dividend moments in the dual risk model: exact and ... · dividend moments in the dual risk model:...

DIVIDEND MOMENTS IN THE DUAL RISK MODEL:EXACT AND APPROXIMATE APPROACHES

BY

ERIC C.K. CHEUNG AND STEVE DREKIC

ABSTRACT

In the classical compound Poisson risk model, it is assumed that a company(typically an insurance company) receives premium at a constant rate and paysincurred claims until ruin occurs. In contrast, for certain companies (typicallythose focusing on invention), it might be more appropriate to assume expensesare paid at a fixed rate and occasional random income is earned. In such cases,the surplus process of the company can be modelled as a dual of the classicalcompound Poisson model, as described in Avanzi et al. (2007). Assumingfurther that a barrier strategy is applied to such a model (i.e., any overshootbeyond a fixed level caused by an upward jump is paid out as a dividend untilruin occurs), we are able to derive integro-differential equations for themoments of the total discounted dividends as well as the Laplace transformof the time of ruin. These integro-differential equations can be solved explic-itly assuming the jump size distribution has a rational Laplace transform.We also propose a discrete-time analogue of the continuous-time dual modeland show that the corresponding quantities can be solved for explicitly leavingthe discrete jump size distribution arbitrary. While the discrete-time model canbe considered as a stand-alone model, it can also serve as an approximationto the continuous-time model. Finally, we consider a generalization of theso-called Dickson-Waters modification in optimal dividends problems by max-imizing the difference between the expected value of discounted dividends andthe present value of a fixed penalty applied at the time of ruin.

KEYWORDS

Dual model, barrier strategy, dividend moments, time of ruin, rational Laplacetransform.

1. INTRODUCTION

In the continuous-time dual risk model, the company’s surplus process {U(t),t $ 0} with initial surplus u = U(0) is given by

Astin Bulletin 38(2), 399-422. doi: 10.2143/AST.38.2.2033347 © 2008 by Astin Bulletin. All rights reserved.

U(t) = u – ct + ,ii

N t

1=

Y!] g

t $ 0, (1.1)

where c is the constant rate of expenses per unit time, {N(t), t $ 0} is a Pois-son process with rate l, and {Yi , i ! �+} is a sequence of independent andidentically distributed (i.i.d.) positive continuous random variables with finitemean, independent of {N(t), t $ 0}. For convenience, we assume all Yi’s havethe same distribution as a generic random variable Y which has probability den-sity function (p.d.f.) p( .), cumulative distribution function (c.d.f.) P( .), sur-vival function (s.f.) P( .), and Laplace transform (of p( .)) p( .). Under the model(1.1), the expected increase in surplus per unit time is m = lE (Y ) – c, and it isassumed that m > 0, or equivalently, E(Y ) = c (1 + q ) / l for some q > 0. For adetailed study of the model (1.1), see Seal (1969, p. 116).

Avanzi et al. (2007) propose that a model with dynamics described by (1.1)might be appropriate for businesses (e.g., brokerage firms) that sell mutualfunds or insurance products having a front-end load, or perhaps for companiesexperiencing occasional gains which can be modelled by a compound Poissonprocess. Pharmaceutical or petroleum companies fall into this latter category,and we can think of each upward jump to be the net present value of futureincome as a result of an invention or discovery. In addition, the model (1.1)might also be applicable in a life insurance setting, such as a company whichregularly pays annuities and which can (randomly) earn a portion of thereserves when, for example, a policyholder dies. For a more detailed discussionof applications, we refer interested readers to Seal (1969, p. 116), Mazza andRullière (2004), Avanzi et al. (2007), and references therein.

Now we further assume a barrier strategy is applied to the model describedby (1.1). This means that when the surplus is less than or equal to a fixed bar-rier b > 0, the modified surplus process behaves exactly like the one without abarrier. Conversely, any overshoot beyond level b caused by an upward jumpis paid out immediately as a dividend. We assume no further dividends arepaid beyond the time of ruin, where ruin is said to occur if the surplus everdrops to 0. A typical sample path of the surplus process under such a barrierstrategy is depicted in Figure 1 of Avanzi et al. (2007).

Under the above description of the dual model with dividend payments, weare mainly interested in two random variables in this paper, namely, Du,b andTu,b, which represent respectively the present value of total dividends paid untilruin under a force of interest d > 0 and the time of ruin, both with initial sur-plus u and dividend barrier b. In what follows, we denote the n-th moment ofDu,b by

Vn(u; b) = E (Dnu,b), n ! � ,

where � = {0} , �+. The 0-th moment, V0(u; b), is assumed to be 1 for any val-ues of u and b. We remark that dividend payments occur at discrete points intime, and this is in contrast to the classical risk model where dividend payments

400 E.C.K. CHEUNG AND S. DREKIC

consist of continuous streams of payments resulting from premium income.Furthermore, the Laplace transform of Tu,b is defined as

f(u;b) = E (e – dTu,b),

where d > 0 represents the Laplace transform argument here. Indeed, the abovequantity also represents the present value of a dollar payable at the time of ruinunder a force of interest d. We point out that this quantity plays a key role inour study of optimal dividends problems, as will be evident in Section 7 of thepaper.

Central to the evaluation of Vn(u; b) is the n-th equilibrium distribution ofthe jump size distribution p( .) (if it exists). For n ! �, we denote the p.d.f.,c.d.f., s.f., and Laplace transform (of the p.d.f.) of the n-th equilibrium distri-bution of the random variable Y by pn( .), Pn( .), Pn( .), and pn( .) respectively,with the usual convention that p0( .) = p( .). Furthermore, we denote the genericrandom variable having p.d.f. pn( .) by nY.

