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arXiv2010

Theoretical and numerical Analysis on Optimal dividend policy of an insurance company with positive transaction cost and higher solvency

Based on a point of view that solvency and security are first, this paper considers regular-singular stochastic optimal control problem of a large insurance company facing positive transaction cost asked by reinsurer under solvency constraint. The company controls proportional reinsurance and dividend pay-out policy to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. The paper aims at deriving the optimal retention ratio, dividend payout level, explicit value function of the insurance company via stochastic analysis and PDE methods. The results present the best equilibrium point between maximization of dividend pay-outs and minimization of risks. The paper also gets a risk-based capital standard to ensure the capital requirement of can cover the total given risk. We present numerical results to make analysis how the model parameters, such as, volatility, premium rate, and risk level, impact on risk-based capital standard, optimal retention ratio, optimal dividend payout level and the company's profit.

q-fin.RM↗

arXiv2010

Optimal dividend and investing control of a insurance company with higher solvency constraints

This paper considers optimal control problem of a large insurance company under a fixed insolvency probability. The company controls proportional reinsurance rate, dividend pay-outs and investing process to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. This paper aims at describing the optimal return function as well as the optimal policy. As a by-product, the paper theoretically sets a risk-based capital standard to ensure the capital requirement of can cover the total risk.

q-fin.RM↗

arXiv2010

Optimization of dividend and reinsurance strategies under ruin probability constraint

This paper considers nonlinear regular-singular stochastic optimal control of large insurance company. The company controls the reinsurance rate and dividend payout process to maximize the expected present value of the dividend pay-outs until the time of bankruptcy. However, if the optimal dividend barrier is too low to be acceptable, it will make the company result in bankruptcy soon. Moreover, although risk and return should be highly correlated, over-risking is not a good recipe for high return, the supervisors of the company have to impose their preferred risk level and additional charge on firm seeking services beyond or lower than the preferred risk level. These indeed are nonlinear regular-singular stochastic optimal problems under ruin probability constraints. This paper aims at solving this kind of the optimal problems, that is, deriving the optimal retention ratio,dividend payout level, optimal return function and optimal control strategy of the insurance company. As a by-product, the paper also sets a risk-based capital standard to ensure the capital requirement of can cover the total given risk, and the effect of the risk level on optimal retention ratio, dividend payout level and optimal control strategy are also presented.

q-fin.RM↗

arXiv2011

Multivariate heavy-tailed models for Value-at-Risk estimation

For purposes of Value-at-Risk estimation, we consider several multivariate families of heavy-tailed distributions, which can be seen as multidimensional versions of Paretian stable and Student's t distributions allowing different marginals to have different tail thickness. After a discussion of relevant estimation and simulation issues, we conduct a backtesting study on a set of portfolios containing derivative instruments, using historical US stock price data.

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arXiv2011

Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management

In this paper we investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending on the applied strategy or the past values of the portfolio. In this setting, a managed investment portfolio serves simultaneously as the underlying security on which the liability/target is contingent and as a replicating portfolio for that liability/target. This is usually the case for capital-protected investments and performance-linked pay-offs. We give examples of pricing, hedging and portfolio management problems (asset-liability management problems) which could be investigated in the framework of time-delayed BSDEs. Our motivation comes from life insurance and we focus on participating contracts and variable annuities. We derive the corresponding time-delayed BSDEs and solve them explicitly or at least provide hints how to solve them numerically. We give a financial interpretation of the theoretical fact that a time-delayed BSDE may not have a solution or may have multiple solutions.

q-fin.PR↗

arXiv2010

Some Remarks on T-copulas

We examine three methods of constructing correlated Student-$t$ random variables. Our motivation arises from simulations that utilise heavy-tailed distributions for the purposes of stress testing and economic capital calculations for financial institutions. We make several observations regarding the suitability of the three methods for this purpose.

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arXiv2010

A Loan Portfolio Model Subject to Random Liabilities and Systemic Jump Risk

We extend the Vasiček loan portfolio model to a setting where liabilities fluctuate randomly and asset values may be subject to systemic jump risk. We derive the probability distribution of the percentage loss of a uniform portfolio and analyze its properties. We find that the impact of liability risk is ambiguous and depends on the correlation between the continuous aggregate factor and the asset-liability ratio as well as on the default intensity. We also find that systemic jump risk has a significant impact on the upper percentiles of the loss distribution and, therefore, on both the VaR-measure as well as on the expected shortfall.

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arXiv2010

Alarm System for Insurance Companies: A Strategy for Capital Allocation

One possible way of risk management for an insurance company is to develop an early and appropriate alarm system before the possible ruin. The ruin is defined through the status of the aggregate risk process, which in turn is determined by premium accumulation as well as claim settlement outgo for the insurance company. The main purpose of this work is to design an effective alarm system, i.e. to define alarm times and to recommend augmentation of capital of suitable magnitude at those points to prevent or reduce the chance of ruin. To draw a fair measure of effectiveness of alarm system, comparison is drawn between an alarm system, with capital being added at the sound of every alarm, and the corresponding system without any alarm, but an equivalently higher initial capital. Analytical results are obtained in general setup and this is backed up by simulated performances with various types of loss severity distributions. This provides a strategy for suitably spreading out the capital and yet addressing survivability concerns at satisfactory level.

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arXiv2010

What risk measures are time consistent for all filtrations?

