Assessing the cost of capital for longevity risk

The cost of capital is a key element of the embedded value methodology for the valuation of a life business. Further, under some solvency approaches (in particular, the Swiss Solvency Test and the developing Solvency 2 project) assessing the cost of capital constitutes a step in determining the required capital allocation.Whilst the cost of capital is usually meant as a reward for the risks encumbering a given life portfolio, in actuarial practice the relevant parameter has been traditionally chosen, at least to some extent, inconsistently with such risks. The adoption of market-consistent valuations has then been advocated to reach a common standard.A market-consistent value usually acknowledges a reward to shareholders’ capital as long as the market does, namely if the risk is systematic or undiversifiable. When dealing with a life annuity portfolio (or a pension plan), an important example of systematic risk is provided by the longevity risk, i.e. the risk of systematic deviations from the forecasted mortality trend. Hence, a market-consistent approach should provide appropriate valuation tools.In this paper we refer to a portfolio of immediate life annuities and we focus on longevity risk. Our purpose is to design a framework for a valuation of the portfolio which is market-consistent, and therefore based on a risk-neutral argument, while involving some of the basic items of a traditional valuation, viz best estimate future flows and allocated capital. This way, we try to reconcile the traditional with a market-consistent (or risk-neutral) approach. This allows us, in particular, to translate the results obtained under the risk-neutral approach in terms of a properly redefined embedded value.     

新型风险投资组合选择模型

为克服现有关于风险投资的投资组合选择研究所存在的诸如相关参量在实际中不易确定、对投资风险的刻画过于粗糙等不足,本文依据风险投资的特点,提出了直接基于对风险投资项目的综合评价值来预测其未来投资效益与风险的新方法,给出了两种更合理的风险度量,并由此导出了相应的新型风险投资的投资组合选择模型,模拟结果说明了新度量和模型的合理性和实用价值.     

Capital allocation for portfolios with non-linear risk aggregation

Existing risk capital allocation methods, such as the Euler rule, work under the explicit assumption that portfolios are formed as linear combinations of random loss/profit variables, with the firm being able to choose the portfolio weights. This assumption is unrealistic in an insurance context, where arbitrary scaling of risks is generally not possible. Here, we model risks as being partially generated by Lévy processes, capturing the non-linear aggregation of risk. The model leads to non-homogeneous fuzzy games, for which the Euler rule is not applicable. For such games, we seek capital allocations that are in the core, that is, do not provide incentives for splitting portfolios. We show that the Euler rule of an auxiliary linearised fuzzy game (non-uniquely) satisfies the core property and, thus, provides a plausible and easily implemented capital allocation. In contrast, the Aumann–Shapley allocation does not generally belong to the core. For the non-homogeneous fuzzy games studied, Tasche’s (1999) criterion of suitability for performance measurement is adapted and it is shown that the proposed allocation method gives appropriate signals for improving the portfolio underwriting profit.     

Allocation of risk capital on an internal market

We propose an allocation process for economic risk capital using an internal sequential auction in which investment allowances are based on marginal risk contributions. Division managers have incentive to give truthful bids because of bonus payments, which are linear in the division’s profit and linked to the auction bids. With our model, the auction process reaches an equilibrium identical to the optimal allocation if division managers have no diverging interests. When division managers do have diverging preferences in terms of empire building, headquarters faces a trade-off between incurring opportunity costs for achieving a suboptimal allocation and bonus costs paid to division managers to overcome their diverging interests. However, bonus costs are partially offset by proceeds from the auction. Depending on the model parameters, total agency costs can become negative. We show that for large values of new risk capital to be allocated, headquarters can always choose a level of bonus payments so that total costs are negative.     

Factor risk quantification in annuity models

Calculation of risk contributions of sub-portfolios to total portfolio risk is essential for risk management in insurance companies. Thanks to risk capital allocation methods and linearity of the loss model, sub-portfolio (or position) contributions can be calculated efficiently. However, factor risk contribution theory in non-linear loss models has received little interest. Our concern is the determination of factor risk contributions to total portfolio risk where portfolio risk is a non-linear function of factor risks. We employ different approximations in order to convert the non-linear loss model into a linear one. We illustrate the theory on an annuity portfolio where the main factor risks are interest-rate risk and mortality risk.     

