Stable Dividends under Linear-Quadratic Optimization Benjamin Avanzia Debbie Kusch Faldenab Mogens Steensenb aCentre for Actuarial Studies Department of Economics Faculty of Business and Economics

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Stable Dividends under Linear-Quadratic Optimization

Benjamin Avanzia, Debbie Kusch Faldena,b,∗, Mogens Steﬀensenb

aCentre for Actuarial Studies, Department of Economics, Faculty of Business and Economics

University of Melbourne, Melbourne VIC 3010, Australia

bDepartment of Mathematical Sciences, University of Copenhagen

DK-2100 Copenhagen, Denmark

Abstract

The optimization criterion for dividends from a risky business is most often formalized in terms of the

expected present value of future dividends. That criterion disregards a potential, explicit demand for

stability of dividends. In particular, within actuarial risk theory, maximization of future dividends have

been intensively studied as the so-called de Finetti problem. However, there the optimal strategies typically

become so-called barrier strategies. These are far from stable and suboptimal aﬃne dividend strategies

have therefore received attention recently. In contrast, in the class of linear-quadratic problems a demand

for stability if explicitly stressed. These have most often been studied in diﬀusion models diﬀerent from

the actuarial risk models. We bridge the gap between these patterns of thinking by deriving optimal aﬃne

dividend strategies under a linear-quadratic criterion for a general L´evy process. We characterize the value

function by the Hamilton-Jacobi-Bellman equation, solve it, and compare the objective and the optimal

controls to the classical objective of maximizing expected present value of future dividends. Thereby we

provide a framework within which stability of dividends from a risky business, as e.g. in classical risk theory,

is explicitly demanded and explicitly obtained.

Keywords: Risk theory, Stability, Linearity, Stochastic control, Dividends

JEL codes: C44, C61, G22, G32, G35

MSC classes: 60G51, 93E20, 91G70, 62P05 91B30

1. Introduction

We formalize a dividend optimization problem within a risk theoretical framework where a demand for

stability is explicitly expressed via a linear-quadratic objective function. The solution is aﬃne dividends

and, thus, we contribute with an understanding of the class of objectives where the recently studied aﬃne

dividend processes are actually optimal. We realize that the diﬀusion approximation of a general L´evy

process is adequate under this criterion. Finally, we illustrate and discuss the performance of our optimal

dividends under the standard de Finetti (1957) criterion where stability is neither expressed nor obtained.

B¨uhlmann (1970) discussed, in the context of classical actuarial risk theory, the problem of oﬀering stability

to investors providing capital to a risky business. The insurance business is a core example where risk

transfer is the rational for making business the discussion goes beyond that business. Historically, in risk

theory the criteria have been to minimize the probability of ruin as in Asmussen and Albrecher (2010) or to

∗Corresponding author.

Email addresses: b.avanzi@unimelb.edu.au (Benjamin Avanzi), dkf@math.ku.dk (Debbie Kusch Falden ),

mogens@math.ku.dk (Mogens Steﬀensen)

October 10, 2022

arXiv:2210.03494v1 [math.OC] 7 Oct 2022

maximize the expected present value of dividends until ruin as in Albrecher and Thonhauser (2009); Avanzi

(2009). The general endeavour of the stability problem is still a contemporary one and can be seen through

the lens of modern Enterprise Risk Management, described in Taylor (2013). Essentially, Enterprise Risk

Management is a process by which certain decisions are made (the controls) to achieve certain outcomes (the

objective), possibly under certain constraints that are either external to the decision maker (e.g., coming

from a regulatory environment) or internal to the decision maker (and in that case not necessarily truly

distinguishable from other aspects of the objective). Such a management procedure can be advantageously

presented by stylised modelling as the one developed in the proliﬁc risk theoretical literature; see also Cairns

(2000, Section 1.4, on the value of simple but tractable models) and Gerber and Loisel (2012, on the value

of ruin theory for risk managers, in particular for capital modelling)

Stability criteria are most often linked to the surplus of a company, for instance, and, originally, related

to minimizing the probability of ruin corresponding to the probability of a negative surplus. Historically,

risk theoretical surplus models focused on insurance type dynamics where the risk is downside risk with

deterministic income and stochastic losses. The classical formulation is the Cram´er-Lundberg model, which

is a compound Poisson surplus model formalized by Lundberg (1909); Cram´er (1930). Recently, more general

risky business types have been considered, for instance, where the stochastic nature is mostly on the upside

as gains (Avanzi et al., 2007; Bayraktar and Egami, 2008). The most general formulations are in terms of

spectrally negative or positive L´evy processes (Loeﬀen, 2008; Bayraktar et al., 2014). The generalization

from the classical compound Poisson model to a more general surplus process opens up for applying the

patterns of thinking far beyond the insurance business.

