De Schepper, A.

Dhaene, Jan

Goovaerts, Marc

Kaas, R.

Vyncke, D.

English

Research Papers in Economics

Stable laws and the distribution
of cash-ﬂows
Marc Goovaerts1,2, Ann De Schepper3,D a v i dV y n c k e 1,
Jan Dhaene1,2, Rob Kaas1,2
1 University of Leuven
Minderbroedersstraat 5
B-3000 Leuven, Belgium
t e l .+ 3 21 63 23 74 6
f a x .+ 3 21 63 23 74 0
Marc.Goovaerts@econ.kuleuven.ac.be
2 University of Amsterdam, the Netherlands
3 University of Antwerp, Belgium
Abstract
In a paper of 1994, Kozubowski and Rachev investigated the performance
of stable laws and in particular geometric stable laws when modeling asset
returns.One of their most important ﬁndings was that these models provide
a good ﬁt for ﬁnancial data sets.
In the present contribution, we consider the present value of a series of cash
ﬂows under stochastic interest rates and we make use of stable laws to model
these interest rates.Analogously to former papers, we will not try to cal-
culate the exact analytical distribution for the cash-ﬂow since this seems to
be impossible.Instead we determine upper bounds which are easier to com-
pute, and we derive results for the stop loss premium and distribution of these
bounds.1 Introduction
In some former contributions (see [1, 2, 5]) we investigated the present value
of a series of non-negative payments at times 1 up to n
A =
n  
j=1
cje−Y (tj), (1)
where Y (tj) represents the stochastic continuous compounded rate of return
over the period [0,t j].
Up to now, the results based on comonotonic risks have been applied to the
case of Wiener processes.In fact, we wrote the rate of return as the sum of
increments over the previous periods,
Y (tj)=
j  
i=1
(Y (ti) − Y (ti−1)) (2)
where 0 = t0 <t 1 < ... < tn = t, and where each component Y (ti) − Y (ti−1)
denotes the rate of return for the period [ti−1,t i].Remark that due to this
characterization, the variables Y (tj) are mutually dependent.
Since we assumed the stochastic process {Y (s)}s≥0 to be a Wiener process,
the increments Y (ti) − Y (ti−1) are independent and normally distributed
Y (ti) − Y (ti−1) ∼ N(µ(ti − ti−1),σ2(ti − ti−1)) (3)
or
Y (ti) − Y (ti−1) d
= µ(ti − ti−1)+( ti − ti−1)1/2σZi (4)
with Zi ∼ N(0,1) independent standard normal variates.
As a consequence of the properties of Wiener processes, for the compounded
rates of return we have
Y (tj) ∼ N(µtj,σ2tj), (5)
or
Y (tj) d
= µtj + t
1/2
j σXj (6)
with Xj ∼ N(0,1) again a standard normal variate.Note that the variables
X1,...,Xn are no longer independent.
In the present contribution, we will make two generalizations.1. A normal distribution is a special case of a stable law, which in addition to
the mean µ and deviation σ has two more parameters : α ∈ (0,2] being the
index of stability, determining the tail of the distribution, and β ∈ [−1,+1]
being a skewness parameter.Using the (more general) stable law instead of
a normal one, the distribution of the increments is a stable law, denoted as
Y (ti) − Y (ti−1) ∼ Sα(σ,β,µ;ti − ti−1)( 7 )
which means that equation (6) is changed into
Y (ti) − Y (ti−1) d
= µ(ti − ti−1)+( ti − ti−1)1/ασZi (8)
with Zi ∼ Sα(1,β,0;1) independent standard stable variates.
Since we work with stable processes, for the total rate of return we have
Y (tj) ∼ Sα(σ,β,µ;tj), (9)
or
Y (tj) d
= µtj + t
1/α
j σXj (10)
with Xj ∼ Sα(1,β,0;1) again a standard stable variate.
Just as in the Wiener case, the variables X1,...,Xn are dependent.
A justiﬁcation of this result as well as more details with respect to this dis-
tribution and the meaning of the parameters are given in section 2.
For a choice of α = 2, the normal model emerges.
2. Next, we introduce a risk parameter Θ.Conditionally on this risk para-
meter, the distribution of the increments is the one of a stable law.Thus for
the compounded rate of return we have
Y (tj)|Θ=θ ∼ Sα(σ,β,µ;tjθ) ; (11)
equation (10) then yields
Y (tj)|Θ
d
= µtjΘ+( tjΘ)1/ασXj (12)
with Xj ∼ Sα(1,β,0;1), which is again a standard stable variate.
Remark that in case Θ has all its mass in one, i.e. Prob[Θ = 1] = 1, we
recognize the ﬁrst stable model.The aim of this paper is the investigation of the present value of equation (1)
when rates of return are modeled by means of the general model of equa-
tion (12).The paper is organized as follows.First we will give a summary
of the concepts, properties and methods that are needed to reach our goal.
In section 2 we brieﬂy describe the stable laws, while section 3 is used to
go through the methodology of bounds in convexity order.Afterwards in
section 4, we will be able to present the results about the present value of
equation (1).Finally section 5 is meant to numerically illustrate the results
of section 4.
2 Stable laws
The most easy way to deﬁne a stable law starts from the standard stable law,
which is best described through the characteristic function.
Deﬁnition 2.1 Av a r i a b l eX is a standard stable variate, or
X ∼ Sα(1,β,0;1) (13)
if the characteristic function equals
ϕ(t)=E
 