Time-invariant portfolio strategies in structured products with guaranteed minimum equity exposure

We introduce a new exotic option to be used within structured products to address a key disadvantage of standard time-invariant portfolio protection: the well-known cash-lock risk. Our approach suggests enriching the framework by including a threshold in the allocation mechanism so that a guaranteed minimum equity exposure (GMEE) is ensured at any point in time. To be able to offer such a solution still with hard capital protection, we apply an option-based structure with a dynamic allocation logic as underlying. We provide an in-depth analysis of the prices of such new exotic options, assuming a Heston–Vasicek-type financial market model, and compare our results with other options used within structured products. Our approach represents an interesting alternative for investors aiming at downsizing protection via time-invariant portfolio protection strategies, meanwhile being also afraid to experience a cash-lock event triggered by market turmoils.     

Regulatory capital decisions in the Context of consumer loan portfolios

A topic of interest in recent literature is regulatory capital requirements for consumer loan portfolios. Banks are required to hold regulatory capital for unexpected losses, while expected losses are to be covered by either provisions or future income. In this paper, we show the set of efficient operating points in the market share and profit space for a portfolio manager operating under Basel II capital requirement and under capital constraints are a union of single-cutoff-score and double-cutoff-score operating points. For a portfolio manager to increase market-share beyond the maximum allowable under a single-cutoff score policy (eg, with binding capital constraints) requires granting loans to higher than optimal risk applicants. We show this result in greater portfolio risk but without an increase in regulatory capital requirement amount. The increase in forecasted losses is assumed to be absorbed by provisions or future margin income. Given portfolio managers take on higher risk under the same regulatory capital amount, our findings call for greater focus on provision amounts and future margin income under the supervisory review pillar of Basel II. This research raises the issue of whether the design of the regulatory formula for consumer loan portfolios is flawed.     

A note on allocation of portfolio shares of random assets with Archimedean copula

This paper further studies the single-period portfolio allocation of risk assets under the assumption that random returns having increasing utility and Archimedean copula. The shares of risk assets in the optimal allocation are proved to be ordered when marginal returns have the likelihood ratio order, and sufficient conditions for the joint density of returns of a multivariate risk to be arrangement increasing is built as well.     

Extremes for coherent risk measures

Various concepts appeared in the existing literature to evaluate the risk exposure of a financial or insurance firm/subsidiary/line of business due to the occurrence of some extreme scenarios. Many of those concepts, such as Marginal Expected Shortfall or Tail Conditional Expectation, are simply some conditional expectations that evaluate the risk in adverse scenarios and are useful for signaling to a decision-maker the poor performance of its risk portfolio or to identify which sub-portfolio is likely to exhibit a massive downside risk. We investigate the latter risk under the assumption that it is measured via a coherent risk measure, which obviously generalizes the idea of only taking the expectation of the downside risk. Multiple examples are given and our numerical illustrations show how the asymptotic approximations can be used in the capital allocation exercise. We have concluded that the expectation of the downside risk does not fairly take into account the individual risk contribution when allocating the VaR-based regulatory capital, and thus, more conservative risk measurements are recommended. Finally, we have found that more conservative risk measurements do not improve the fairness of the cost of capital allocation when the uncertainty with parameter estimation is present, even at a very high level.     

双重网络嵌入对公司IPO后市场表现的影响研究

将社会网络嵌入理论从单一网络拓展至双重网络,分析上市公司同时嵌入风险投资网络和承销商网络对公司IPO上市后市场表现的影响。利用2004～2017年有风险投资支持的IPO公司数据,实证检验结果表明：(1)单独嵌入风险投资网络中心位置、单独嵌入承销商网络中心位置,均会提高公司IPO后市场表现;(2)同时嵌入风险投资网络中心位置和承销商网络中心位置,对公司IPO后短期市场表现的正向影响会相互替代;(3)同时嵌入风险投资网络中心位置和承销商网络中心位置,对公司IPO后长期市场表现的正向影响会互相补充。     

Multi-parameter maintenance optimisation via the marginal cost approach

In this paper we show how the marginal-cost approach can be used to optimise multi-parameter replacement rules. We will illustrate this for an opportunity-based age replacement rule that consists of two parameters. The first parameter is a control limit t, which indicates from what age on a unit is replaced preventively at the first arising opportunity. The second parameter is a planned replacement age T, which indicates at what age the unit is replaced if it has not been replaced yet. The unit can fail and is immediately replaced upon failure. It can be shown that this replacement rule belongs to a class of policies for which the long-run average-cost function is unimodal. The marginal cost approach is based on the following assertion: any point, in which the marginal cost(s) of deferring maintenance equals the average-cost, is an average-cost minimum. Assuming unimodality the minimisation problem can be solved as a root-finding problem, for which there are numerous efficient routines. It appears that the marginal cost approach is very practical for the optimisation of the considered replacement rule, especially because a quick assessment can be made of the optimal parameter values. The marginal cost approach can be used for many other multi-parameter problems, insofar as they can be modelled as a regenerative process.     

