This paper studies a hierarchical risk model where an accident can cause a combination of different types of claims, whose sizes could be dependent. In addition, the frequencies of accidents that cause the different combinations of claims are dependent. We first derive formulas for computing risk measures, such as the Tail Conditional Expectation and Tail Variance of the aggregate losses for a portfolio of businesses. Then, we present formulas for performing the associated capital allocation to different types of claims in the portfolio. The main tool we used is the moment (or size-biased) transform of the multivariate distributions.

In this paper, we study the tail risk measures for several commonly used multivariate aggregate loss models where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies. We first develop formulas for the moment (or size biased) transforms of the multivariate aggregate losses, showing their relationship with the moment transforms of the claim frequencies and claim sizes. Then, we apply the formulas to compute some popular risk measures such as the tail conditional expectation and tail variance of the multivariate aggregated losses and to perform capital allocation analysis.

This paper investigates risk aggregation and capital allocation problems for an insurance portfolio consisting of several lines of business. The class of multivariate INAR(1) processes is proposed to model different sources of dependence between the number of claims of the portfolio. The total capital required for the whole portfolio is evaluated under the TVaR risk measure, and the contribution of each line of business is derived under the TVaR-based allocation rule. We provide the risk aggregation and capital allocation formulas in the general case of continuous and strictly positive claim sizes and then in the case of mixed Erlang claim sizes. The impact of both time dependence and cross-dependence on the behavior of risk aggregation and capital allocation is numerically illustrated.

Unlike most existing studies on the endogenous institutions, literature on theoretical growth has traditionally considered institutions as exogenous. In this paper, a learn-by-doing-based growth model is adopted and integrated with endogenous institutions to study how economic agents’ incentives engage in institutional improvements or exploit institutional imperfections. From maximization of identical agent utility, the economic growth model includes capital, labor, technology, and institutions. The study is to analyze the effect of institutions on the stability of equilibrium, balanced economic growth path, and convergence rate in the process of economic growth. It is concluded that, firstly, improving institutions is a decisive factor for China’s high economic growth rate for the past years; secondly, improving institutions can increase the capital stock per unit of effective labor in steady state; thirdly, imperfect institutions can explain income difference among countries; and finally, technology plays a key role only under the conditions adapting to institutions.

In this paper we consider an extension to the aggregation of the FGM mixed Erlang risks, proposed by Cossette et al. (2013 Insurance: Mathematics and Economics, 52, 560–572), in which we introduce the Sarmanov distribution to model the dependence structure. For our framework, we demonstrate that the aggregated risk belongs to the class of Erlang mixtures. Following results from S. C. K. Lee and X. S. Lin (2010 North American Actuarial Journal, 14(1) 107–130), G. E. Willmot and X. S. Lin (2011 Applied Stochastic Models in Business and Industry, 27(1) 8–22), analytical expressions of the contribution of each individual risk to the economic capital for the entire portfolio are derived under both the TVaR and the covariance capital allocation principle. By analysing the commonly used dependence measures, we also show that the dependence structure is wide and flexible. Numerical examples and simulation studies illustrate the tractability of our approach.

We discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.

In this paper, we study a continuous-time bivariate risk process in which each individual line of business implements a dividend barrier strategy. The insurance portfolios of the two insurers are correlated as they are subject to common shocks that induce dependent claims. To analyse the expected discounted dividends until the joint ruin time of the bivariate process (i.e. exit from the positive quadrant), we propose a discrete-time counterpart of the model and apply a bivariate extension of the Dickson−Waters discretisation with the use of a bivariate Panjer-type recursion. Detailed numerical examples under different dependencies via common shocks, copulas and proportional reinsurance are discussed, and applications to optimal problems in reinsurance, capital allocation and dividends are given. It is also illustrated that the optimal pair of dividend barriers maximising the dividend function is dependent on the initial surplus levels. A modified type of dividend barrier strategy is proposed towards the end.

The level of private sector labour productivity has been particularly weak since the start of the crisis. In this paper we explore whether impairment to capital reallocation has been contributing to this weakness. The recent increase in the dispersion of output, prices and rates of return across firms and sectors is stark, and suggests that resources have had incentives to move. Efficient allocation requires that capital moves to firms and sectors where rates of return are relatively high. And the change in capital levels across sectors has been particularly low, suggesting there has been an unusually slow process of capital reallocation since 2008 compared to previous UK recessions and other banking crises. This result is also apparent within sectors. We use a simple and general model to show that increased price dispersion can be a consequence of frictions to efficient capital allocation. And the size of this dispersion can usefully inform us about the size of the associated output and productivity loss. We then find that – using firm level data – the relationship between rates of return and subsequent capital movements has changed since the financial crisis. Overall, our results suggest that impaired capital reallocation across the UK economy is likely to have been one factor contributing to the recent weakness in productivity growth.

A critical problem in financial and insurance risk analysis is the calculation of risk margins. When there are a number of risks, the total risk margin is often reduced to reflect diversification. How large should the “diversification benefit” be? And how should the benefit be allocated to the individual risks? We propose a simple statistical solution. While providing a theoretical analysis, the final expressions are readily implemented in practice.

We propose a capital allocation method for insurance companies. The amount of capital is directly related to the default risk. The expected value of default can be distributed among the liabilities based on the rule of asset payoff at the time of default. We derive a capital allocation scheme from this allocation of the expected default. Assets, liabilities, and other risky items on the balance sheet are treated in a uniform framework. The insurer’s capital is allocated among all these risk contributors. The allocated capitals are given in closed-form formulas, which have straightforward interpretations and are easy to compute. Connections with other allocation methods are also discussed.

Merton and Perold (1993) offered a framework for determining risk capital in a financial firm based on the cost of the implicit guarantee the firm provides to its subsidiaries to make up any operating shortfall. Merton and Perold assume the price of such guarantees is observable from the market at large. For an insurer, this may not be a realistic assumption. This paper proposes an insurance-specific framework for determining the cost of those parental guarantees, and utilizing that cost in pricing and portfolio mix evaluation. An insurer’s capital is treated as a shared asset, with the insurance contracts in the portfolio having simultaneous rights to access potentially all that shared capital. By granting underwriting capacity, an insurer’s management team is implicitly issuing a set of options to draw upon the common capital pool — similar in structure to letters of credit (LOC), except they are not loans but grants. The paper will (i) discuss the valuation of parental guarantees, beginning with Merton and Perold; (ii) treat insurer capital as a shared asset and explore the dual nature of insurer capital usage; (iii) offer a method for determining insurer capital usage cost; and (iv) demonstrate how this method could be used for product pricing and portfolio mix evaluation using economic value added concepts.