The paper proposes an original methodology for constructing quantitative statistical models based on multidimensional distribution functions constructed on the basis of the insurance companies' data on inshurance policies (including policies with deductible) and claims incurred. Real data of some Russian insurance companies on non-life insurance contracts illustrate some opportunities of the proposed approach. The point and interval estimates of net premium, claims frequency, claims reserves including IBNR and OCR, are thus obtained. The resulting estimate of claims reserves falls in the range of reasonable estimates calculated on the basis of traditional reserving methods (the chain-ladder method, the frequency-severity method and the Bornhuetter-Ferguson method).
The proposed methodology is based on additive estimates of a company's financial indicators, in the sense that they are calculated as a sum of estimates built separately for each element of the sample (claim). This allows using the proposed methodology to model insurance companies' financial flows and, in particular, to solve the problems of reserve redistribution between particular segments of insurance portfolio and/or time intervals; to adjust risk as part of financial reporting under IAS 17 Insurance Contracts; and to deal with many other tasks.
The accuracy of insurance companies' financial parameters estimate based on the proposed methods was tested by statistical modeling. IBNR was used as the test parameter. The modeling results showed a satisfactory accuracy of the proposed reserve estimates.
We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-of-capital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capi…
The main purpose of this work is to derive a partial differential equation for the reserves of life insurance liabilities subject to stochastic interest rates where the benefits and premiums depend directly on changes in the interest rate curve. In particular, we allow the payment streams to depend on the performance of an overnight technical interest rate, making them stochastic as well. This opens up for considering new types of contracts based on the performance of the insurer's returns on their own investments. We provide explicit solutions for the reserves when the premiums and benefits v…
This paper presents the hierarchical generalized linear model (HGLM) for loss reserving in a non-life insurance company. Because in this case the error of prediction is expressed by a complex analytical formula, the error bootstrap estimator is proposed instead. Moreover, the bootstrap procedure is used to obtain full information about the error by applying quantiles of the absolute prediction error. The full R code is available on the Github https://github.com/woali/BootErrorLossReserveHGLM.
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 c…
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.
In this paper, we study two optimisation settings for an insurance company, under the constraint that the terminal surplus at a deterministic and finite time $T$ follows a normal distribution with a given mean and a given variance. In both cases, the surplus of the insurance company is assumed to follow a Brownian motion with drift. First, we allow the insurance company to pay dividends and seek to maximise the expected discounted dividend payments or to minimise the ruin probability under the terminal distribution constraint. Here, we find explicit expressions for the optimal strategies in bo…
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 aim of this paper is to define the market-consistent multi-period value of an insurance liability cash flow in discrete time subject to repeated capital requirements, and explore its properties. In line with current regulatory frameworks, the approach presented is based on a hypothetical transfer of the original liability and a replicating portfolio to an empty corporate entity whose owner must comply with repeated one-period capital requirements but has the option to terminate the ownership at any time. The value of the liability is defined as the no-arbitrage price of the cash flow to th…
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, comp…
In this paper we consider two problems on optimal implementation delay of taxation with trade-off for spectrally negative Lévy insurance risk processes. In the first case, we assume that an insurance company starts to pay tax when its surplus reaches a certain level $b$ and at the termination time of the business there is a terminal value incurred to the company. The total expected discounted value of tax payments plus the terminal value is maximized to obtain the optimal implementation level $b^*$. In the second case, the company still pays tax subject to an implementation level $a$ but with …
The economic equities maximization criterion (MFPE) leads to the choice of financial portfolio, which maximizes the ratio of the expected value of the insurance company on the capital. This criterion is presented in the framework of a non-life insurance company and is applied within the framework of the French legislation and in a lawful context inspired of the works in progress about the European project Solvency 2. In the French regulation case, the required solvency margin does not depend of the asset allocation. It is quite different in the Solvency 2 framework because the target capital h…
In the paper we develop mathematical tools of quantile hedging in incomplete market. Those could be used for two significant applications: o calculating the \textbf{optimal capital requirement imposed by Solvency II} (Directive 2009/138/EC of the European Parliament and of the Council) when the market and non-market risk is present in insurance company. We show hot to find the minimal capital $V_0$ to provide with the one-year hedging strategy for insurance company satisfying $E\left[{\mathbf 1}_{\{V_1 \geq D\}}\right]=0.995$, where $V_1$ denotes the value of insurance company in one year time…
The net-premium principle is considered to be the most genuine and fair premium principle in actuarial applications. However, an insurance company, applying the net-premium principle, goes bankrupt with probability one in the long run, even if the company covers its entire costs by collecting the respective fees from its customers. It is therefore an intrinsic necessity for the insurance industry to apply premium principles, which guarantee at least further existence of the company itself; otherwise, the company naturally could not insure its clients to cover their potential, future claims. Be…
This paper is concerned with cost optimization of an insurance company. The surplus of the insurance company is modeled by a controlled regime switching diffusion, where the regime switching mechanism provides the fluctuations of the random environment. The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of (Fleming and Soner 2006, Section V.2) for the continuity of the value function is obtained. Further, the value function is shown to be a viscosity solution of a Hamilton-Jacobian-Bellman equation.