We study coherent risk measures which are time-consistent for multiple filtrations. We show that a coherent risk measure is time-consistent for every filtration if and only if it is one of four main types. Furthermore, if the risk measure is strictly monotone it is linear, and if the reference probability space is not atomic then it is either linear or an essential supremum.

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arXiv2010

Target market risk evaluation

After the shocking series of bankruptcies started in 2008, the public does not trust anymore the classical methods of assessing business risks. The global economic severe downturn caused demand for both developed and emerging economies' exports to drop and the crisis became truly global. However, this current crisis offers opportunities for those companies able to play well their cards. Entering new markets has always been a hazardous entrepreneurial attempt, but also a rewarding one, in the case of success. The paper presents a new indicator meant for assessing the prospective of success or failure for a company trying to enter a new market by using an associative strategy. In order to take the right decision concerning the optimal market entry strategy, marketers may use a software application, "AnBilant", created by a research team from Hyperion University.

q-fin.RM↗

arXiv2010

Optimal control of a big financial company with debt liability under bankrupt probability constraints

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company's bankruptcy soon. In order to protect the shareholders' profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, PDE methods and variational inequality approach. Moreover, we establish a risk-based capital standard to ensure the capital requirement of can cover the total given risk by numerical analysis and give reasonable economic interpretation for the results.

q-fin.RM↗

arXiv2010

Analytical Framework for Credit Portfolios

Analytical, free of time consuming Monte Carlo simulations, framework for credit portfolio systematic risk metrics calculations is presented. Techniques are described that allow calculation of portfolio-level systematic risk measures (standard deviation, VaR and Expected Shortfall) as well as allocation of risk down to individual transactions. The underlying model is the industry standard multi-factor Merton-type model with arbitrary valuation function at horizon (in contrast to the simplistic default-only case). High accuracy of the proposed analytical technique is demonstrated by benchmarking against Monte Carlo simulations.

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arXiv2010

Modeling total expenditure on warranty claims

We approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter. We consider two kinds of warranty policies, namely, the non-renewing free replacement warranty policy and the non-renewing pro-rata warranty policy. Our approximation holds under modest assumptions on the distribution of the sales process of the warranted item and the nature of arrivals of warranty claims. We propose a method of using historical data to statistically estimate the parameters of the approximate distribution. Our methodology is applied to the warranty claims data from a large car manufacturer for a single car model and model year.

q-fin.GN↗

arXiv2010

Calculation of aggregate loss distributions

Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed.

q-fin.CP↗

arXiv2010

Log-supermodularity of weight functions and the loading monotonicity of weighted insurance premiums

The paper is motivated by a problem concerning the monotonicity of insurance premiums with respect to their loading parameter: the larger the parameter, the larger the insurance premium is expected to be. This property, usually called loading monotonicity, is satisfied by premiums that appear in the literature. The increased interest in constructing new insurance premiums has raised a question as to what weight functions would produce loading-monotonic premiums. In this paper we demonstrate a decisive role of log-supermodularity in answering this question. As a consequence, we establish - at a stroke - the loading monotonicity of a number of well-known insurance premiums and offer a host of further weight functions, and consequently of premiums, thus illustrating the power of the herein suggested methodology for constructing loading-monotonic insurance premiums.

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arXiv2010

Monte Carlo Portfolio Optimization for General Investor Risk-Return Objectives and Arbitrary Return Distributions: a Solution for Long-only Portfolios

We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. In this first paper we explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. A practical scheme for the long-only fully-invested problem is developed and tested for the classic QP application. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, and that the underlying return distribution may be computed without the classical Gaussian limitations. The optimization of quadratic risk-return functions, VaR, CVaR, may be handled in a similar manner to variability ratios such as Sortino and Omega, or mathematical constructions such as expected utility and its behavioural finance extensions. Robustification is also possible. Grid computing technology is an excellent platform for the development of such computations due to the intrinsically parallel nature of the computation, coupled to the requirement to transmit only small packets of data over the grid. We give some examples deploying GridMathematica, in which various investor risk preferences are optimized with differing multivariate distributions. Good comparisons with established results in Mean-Variance and CVaR optimization are obtained when ``edge-vertex-biased'' sampling methods are employed to create random portfolios. We also give an application to Omega optimization.

q-fin.PM↗

arXiv2010

Optimal Dividend and reinsurance strategy of a Property Insurance Company under Catastrophe Risk

We consider an optimal control problem of a property insurance company with proportional reinsurance strategy. The insurance business brings in catastrophe risk, such as earthquake and flood. The catastrophe risk could be partly reduced by reinsurance. The management of the company controls the reinsurance rate and dividend payments process to maximize the expected present value of the dividends before bankruptcy. This is the first time to consider the catastrophe risk in property insurance model, which is more realistic. We establish the solution of the problem by the mixed singular-regular control of jump diffusions. We first derive the optimal retention ratio, the optimal dividend payments level, the optimal return function and the optimal control strategy of the property insurance company, then the impacts of the catastrophe risk and key model parameters on the optimal return function and the optimal control strategy of the company are discussed.

q-fin.RM↗

arXiv2012

On the nature of financial leverage

The article presents a translation of some widespread financial terminology into the language of decision theory. For instance, financial leverage can be regarded as an object of choice or a decision. We show how the optics of decision theory allows perceiving the recently introduced metrics of see-through-leverage, which proved to be very useful in understanding the phenomenology of the recent economic crisis. The importance for practical decision making of specification of the statistical regularity of the random phenomena at hand as well as of the rationality class of the decision maker is discussed.

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