The remainder of the paper is organized as follows. In Section 2, integro-differential equations satisfied by Vn(u; b) and f(u; b) are derived. These inte-gro-differential equations are then solved explicitly in Section 3 assuming thatthe jump size random variable Y is distributed as a combination of exponen-tials, whereas in Section 4 more general solutions are given under the assump-tion that the Laplace transform p( .) is a rational function. In Section 5, a dis-crete-time analogue of the continuous-time dual model is proposed and studied.In particular, we show that the corresponding dividend moments and theLaplace transform of the time of ruin in the discrete-time model can be solvedexplicitly while leaving the discrete jump size distribution arbitrary. The discrete-time model can also serve as an approximation to the continuous-time model,and this is the focus of Section 6. In Section 7, we consider a generalizationof the so-called Dickson-Waters modification in optimal dividends problemsby maximizing the difference between the expected value of discounted divi-dends and the present value of a fixed penalty applied at the time of ruin.Some results in Avanzi et al. (2007) are extended here. We conclude the paperwith Section 8 demonstrating some interesting numerical results and findings.

2. INTEGRO-DIFFERENTIAL EQUATIONS FOR Vn(u; b) AND f(u; b)

2.1. Dividend Moments

Since ruin occurs immediately for an initial surplus of zero, we have

Vn(0; b) = 0, n ! �+. (2.1)

Furthermore, if u > b, an initial dividend of u – b is immediately paid out,with any future dividend payments being modelled by the quantity Db,b. There-fore, applying a binomial expansion, we arrive at

DIVIDEND MOMENTS IN THE DUAL RISK MODEL 401

Vn(u; b) =njj

n

0=

! d n (u – b)n – jVj (b; b), u > b; n ! �+. (2.2)

For 0 < u # b, we consider all possible events over a small time interval [0,h]and obtain

Vn(u; b) = (1 – lh)e–ndhVn(u – ch;b) + lhe–ndhn

b u ch

0

- -V# ] g

(u – ch + y; b)p(y)dy

+ lhe–ndh njj

n

b u ch 0

3

=- -# ! d] ng (u – ch + y – b)n – jVj (b;b)p(y)dy + o(h).

Since e– ndh = 1 – ndh + o(h), the first term on the right-hand side can beexpanded as (1 – lh)e–ndhVn(u – ch; b) = Vn(u; b) – chVn�(u; b) – (l + nd) hVn(u; b) +o(h) via a Taylor series expansion of the quantity Vn(u – ch; b). Thus, cancellingVn(u; b) on both sides of the above equation, dividing by h, and letting h goto 0, we get

cVn�(u; b) + (l + nd )Vn(u; b) – l n

b u

0

-V# (u + y; b) p(y)dy

(2.3)

–lnj

y b u p y dyj

nn j

b u0

- -3

=

-

-#! d ] ^cn g h m" , Vj (b; b) = 0, 0 < u # b; n ! �+.

We remark that when n = 1, (2.3) reduces to equation (2.3) of Avanzi et al.(2007).

The right-hand side of (2.3) involves stop-loss moments. By the well-knownlink between stop-loss moments and equilibrium distributions (e.g., see Willmotet al. (2005, equation (1.5))), (2.3) can be re-expressed as

cVn�(u; b) + (l + nd )Vn(u; b) – l nu

bV# (y; b) p(y – u)dy

(2.4)

–lnjj

n

0=

! d n Pn – j (b – u) E (Y n – j)Vj (b; b) = 0, 0 < u # b; n ! �+.

Note that (2.4) together with the boundary conditions given by (2.1) deter-mine Vn (u; b) for 0 # u # b. Thus, for u > b, Vn (u; b) can be computed via(2.2).

From (2.4), we observe that the existence of the n-th moment of Du,b

depends upon the existence of the n-th equilibrium distribution of the jumpsize random variable Y. This intuitively makes sense since in the dual model itis the jump which causes dividends to be paid and the size of the jumps deter-mines the total dividend payout. This is in contrast to the classical model wherethe dividend rate is bounded by the premium rate c and the n-th moment ofthe discounted dividends cannot exceed (c/d ) n.


2.2. Laplace Transform of the Time of Ruin

With zero initial surplus, ruin occurs immediately and therefore

f(0; b) = 1. (2.5)

In addition, when u > b, the surplus drops to level b immediately due to the ini-tial payment of dividends, and thus

f(u; b) = f(b;b), u > b. (2.6)

For 0 < u # b, considering again a small increment h > 0 leads to

f(u; b) = (1 – lh)e–dh f (u – ch;b) + lhe– dh fb u ch

0

- -# ] g(u – ch + y;b) p(y)dy

+ lhe– dh fb u ch

3

- -# ] g (b;b)p(y)dy + o(h).

In the same manner as (2.3) was obtained, we arrive at

cf�(u; b) + (l + d)f(u; b) – l fb u

0

-# (u + y;b)p(y)dy – lP(b – u)f(b;b) = 0,(2.7)

0 < u # b.

The integro-differential equation (2.7), together with the boundary condition(2.5), determines f(u; b) for 0 # u # b.

