to the pure premium rate,
c = λE(J)
. If the premium rate is higher
or lower than this pure premium, the reinsurance rate is inversely proportional to the second moment of the claims size. Finally, the optimal reinsurance is independent from the insurer's 1 E(J) − λc wealth. It is a pure deterministic function of time that converges toward γ2 E(J 2 ) at expiry. On the other hand, the optimal investment policy consists to invest the following time varying percentage of the total asset in stocks:
(µ − r) 1 1 2 σ C(t, T ) Xt 1 −r(T −t) 1 (µ − r) 1 1 −r(T −t) + e = 1−e σ2 γ1 r γ2 Xt
πt∗ = −
(34)
and is independent from clocks of nancial and insurance markets. At the end of the time (µ−r) horizon, the insurance company holds 2 of stocks. As for the reinsurance rate, the optimal σ γ2 investment policy is based on the drift and variance measured on the nancial market scale and not on the chronological scale. This is not the case of the optimal dividend that explicitly depends on intensities of clocks:
d∗t
1 1 S L ln − C(t, T ) + A(t, T ) + C(t, T ) Xt + BS (t, T )λt + BL (t, T )λt = − γ1 γ2
S However, as BS (t, T ) and BL (t, T ) converge toward zero when t → T , the in uence of λt and λLt is lessened with the passage of time. Even if the optimal management policy is determined, we don't have at this stage any information about the distribution of the optimal wealth, Xt∗ . However, we can approach numerically its moments. Under the assumption that the optimal reinsurance ratio is close to the one in equation (33), the dynamics of the surplus is given by the next SDE
dXt∗ = µ∗1 (t, T ) dt + µ∗2 (t, T )Xt∗ dt + µ∗3 (t, T )λSt dt + µ∗4 (t, T )λLt dt q E(J) − λc (µ − r) S S λt dWt − JdNtb . − σC(t, T ) C(t, T )E(J 2 ) with
1 ln − C(t, T ) + A(t, T ) γ2
µ∗1 (t, T )
1 = γ1
µ∗3 (t, T )
BS (t, T ) (µ − r)2 = − 2 γ1 σ C(t, T )
and where functions
C(t, T ), BL (t, T )
1 = r + C(t, T ) γ1 BL (t, T ) c E(J) − λc ∗ µ4 (t, T ) = + γ1 C(t, T )E(J 2 )
and
BS (t, T )
µ∗2 (t, T )
are de ned in proposition 4.1 by the
system of ODE's (30).
The mgf of the optimal wealth under the assumption that the optimal reinsurance ratio is approached by (33) is given by the following expression
Proposition 4.2.
∗
E eωXs |Ft
= exp A∗ (t, s) +
BS∗ (t, s) BL∗ (t, s)
24
>
λSt λLt
! + C ∗ (t, s)Xt
,
(35)
where the functions A∗ , BS∗ , BL∗ and C ∗ satis es the next system of ODE's: ∂ ∗ A (t, s) = −µ1 (t, T )C ∗ (t, s) − αS θS BS∗ (t, s) − αL θL BL∗ (t, s) ∂t ∂ ∗ C (t, s) = −µ2 (t, T )C ∗ (t, s) ∂t ! 2 ∂ ∗ 1 (µ − r)2 C ∗ (t, s) ∗ BS (t, s) = −µ3 (t, T )C (t, s) − + αS BS∗ (t, s) 2 ∂t 2 σ C(t, s) − φJS (BS∗ (t, s)ηSS + BL∗ (t, s)ηLS ) − 1 ∂ ∗ BL (t, s) = −µ4 (t, T )C ∗ (t, s) + αL BL∗ (t, s) − φJL (BS∗ (t, s)ηSL + BL∗ (t, s)ηLL ) − 1 ∂t −λ φJ (−C ∗ (t, s) qt∗ z) − 1
with the terminal conditions A∗ (s, s) = 0, C ∗ (s, s) = ω , BS∗ (s, s) = 0 and BL∗ (s, s) = 0. Proof.
Let us denote by
f (t, Xt , λSt , λLt ),
the moment generating function of
Xs
for
s ≥ t.