The objective of minimizing the probability of ruin over inﬁnite time implies that the surplus increases

without a limit. In order to resolve this issue de Finetti (1957) allowed for a surplus leakage to the share-

holders of the company, referred to as dividends, and formed stability criteria based on the future dividend

payouts. The classical objective is to maximize the expected present value of future dividends until the

company is ruined. This coincides with the Dividend Discount Model by Williams (1938), also known as the

Gordon Model in ﬁnance (Gordon, 1962). Loeﬀen (2009) and Yin and Wen (2013) studied optimal dividend

problems within the classical objective for spectrally negative L´evy processes. Realistic features of both

the controlled process, the control, and the objective are, still, being added to the dividend optimization

criterion. Avanzi et al. (2016a) developed a list of realistic features one might want to include, in particular

based on the corporate ﬁnance literature. One important and well-known aspect is that companies and

investors like stable dividends (see, e.g., Lintner, 1956; Fama and Babiak, 1968; Avanzi et al., 2016a). Un-

fortunately, the optimal strategies for the dividend optimization problem turn out to be relatively irregular,

namely the so-called barrier strategy, hardly acceptable in practice. This was ﬁrst pointed out by Gerber

(1974), but received little attention until recently. In an attempt to address this issue, Avanzi and Wong

(2012) introduced a linear dividend strategy, in a diﬀusion framework, leading to mean reversion of the

surplus process and much improved stability. This was generalised to aﬃne dividend strategies by Albrecher

and Cani (2017), in a Cram´er-Lundberg framework, who also derived a closed-form Laplace transform of

the time to ruin. Importantly, both Avanzi and Wong (2012) and Albrecher and Cani (2017) illustrate

that aﬃne dividend strategies perform closely to the optimal barrier strategies, but they make the surplus

process much more stable. This latter point is more rigorously explored in Albrecher and Cani (2017) by

their theoretical analysis of ruin.

In both Avanzi and Wong (2012) and Albrecher and Cani (2017) the linear and dividend strategies, re-

spectively, are introduced ad hoc and not discussed as solutions to any optimal control problem. Optimal

parameters, within the ad hoc speciﬁed strategy classes, that maximise the expected present value of div-

idends are obtained. However, aﬃne strategies have been found optimal in diﬀerent but related contexts

by Cairns (2000) and Steﬀensen (2006), where objectives of linear-quadratic form (LQ optimization) are

studied in the context of life insurance and pensions. Aﬃne dividend strategies are arguably much more

realistic than the usually optimal ones such as barrier strategy. In this paper, we establish a connection risk

theory and the class of models typically considered there, and linear-quadratic optimization. In order to

show that the linear-quadratic objective entails aﬃne optimal strategies also in rather general risk models,

we characterize the value function by the so called Hamilton-Jacobi-Bellman equation. The quadratic form

of the value function in terms of the surplus leads to aﬃnity of the optimal dividends payouts. These strate-

gies are obviously suboptimal in relation to dividend optimization but we are able to calculate explicitly

the value of the optimal aﬃne dividends coming out of our problem. This allows for a comparison of the

performance of our aﬃne dividends in the dividens optimization problem, however, with particular attention

to the fact that the classical dividend optimization problem is stopped upon ruin whereas ours is not.

The paper is organised as follows. Section 2 introduces the surplus process and the LQ objective that

we propose in this paper is analysed and motivated. We derive the Hamilton-Jacobi-Bellman equation

and an appropriate veriﬁcation lemma in Section 3, along with an expression that characterizes the value

function. The LQ objective is compared to the classical objective in Section 4, where we also study choices

of benchmarks in the LQ problem and the resulting optimal dividend strategy. Numerical illustrations are

provided in Section 5.

