Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns

The existing literature on investment and reinsurance is limited to the study of continuous-time problems, while discrete-time problems are always ignored by researchers. In this study, we first discuss a multi-period investment and reinsurance optimization problem under the classical mean-variance framework. When the asset returns with a serially correlated structure, the time-consistent investment and reinsurance strategies are acquired via backward induction. In addition, we propose an alternative time-consistent mean-variance optimization model that contrasts with the classical mean-variance model, and the corresponding optimal strategy and value function are also derived. We find that the investment and reinsurance strategies are both independent of the current wealth for the above two optimization problems, which coincides with the conclusion presented in the continuous-time problems. Most importantly, the above investment strategies with serially correlated structures are both conditional mean-based strategies, rather than unconditional ones. Finally, we compare the investment and reinsurance strategies suggested above based on the simulation approach, to shed light on which investment-reinsurance strategies are more suitable for insurers.     

Time-consistent investment policies in Markovian markets: A case of mean–variance analysis

The optimal investment policy for a standard multi-period mean–variance model is not time-consistent because the variance operator is not separable in the sense of the dynamic programming principle. With a nested conditional expectation mapping, we develop an investment model with time consistency in Markovian markets. Furthermore, we examine the differences of the investment policies with a riskless asset from those without a riskless asset. Analytical solutions for time-consistent optimal investment policies and the resulting mean–variance efficient frontier are obtained. Finally, using numerical examples, we show that the optimal investment policy derived from our model is more efficient than that of the standard mean–variance model in which the trade-off is determined between the mean and variance of the terminal wealth.     

Financial accumulation implies ever-increasing wealth inequality

Wealth inequality is an important matter for economic theory and policy. The recent rise in wealth inequality has been discussed in connection with the recent development of active global financial markets. The existing literature on wealth distribution links wealth inequality to a variety of drivers. Our approach develops a minimalist modelling strategy that combines three featuring mechanisms: active financial markets, individual wealth accumulation and compound interest structure. We provide mathematical proof that accumulated financial investment returns involve ever-increasing wealth concentration and inequality across individual investors most of the time. This cumulative effect over space and time depends on financial accumulation processes, including under efficient financial markets, which generate a fair investment game that individual investors repeatedly play through time.

     

Time preference and real investment

This paper studies the investment timing problem of an entrepreneur with a non-tradable real option with undiversifiable risk. We find that the time preference can have a significant impact on the risk attitude toward the idiosyncratic risk, which results from the wealth effect on the implied option value. If the agent is impatient (patient), an increase in idiosyncratic volatility increases (decreases) the agent’s value and delays (hastens) investment. This finding suggests several important implications and empirical predictions for investment decisions in private firms and public firms with concentrated ownership.     

The mean-variance investment problem in a constrained financial market

This paper extends the mean-variance analysis and the two-fund separation theorem to a market with some constraints, such as, the incompleteness, prohibition of short-selling, and partial information, with stochastic interest rate, and with stochastic volatility for risky assets. By maximizing a quadratic utility of terminal wealth, we show that the efficient frontier for the problem is a straight line in the mean-standard-deviation diagram. The quadratic utility function exhibits mean-variance efficiency. Our results apply to portfolios of claims in a single period, multiperiod, and continuous time.     

Alpha-robust mean-variance reinsurance-investment strategy

Inspired by the α-maxmin expected utility, we propose a new class of mean-variance criterion, called α-maxmin mean-variance criterion, and apply it to the reinsurance-investment problem. Our model allows the insurer to have different levels of ambiguity aversion (rather than only consider the extremely ambiguity-averse attitude as in the literature). The insurer can purchase proportional reinsurance and also invest the surplus in a financial market consisting of a risk-free asset and a risky asset, whose dynamics is correlated with the insurance surplus. Closed-form equilibrium reinsurance-investment strategy is derived by solving the extended Hamilton–Jacobi–Bellman equation. Our results show that the equilibrium reinsurance strategy is always more conservative if the insurer is more ambiguity-averse. When the dependence between insurance and financial risks are weak, the equilibrium investment strategy is also more conservative if the insurer is more ambiguity-averse. However, in order to diversify the portfolio, a more ambiguity-averse insurer may adopt a more aggressive investment strategy if the insurance market is very ambiguous. For an ambiguity-neutral insurer, the investment strategy is identical to the non-robust investment strategy.     