3. EXACT SOLUTIONS WHEN THE JUMP SIZE DISTRIBUTION

IS A COMBINATION OF EXPONENTIALS


Assume the jump size random variable Y is distributed as a combination ofexponentials, so that its p.d.f. is given by

p(y) = iAii

ryb

1=

-b e i! , y > 0, (3.1)

where bi > 0 for i = 1,2,…, r and A 1iir

1 ==

! . Clearly, the n-th moment of Y isgiven by

E (Yn) = n !i

.Ab n

i

i

r

1=

! (3.2)


Furthermore, it follows by induction that

Pn(y) = A ,i ni

ryb

1

i

=

-e! , y $ 0; n ! � , (3.3)

where the Ai,n’s can be computed recursively as

Ai,n =j

i

/

/,

A

b

b

,

,

j njr

i n

11

1

-=

-

A! i = 1, 2, …, r; n ! �+,

with starting values Ai,0 = Ai for i = 1, 2, …, r. With (3.1), (3.2), and (3.3), wecan write (2.4) as

cVn�(u; b) + (l + nd)Vn(u; b) – l i ;A y b dyii

ru

nu

b yb b

1

i

=

-b e ei V#! ^ h– l !

nj

A n jA

b,j

n

i n ji

rb u

kn j

k

k

rb

0 1 1

-=

-=

- -

-=

e i! ! !d ] ^n g h Vj (b; b) = 0, 0 < u # b; n ! �+,

or equivalently,

cVn�(u; b) + (l + nd)Vn(u; b) – l i ;A y b dyii

ru

nu

b yb b

1=

-b e ei iV#! ^ h(3.4)

– ln! jk

! ; ,jA A

b bb

0,i n jn j

k

k

r

j

n

i

ru bb

101

=-

-===

-e iV!!! ] ]g g* 4 0 < u # b; n ! �+.

We now follow the approach of Avanzi et al. (2007) and apply the operator

dud

b ii

r

1

-=

% c m

to (3.4), thereby resulting in Vn(u; b) satisfying an (r + 1)-th order linear homo-geneous differential equation (with constant coefficients which are independentof u). Therefore, we have that

Vn(u; b) = r ,e,k nu

k

r

0

,k n

=

C! 0 # u # b; n ! �+, (3.5)

where the coefficients Ck,n = Ck,n(b) and rk,n are to be determined.

Upon substitution of (3.5) into (3.4), we find, after some algebra, that


kk

r r

r! ! ! ,

< ; .�

c nA

e A e

n jA A

e n AA

e

u b n

r l d l r l rr

b b0

0

, ,,

,,

,

,, ,

k nk

r

k ni k n

i i

i

ru

i k ni k n

k n

k

r

i

rb

i n j

j

n

n jk

k

r

l jl

rb

i n nk

k

ru bb

0 1 01

1

1

1 0 1

, ,

,

k n k n

l j i

# !

+ + --

+-

- - =

= = ==

-

=

-

-= = =

-

+

bb

bCC

C

! ! !!

! ! ! !

e e]

op g

(3.6)

From the coefficients of re u,k n in (3.6), we have that

crk,n + l + nd – l ,A

r 0,i k n

i i

i

r

1-

==

bb! k = 0, 1, …, r; n ! �+.

In other words, for each fixed n ! �+, the rk,n’s, k = 0, 1, …, r, are the roots ofLundberg’s fundamental equation (in z) under the dual model at force of inter-est nd, namely (e.g., see Avanzi et al. (2007, equation (4.9)))

cz + l + nd – lp(– z) = 0.

Similarly, looking at the coefficients of e bi (u – b) in (3.6) leads to

r r! ! ! ,

, , ..., ; .�

A e n jA A

C e n AA

i r n

rr

b b

1 2

,,

, ,, ,i k n

k

r

i k n

k n b i n j

j

n

kn j

k

k

r

l jb

l

r

i nknk

k

r

0 1

1

1 0 1

, ,k n l j

!

-= +

=

=

-

=

-

-= = =

+

bC! ! ! ! !

(3.7)

Furthermore, it follows from (2.1) and (3.5) that

,0,k nk

r

0

==

C! n ! �+. (3.8)

Therefore, for each fixed n ! �+, (3.7) and (3.8) form a system of r + 1 linearequations for the recursive (in terms of n) evaluation of the Ck,n’s, k = 0,1, …, r.We remark that when n = 1, this system is identical to that formed by equa-tions (4.10) and (4.11) in Avanzi et al. (2007), and as such, becomes a startingpoint for the recursion.


When the jump size distribution is governed by (3.1), one can easily follow thederivations in Section 3.1 to show that (2.7) gives rise to


f(u; b) = rk ,B e

k

ru

0

,k 1

=

! 0 # u # b,

where the Bk’s satisfy

rk ,B er

r0

,

,

k

r

i k

k b

0 1

1 ,k 1

-=

=b! i = 1, 2, …, r,

and

k .B 1k

r

0

==

!

4. ALTERNATIVE APPROACH FOR JUMP SIZE DISTRIBUTION

WITH RATIONAL LAPLACE TRANSFORM

In this section, we assume that the jump size random variable Y has a ratio-nal Laplace transform, i.e., we can write p( .) as

p(s) = ,h sl s]] gg

where h ( .) and l ( .) are polynomials of degree m and (at most) m – 1 respec-tively, with no common zeros. Furthermore, h ( .) has no roots in the positivehalf plane, and h (0) = l (0) (since p(0) = 1). Without loss of generality, weassume the leading coefficient of h ( .) is 1. Then, the Laplace transform of thep.d.f. of the equilibrium random variable 1Y is given by

p1(s) = ,sE Y

sh sl sp1

0

1-=^ ] ]]hg gg

where

l1(s) =sE Y

h s l s

0

-^] ]hg g

is a polynomial of degree (at most) m – 1 since s = 0 is a root of h(s) – l (s).Therefore, inductively, we write the Laplace transform of pk( .) as

pk(s) = ,h sl sk]] gg k ! � ,

where the lk( .)’s are given recursively by

lk(s) = ,sE Y

h s s

k

k

1

1-

-

-l^] ]hg gk ! �+,

with l0( .) = l ( .).