2. The optimization problem

2.1. The surplus model

We model the surplus of a company at time tafter distribution of dividends by the dynamics

dX(t) = c(t)dt +dS(t)−dD(t),(2.1)

where c(t) is deterministic and represents the predictable modiﬁcation component of the surplus due to

income and expenses, S(t) is stochastic and represents the aggregate random variations of the surplus due

to, for instance, losses with S(0) = 0, and D(t) is the aggregated net dividends with D(0) = 0.

If c(t) is a positive constant and S(t) is a compound Poisson process with negative jumps only, then (2.1)

has the dynamics of a Cram´er-Lundberg process. Conversely, if c(t) is a negative constant and S(t) is a

compound Poisson process with positive jumps only, then (2.1) has the dynamics of a so-called dual model

(Mazza and Rulli`ere, 2004).

We assume S(t) is the following process

S(t) =

N(t)

i=1

Yi+ςW (t), N(0) = W(0) = 0,(2.2)

where (Yi)i∈Nis i.i.d. and Yican follow any distribution on R, with ﬁnite ﬁrst two moments,

E[Yj

i] = pj, j = 1,2,(2.3)

where W(t) is a Brownian motion, and where N(t) is an inhomogeneous Poisson process with intensity λ(t),

t≥0. Note that S(t) is a L´evy process for λ(t) = λ, where N(t) is a homogeneous Poisson process. Such

two-sided formulations are rare in the actuarial literature, but they exist; see Cheung (2011); Labb´e et al.

(2011, for references with negative and positive c(t), respectively) or Cheung et al. (2018).

The dividend process, D(t), is not strictly increasing. Hence, we allow negative dividends, spoken of as

capital injections. Furthermore, dividends and capital injections can be paid continuously or as lump sums

upon jumps in S(t), such that the dynamics of Dis given by

dD(t) = l(t, X(t))dt+i(t, X(t−))dN(t).

2.2. The Linear-Quadratic (LQ) objective

We consider a ﬁnite time frame T≥0 and would like to consider a general objective of the form

min Et,xhdiscounted penalties for continuous dividends away from a benchmark

+ discounted penalties for lump sum (discrete) dividends

+ discounted penalties for the wealth process away from a benchmark

+ subject to a constraint on terminal wealth X(T)i,

where the subscript of the expectation refers to the expectation conditional of X(t) = x. This is opera-

tionalised into the following value function,

V(t, x) = min

l,i Et,x1

2ZT

e−δ(s−t)l(s, X(s)) −l0(s)−l1(s)X(s)2ds

2ZT

e−δ(s−t)γi(s)i(s, X(s−))2dN(s)

2ZT

e−δ(s−t)X(s)−x0(s)2dΓ(s)

+κe−δ(T−t)(X(T)−xT)τ,(2.4)

for t≤T, where δis a ﬁnancial impatience factor. To get a better understanding of the objective behind

this value function, we explain (2.4) line by line.

— The ﬁrst line compares the continuous payout of dividends with an aﬃne benchmark. Dividends are

generally not paid continuously, but we use a continuous model that provides a tractable stylised for-

mulation of a discrete real life situation; for comments about this see Cairns (2000). The benchmark,

l0(s) + l1(s)X(s), consist of two functions, a ﬁxed target, l0(s), and a target which is proportional to

the surplus level, l1(s).

— The second line accounts for lump sum payments. The lump sum payments are interpreted as extra

dividends or capital injections paid on top of the regular dividends. Therefore, the only admissible lump

sum payments are upon jumps in the surplus process, where an abrupt change of surplus level due to a

jump may require a discrete adjustment of the surplus. The benchmark is zero, since we prefer not to

have lump sum dividends, and we introduce a weight function γi(s)≥0, s≥0 to adjust this preference.

The undesirable signals of lump sum dividends are discussed in Avanzi et al. (2016b) and Avanzi et al.

(2017). The squared function means we equally dislike lump sum dividends and capital injections, and

that we prefer a series of small dividend payouts to one single large one.

— The last two lines consider the surplus process. The third line compares the surplus with a surplus

benchmark. The benchmark, x0(s), could be a result of regulatory constraints or correspond to the

explicit target capitalisation of the company. Companies often set and publish such targets; see, e.g.

Australian Actuaries Institute (2016, for insurance companies). It is reasonable to assumes the function

x0is non-negative, to not aim for the company to ruin. In order to balance this objective with the ﬁrst

two lines, the third line contains a mixed aggregate weight function Γ(t) = Rt

0γ(s)ds, 0 ≤t<T. It is

written using the Riemann-Stietjes notation to allow for a ﬁnal mass at termination ∆Γ(T)≥0.

