Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria

In this paper, an ambiguity-averse insurer (AAI) whose surplus process is approximated by a Brownian motion with drift, hopes to manage risk by both investing in a Black–Scholes financial market and transferring some risk to a reinsurer, but worries about uncertainty in model parameters. She chooses to find investment and reinsurance strategies that are robust with respect to this uncertainty, and to optimize her decisions in a mean-variance framework. By the stochastic dynamic programming approach, we derive closed-form expressions for a robust optimal benchmark strategy and its corresponding value function, in the sense of viscosity solutions, which allows us to find a mean-variance efficient strategy and the efficient frontier. Furthermore, economic implications are analyzed via numerical examples. In particular, our conclusion in the mean-variance framework differs qualitatively, for certain parameter ranges, with model-uncertainty robustness conclusions in the framework of utility functions: model uncertainty does not always result in an agent deciding to reduce risk exposure under mean-variance criteria, opposite to the conclusions for utility functions in Maenhout and Liu. Our conclusion can be interpreted as saying that the mean-variance problem for the AAI explains certain counter-intuitive investor behaviors, by which the attitude to risk exposure, for an AAI facing model uncertainty, depends on positive past experience.     

The 1/N investment strategy is optimal under high model ambiguity

The 1/N investment strategy, i.e. the strategy to split one’s wealth uniformly between the available investment possibilities, recently received plenty of attention in the literature. In this paper, we demonstrate that the uniform investment strategy is rational in situations where an agent is faced with a sufficiently high degree of model uncertainty in the form of ambiguous loss distributions. More specifically, we use a classical risk minimization framework to show that, for a broad class of risk measures, as the uncertainty concerning the probabilistic model increases, the optimal decisions tend to the uniform investment strategy.To illustrate the theoretical results of the paper, we investigate the Markowitz portfolio selection model as well as Conditional Value-at-Risk minimization with ambiguous loss distributions. Subsequently, we set up a numerical study using real market data to demonstrate the convergence of optimal portfolio decisions to the uniform investment strategy.     

The Value and Risk of Defined Contribution Pension Schemes: International Evidence

We use historical data on investment returns and labor income from 16 countries to quantify the value and risk of defined contribution pension plans, building frequency distributions of pension fund and pension replacement ratios for each country. We show that pension risk is substantial and find that pension fund ratios are lower and less variable than when the correlation between wage growth and investment returns is ignored, typically halving the median pension fund ratio. We also show that an all‐equity fund is the dominant investment strategy across all countries, although sometimes a life‐cycle strategy insures against downside risk.     

A note on institutional hierarchy and volatility in financial markets

From a statistical point of view, the prevalence of non-Gaussian distributions in financial returns and their volatilities shows that the Central Limit Theorem (CLT) often does not apply in financial markets. In this article, we take the position that the independence assumption of the CLT is violated by herding tendencies among market participants, and investigate whether a generic probabilistic herding model can reproduce non-Gaussian statistics in systems with a large number of agents. It is well known that the presence of a herding mechanism in the model is not sufficient for non-Gaussian properties, which crucially depend on the details of the communication network among agents. The main contribution of this article is to show that certain hierarchical networks, which portray the institutional structure of fund investment, warrant non-Gaussian properties for any system size and even lead to an increase in system-wide volatility. Viewed from this perspective, the mere existence of financial institutions with socially interacting managers contributes considerably to financial volatility.     

Optimal subsidies and guarantees in public–private partnerships

In this paper, we analyse how certain subsidies and guarantees given to private firms in public–private partnerships should be optimally arranged to promote immediate investment in a real options framework. We show how an investment subsidy, a revenue subsidy, a minimum demand guarantee, and a rescue option could be optimally arranged to induce immediate investment, compensating for the value of the option to defer. These four types of incentives produce significantly different results when we compare the value of the project after the incentive structure is devised and also when we compare the timing of the resulting cash flows.     