Here, we follow the ideas in Section 6 of Avanzi et al. (2007). Letting z = b – u,we introduce the function

Wn (z; b) = Vn(b – z; b) = Vn(u; b), 0 # z # b; n ! � .

Note that

W0(z; b) = 1, 0 # z # b,

and

Wn(b; b) = Vn(0; b) = 0, n ! �+.

With the above definition, (2.4) can be re-expressed as

– cWn�(z; b) + (l + nd)Wn(z; b) – l n

z

0W# (z – y; b) p(y) dy

– lnjj

n

0=

! d n Pn – j (z) E (Y n – j)Wj (0; b) = 0, 0 # z < b; n ! �+,

or equivalently,

– cWn�(z; b) + (l + nd)Wn(z; b) – l n

z

0W# (z – y; b) p(y) dy

– lnjj

n

0=

! d n pn – j + 1(z) E( n – jY )E (Y n – j)Wj (0; b) = 0, 0 # z < b; n ! �+. (4.1)

As in Avanzi et al. (2007), we extend the domain for the definition of Wn(z; b)from 0 # z # b to z $ 0, and denote the resulting function by wn(z). We thenobtain the conditions

w0(z) = 1, z $ 0,

and

wn(b) = 0, n ! �+. (4.2)

Taking Laplace transforms on both sides of (4.1) with wn(z) in place of Wn(z; b)yields

cwn(0) – cswn(s) + (l + nd)wn(s) – lwn(s) p(s)

– lnjj

n

0=

! d n pn – j + 1(s) E( n – jY )E (Y n – j)wj (0) = 0, n ! �+.


Solving for wn(s) in the above equation, we obtain

,

.�

ss s

E Y s s E E Y

n

p

p p1 0 0n

cn

c

c n cnj n j n j

n jjj

n

l d l

l l1 10

1

!

=- +

- -

+

- + --

=

-

+

wYw w!] ]

] ] ] a ] _ a ]g gg g g k g i k g% /

(4.3)

We remark that (4.3) holds true for an arbitrary jump size distribution. Withthe additional assumption of a rational Laplace transform, (4.3) then becomes

, ,�ss h s l s

q snn

cn

c

nl d l

!=- +

+

+w ] ` ] ]]g j g gg

(4.4)

where

qn(s) = h s c E Y l sl1-] ] ]g g g' 1wn(0) – c

nj

l

j

n

0

1

=

-

! d n ln – j + 1(s)E(n – jY )E(Y n – j)wj (0),

n ! �+. (4.5)

We observe that the numerator of (4.4) is a polynomial of degree m while thedenominator is of degree m + 1. As a result, we can apply a partial fractiondecomposition to (4.4) and obtain

wn(s) = ,s R ,

,

k n

k n

k

m

0-

=

Z! n ! �+, (4.6)

where for each fixed n ! �+, the Rk,n’s, k = 0,1, …, m, are the roots of Lund-berg’s fundamental equation (in z) under the classical risk model at force ofinterest nd, i.e.,

cz – (l + nd) + lp(z) = 0,

and the Zk,n’s are given by

,ZR R

q R

!

,, ,,

,k n

k n i ni i km

n k n

0

=-

=% _

_i

ik = 0, 1, …, m; n ! �+. (4.7)

Inversion of (4.6) immediately yields

R ,z e,n k nk

mz

0

,k n==

Zw !] g z $ 0; n ! �+.


For each fixed n ! �+, we note from (4.7) that each of the Zk,n’s involves thefunction qn( .), which in turn involves the constants w1(0), w2(0), …, wn – 1(0).To determine these constants, we use condition (4.2) to get

R ,e 0,k nk

mb

0

,k n ==

Z! n ! �+,

which, by application of (4.5) and (4.7), leads to

j

R

R,

.�

we

E Y E Y e w

n

00

n

R

h R c E Y l R R bkm

cnj n j

n j

R

l R R bkm

j

n

l

l

0

00

1

!

!

, ,,

, , ,

, ,,

, ,

k n i ni i k

mk n k n k n

k n i ni i k

mn j k n k n

0

1

0

1

!

=

-

-

=

--

-==

-

+

=

=

- +

!

!!

%

%]]] ] ]

a _ ` ]] ]ggg g g

k i j gg g) 3

We remark that the above equation enables the recursive computation of wn(0).Finally, the dividend moments are given by

Vn(u; b) = wn(b – u) = R ,e,k nk

mb u

0

,k n

=

-Z! ] g 0 # u # b; n ! �+.


Following virtually the same set of procedures as in Section 4.1, we omit thetedious algebraic details and simply state that

f(u; b) = f(b – u), 0 # u # b,

where

, ,

, , , ..., ,

z L e z

LR R

h R E Y l Rk m

f

f

0

00 1

! , ,,

, ,

kR z

k

m

kk ii i k

m

k c kl

0

1 10

1 1 1

,k 1 $=

=-

-=

=

=

!

%

]

__ ] _ ]

g

ii g i g$ .

and

R

.e

f 0 1

R

h R c E Y l R R bkm

l

0! , ,,

, , ,

k ii i k

mk k k

1 10

1 1 1 1

=

-

-

==

! %

]]] ] ]g

gg g g


5. DISCRETE-TIME MODEL

In the discrete-time model, we assume, without loss of generality, that the com-pany incurs an expense of 1 per unit time, and there is an upward jump of sizeXk at the end of the k-th time period, k ! �+. We further assume that the jumpsizes are i.i.d. having common distribution X with mean E (X ) > 1. Let theprobability mass function (p.m.f.) of X be denoted by gi = Pr(X = i ), i ! �.Note that there is a probability g0 of no upward jump at the end of each timeperiod. Under the above description, the surplus process {Ud (k), k ! �} canbe described by

Ud (k) = u – k + ,ii

k

1=

X!

where u = Ud (0) ! � is the initial surplus of the company. Ruin is said to occurif the surplus reaches level 0.