— The last line serves to control the terminal value of the surplus to a benchmark, xT. The parameter κ

is a Lagrangian multiplier, and the parameter τallows for three levels of constraints on the terminal

value X(T). The case τ= 0 corresponds to absence of constraint. If τ= 1, the expected value of the

terminal value is xT, where κis solved to satisfy this constraint. For τ= 2 the constraint is stronger

and the process is forced to reach xTat time Tby letting κgo to inﬁnity. See Steﬀensen (2006) and

Steﬀensen (2001) for details.

The ﬁnal weight function mass ∆Γ(T) is not redundant with the last row for τ= 2, since the third

row expresses a preference and the fourth row expresses a constraint. Therefore, they are operationalised

diﬀerently, where the weight at ∆Γ(T) remains a ﬁnite constant, while κis meant to diverge in the constraint,

such that X(T) is exactly xT. Furthermore, we can have the constraint with τ= 1, and use the weight

∆Γ(T) to express a strong preference for the terminal value of the surplus without the binding constraint

of τ= 2.

Except for the last line, all distances from the dividends and the surplus to the benchmarks, respectively, are

penalised by a quadratic loss function. Objectives on this form are well known in the quantitative ﬁnance

literature such as Wonham (1968) and Bj¨ork (2009), and optimization in this context is commonly referred to

as “linear-quadratic (LQ) optimization”. LQ optimization is most commonly formalized with an underlying

diﬀusion process without jumps and mainly considered in the context of pensions funds within actuarial

risk theory (Cairns, 2000; Steﬀensen, 2006; Avanzi et al., 2022). It is also well known that LQ optimization

results in aﬃne optimal strategies, which induces the desire to understand the objective and the resulting

dividend strategy in a broader actuarial context. The objective and optimal controls are relevant to compare

to the classical objective of maximizing expected present value of future dividends. In order to show that the

LQ objective leads to aﬃne optimal strategies, we characterize the value function by diﬀerential equations,

and express the value function as a quadratic function in the surplus.

3. HJB equation and veriﬁcation lemma

3.1. HJB equation and veriﬁcation lemma for the LQ objective

Under the assumption that the optimal control strategies exist, the value function satisﬁes a system of

diﬀerential equations, referred to as the Hamilton-Jacobi-Bellman equation (HJB equation). The optimal

control strategies are the functions t7→ l(t, X(t)) and t7→ i(t, X(t)) that minimize the value function and

are predictable with respect to the ﬁltration generated by the surplus process. The subscript of a function

refers to the partial derivative with respect to that subscript i.e. Vt(t, x) = ∂

∂t V(t, x).

Proposition 3.1. Assume the value function is twice continuously diﬀerentiable, V∈C1,2and the optimal

control strategies exist. Then the value function satisﬁes the HJB equation

0 = Vt(t, x)−δV (t, x) + inf

l,i (1

2l(t, x)−l0(t)−l1(t)x2

2γi(t)i(t, x)2λ(t)

2γ(t)x−x0(t)2

+Vx(t, x)c(t)−l(t, x)

2Vxx(t, x)ς2

+λ(t)EhV(t, x +Y1−i(t, x)) −V(t, x)i),(3.1)

with boundary condition

V(T, x) = κ(x−xT)τ+ ∆Γ(T)(x−x0(T))2.(3.2)

For each (t, x)∈[0, T ]×Rthe inﬁmum is attained by the optimal control strategies, and Y1is one of the

stochastic variables in the jump process representing a jump size.

Proof. See Appendix A

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StableDividendsunderLinear-QuadraticOptimizationBenjaminAvanzia,DebbieKuschFaldena,b,,MogensSteensenbaCentreforActuarialStudies,DepartmentofEconomics,FacultyofBusinessandEconomicsUniversityofMelbourne,MelbourneVIC3010,AustraliabDepartmentofMathematicalSciences,UniversityofCopenhagenDK-2100Copenhag...

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Stable Dividends under Linear-Quadratic Optimization Benjamin Avanzia Debbie Kusch Faldenab Mogens Steensenb aCentre for Actuarial Studies Department of Economics Faculty of Business and Economics

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