Behavior patterns of investment strategies under Roy’s safety-first principle

The safety-first principle is a natural motivational factor in decision making, and is closely related to certain popular heuristics such as satisficing. We provide a systematic analysis of optimal portfolio choice under Roy’s safety-first principle by examining and comparing the behavior patterns of three popular investment strategies: the optimal constant-rebalanced portfolio, dynamic-rebalanced portfolio and buy-and-hold strategies. Our results indicate the importance of a match between the investment strategy, the investment goal, and the investment horizon. We also develop a geometric approach to investigate the relationships among the safety-first, expected utility, and mean-variance models and offer an explanation for the long-standing debate concerning different patterns of time-diversification effects.     

Rationalizing investors’ choices

Assuming that agents’ preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.     

A continuous-time portfolio turnpike theorem

We prove a continuous-time portfolio turnpike theorem. The proof uses the theory of martin-gales and is more intuitively appealing than the usual discrete-time mode of proof using dynamic programming. When the interest rate is strictly positive, the present value of any contingent claim having payoffs bounded from above can be made arbitrarily small when the investment horizon increases. Thus an investor concentrates his wealth in buying contingent claims that have payoffs unbounded from above at the very beginning of his horizon. As a consequence, it is the asymptotic property of his utility function as wealth goes to infinity that determines his optimal investment strategy at the very beginning of his horizon.     

Implied Sharpe ratios of portfolios with options: Application to Nikkei futures and listed options

We propose a criterion for portfolio selection, implied excess Sharpe ratio. The implied excess Sharpe ratio is intended as an excess Sharpe ratio (versus the underlying stock) that investors can expect to enjoy from portfolios that include options and is a useful ex ante indicator that can be easily calculated. There are a variety of ways to include options in a portfolio, but we theoretically show that the combination that produces the largest implied excess Sharpe ratio is the best way to maximize the short-term Sharpe ratio. The selection process uses implied excess Sharpe ratio, which is easily calculated from stock lending fees implied by stock prices and actual stock lending fee. It does not require historical simulation or prediction of share price average growth rates and is highly transparent as it can be easily reproduced (at a low calculation cost). Hence, the implied excess Sharpe ratio is a simple but effective tool for investors seeking returns in exchange for a certain amount of risk that want to use the options market efficiently. The short-term Sharpe ratio is not necessarily the only criterion, but is a rational benchmark of portfolio performance closely related to criteria such as the long-term Sharpe ratio and maximum drawdown. To examine the benefit of the concept, we construct an investment strategy that automatically selects from multiple candidate portfolios that are made up of combinations of Nikkei futures and Nikkei listed options the portfolio with the largest implied excess Sharpe ratio. Back-testing shows that this investment strategy performs well over the long term as well.     

On optimal investiment strategies

Suppose an investor has a fixed decision horizon and an appropriate utility function for measuring his or her utility of wealth. If there are only two investment vehicles, a risky and a risk-free asset, then the optimal investment strategy is such that, at any time, the amount invested in the risky asset must be the product of his or her “current risk tolerance” and the risk premium on the risky asset, divided by the square of the diffusion coefficient of the risky asset. In the case of more than one risky asset, the optimal investment strategy is similar, with the ratios of the amounts invested in the different risky assets being constant over time.     

The risk-return trade-off in human capital investment

In this paper, we analyze investments in human capital in a way which is standard for financial assets, but not (yet) for human capital assets. We study mean-variance plots of human capital assets. We compare the properties of human capital returns using a performance measure and by using tests for mean-variance spanning. Fields differ strongly not only in common rates of return, but also in return per unit of risk. We identify a range of educations that are efficient in terms of investment goods, and a range of educations that may be chosen for consumption purposes.     

Market selection of constant proportions investment strategies in continuous time

This paper studies the wealth dynamics of investors holding self-financing portfolios in a continuous-time model of a financial market. Asset prices are endogenously determined by market clearing. We derive results on the asymptotic dynamics of the wealth distribution and asset prices for constant proportions investment strategies. This study is the first step towards a theory of continuous-time asset pricing that combines concepts from mathematical finance and economics by drawing on evolutionary ideas.     

Instability and concentration in the distribution of wealth

We consider a setup in which infinitely lived households face idiosyncratic investment risk and show that in this case the equilibrium distribution of wealth becomes increasingly right-skewed over time until wealth concentrates entirely at the top. The households in our setup are identical in terms of their patience and their abilities, and we assume that there are no redistributive mechanisms—neither explicit in the form of government tax or fiscal policies, nor implicit in the form of limited intergenerational transfers. Our results demonstrate that the presence of such redistributive mechanisms alone ensures the stability of the distribution of wealth over time.     