This mgf is solution of the Itô SDE:
1 1 (µ − r)2 S 0 = f t + µ1 f X + µ2 X f X + µ3 λ f X + µ4 λ f X + λ fXX 2 C 2 σ2 Z +∞ +αS (θS − λS ) fλS + λS f t, Xt , λSt + ηSS z, λLt + ηLS z − f ν S (dz) Z0 +∞ +αL (θL − λL ) fλL + λL f t, Xt , λSt + ηSL z, λLt + ηLL z − f ν L (dz) 0 Z +∞ +λλL f t, Xt − qt∗ z, λSt , λLt − f ν J (dz) S
L
0 Assuming that the mgf has the form of equation (35) and using the same approach as for the proof of proposition 4.1, allows us to prove the proposition. To conclude this section, we solve numerically the asset liability management problem. The parameters of stochastic clocks used for this exercise are those reported in the third column of table 1. The average growth rate and the standard deviation of the asset return,
µ = 5%, σ = 20% whereas the risk free rate is 2%. Claims are exponential random variables with ρ = 1. The frequency of claims is equal to λ = 200. The premium rate includes λ a safety margin of 10% : c = 1.10. The coe cients of risk aversion are respectively γ1 = 10 ρ or γ1 = 20 and γ2 = 5. The discount rate used in utility function is set to β = 1%. Finally the time horizon is 10 years and the initial wealth is X0 = 5.
are set to
25
7
5.5 5
6 5 4
4.5
q*t γ1=10
4
q*t γ1=20
%
3.5 3
3
2.5
2
2 E(Xt) γ1=10
1 0
1.5
std(Xt) γ1=10 0
2
4
6 T
E(X )8γ =20 t
1
1
10
0
2
4
6
8
10
6
8
10
T
std(Xt) γ1=20
0.9
1.8
0.8
π*t Xt γ1=10
0.7
π*t Xt γ1=20
1.6
E(d0t) γ1=10
1.4
E(d0t) γ1=20
0.6 1.2 0.5 1 0.4 0.8
0.3
0.6
0.2 0.1
0
2
4
6
8
0.4
10
0
2
4
T
Figure 3:
Upper left plot:
Upper right plot:
T
expectations and standard deviations of the optimal surplus.
optimal reinsurance ratio.
Lower left plot:
optimal amount of stocks.
Lower right plot: expected dividends.
The optimal ALM strategy presented in gure 3 is obtained with
λS0 = λL0 = θS = θL .
The upper left graph displays the expected wealth for dierent maturities. Its analysis must be related to the lower left graph that shows the term structure of expected dividends. We observe that during a rst period of 4 years, the richness increases on average by 18% to 36%, depending upon
γ1 .
Incomes from insurance activities and investments are on average
higher than distributed dividends that however increase linearly.
After 4 or 5 years, divi-
dends become too high to be nanced exclusively by incomes and a part of the surplus is redistributed to shareholders. On the other hand, positions in risky assets and reinsurance are reduced with time. The upper right graph of gure 3 presents the optimal reinsurance rate that is exclusively a function of time. This ratio falls nearly linearly from 5.4% or 3.1% ∗ for γ1 = 10 or γ1 = 20 to 1%. The optimal amount of stocks (πt Xt ) is also independent from the size of the surplus and decreases linearly from 0.8 or 0.47 for γ1 = 10 or γ1 = 20 to 0.15. The graphs in gure 4 show the in uence of the initial values of
λS0
and
λL0
on expecta-
tions and standard deviations of the future expected wealth. Three scenarii are compared: λS0 = λL0 = θS = θL , (λS0 , λL0 )> = (θS , θL )> + Ξ( ρ3S , 0)> and (λS0 , λL0 )> = (θS , θL )> + Ξ(0, ρ3L )> . S L Stepping up λ0 or λ0 respectively accelerates the asset and liability clocks. As on average the risk process and investments are pro table, any acceleration of business time increases
26
the gain but also the risk, measured on the chronological time scale. As most of gains are S L capitalized, high values for λ0 or λ0 raise the expected wealth over the rst six months. Figure 5 presents the term structure of expected dividends, in three scenarii. As high values L S for λ0 or λ0 generate an extra pro t over the rst six months, the initial expected dividend L S is bigger than when λ0 = λ0 = θS = θL .
8 7 6 5 4 3
E(Xt) λS =λL0=θS 0
2
std(Xt) λS =λL0=θS 0
1
E(Xt) [λS λL]’=[θS θL]’+ Ξ×[3/ρS 0/ρL] 0 0
0
std(Xt) [λS λL]’=[θS θL]’+ Ξ×[3/ρS 0/ρL] 0 0 0
1
2
3
4
5 T
6
7
8
9
10
E(X ) λS=λL=θ t
0
0
S
std(Xt) λS =λL0=θS 0
12
E(Xt) [λS λL]’=[θS θL]’+ Ξ×[0/ρS 3/ρL] 0 0 std(Xt) [λS λL]’=[θS θL]’+ Ξ×[0/ρS 3/ρL] 0 0
10 8 6 4 2 0
0
1
2
3
4
5 T
6
7
8
9
10
Figure 4: The upper graph compares the expectations and standard deviations of the wealth 3 > L > S L S , 0)> . The process in two scenarii: 1) λ0 = λ0 = θS = θL 2) (λ0 , λ0 ) = (θS , θL ) + Ξ( ρS L S lower graph compares the expectations and standard deviations in two scenarii 1) λ0 = λ0 = 3 > > L > S θS = θL and 2) (λ0 , λ0 ) = (θS , θL ) + Ξ(0, ρL ) .