A NOTE ON INVESTMENT DECISION RULES BASED ON UTILITY FUNCTIONS

Many investment decision models are formulated on the basis of certain assumptions regarding investor's tastes combined with the assumed objective of expected utility maximization. The rules are often expressed in the form of a finite number of moments of the returns distributions, depending on the specific utility function restrictions imposed. For example, mean/variance decisions have been derived using the assumption of a quadratic preference function. Borch (1969) and many others have demonstrated the ambiguity of mean/variance decision rules based on this specific utility functional form. In this note, Borch's findings are extended t o all investment decision rules based on the assumption of any finite order utility function for a general class of risk averters.     

Investment timing and optimal capital structure under liquidity risk

Deterioration in debt market liquidity reduces debt values and affects firms' decisions. Considering such risk, we develop an investment timing model and obtain analytic solutions. We carry out a comprehensive analysis in optimal financing, default, and investment strategies, and stockholder–bondholder conflicts. Our model explains stylized facts and replicates empirical findings in credit spreads. We obtain six new insights for decision makers. We propose a ‘new trade-off theory’ for optimal capital structure, a new tax effect, and new explanations of ‘debt conservatism puzzle’ and ‘zero-leverage puzzle’. Failure in recognizing liquidity risk results in substantially over-leveraging, early bankruptcy or investment, overpriced options, and undervalued coupons and credit spreads. In addition, agency costs are surprisingly small for a high liquidity risk or a low project risk. Interestingly, the risk shifting incentive and debt overhang problem decrease with liquidity risk under moderate tax rates while they increase under high tax rates.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

Optimal investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.     

Asset–liability modelling and pension schemes: the application of robust optimization to USS

This paper uses a novel numerical optimization technique – robust optimization – that is well suited to solving the asset–liability management (ALM) problem for pension schemes. It requires the estimation of fewer stochastic parameters, reduces estimation risk and adopts a prudent approach to asset allocation. This study is the first to apply it to a real-world pension scheme, and the first ALM model of a pension scheme to maximize the Sharpe ratio. We disaggregate pension liabilities into three components – active members, deferred members and pensioners, and transform the optimal asset allocation into the scheme's projected contribution rate. The robust optimization model is extended to include liabilities and used to derive optimal investment policies for the Universities Superannuation Scheme (USS), benchmarked against the Sharpe and Tint, Bayes–Stein and Black–Litterman models as well as the actual USS investment decisions. Over a 144-month out-of-sample period, robust optimization is superior to the four benchmarks across 20 performance criteria and has a remarkably stable asset allocation – essentially fix-mix. These conclusions are supported by six robustness checks.     

How Are Derivatives Used? Evidence from the Mutual Fund Industry

We investigate investment managers' use of derivatives by comparing return distributions for equity mutual funds that use and do not use derivatives. In contrast to public perception, derivative users have risk exposure and return performance that are similar to nonusers. We also analyze changes in fund risk in response to prior fund performance. Changes in risk are substantially less severe for funds using derivatives, consistent with the explanation that managers use derivatives to reduce the impact of performance on risk. We provide new evidence regarding the implications of cash flows and managerial gaming for the relation between performance and risk.     

Dynamic portfolio selection with process control

The risk inherent in the accumulation of investment capital depends on the true return distributions of the risky assets, the accuracy of estimated returns, and the investment strategy. This paper considers risk control with Value-at-Risk and Conditional Value-at-Risk, using control limits to determine times for portfolio rebalancing. Optimal strategies and control limits are determined for a geometric Brownian motion asset pricing model with random parameters. The approaches to risk control are applied to the fundamental problem of investment in stocks, bonds, and cash over time.     