Mean-variance portfolio selection of cointegrated assets

This paper considers the continuous-time mean-variance portfolio selection problem in a financial market in which asset prices are cointegrated. The asset price dynamics are then postulated as the diffusion limit of the corresponding discrete-time error-correction model of cointegrated time series. The problem is completely solved in the sense that solutions of the continuous-time portfolio policy and the efficient frontier are obtained as explicit and closed-form formulas. The analytical results are applied to pairs trading using cointegration techniques. Numerical examples show that identifying a cointegrated pair with a high mean-reversion rate can generate significant statistical arbitrage profits once the current state of the economy sufficiently departs from the long-term equilibrium. We propose an index to simultaneously measure the departure level of a cointegrated pair from equilibrium and the mean-reversion speed based on the mean-variance paradigm. An empirical example is given to illustrate the use of the theory in practice.     

A closed-form solution for the continuous-time consumption model with endogenous labor income

In this paper, we study the consumption, labor supply, and portfolio decisions of an infinitely lived individual who receives a certain wage rate and income from investment into a risky asset and a risk-free bond. Uncertainty about labor income arises endogenously, because labor supply evolves randomly over time in response to changes in financial wealth. We derive closed-form solutions for optimal consumption, labor supply and investment strategy. We find that deferring the retirement age stimulates optimal consumption over time and discourages optimal labor supply during the working life. We also find explicitly that optimal portfolio allocation becomes more ‘conservative’ when the individual approaches his prescribed retirement age. The effects of risk-aversion coefficients on optimal decisions are examined.     

Investment and concern for relative position

Economists typically analyze individuals' market behavior in isolation from their nonmarket decisions. While this research strategy has generally been successful, it can lead to systematic errors when agents' nonmarket behavior affects their market choices. In this paper we analyze how individuals' investment behavior changes as a result of nonmarket behavior. Specifically, we analyze a model in which individuals must decide how to allocate their initial endowment between two random investments, where the returns are perfectly correlated across individuals for the first investment but independent across individuals for the second. We consider an environment in which men and women match, with wealthier individuals more successful in matching. We show how individuals' concern about relative wealth can affect their investment decisions, and we provide conditions under which individuals bias their investments either toward or away from the investment with correlated returns. A modification of the model is used to explain why agents' investments might exhibit a home country bias.     

Homeownership, wealth accumulation and income status

This paper examines the extent to which homeownership had an independent effect on the ability of low- and moderate-income (LMI) households to accumulate wealth during the mid-to-late 1990s. Using household data from the PSID, we generate a panel of households whose homeownership we observe over a 15 year period and whose wealth accumulation we observe at three points in time: 1994, 1999 and 2001. We investigate the extent to which homeownership has an independent impact on the wealth accumulation of LMI households, controlling for a host of other variables and unobserved heterogeneity. Accounting for the skewed nature of the wealth distribution, we find that each additional year of homeownership increases total net wealth by $13.7 K on average for the full sample. Interacting income status with years of homeownership indicates that the impact of homeownership varies by income status, with each additional year of homeownership being associated with $15 K more in wealth holdings for high-income households and roughly $6 to 10 K more in wealth holdings for LMI households.     

Irreversible investment and R&D spillovers in a dynamic duopoly

Investments in new production processes usually involve a significant amount of R&D, generating spillovers that lowers the second comer's investment cost. We show that these spillovers substantially affect the equilibrium of the dynamic game. Even for low spillover values, the leader delays her investment until the stochastic fundamental has gone past the level such that the follower's optimal strategy is to invest as soon as he attains the spillover. This bears several interesting implications. First, because the follower invests as he benefits from the spillover, in equilibrium the average time delay between the two investments is short, as it should be expected. Second, in case of a major innovation, an optimal public policy requires an intervention in favor of the investment activity; an increase in uncertainty - delaying the equilibrium - calls for higher subsidization rates. Third, numerical simulations show that the spillover reduces the difference between the leader's and the follower's maximum value functions. Accordingly, our model can help generate realistic market betas.     

Mean-variance hedging for interest rate models¶with stochastic volatility

The mean-variance hedging approach for pricing and hedging claims in incomplete markets was originally introduced for risky assets. The aim of this paper is to apply this approach to interest rate models in the presence of stochastic volatility, seen as a consequence of incomplete information. We fix a finite number of bonds such that the volatility matrix is invertible and provide an explicit formula for the density of the variance-optimal measure which is independent of the chosen times of maturity. Finally, we compute the mean-variance hedging strategy for a caplet and compare it with the optimal stategy according to the local risk minimizing approach. Received: 14 July 2000 / Accepted: 10 April 2001