5 Optimal asset allocation, reinsurance and dividends with a Brownian approximation In many circumstances, working with Brownian motions rather than jump processes allows to obtain analytical results. On the other hand, approaching a claims process by an equivalent Brownian dynamics is often a good approximation, particularly if the number of claims is high. Theses reasons motivate us to study the case in which the liabilities of the insurance company are driven by the next SDE:
dLt = c dτtL − λE(J) dτtL +
27
p λE(J 2 )dWτLL t
1.3 1.2 1.1
E(dt) λS =λL0=θS 0 E(dt) [λS λL]’=[θS θL]’+ Ξ×[3/ρS 0/ρL] 0 0
1
E(dt) [λS λL]’=[θS θL]’+ Ξ×[0/ρS 3/ρL] 0 0 0.9 0.8 0.7 0.6 0.5
0
0.2
0.4
0.6
0.8
1
T
Figure 5: This graph presents the term structure of expected dividends in three scenarii: 1) λS0 = λL0 = θS = θL 2) (λS0 , λL0 )> = (θS , θL )> + Ξ( ρ3S , 0)> and 3) (λS0 , λL0 )> = (θS , θL )> + Ξ(0, ρ3L )> .
where
WτLL t
is a Brownian motion. The scaling property of the Brownian motion allows us to
rewrite the liabilities process as follows:
dLt = (c − λE(J))
λLt dt
+
from which we infer that the risk process at time
Lt = (c −
λE(J)) τtL
t
p
q λLt dWtL
λE(J 2 )
(36)
is the following sum:
Z tp p 2 + λE(J ) λLs dWsL . 0
This expressions reveals that both the average and variance of Lt are proportional to the L 2 L chronometer of the insurance market: (c − λE(J)) E τt |F0 and λE(J )E τt |F0 . It is possible to show that the covariance between liabilities and the log prices of stocks is induced by the dependence between clocks of nancial and insurance markets. And this covariance is still provided by the proposition 3.1. We will not present all features of this process like the joint mgf of
Lt
and
log St .
However, most of proofs presented in previous sections are
easily adaptable to the Brownian case. As previously,
πt , d t
and
qt
denote respectively the
percentage of the stocks hold by the insurer, the dividend and the retention level.
In the
Brownian framework, the dynamics of the surplus is driven by the next relation:
dSt − dt + dLt St Z tp p L L 2 L −(1 − qt ) c dτt + (1 − qt ) λE(J)τt − λE(J ) λs dWs
dXt = (1 − πt )Xt r dt + πt Xt
0 dSt If we replace in this last equation by its expression (15) and St (36), we infer that Xt is now ruled by the SDE:
dXt =
dLt
by its approximation
rXt + πt (µ − r)λSt Xt − dt + c qt λLt − λ qt E(J) λLt dt q q p S S 2 +πt Xt σ λt dWt + qt λE(J ) λLt dWtL . 28
By construction, the Brownian motions
WtL
and
WtS
are independent and the correlation is
only induced by the stochastic clocks. Then
q q q p √ S L S L 2 2 2 S 2 L 2 2 πt Xt σ λt dWt + qt λE(J ) λt dWt ∼ N 0, πt Xt σ λt + qt λE(J )λt dt We can then replace these two Brownian motions by a single one
Wt
de ned on the same
ltration as follows
dXt =
r + πt (µ − r)λSt Xt − dt + qt (c − λE(J)) λLt dt q + πt2 Xt2 σ 2 λSt + qt2 λE(J 2 )λLt dWt
The insurer adjusts the investment, dividend and reinsurance policy so as to maximize the following objective:
V Where
(t, Xt , λSt , λLt , τtS , τtL ) U1 (y) = − γ11 e−γ1 y
Z
and
−β(s−t)
e
= max E πt ,dt qt
T
U1 (ds )ds + e
−β(T −t)
U2 (XT ) |Ft .