Prudent management of insurance investment portfolios requires competent asset pricing of fixed-income assets with time-to-event contingent cash flows, such as consumer asset-backed securities (ABS). Current market pricing techniques for these assets either rely on a non-random time-to-event model or may not utilize detailed asset-level data that is now available with most public transactions. We first establish a framework capable of yielding estimates of the time-to-event random variable from securitization data, which is discrete and often subject to left-truncation and right-censoring. We …
We consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_\zs{t}$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return $a$ and volatility $σ>0$. If $β:=2a/σ^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $Ψ(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_*u^{-β}\leΨ(u)\le C^*u^{-β}$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{γt}$ with $γ\le 0$ we find the exact asymptotics of the ruin p…
We study the application of dynamic pricing to insurance. We view this as an online revenue management problem where the insurance company looks to set prices to optimize the long-run revenue from selling a new insurance product. We develop two pricing models: an adaptive Generalized Linear Model (GLM) and an adaptive Gaussian Process (GP) regression model. Both balance between exploration, where we choose prices in order to learn the distribution of demands & claims for the insurance product, and exploitation, where we myopically choose the best price from the information gathered so far. The…
Within the Solvency II framework the insurance industry requires a realistic modelling of the risk processes relevant for its business. Every insurance company should be capable of running a holistic risk management process to meet this challenge. For property and casualty (P&C) insurance companies the risk adequate modelling of the claim reserves is a very important topic as this liabilities determine up to 70% percent of the balance sum. We propose a three dimensional (3D) stochastic model for claim reserving. It delivers consistently the reserve's distribution function as well as the distri…
The aim of this study is to investigate quantitatively whether share prices deviated from company fundamentals in the stock market crash of 2008. For this purpose, we use a large database containing the balance sheets and share prices of 7,796 worldwide companies for the period 2004 through 2013. We develop a panel regression model using three financial indicators--dividends per share, cash flow per share, and book value per share--as explanatory variables for share price. We then estimate individual company fundamentals for each year by removing the time fixed effects from the two-way fixed e…
We model the influence of sharing large exogeneous losses to the reinsurance market by a bipartite graph. Using Pareto-tailed claims and multivariate regular variation we obtain asymptotic results for the Value-at-Risk and the Conditional Tail Expectation. We show that the dependence on the network structure plays a fundamental role in their asymptotic behaviour. As is well-known in a non-network setting, if the Pareto exponent is larger than 1, then for the individual agent (reinsurance company) diversification is beneficial, whereas when it is less than 1, concentration on a few objects is t…
In general insurance companies, a correct estimation of liabilities plays a key role due to its impact on management and investing decisions. Since the Financial Crisis of 2007-2008 and the strengthening of regulation, the focus is not only on the total reserve but also on its variability, which is an indicator of the risk assumed by the company. Thus, measures that relate profitability with risk are crucial in order to understand the financial position of insurance firms. Taking advantage of the increasing computational power, this paper introduces a stochastic reserving model whose aim is to…
Within the Own Risk and Solvency Assessment framework, the Solvency II directive introduces the need for insurance undertakings to have efficient tools enabling the companies to assess the continuous compliance with regulatory solvency requirements. Because of the great operational complexity resulting from each complete evaluation of the Solvency Ratio, this monitoring is often complicated to implement in practice. This issue is particularly important for life insurance companies due to the high complexity to project life insurance liabilities. It appears relevant in such a context to use par…
Alignment of financial market incentives and carbon emissions disincentives is key to limiting global warming. Regulators and standards bodies have made a start by requiring some carbon-related disclosures and proposing others. Here we go further and propose a Carbon Equivalence Principle: all financial products shall contain a description of the equivalent carbon flows from greenhouse gases that the products enable, as well as their existing description in terms of cash flows. This description of the carbon flows enabled by the project shall be compatible with existing bank systems that track…
