On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model With Phase-Type Gains

Ng, Andrew

doi:10.2143/ast.40.1.2049230

Cited by 25 publications

(15 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Besides, Avanzi et al (2007) found solutions for equation (2.5) when exponential or mixtures of exponential gains size distributions are considered. Ng (2010) shows solutions when individual gains are phasetype distributed. For higher moments of discounted dividends, Cheung and Drekic (2008) with a similar procedure show integro-differential equations similar to (2.5), for n = 1, 2, ...,…”

Section: Paper Review and Resultsmentioning

confidence: 99%

“…Higher moments have also been considered. See Avanzi et al (2007), Avanzi and Gerber (2008), Cheung and Drekic (2008), Gerber and Smith (2008), Ng (2009) and Ng (2010). Yang and Zhu (2008) compute bounds for the ruin probability.…”

Section: Introductionmentioning

confidence: 99%

“…ψ(u, δ) is the Laplace transform of time to ruin τ x . If δ = 0 it reduces to the probability of ultimate ruin of the process free of the dividend barrier, when δ > 0 we can see ψ(u, δ) as the present value of a contingent claim of 1 payable at τ x , evaluated under a given valuation force of interest δ [see Ng (2010)]. Let's now consider an arbitrary upper level β ≥ u ≥ 0 in the model, see the upper graph of Figure 1, we don't call it yet a dividend barrier.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dividend problems in the dual risk model

Afonso

Cardoso

Reis

2013

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

Section: Paper Review and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dividend problems in the dual risk model

Afonso

Cardoso

Reis

2013

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

“…The two-layer (a, b) strategy was first considered by Ng (2010), who investigated the present value of dividends under a combined threshold and barrier strategy for the compound Poisson dual model, although the issue of optimality was not considered. This is also related to the multilayer dividend strategy (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models

Avanzi

Pérez

Wong

et al. 2017

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

The expected present value of dividends is one of the classical stability criteria in actuarial risk theory. In this context, numerous papers considered threshold (refractive) and barrier (reflective) dividend strategies. These were shown to be optimal in a number of different contexts for bounded and unbounded payout rates, respectively.In this paper, motivated by the behaviour of some dividend paying stock exchange companies, we determine the optimal dividend strategy when both continuous (refractive) and lump sum (reflective) dividends can be paid at any time, and if they are subject to different transaction rates.We consider the general family of spectrally positive Lévy processes. Using scale functions, we obtain explicit formulas for the expected present value of dividends until ruin, with a penalty at ruin. We develop a verification lemma, and show that a two-layer (a, b) strategy is optimal. Such a strategy pays continuous dividends when the surplus exceeds level a > 0, and all of the excess over b > a as lump sum dividend payments. Results are illustrated.One has to determine when and how much dividends to pay-the answer to which is referred to as a dividend strategy. Traditionally, researchers focused on determining the dividend strategy that maximises the expected present value of dividends (without modifications). Main references are Gerber (1969); Loeffen (2008) for the Cramér-Lundberg model, and Bayraktar and Egami (2008); Avanzi, Shen, and Wong (2011); Bayraktar, Kyprianou, and Yamazaki (2013) for the dual model. In absence of fixed transaction costs, such optimal strategies are often of the barrier (more generally, band) type if dividend payouts are not constrained (see Bayraktar, Kyprianou, and Yamazaki, 2013, for the dual model), and of threshold type if dividend payouts are subject to a maximum payment rate (see Yin and Wen, 2013). These two main types lead to reflection or refraction strategies, respectively; see also Gerber and Shiu (2006). In presence of fixed transaction costs, impulse strategies appear (see, e.g. Bayraktar, Kyprianou, and Yamazaki, 2014).Often, reflection and refraction strategies are studied separately. However, a combination of both can be observed sometimes in practice, and can be justified from an economic point of view; see also Gerber and Loisel (2012), who argue that companies often have distribution strategies that have barrier and threshold components. In essence, the dividend optimisation determines whether surplus dollars are best distributed now, or kept for generating dividends in the future. The existence of a surplus (a buffer) is justified by the existence of frictional costs between capital injections and dividends. In other words, the existence of transaction costs (taxes, financial intermediaries, delays) mean that one should minimise any back and forth movements between companies and their investors. Because of this, if there is not much free money available (if the surplus is low) then it makes sense to minimise distribution. If there is a healthy cush...

show abstract

“…Under such framework, Ng [18] combined barrier strategy and threshold strategy for the first time and then proposed a hybrid dividend strategy, who considered a dual risk model with phase-type gains under a hybrid dividend strategy and derived the explicit formula for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin. In this paper, we consider the hybrid dividend strategy for the classical risk model.…”

Section: Introductionmentioning

confidence: 99%

Dividend Payments with a Hybrid Strategy in the Compound Poisson Risk Model

Yin

Zhou²

2014

View full text Add to dashboard Cite

In this paper, a hybrid dividend strategy in the compound Poisson risk model is considered. In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process. Dividends are paid at a constant rate whenever the modified surplus is in a interval; the premium income no longer goes into the surplus but is paid out as dividends whenever the modified surplus exceeds the upper bound of the interval, otherwise no dividends are paid. Integrodifferential equations with boundary conditions satisfied by the expected total discounted dividends until ruin are derived; for example, closed-form solutions are given when claims are exponentially distributed. Accordingly, the moments and moment-generating functions of total discounted dividends until ruin are considered. Finally, the Gerber-Shiu function and Laplace transform of the ruin time are discussed.

show abstract

On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model With Phase-Type Gains

Cited by 25 publications

References 15 publications

Dividend problems in the dual risk model

Dividend problems in the dual risk model

On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models

Dividend Payments with a Hybrid Strategy in the Compound Poisson Risk Model

Contact Info

Product

Resources

About