(37)
t
U2 (y) = − γ12 e−γ2 y
are the utility from dividends and from the
terminal surplus. The value function of this optimization problem solves the next Hamilton Jacobi Bellman equation (HJB):
0 = Vt − βV + max
π,d,q
rX + π(µ − r)λS X − d + c q λL − λqE(J) λL VX + U1 (d)
1 2 2 2 S π X σ λ + q 2 λE(J 2 )λLt VXX + Vτ S λS + Vτ L λL 2 Z S S S S +α θ − λt VλS + λt V (λSt + ηSS z, λLt + ηLS z) − V ν S (dz) Z L L L L S L L +α θ − λt VλL + λt V (λt + ηSL z, λt + ηLL z) − V ν (dz) +
with the terminal conditions
V (T, XT , λST , λLT , τTS , τTL ) = U2 (XT ).
Using the same approach
as for proposition 4.1 allows us to establish the next result:
The value function de ned by equation (37) in a Brownian setting, is the exponential of an a ne function of risk factors
Proposition 5.1.
1 V (T, XT , λST , λLT ) = − exp A(t, T ) + γ2
BS (t, T ) BL (t, T )
>
λSt λLt
! + C(t, T )Xt ,
where A(t, T ), BS (t, T ), BL (t, T ) and C(t, T ) are functions of time, solutions of the following
29
ODE's ∂ 1 1 A(t, T ) = β − C(t, T ) ln − C(t, T ) + A(t, T ) − 1 ∂t γ1 γ2 S S L L −α θ BS (t, T ) − α θ BL (t, T ) ∂ 1 C(t, T ) = −rC(t, T ) − C(t, T )2 ∂t γ1 2 ∂ 1 (µ − r) 1 S BS (t, T ) = C(t, T ) − α BS (t, T ) − ∂t 2 σ2 γ1 − φS (ηSS BS (t, T ) + ηLS BL (t, T )) − 1 1 (c − λE(J))2 1 ∂ L BL (t, T ) = − C(t, T ) − α BL (t, T ) ∂t 2 λE(J 2 ) γ1 − φL (ηSL BS (t, T ) + ηLL BL (t, T )) − 1
with the terminal conditions A(T, T ) = 0, C(T, T ) = −γ2 , BS (T, T ) = BL (T, T ) = 0. The optimal investment policy is given by πt∗ = −
(µ − r) 1 . σ 2 C(t, T ) Xt
(38)
The optimal dividend is equal to d∗t
1 1 S L ln − C + A + C X + BS λ + BL λ = − γ1 γ2
(39)
and the optimal reinsurance ratio is q∗ = −
(c − λE(J)) . λE(J 2 )C(t, T )
(40)
This last proposition emphasizes that the investment strategy remains unchanged compared to the one obtained with the original claims process. The expression (39) of the optimal dividend is also identical to the one in the previous model. However, as functions
BL
A, BS
and
dier from these de ned in proposition 4.1, dividends eectively depend upon the claims
model. Finally, we notice that the optimal reinsurance rate is equal to the approached ratio proposed in equation (33) for the original claims dynamics.
6 Conclusions This study develops a model in which the contagion between insurance and nancial markets is induced by time-changed processes. This framework presents several interesting features. Firstly, the moment generating functions of market clocks, assets and liabilities have a semiclosed form expression. admits an upper bound.
Secondly, the asymptotic probability of ruin for the risk process Thirdly, the model may be used for asset-liability management
purposes.
30
Numerical tests emphasize the ability of the model to generate a wide variety of term structures of correlations between assets and liabilities.
On the other hand, the correlation is
induced by earnings of the insurance business that are reinvested in the nancial market. If the insurer does not charge any fee above the pure premium, there is not any linear dependence between the asset and liability despite the fact that the asset price and the risk process are not independent by construction. Another interesting feature is that the short term correlation between markets is negligible. In our approach, a delay is induced between the occurrence of an event in one market and the reaction of the other market.
In other
words, there is well contagion between the insurance and nancial markets but the impact is not instantaneous. When used in a ALM framework, the model remains analytically tractable. Optimal reinsurance and investment rates admit closed form expressions and are independent from stochastic clocks. The optimal dividend is a linear function of the wealth and of intensities of chronometers. Finally, the optimal policy depends on parameters de ning asset and liability dynamics on the market time scale and not on the chronological time scale.
Acknowledgment I thank for its support the Chair Data Analytics and Models for insurance of BNP Paribas Cardi, hosted by ISFA (Université Claude Bernard, Lyon France).
References [1] Aït-Sahalia, Y., Cacho-Diaz, J., Laeven, R.J.A., 2015 (a). Modeling nancial contagion using mutually exciting jump processes.