eitX
 
=e x p{−|t|α ωα,β(t)} (14)
where
ωα,β(t)=



1 − iβ sign(t)tan(πα/2) if α  =1
1+iβ 2
π sign(t)log|t| if α =1.
(15)
Deﬁnition 2.2 Av a r i a b l eY is a (general) stable variate, or
Y ∼ Sα(σ,β,µ;θ) (16)
if we have the equality in distribution
Y d
=



µθ + θ1/ασX if α  =1
µθ + θσX + θσβ 2
π log|θσ| if α =1
(17)
with X a standard stable variate.
Remark that for α =2t h ev a r i a b l eX is standard normally distributed, and
the variable Y is normally distributed with mean µθ and variance σ2θ.Without loss of generality, from now on we will assume that α  =1i no r d e r
not to complicate the formulas.The case where α =1c a nb ew r i t t e nd o w n
in an analogous way.
The next lemma illustrates the “stability property” of random variables with
stable distribution as deﬁned in equation (12), and at the same time proves
the result of equation (10).
Lemma 2.1 Let the variables Y1 and Y2 be deﬁned as
Y1|Θ = µτΘ+( τΘ)1/ασX1 (18)
Y2|Θ = µ(t − τ) Θ+( ( t − τ)Θ)1/ασX2 (19)
with 0 ≤ τ ≤ t and with X1 and X2 independent standard stable variates.
Then conditionally on Θ,t h es u m˜ Y = Y1 + Y2 in distribution equals
˜ Y |Θ
d
= µtΘ+( tΘ)1/ασ ˜ X (20)
with ˜ X a new standard stable variate.
Proof. Although this result is well known, we give a proof for the complete-
ness.
Conditionally on Θ, the characteristic function can be written as
E
 
eik˜ Y |Θ
 
= E
 
e
ik
 
µτΘ+( τΘ)1/ασX1 + µ(t − τ) Θ+( ( t − τ)Θ)1/ασX2
  
= eikµtΘ · E
 
eik(τΘ)1/ασX1
 
· E
 
eik((t − τ)Θ)1/ασX2
 
.
(21)
Making use of equations (14) and (15) for both X1 and X2, we ﬁnd
E
 
eik˜ Y |Θ
 
= eikµtΘ · E
 
e
−kατΘσα
 
1 − iβ sign(k(τΘ)1/ασ)tan(πα/2)
  
·E
 
e
−kα(t − τ)Θσα
 
1 − iβ sign(k((t − τ)Θ)1/ασ)tan(πα/2)
  
= eikµtΘ · E
 
e
−kαtΘσα
 
1 − iβ sign(k(tΘ)1/ασ)tan(πα/2)
  