Modelling long-term investment returns via Bayesian infinite mixture time series models

This paper introduces the class of Bayesian infinite mixture time series models first proposed in Lau & So (2004) for modelling long-term investment returns. It is a flexible class of time series models and provides a flexible way to incorporate full information contained in all autoregressive components with various orders by utilizing the idea of Bayesian averaging or mixing. We adopt a Bayesian sampling scheme based on a weighted Chinese restaurant process for generating partitions of investment returns to estimate the Bayesian infinite mixture time series models. Instead of using the point estimates, as in the classical or non-Bayesian approach, the estimation in this paper is performed by the full Bayesian approach, utilizing the idea of Bayesian averaging to incorporate all information contained in the posterior distributions of the random parameters. This provides a natural way to incorporate model risk or uncertainty. The proposed models can also be used to perform clustering of investment returns and detect outliers of returns. We employ the monthly data from the Toronto Stock Exchange 300 (TSE 300) indices to illustrate the implementation of our models and compare the simulated results from the estimated models with the empirical characteristics of the TSE 300 data. We apply the Bayesian predictive distribution of the logarithmic returns obtained by the Bayesian averaging or mixing to evaluate the quantile-based and conditional tail expectation risk measures for segregated fund contracts via stochastic simulation. We compare the risk measures evaluated from our models with those from some well-known and important models in the literature, and highlight some features that can be obtained from our models.     

The maximum surplus before ruin for dependent risk models through Farlie–Gumbel–Morgenstern copula

We extend the classical compound Poisson risk model to consider the distribution of the maximum surplus before ruin where the claim sizes depend on inter-claim times via the Farlie–Gumbel–Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for this distribution, of which the Laplace transform is provided. We obtain the renewal equation and explicit expressions for this distribution are derived when the claim amounts are exponentially distributed. Finally, we present numerical examples.     

Insurance demand in the presence of loss-dependent background risk

We analyze insurance demand when insurable losses come with an uninsurable zero-mean background risk that increases in the loss size. If the individual is risk vulnerable, loss-dependent background risk triggers a precautionary insurance motive and increases optimal insurance demand. Prudence alone is sufficient for insurance demand to increase in two cases: the case of fair insurance and the case where the smallest possible loss exceeds a certain threshold value (referred to as the large loss case). We derive conditions under which insurance demand increases or decreases in initial wealth. In the large loss case, prudence determines whether changes in the background risk lead to more insurance demand. We generalize this result to arbitrary loss distributions and find conditions based on decreasing third-degree Ross risk aversion, Arrow–Pratt risk aversion, and Arrow–Pratt temperance.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

Optimal dynamic reinsurance with dependent risks: variance premium principle

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the variance premium principle, we adopt a nonstandard approach to examining the existence and uniqueness of the optimal reinsurance strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived for the compound Poisson risk model as well as for the Brownian motion risk model. From the numerical examples, we see that the optimal results for the compound Poisson risk model are very different from those for the diffusion model. The former depends not only on the safety loading, time, and the interest rate, but also on the claim size distributions and the claim number processes, while the latter depends only on the safety loading, time, and the interest rate.     

The Role of Payback in the Investment Process

This study examines the use of the payback (PB) method as a means of evaluating a proposed asset's risk and its joint application with profit-oriented capital budgeting models. Previous research studies indicating a linkage between the PB method and risk analysis are reviewed. A certainty-equivalent model is used to demonstrate this relationship and the properties of the relationship exploited by PB when used as a heuristic. Results of the analysis indicate that using a hurdle PB as a filter for identifying proposals with acceptable risk and return attributes is consistent with more quantitatively oriented investment techniques under certain conditions. The study then examines the conceptual relationship between PB and profit-oriented capital budgeting models. Results suggest that PB and profit-oriented capital budgeting techniques measure different attributes of an investment and complement one another in describing and analysing its cash flows.     

The role of perceived costs and perceived benefits in the relationship between personality and risk‐related choices

This paper considers how perceptions of costs and benefits can influence the association between personality and risky choice behaviour. We assessed perceptions and behaviours in six domains (ethical; investment; gambling; health and safety; recreational; social) using the DOSPERT and measured personality using the NEO PI‐R. Results from structural equation modelling showed that personality had a direct effect on risky choice behaviour in four domains (social, ethical, gambling and recreational risk‐taking). In addition, perceived costs and benefits mediated the relations between personality and risk‐taking in the five domains (social, ethical, gambling, recreational and investment risk‐taking). Evidence for a mechanism that integrates both direct and indirect effects of personality on behaviour is discussed.