J. of Fin. Econ. 117(3), 586-606.
[2] Aït-Sahalia Y., Laeven R.J.A, Pelizzon L., 2014. Mutual excitation in Eurozone sovereign CDS.
J. of Econ.
, 183, 151-167.
[3] Andrieu C., Doucet A. , Holenstein R., 2010. Particle Markov chain Monte Carlo methods.
J. R. Statist. Soc. B. 72(3), 269342.
[4] Ané T., Geman H. 2000. Transaction Clock, and Normality of Asset Returns.
Finance, 55(5), pp. 2259-2284.
J. of
[5] Asmussen, S., Taksar M. 1997. Controlled diusion models for optimal dividend payout.
Insurance: Math. and Eco. 20, 1-15.
[6] Albrecher, H., Asmussen, S., 2006. Ruin probabilities and aggregate claims distributions for shot noise Cox processes.
Scand. Actuarial J. 2, 86110.
[7] Björk, T., Grandell, J., 1988. Exponential inequalities for ruin probabilities in the Cox case.
Scand. Actuarial J. 77111.
31
[8] Browne, S., 1995. Optimal investment policies for a rm with a random risk process: exponential utility and minimizing the probability of ruin.
Mathematics of Op. Res. 20,
937958. [9] Dassios, A., Zhao, H., 2011. A dynamic contagion process.
Advances in Applied Prob.
43 (3), 814846. [10] Dassios A., Zhao H. 2012. Ruin by dynamic contagion claims.
Eco. 51, 93106.
Insurance: Math. and
[11] Embrechts, P., Grandell, J., Schmidli, H., 1993. Finite-time Lundberg inequalities in the Cox case.
Scand. Actuarial J. 1, 1741.
[12] Embrechts, P., Liniger, T., Lu, L., 2011. Multivariate Hawkes processes: an application to nancial data.
J. of Applied Proba., 48 (A), 367378.
[13] Hainaut D. 2016 a. A model for interest rates with clustering eects.
Quant. Finance,
16, pp 1203-1218. [14] Hainaut D. 2016 b. A bivariate Hawkes process for interest rates modelling.
elling
Econ. Mod-
57, pp 180-196.
[15] Hainaut D. 2016 c. Impact of volatility clustering on equity indexed annuities.
Math. and econ.
Ins.:
71, pp 367-381.
[16] Hainaut D. 2016 d. Clustered Lévy processes and their nancial applications. SSRN working paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2787603 [17] Hawkes, A., Point sprectra of some mutually exciting point processes.
Stat. Soc. B
[18] Hawkes,
A.,
Biometrika
J. of the Royal
1971 (a), 33, 438-443. Spectra of some self-exciting and mutually exciting point processes.
1971 (b), 58, 8390.
[19] Hawkes A. and Oakes D., A cluster representation of a self-exciting process.
Proba. 1974, 11, 493-503.
[20] Hipp C. Plum M. 2000. Optimal investment for insurers.
J. of Applied
Insurance: Math. and Eco. 27,
215228. [21] Kaluszka, M., 2001. Optimal reinsurance under meanvariance premium principles.
surance: Math. and Eco. 28, 6167.
[22] Kaluszka, M., 2004. Meanvariance optimal reinsurance arrangements.
J. 2841.
[23] Lundberg,
P.
Aterf¨ors¨akring
1903 av
framst¨alling
Approximerad
kollektivrisker.
Akad.
salla.
32
Afhandling.
av
In-
Scand. Actuarial
sannolikhetsfunktionen.
Almqvist
&
Wiksell,
Up-
[24] Salmon M. , Tham W.W., 2007. Time Deformation and the Yield Curve. SSRN working paper. http://dx.doi.org/10.2139/ssrn.999841 [25] Schmidli, H. 2002. On minimising the ruin probability by investment and reinsurance.
Ann. Appl. Probab. 12, 890-907.
[26] Schmidli, H. 2006. Optimisation in non-life insurance.
Stoch. Models
22, 689-722.
[27] Yuen K.C. , Liang Z., Zhou M. 2015. Optimal proportional reinsurance with common shock dependence.
Insurance: Math. and Eco. 64, 113
[28] Yin C., Yuen K.C. 2015. Optimal dividend problems for a jump diusion model with capital injections and proportional transaction costs.
optim. 11(4), 1247-1262.
J. of industrial and management
[29] Zheng X., Zhou J., Sun Z. 2016. Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model.
Insurance: Math. and Eco. 67, 77-87.
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