Under a barrier strategy, a fixed dividend barrier b ! �+ is introduced tothe above model. If u > b, a dividend of u – b is payable immediately at time 0.For k ! �+, however, a dividend of Ud (k – 1) – 1 + j – b is payable at the end ofthe k-th time period if

(i) Ud (k – 1) ! �+, and

(ii) Xk = j $ b – Ud (k – 1) + 2.

We assume no further dividends are payable beyond the time of ruin. We arethen interested in the moments of the total discounted dividends undera force of interest a > 0 per unit time, and we denote the n-th moment byVn,d (u; b). The 0-th moment, V0,d (u; b), is taken to be 1 for any u and b. Mir-roring the analysis in the continuous-time analogue, we denote the Laplacetransform of the time of ruin (with argument a > 0) in the discrete-timemodel by fd (u; b).


Analogous to (2.1) and (2.2), we have that

Vn,d (0; b) = 0, n ! �+, (5.1)

and

Vn,d (u; b) =njj

n

0=

! d n (u – b)n – jVj,d(b; b), u = b + 1, b + 2, …; n ! �+. (5.2)

Then, by conditioning on X1, we arrive at


n k-

; ;

; , , , ..., ; .�

u b e u j b

enk

u j b b b u b n

1

1 1 2

, ,

,

a

a

n dn

j n dj

b u

nj

k

n

j b uk d

0

1

02

!

= - +

+ - + - =3

-

=

- +

-

== - +

+

V g V

g V

!

!!

] ^d ^ ]

g hn h g

(5.3)

Therefore, for each fixed n ! �+, Vn,d(1; b), Vn,d(2; b), …, Vn,d(b; b) can be solvedrecursively from the above system of b linear equations in terms of V1,d(b; b),V2,d(b; b), …, Vn – 1,d(b; b). Then, for u = b + 1, b + 2, …, Vn,d(u; b) can be com-puted via (5.2).

When the jump size random variable X has infinite support, note that thesystem (5.3) involves infinite summation and direct application might posesome computational difficulties. However, it is possible to express (5.3) only interms of finite summation when the moments of X are known. Specifically, (5.3)can also be written as

n k-

; ; ;

; , , , ..., ; ,�

u b e u j b e b b

enk

E X b u b b u b n

1 1

1 1 2

, , ,

,

a a

a

n dn

j n dj

b un

jj

b u

n d

n

k

n

k d

0 0

0

1

!

= - + + -

+ - - + =

-

=

--

=

-

-

=

-

+

+

V g V g V

V

! !

!

J

LKK] ^ ]

d ]_ ]

N

POOg h g

n g i g: D" ,(5.4)

where the stop-loss moments can be computed as

j j

j

n k

n k n k

-

- -

n k-

n

n

,

, , ..., ; .�

E X b u

E X X b u

nk

E X X b u

nk

j g b u j g

nk

b u E X b u j j g

u b n

1

1

1 1

1 1

1 1 1

1 2

k

nn k k

k

nn k n k

j b uj

b u

k

nn k k k n

j

b u

0

0 20

1

0 0

1

/

/

!

- - +

= - - +

= - - +

= - + - +

= - - + - - + -

=

3

+

=

-

=

-

= - +=

- +

=

-

=

- +

+

!

! !!

! !J

LKK

]_]

d ] ]d ] ]d ] ] ` ] N

POO

g ig

n g gn g gn g g j g

:8

9

DB

C

""

"*

$

,,

,4

.(5.5)

Note that (5.5) only involves finite summation if the corresponding momentsof X are known. In particular, when n = 1, (5.5) reduces to

E [{X – (b – u + 1)}+] = E (X ) – (b – u + 1) + b u j1j

b u

0

1

- + -=

- +

! ^ hgj,

u = 1, 2, …, b.(5.6)


Returning to (5.4), we further distinguish between the case b = 1 and b > 1.For the case b = 1, with the use of (5.1), (5.4) becomes

Vn,d (1;1) = e– na(1 – g0)Vn,d (1;1) + e– nankk

n

0

1

=

-

! d nE [{(X – 1)+}n – k] Vk,d (1;1), n ! �+.

Solving for Vn,d (1;1) yields

Vn,d (1;1) =0e g

e nk1 1a

a

n

n

k

n

0

1

- --

-

=

-

!^ dh nE [{(X – 1)+}n – k] Vk,d (1;1), n ! �+. (5.7)

For n = 1, application of (5.6) to (5.7) leads to

V1,d (1;1) =0

0

0

.e g

e E X g

e ge E X

1 1

11

1 11

a

a

a

a

- -

- += +

- -

--

-

-

-

^] ]̂

hg

hg" ,

(5.8)

We note that (5.8) has the following interesting implication, namely, the signof V1,d (1;1) – 1 is identical to that of e–aE (X ) – 1. In other words, V1,d (1;1) isgreater (less) than 1 if and only if E (X ) is greater (less) than ea.

