PORTFOLIO SELECTION BY MINIMIZING THE PRESENT VALUE OF CAPITAL INJECTION COSTS

2014 ◽  
Vol 45 (1) ◽  
pp. 207-238 ◽  
Author(s):  
Ming Zhou ◽  
Kam C. Yuen
Keyword(s):  
Portfolio Selection ◽  
Risk Model ◽  
Numerical Study ◽  
Optimal Investment ◽  
Investment Policy ◽  
Model Parameters ◽  
Capital Injection ◽  
Surplus Process ◽  
Insurance Portfolio ◽  
Special Cases

AbstractThis paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

scholarly journals ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS

2014 ◽  
Vol 44 (3) ◽  
pp. 635-651 ◽  
Author(s):  
Chuancun Yin ◽  
Yuzhen Wen ◽  
Yongxia Zhao
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Levy Process ◽  
Dual Model ◽  
Threshold Strategy ◽  
Classical Risk Model ◽  
Surplus Process ◽  
Optimal Dividend ◽  
Special Cases ◽  
The Classical Risk Model

AbstractIn this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Lévy process before dividends are deducted. This model includes the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to an admissible strategy whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that the optimal dividend strategy is formed by a threshold strategy.

Mean-variance portfolio selection with an uncertain exit-time in a regime-switching market

2019 ◽  
Vol 53 (4) ◽  
pp. 1171-1186
Author(s):  
Reza Keykhaei
Keyword(s):  
Portfolio Selection ◽  
Regime Switching ◽  
Exit Time ◽  
Efficient Frontier ◽  
Optimal Investment ◽  
Homogeneous Markov Chain ◽  
Investment Strategies ◽  
Special Cases ◽  
Mean Variance Portfolio ◽  
Mean Variance

In this paper, we deal with multi-period mean-variance portfolio selection problems with an exogenous uncertain exit-time in a regime-switching market. The market is modelled by a non-homogeneous Markov chain in which the random returns of assets depend on the states of the market and investment time periods. Applying the Lagrange duality method, we derive explicit closed-form expressions for the optimal investment strategies and the efficient frontier. Also, we show that some known results in the literature can be obtained as special cases of our results. A numerical example is provided to illustrate the results.

A CHANCE-CONSTRAINED PORTFOLIO SELECTION PROBLEM UNDER t-DISTRIBUTION

2007 ◽  
Vol 24 (04) ◽  
pp. 535-556 ◽  
Author(s):  
YI WANG ◽  
ZHIPING CHEN ◽  
KECUN ZHANG
Keyword(s):  
Stock Returns ◽  
Portfolio Selection ◽  
Asset Allocation ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Chance Constraint ◽  
Model Parameters ◽  
Allocation Model ◽  
Second Order Cone ◽  
T Distribution

Aimed at better modeling stock returns and finding robustly optimal investment decisions, a new portfolio selection model is proposed in this paper. The model differs from existing ones in following ways: multiple market frictions are taken into account simultaneously; the adopted multivariate t-distribution can capture the well-recognized fat tails in the return data by adding only one more parameter relative to the normal; the downside loss risk is controlled by a chance constraint which, including VaR as a special case, is flexible in terms of adjusting the threshold return and the loss probability level; one important advantage about the combination of the latter two innovations is that the derived asset allocation model can be transformed into a second-order cone program or a linear program, which can be easily solved in polynomial time. Empirical results based on some S&P 500 component stocks not only demonstrate the practicality of our new model, but show how different model parameters could affect the optimal portfolio selection. This is very useful in guiding investors to choose a correct model and to find the investment strategy most suitable for their specific purpose.

scholarly journals Optimal Dynamic Risk Control for Insurers with State-Dependent Income

2014 ◽  
Vol 51 (2) ◽  
pp. 417-435 ◽  
Author(s):  
Ming Zhou ◽  
Jun Cai
Keyword(s):  
Risk Model ◽  
Closed Form Solution ◽  
Jump Process ◽  
Form Solution ◽  
Markov Jump ◽  
Surplus Process ◽  
State Dependent ◽  
Absolute Ruin ◽  
Insurance Portfolio ◽  
Optimal Dynamic

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles

2019 ◽  
Vol 53 (1) ◽  
pp. 179-206
Author(s):  
Junna Bi ◽  
Kailing Chen
Keyword(s):  
Risk Model ◽  
Optimal Strategies ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Model Parameters ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Common Shock ◽  
The Impact ◽  
Poisson Risk Model

This paper considers the optimal investment-reinsurance strategy in a risk model with two dependent classes of insurance business under two kinds of premium principles, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the expected value premium principle and the variance premium principle, we use the stochastic optimal control theory to derive the optimal strategy and the value function for the compound Poisson risk model as well as for the Brownian motion diffusion risk model. In particular, we find that the optimal investment strategy on the risky asset is independent to the reinsurance strategy and the reinsurance strategy for the compound Poisson risk model are very different from those for the diffusion model under both two kinds of premium principles, but the investment strategies are the same in this two risk models. Finally, numerical examples are presented to show the impact of model parameters in the optimal strategies.

scholarly journals Randomized observation periods for compound Poisson risk model with capital injection and barrier dividend