.
(22)From this intermediate result it is immediately clear that
E
 
eik˜ Y |Θ
 
= E
 
e
ik
 
µtΘ+( tΘ)1/ασ ˜ X
  
(23)
with ˜ X a standard stable variate.
Q.E.D.
3 Convex upper bounds
In many ﬁnancial and actuarial applications, the distribution of the (stochas-
tic) quantity under investigation is too diﬃcult to obtain.In the present case
for example (see equation (1)), the stochastic variables Y (tj) are dependent,
since they are constructed as successive sequences of several independent va-
riables (see equation (2)).
In such cases, the method of convex upper bounds is extremely helpful.The
idea consists of replacing the incalculable exact distribution by a simpler ap-
proximate distribution that is known to be associated with a quantity which
is more dangerous than the original one.
The following theorem summarizes the most important result regarding this
idea.
Proposition 3.1 Consider a sum of functions of random variables
V = φ1(X1)+φ2(X2)+... + φn(Xn), (24)
where the functions φt :  → : x  → φt(x) are all increasing or all decreasing.
The variable
W = φ1(F−1
X1 (U)) + φ2(F−1
X2 (U)) + ... + φn(F−1
Xn(U)) (25)
with U an arbitrary random variable that is uniformly distributed on [0,1] then
deﬁnes an upper bound in convexity order, or
V ≤cx W. (26)In the previous result, the notation FXj(x) is used for the distribution function
of Xj,
FXj(x)=Prob(Xj ≤ x) ; (27)
the inverse function is deﬁned as (p ∈ [0,1])
F−1
Xj (p)=i n f {x ∈ : FXj(x) ≥ p} . (28)
For a proof of proposition 3.1, we refer to [1].
The meaning of the convex order becomes clear if we mention three diﬀerent
and equivalent characterizations of this concept.
We say that W is an upper bound for V in convexity order, V ≤cx W,i f
a) E [u(V )] ≤ E [u(W)] for each convex function u ;
since convex functions take on their largest values in the tails, the vari-
able W is more likely to take on extreme values than the variable V and
thus more dangerous.
b) E [u(−V )] ≥ E [u(−W)] for each concave function u ;
each risk averse decision maker prefers a loss V over a loss W, and thus
the variable W is more dangerous.
c) E[V ]=E[W]a n dE[(V − k)+] ≤ E[(W − k)+]f o re a c hv a l u eo fk ;
the ﬁnancial loss of realizations exceeding a number k (the so-called stop-
loss premium) is always larger for W than for V and thus the variable
W is more dangerous.
As a consequence, replacing a variable V with unknown distribution by a
variable W (satisfying one of the previous properties) with known distribution,
can be seen as a prudent strategy.
4 Results for cash-ﬂows
We now return to the present value under investigation, or
A =
n  
j=1
cje−Y (tj). (29)The variables Y (tj)( t =1 ,...,n), representing the stochastic continuous com-
pounded rate of return over the period [0,t j], are modeled as
Y (tj)=
j  
i=1
(Y (ti) − Y (ti−1)) (0 = t0 <t 1 < ... < tn = t) (30)
with
Y (ti) − Y (ti−1)|Θ
d
= µ(ti − ti−1) Θ+( ( ti − ti−1)Θ)1/ασZi ; (31)
the random variables Zi are independent standard stable variates with distri-
bution Sα(1,β,0;1), and the risk parameter Θ is independent of the variables
Zi.
As mentioned before, it follows from the model that
Y (tj)|Θ
d
= µtjΘ+( tjΘ)1/ασXj (32)
w h e r en o wt h ev a r i a b l e sXj are dependent standard stable variates.
4.1 General results
Proposition 4.1 Let U be a random variable which is uniformly distributed
on [0,1]. For the present value A in equation (29), the variable
Aupp =
n  
j=1
cje−µtjΘ − (tjΘ)1/ασF−1(U;α,β) (33)
where F(x;α,β)=Prob(Xj ≤ x) denotes the distribution function of a
standard stable variate, deﬁnes an upper bound in convexity order, or
A ≤cx Aupp . (34)
Proof. This follows in a straightforward way from proposition 3.1.
Q.E.D.
Starting from this result for the boundary variable, we arrive at an expression
for the stop-loss premiums.Proposition 4.2 The stop-loss premiums of the present value A in (29) are
bounded from above by
E [(A − k)+]
≤
  +∞
0 dFΘ(θ)
  xθ(d)
0 dF(x;α,β)
 