We now turn our attention to the case where b > 1. In such instances, a sys-tem of linear equations has to be solved and as such, results are stated in termsof matrices. The system of linear equations (5.4) can be expressed in matrixform as

An(b)Vn,d (b) = Pn(b) , b > 1; n ! �+, (5.9)

where the involved quantities are defined as

An(b) = e–na Q – Ib,

,Q 0

00

00

00

1

1

1

11

b

b

b

jj

b

jj

b

jj

b

1

0

2

1

0

3

2

1

1

2

3

1

0

0

1

0

2

0

3

0 1

0

h h h

g

g

g

j

g

g

h h

=

-

-

-

- -

-

-

-

-

=

-

=

-

=

-

g

g

g

g

g

g

g

g

g

g

g

gg

g gg

g

g

g

!!!

J

L

KKKKKKKKK

N

P

OOOOOOOOO

Vn,d (b) = (Vn,d (1; b), Vn,d (2; b), …, Vn,d (b; b))T,

Pn(b) = (pn(1; b), pn(2; b), …, pn(b; b))T,

pn(u; b) = – e–nankk

n

0

1

=

-

! d nE [({X – (b – u + 1)}+)n – k ] Vk,d(b; b), u = 1, 2, …, b,


and Ib is a b ≈ b identity matrix. Note that the stop-loss moments appearingin pn(u; b) can be computed using (5.5). Then, from (5.9), Vn,d (b) is given by

Vn,d (b) = [An(b)] –1 Pn(b), b > 1; n ! �+.

We remark that the above inverse [An(b)] –1 always exists. To see this, we canimagine e–na Q to be the transition probability matrix among the transientstates of a discrete-time Markov chain, and therefore (Ib – e–na Q)–1 is alwaysknown to exist.


Identical to (2.5) and (2.6), we have

fd (0; b) = 1 (5.10)

andfd (u; b) = fd (b; b), u = b + 1, b + 2, ….

Conditioning on X1, we obtain

fd (u; b) = e–aj

j

b u

0=

-

g! fd (u – 1 + j; b) + e–a 1 jj

b u

0

-=

-

g!J

LKK

N

POO fd (b; b), u = 1, 2, …, b.

(5.11)

We again distinguish between the cases b = 1 and b > 1. For b = 1, with the appli-cation of (5.10), (5.11) reduces to

fd (1;1) = e–ag0 + e–a(1 – g0) fd (1;1),

which immediately yields

fd (1;1) = .ee

1 1a

a

0

0

- --

-

gg

^ h (5.12)

Note that (5.12) represents the Laplace transform of a zero-truncated geo-metric random variable. This makes sense because when u = b = 1, the time ofruin is k, k ! �+, if there is a positive jump in each of the first k – 1 periodsand no jump at the end of the k-th period.

Regarding the case b > 1, similar to (5.9), (5.11) can be expressed as

A1(b) Fd (b) = �, b > 1, (5.13)

where

Fd (b) = (fd(1; b), fd(2; b), …, fd(b; b))T


and

� = (–g0 e–a, 0, …, 0)T.

Clearly, (5.13) implies

Fd (b) = [A1(b)] –1�, b > 1.

6. APPROXIMATION OF THE CONTINUOUS-TIME MODEL

In this section, we follow the essential ideas in Dickson and Waters (1991,2004) to establish a connection between the continuous-time model and thediscrete-time model in Section 5. In other words, the quantities Vn(u; b) andf(u; b) in the continuous-time model can be approximated by the correspond-ing quantities in the discrete-time model when exact solutions are not available.

Without loss of generality, we assume, in the continuous-time model, that thejump size random variable Y has mean E(Y) = 1, and the Poisson rate of jumparrival is l = 1. Therefore, the rate of expenses per unit time is c = 1 / (1 + q ).Our goal is to approximate Vn(u; b) and f(u; b) under a force of interest d > 0.Similar to the steps outlined in Dickson (2005, pp. 147-148), our approxima-tion procedure involves three steps.

Step 1: Discretization of YDiscretize the random variable Y on {0,1 / b,2 / b, …} for some chosen para-meter b > 0, which is referred to as the scaling factor. If we call the discretizedrandom variable (1)Y, the discretization procedure should be such that (1)Yapproximates Y. The mean preserving method (e.g., see Dickson (2005, p. 80),or De Vylder and Goovaerts (1988, Section 7) for further details) is used sothat the mean of (1)Y is also 1. Denoting the n-th moment of the total dis-counted dividends by 1Vn(u; b) and the Laplace transform of the time of ruinby 1f(u; b) in the same compound Poisson model but with jump size randomvariable (1)Y, ideally we have

Vn(u; b) - 1Vn(u; b), n ! �+, (6.1)

andf(u; b) - 1f(u; b) (6.2)

if (1)Y is a good approximation of Y.

Step 2: Change of monetary unitDefine the random variable (2)Y = b · (1)Y with mean b. Note that (2)Y is distri-buted on �. If 2Vn(u; b) denotes the n-th moment of the total discounted div-idends and 2f(u; b) denotes the Laplace transform of the time of ruin in thecompound Poisson model with jump size random variable (2)Y, jump rate 1,


force of interest d, constant rate of expenses per unit time b / (1 + q), initialsurplus u, and barrier b, it is immediate that

1Vn(u; b) =b1

n 2Vn(bu; bb) , n ! �+,

and

1f(u; b) = 2f(bu; bb).

Then, from (6.1) and (6.2), ideally we have

Vn(u; b) -b1

n 2Vn(bu; bb), n ! �+,

andf(u; b) - 2f(bu; bb).

Step 3: Change of time unitIf we change the time unit of the model in Step 2 such that the rate of expensesper unit time is 1, then the model in Step 2 is equivalent to a model where theforce of interest per unit time is d(1 + q) /b, and the jump size random variableper unit time, defined as (3)Y, has a compound Poisson distribution with rate(1 + q) /b and secondary distribution (2)Y. This is precisely the model describedin Section 5 where a = d(1 + q) /b and the jump size random variable X has thesame distribution as (3)Y. Therefore, we arrive at the approximation

Vn(u; b) -b1

n Vn,d (bu; bb), n ! �+,

and

f(u; b) - fd(bu; bb).