Multivariate skew-normal distributions with applications in insurance

In this paper, we discuss the skew-normal distribution as an alternative to the classical normal one in the context of both risk measurement and capital allocation. As main risk measure, we consider the tail conditional expectation (TCE). Hence, we investigate an allocation formula based on the TCE, but we also consider Wang’s [Wang, S., 2002. A set of new methods and tools for enterprise risk capital management and portfolio optimization. Working paper. SCOR reinsurance company (www.casact.com/pubs/forum/02sforum/02sf043.pdf)] allocation formula.     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.     

Optimal and coherent economic-capital structures: evidence from long and short-sales trading positions under illiquid market perspectives

This paper broadens research literature associated with the assessment of modern portfolio risk management techniques by presenting a thorough modeling of nonlinear dynamic asset allocation and management under the supposition of illiquid and adverse market settings. Specifically, the paper proposes a re-engineered and robust approach to optimal economic capital allocation, in a Liquidity-Adjusted Value at Risk (L-VaR) framework, and particularly from the perspective of trading portfolios that have both long and short-sales trading positions. This paper expands previous approaches by explicitly modeling the liquidation of trading portfolios, over the holding period, with the aid of an appropriate scaling of the multiple-assets’ L-VaR matrix along with GARCH-M technique to forecast conditional volatility and expected return. Moreover, in this paper, the authors develop a dynamic nonlinear portfolio selection model and an optimization algorithm which allocates both economic capital and trading assets subject to some selected financial and operational rational constraints. The empirical results strongly confirm the importance of enforcing financially and operationally meaningful nonlinear and dynamic constraints, when they are available, on economic capital optimization procedure. The empirical results are interesting in terms of theory as well as practical applications and can aid in developing robust portfolio management algorithms that financial entities could consider in light of the aftermath of the latest financial crisis.     

Supplier selection in make-to-order environment with risks

The problem of allocation of orders for parts among part suppliers in a customer driven supply chain with operational risk is formulated as a stochastic single- or bi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase parts required for each customer order to minimize total cost and to mitigate the impact of delay risk. The selection of suppliers and the allocation of orders is based on price and quality of purchased parts and reliability of on time delivery. To control the risk of delayed supplies, the two popular percentile measures of risk are applied: value-at-risk and conditional value-at-risk. The proposed approach is capable of optimizing the supply portfolio by calculating value-at-risk of cost per part and minimizing mean worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

Consumption and portfolio selection with labor income: A discrete-time approach

This paper studies the consumption and portfolio selection problem of an agent who is liquidity constrained and has uninsurable income risk in a discrete time setting. It gives properties of optimal policies and presents numerical solutions. The paper, in particular, shows that liquidity constraints and uninsurable income risk reduce consumption and investment in the risky asset substantially from the levels for the case where no market imperfections exist. This paper also shows how the agent evaluates his or her human capital and relates the evaluation to optimal decisions.     

Technology selection and capacity investment under uncertainty

We analyze the problem of technology selection and capacity investment for electricity generation in a competitive environment under uncertainty. Adopting a Nash-Cournot competition model, we consider the marginal cost as the uncertain parameter, although the results can be easily generalized to other sources of uncertainty such as a load curve. In the model, firms make three different decisions: (i) the portfolio of technologies, (ii) each technology’s capacity and (iii) the technology’s production level for every scenario. The decisions related to the portfolio and capacity are ex-ante and the production level is ex-post to the realization of uncertainty. We discuss open and closed-loop models, with the aim to understand the relationship between different technologies’ cost structures and the portfolio of generation technologies adopted by firms in equilibrium. For a competitive setting, to the best of our knowledge, this paper is the first not only to explicitly discuss the relation between costs and generation portfolio but also to allow firms to choose a portfolio of technologies. We show that portfolio diversification arises even with risk-neutral firms and technologies with different cost expectations. We also investigate conditions on the probability and cost under which different equilibria of the game arise.     

Risk measurement in the presence of background risk

A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolio’s re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measure’s aggregation properties are obtained.