 n
j=1 cje−µtjθ − (tjθ)1/ασx − k
 
(35)
where for each value of k and θ the value xθ(k) is deﬁned implicitly through
the equation
n  
j=1
cje−µtjθ − (tjθ)1/ασxθ(k) = k. (36)
The function FΘ(θ) denotes the distribution function of the risk parameter Θ.
Proof. Because of the result of proposition 4.1, we know that
E [(A − k)+] ≤ E [(Aupp − k)+] (37)
with
E [(Aupp − k)+]
=
  +∞
0 dFΘ(θ)
  1
0 du
 
 n
j=1cje−µtjθ − (tjθ)1/ασF−1(u;α,β) − k
 
+
.
(38)
Deﬁning xθ(k) as in equation (36), and making use of a substitution u =
F(x;α,β) in the second integral, the desired result follows. Q.E.D.
Once the stop-loss premiums are found, the distribution function can be easily
determined.Indeed, there is a well-known link between stop-loss premiums
and distribution, stating that
d
dk
E
 
(A − k)+
 
= FA(k) − 1 , (39)
where the notations are obvious.
Proposition 4.3 The cumulative distribution for the quantity Aupp menti-
oned in proposition 4.1 can be calculated as
Fupp(k)=Prob[Aupp ≤ k]=1−
  +∞
0
dFΘ(θ)F(xθ(k);α,β) (40)
with xθ(k) deﬁned implicitly in equation (36).
Proof. This follows immediately when applying (39) to (35). Q.E.D.4.2 Special cases
After presenting the general results, we want to specify the results for three
special cases for the distribution of the variable Θ.We will use the same three
cases for the numerical illustrations in the next section.
1. The risk parameter Θ has all its mass in one, or Prob[ Θ=1 ]=1 .
The model degenerates to the ordinary and unconditional stable model.
The distribution function of the upper bound can be written as
F(1)
upp(k)=1− F(x(k);α,β) (41)
with the values x(k) deﬁned implicitly through the equation
n  
j=1
cje−µtj − t
1/α
j σx(k) = k. (42)
If α is chosen equal to 2, we recover the results as mentioned e.g. in [2].
2. The risk parameter Θ is exponentially distributed, with unit mean.
The model is said to follow a geometric stable law.The variable Y (t)
can be seen as the sum of a stochastic number of independent standard
stable variables, where the total number of terms follows a geometric
distribution (see [3]).
Now the distribution function of the upper bound can be written as
F(2)
upp(k)=1−
  +∞
0
e−θF(xθ(k);α,β) dθ (43)
with the values xθ(k) deﬁned in equation (36).
3. The risk parameter Θ only appears in the volatility term.
In this case, the model slightly diﬀers, and the rate of return Y (tj)i s
written as
Y (tj)|Θ
d
= µtj +( tjΘ)1/ασXj . (44)
The distribution function of the upper bound then equals
F(3)
upp(k)=1−
  +∞
0
dFΘ(θ)F(yθ(k);α,β) (45)
with yθ(k) deﬁned implicitly through
n  
j=1
cje−µtj − (tjθ)1/ασyθ(k) = k. (46)5Numerical illustration
In this section, we will present a few ﬁgures with graphs of the distribution
functions of the upper bounds for the present value (29), as given in equati-
ons (41), (43) and (45).
The use of stable laws brings about a diﬃculty, which has to be found in the
fact that we do not have a close formulation for the distribution function of
standard stable variates, denoted as F(x;α,β).In order to solve this problem,
we will make use of a numerical algorithm by Nolan (see [4]).
In Figure 1 we plot the distribution function of Aupp,i nc a s eo fac a s h - ﬂ o w
ct = 10, t =1 ,...,10, and with Prob[Θ = 1] = 1.The parameters of the
stable distribution are α =1 .8a n dβ = −0.05, while µ and σ equal 0.07 and
0.10 respectively. The distribution function appears to be rather close to the
distribution function of A, which was obtained by Monte Carlo simulation.
In order to compare the accuracy in the tails, we construct a QQ-plot of the
corresponding distributions.Figure 2 conﬁrms the heavy-tailedness of the
upper bound and indicates that the right quantiles are slightly overestimated.
For instance, the relative error of the 99% quantile is approximately 8%.
Replacing the distribution of the risk parameter Θ by the Exp(1) distribution
y i e l d sF i g u r e3a n di nF i g u r e4w et u r nt os p e c i a lc a s e3w i t hΘ∼ χ2
1.Again,
both upper bounds prove to be good approximations for the corresponding
exact distributions.
In Figures 5 and 6, we use the same model as in Figure 1, but we change
the cash-ﬂow to ct =1 ,...,10 and ct =1 0 ,...,1 respectively.In case of
an increasing cash-ﬂow, the upper bound seems to approximate the exact
distribution slightly better than in case of a decreasing cash-ﬂow.
References
[1] Goovaerts M., Dhaene J., & De Schepper A. (2000). “Stochastic up-
per bounds for present value functions”, Journal of Risk and Insurance,
vol.67(1), p.1-14.
[2] Kaas R., Dhaene J., & Goovaerts M. (2000). “Upper and lower bounds
for sums of random variables”, Insurance: Mathematics and Economics,
vol.27(2), p.151-168.[3] Kozubowski T.J., & Rachev S.T. (1994). “The theory of geometric stable
distributions and its use in modeling ﬁnancial data”, European Journal
of Operational Research, vol.74, p.310-324.
[4] Nolan J.P. (1997). “Numerical computation of stable densities and dis-
tribution functions”, Communications in Statistics - Stochastic Models,
vol.13(4), p.759-774.
[5] Vyncke D., Goovaerts M. & Dhaene J. (2000). “Convex upper and lower
bounds for present value functions”, Onderzoeksrapport KUL. Departe-
ment toegepaste economische wetenschappen, vol.0025, pp.17.Present Value
D
i
s
t
r
i
b
u
t
i
o
n
 