We remark that in order to evaluate the p.m.f. of (3)Y, the well-known (a, b, 0)Panjer recursion can be used (e.g., see Klugman et al. (2004, pp. 91-92)). More-over, we expect the approximation to improve as the value of b increases.

7. OPTIMAL DIVIDENDS: GERBER-LIN-YANG MODIFICATION

Avanzi et al. (2007) consider the problem of maximizing the expected value oftotal discounted dividends in the dual model. In this section, we consider amodified problem regarding the choice of the optimal dividend barrier.

Our modified problem is inspired by Dickson and Waters (2004) as well asGerber et al. (2006a). Both papers consider dividend problems in the classicalrisk model. Dickson and Waters (2004) argue that the shareholders shouldbe responsible for the deficit at ruin, and therefore one should maximize the


difference between the expected discounted dividends and the discounted deficitat ruin. This is referred to as the ‘Dickson-Waters modification’ by Gerber et al.(2006b). In the dual model, however, this is no longer applicable since thedeficit at ruin always equals 0. Gerber et al. (2006a) consider a more generalproblem of maximizing the difference between the expected discounted divi-dends and the expected discounted penalty at ruin. Therefore, if we assume thata constant penalty of w $ 0 should be paid by the company at the time of ruin,then we aim at maximizing (with respect to b) the quantity

g (u; b,w) = V1(u; b) – wf(u; b). (7.1)

We refer to this maximization problem as the ‘Gerber-Lin-Yang’ modification.Note that the value w can be interpreted as a ‘weight’ assigned to the Laplacetransform of the time of ruin. This means that when w is small, we are moreconcerned about the expected amount of total discounted dividends paid,whereas if w is large, we are more concerned with the possibility of earlyruin.

Our studies reveal that given a fixed value of w $ 0, the function g (u; b,w)is maximized at b = b*

w for any value of u $ 0. In other words, the optimal bar-rier is independent of the initial surplus u but is dependent on the value of thepenalty w at ruin. In the case where w = 0, Avanzi et al. (2007) argue that, ingeneral, the optimal barrier b*

0 is positive. Indeed, if we follow their same lineof logic, it is possible to show that b*

w is positive for any fixed w $ 0. From (2.1),(2.3), (2.5), and (2.7), it is obvious that (7.1) satisfies the integro-differentialequation

cg�(u; b,w) + (l + d) g (u; b,w) –l gb u

0

-# (u + y; b,w) p(y)dy

–l u b yb u

- +3

-# ^ hp(y)dy – lg (b; b,w) P(b – u) = 0, 0 < u # b, (7.2)

with boundary condition

g (0; b,w) = – w. (7.3)

If we now let u and b both go to 0 in (7.2), use (7.3), and then solve for g�(0; 0,w),we obtain

g�(0; 0,w) = c ypl0

3# (y)dy + cd

w = 1 + q + cd

w > 1,

since q > 0 and w $ 0. By continuity, we argue that for each fixed w $ 0 thereexists bw > 0 such that

g�(u; bw,w) > 1, 0 < u < bw.


Replacing u by y in the above condition and integrating (with respect to y) from0 to u, we obtain, by again applying (7.3), that for each fixed w $ 0 there existsbw > 0 such that

g(u; bw,w) > u – w, 0 < u < bw.

Since g(u; 0,w) = u – w from (7.1), it is always more optimal to choose such abarrier of bw than to choose a barrier of 0. Therefore, in general, the optimalbarrier b*

w is positive.Avanzi et al. (2007, equation (5.9)) derive an interesting equation regard-

ing the optimal barrier in the case when w = 0. Expressed in our notation, itis given by

V1(b*0 ; b*

0 ) = dm

,

which happens to be the present value of a perpetuity payable at rate m. In fact,this result can be extended to

g (b*w ; b*

w,w) = dm

,

for any w $ 0. We omit the proof of this result as it is identical to that of Avanziet al. (2007).

8. NUMERICAL ILLUSTRATIONS

This section is devoted to some numerical examples which illustrate the appli-cation of our results from the previous sections. In all examples, we assume thePoisson jump rate l to be 1. We begin by considering the following four jumpsize distributions which all have rational Laplace transform:

Example 1: Damped squared sine distribution

p(y) = 8e–2y sin2y, y > 0.

Example 2: Mixture of an exponential and two Erlangs

p(y) = 21 (22ye–2y) + 8

1 (2.5e–2.5y) + !.

,y e

83

22 5 . y3 2 2 5-J

LKK

N

POO y > 0.

Example 3: Damped sine distribution

p(y) = 2e–y(1 – sin y), y > 0.

Example 4: Mixture of two Erlangs

p(y) = 41 (0.62ye–0.6y) + 4

3 (92ye–9y), y > 0.


418 E.C.K. CHEUNG AND S. DREKICT

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While the above four distributions all have mean 1, they possess differentamounts of variability. Specifically, the coefficients of variation correspondingto the jump size distributions in Examples 1-4 are (rounded to two decimalplaces) 0.50, 0.71, 1.41, and 1.80 respectively.

For each of the four examples above, Table 1 displays both the exact andapproximated (via discretization with a scaling factor of b = 100) values of b*

w,g (10; b*

w, w), and V1(10; b*w) obtained for various combinations of the para-

meters c, d, and w. In addition, since we have explicit formulae for the highermoments of Du, b*

w, Table 1 also presents the coefficient of variation, the

coefficient of skewness, and the coefficient of kurtosis of Du,b*w

for u = 10. Thesethree moment-based quantities are denoted by CV (10; b*

w), CS (10; b*w), and

CK (10; b*w) respectively.