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenguang Yu ◽  
Peng Guo ◽  
Qi Wang ◽  
Guofeng Guan ◽  
Yujuan Huang ◽  
...  
Keyword(s):  
Risk Model ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Capital Injection ◽  
Surplus Process ◽  
Dividend Payments ◽  
Injection Line ◽  
Observation Times ◽  
Poisson Risk Model ◽  
Explicit Expressions

AbstractIn this paper, we model the insurance company’s surplus by a compound Poisson risk model, where the surplus process can only be observed at random observation times. It is assumed that the insurer observes its surplus level periodically to decide on dividend payments and capital injection at the interobservation time having an $\operatorname{Erlang}(n)$ Erlang ( n ) distribution. If the observed surplus level is greater than zero but less than injection line $b_{1} > 0$ b 1 > 0 , the shareholders should immediately inject a certain amount of capital to bring the surplus level back to the injection line $b_{1}$ b 1 . If the observed surplus level is larger than dividend line $b_{2}$ b 2 ($b_{2} > b_{1}$ b 2 > b 1 ), any excess of the surplus over $b_{2}$ b 2 is immediately paid out as dividends to the shareholders of the company. Ruin is declared when the observed surplus level is negative. We derive the explicit expressions of the Gerber–Shiu function, the expected discounted capital injection, and the expected discounted dividend payments. Numerical illustrations are also given to analyze the effect of random observation times on actuarial quantities.

scholarly journals Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1400 ◽  
Author(s):  
Catalina Bolancé ◽  
Raluca Vernic
Keyword(s):  
Random Number ◽  
Negative Binomial ◽  
Risk Model ◽  
Numerical Study ◽  
Random Variables ◽  
Data Set ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Insurance Portfolio ◽  
The Individual

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.

scholarly journals A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Aili Zhang ◽  
Zhang Liu
Keyword(s):  
Risk Model ◽  
Poisson Model ◽  
Risk Process ◽  
Closed Form Expression ◽  
Form Expression ◽  
Ruin Time ◽  
Scale Functions ◽  
Surplus Process ◽  
Dividend Payments ◽  
Special Cases

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

scholarly journals Spatial and Aspectual Optimization of the World Investment Policy for Grain Growing

2020 ◽  
Vol 20 (4) ◽  
pp. 176-194
Author(s):  
D. Gertsekovich ◽  
L. Gorbachevskaya ◽  
O. Podlinyaev ◽  
T. Konstantinova
Keyword(s):  
Risk Ratio ◽  
Risk Model ◽  
Optimal Investment ◽  
Investment Policy ◽  
Attractive Potential ◽  
Risk Level ◽  
Basic Principles ◽  
Effective Decision ◽  
Effective Decision Support ◽  
Necessary And Sufficient

The article is devoted to the problems of creating at the international level the optimal investment policy in agricultural production. The problem is solved by means of the “Return-Risk” model, which is based on the basic principles of the portfolio analysis: return, risk and the return-risk ratio. The model is easy to implement and does not require special skills; for important estimates it is necessary and sufficient to use MS EXCEL Date Mining Add-in. This article analyses the historical data of 2010–2017 by productivity of grain crops in general: corn, wheat, barley and rice in different countries. The time span is 1 year. The source data are exported from knoema.com. The findings are as follows: 1) the leading countries with the most attractive potential investment results have been formed for each sample subgroup promise; 2) the benchmarking quantitative analysis on return, risk and the return-risk ratio revealed both the most investment-attractive crops depending on the investor’s attitude to the acceptable risk level, and the countries where the cultivation will be most profitable; 3) the results of the spatial, aspectual and global optimization afford synthesis of effective decision support investment systems. It is proved that the 3 % increase in risk leads to only 1 % return.

Optimal Dynamic Risk Control for Insurers with State-Dependent Income

2014 ◽  
Vol 51 (02) ◽  
pp. 417-435
Author(s):  
Ming Zhou ◽  
Jun Cai
Keyword(s):  
Risk Model ◽  
Closed Form Solution ◽  
Jump Process ◽  
Form Solution ◽  
Markov Jump ◽  
Surplus Process ◽  
State Dependent ◽  
Absolute Ruin ◽  
Insurance Portfolio ◽  
Optimal Dynamic

In this paper we investigate optimal forms of dynamic reinsurance polices among a class of general reinsurance strategies. The original surplus process of an insurance portfolio is assumed to follow a Markov jump process with state-dependent income. We assume that the insurer uses a dynamic reinsurance policy to minimize the probability of absolute ruin, where the traditional ruin can be viewed as a special case of absolute ruin. In terms of approximation theory of stochastic process, the controlled diffusion model with a general reinsurance policy is established strictly. In such a risk model, absolute ruin is said to occur when the drift coefficient of the surplus process turns negative, when the insurer has no profitability any more. Under the expected value premium principle, we rigorously prove that a dynamic excess-of-loss reinsurance is the optimal form of reinsurance among a class of general reinsurance strategies in a dynamic control framework. Moreover, by solving the Hamilton-Jacobi-Bellman equation, we derive both the explicit expression of the optimal dynamic excess-of-loss reinsurance strategy and the closed-form solution to the absolute ruin probability under the optimal reinsurance strategy. We also illustrate these explicit solutions using numerical examples.

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