f
u
n
c
t
i
o
n
0 50 100 150 200 250
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Figure 1: Distribution function of Aupp (black) for ct =1 0( t =1 ,...,10) and
Prob[Θ = 1] = 1, compared to a simulated distribution function of A (grey).
Simulated quantiles
Q
u
a
n
t
i
l
e
s
 
o
f
 
t
h
e
 
u
p
p
e
r
 
b
o
u
n
d
50 100 150 200
5
0
1
0
0
1
5
0
2
0
0
Figure 2: QQ-plot of Aupp versus A,f o rct =1 0( t =1 ,...,10) and Prob[Θ = 1] = 1.Present Value
D
i
s
t
r
i
b
u
t
i
o
n
 
f
u
n
c
t
i
o
n
0 50 100 150 200
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Figure 3: Distribution function of Aupp (black) for ct =1 0( t =1 ,...,10) and Θ ∼
Exp(1), compared to a simulated distribution function of A (grey).
Present Value
D
i
s
t
r
i
b
u
t
i
o
n
 
f
u
n
c
t
i
o
n
0 100 200 300
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Figure 4: Distribution function of Aupp (black) for ct =1 0( t =1 ,...,10) in special
case 3 with Θ ∼ χ2
1, compared to a simulated distribution function of A (grey).Present Value
D
i
s
t
r
i
b
u
t
i
o
n
 
f
u
n
c
t
i
o
n
0 50 100 150
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Figure 5: Distribution function of Aupp (black) for ct =1 ,...,10 and Prob[ Θ=1 ]=
1, compared to a simulated distribution function of A (grey).
Present Value
D
i
s
t
r
i
b
u
t
i
o
n
 
f
u
n
c
t
i
o
n
0 20 40 60 80 100 120 140
0
.
0
0
.
2
0
.
4
0
.
6
0
.
8
1
.
0
Figure 6: Distribution function of Aupp (black) for ct =1 0 ,...,1a n dProb[ Θ=1 ]=
1, compared to a simulated distribution function of A (grey).

Convex upper and lower bounds for present value functions”, Onderzoeksrapport KUL. Departement toegepaste economische wetenschappen, vol.0025, pp.17.Present Value D i s t r i b u t i o n f u n c t i o n

Numerical computation of stable densities and distribution functions”,

Stochastic upper bounds for present value functions”,

Upper and lower bounds for sums of random variables”,

Stable laws and the present value of fixed cash flows.

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Stable laws and the present value of fixed cash flows.

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