We remark that with b set at 100, all values of b*w on the discretized side

of Table 1 are specified to only two decimal places of accuracy. Nevertheless,we observe that the results from the approximation are very close to thoseobtained via the exact method. We also remark that in cases A, C, and Hwhere only c is varied, there seems to be no monotonic relationship betweenb*

w and c. In particular, b*w appears to first increase and then decrease with c.

On the other hand, in cases B, C, D, E, F, and G, the value of w is only var-ied and, as expected, the value of b*

w increases with w in each example. Thisintuitively makes sense since a larger value of w warrants a greater concern ofearlier ruin, and so b*

w has to increase for safety purposes. We also observethat in cases C, I, J, and K where only d is varied, b*

w decreases as d increasesin all examples.

For all cases of Table 1, the value of b*w appears to have the same order-

ing as the coefficient of variation or variance (since we are fixing the mean tobe 1) of the jump size distribution. This comment is in fact made in Avanzi etal. (2007) for the case w = 0. However, this appears to be true only when weare making the comparison among light-tailed distributions. Specifically, let usnow consider the situation where jump sizes follow a lognormal distributionwith p.d.f.

/ //

, > .expln

p yy

yy

p9 7 21

21

9 781 98

02

= -+^ ] dh g n* 4

This distribution has a mean of 1, but its coefficient of variation is 2.05 andis higher than all the distributions in Examples 1-4. Applying a discretizationwith b = 100, two graphs are plotted assuming c = 0.75 and d = 0.01. Figure 1shows a plot of g(u; b,5) against b for various fixed values of u. From Figure 1,it appears that the optimal value of b which maximizes g (u; b,5) is indepen-dent of u, and we can indeed confirm that for each value of u in Figure 1, theoptimal value of b to two decimal places is 13.93. This is significantly lowerthan 15.12, which is the corresponding optimal value in Example 4 where thejump size distribution has a smaller coefficient of variation of 1.80. Therefore,it would seem that a heavy-tailed distribution such as the lognormal exhibits


FIGURE 1: Plot of g (u; b,5) against b for various fixed values of u.

FIGURE 2: Plot of g (10; b,w) against b for various fixed values of w.

a different sort of behaviour. To add further credence to our claim, we also per-formed the same discretization procedure assuming a Pareto jump size distri-bution with a mean of 1 and coefficient of variation of 1.41, identical to the onein Example 3. We found the value of b*

5 to be 12.41, again significantly lowerthan 13.66, the corresponding value of b*

5 in Example 3. Thus, in general, wecannot say that b*

w increases with the variance of the jump size distribution.Finally, Figure 2 depicts the behaviour of g (10; b,w) as a function of b for

various fixed values of w in the lognormal case. Note that b*w increases with w,

and this is consistent with our findings in Table 1.


0

5

10

15

20

25

30

35

0 5 10 15 20 25 30

b

g (u;

b,5)

u=5

u=10

u=15

u=20

-20

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30

g(10

;b,w

) w=0w=5w=10w=15w=20w=25

b

ACKNOWLEDGEMENTS

The authors would like to thank the referees for their useful remarks andsuggestions which helped to improve this paper. The authors would also liketo thank Professors Harry Panjer and Gordon Willmot for their helpful com-ments and use of references in the literature. This work has been assisted bythe Natural Sciences and Engineering Research Council of Canada.

REFERENCES

AVANZI, B., GERBER, H.U. and SHIU, E.S.W. (2007) Optimal dividends in the dual model. Insur-ance: Mathematics and Economics, 41, 111-123.

DE VYLDER, F. and GOOVAERTS, M.J. (1988) Recursive calculation of finite-time ruin probabilities.Insurance: Mathematics and Economics, 7, 1-7.

DICKSON, D.C.M. (2005) Insurance Risk and Ruin, Cambridge University Press, Cambridge.DICKSON, D.C.M. and WATERS, H.R. (1991) Recursive calculation of survival probabilities.

ASTIN Bulletin, 21, 199-221.DICKSON, D.C.M. and WATERS, H.R. (2004) Some optimal dividends problems. ASTIN Bulletin,

34, 49-74.GERBER, H.U., LIN, X.S. and YANG, H. (2006a) A note on the dividends-penalty identity and

the optimal dividend barrier. ASTIN Bulletin, 36, 489-503.GERBER, H.U., SHIU, E.S.W. and SMITH, N. (2006b) Maximizing dividends without bankruptcy.

ASTIN Bulletin, 36, 5-23.KLUGMAN, S.A., PANJER, H.H. and WILLMOT, G.E. (2004) Loss Models: From Data to Decisions,

2nd edition, Wiley, New York.MAZZA, C. and RULLIÈRE, D. (2004) A link between wave governed random motions and ruin

processes. Insurance: Mathematics and Economics, 35, 205-222.SEAL, H.L. (1969) Stochastic Theory of a Risk Business, Wiley, New York.WILLMOT, G.E., DREKIC, S. and CAI, J. (2005) Equilibrium compound distributions and stop-

loss moments. Scandinavian Actuarial Journal, 6-24.

ERIC C.K. CHEUNG

Department of Statistics and Actuarial ScienceUniversity of Waterloo200 University Avenue WestWaterloo, Ontario, Canada N2L 3G1

STEVE DREKIC (corresponding author)Department of Statistics and Actuarial ScienceUniversity of Waterloo200 University Avenue WestWaterloo, Ontario, Canada N2L 3G1E-mail: [email protected]


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