A BSDE approach to risk-based asset allocation of pension funds with regime switching

An asset allocation problem of a member of a defined contribution (DC) pension fund is discussed in a hidden, Markov regime-switching, economy using backward stochastic differential equations, (BSDEs). A risk-based approach is considered, where the member selects an optimal asset mix with a view to minimizing the risk described by a convex risk measure of his/her terminal wealth. Firstly, filtering theory is adopted to transform the hidden, Markov regime-switching, economy into one with complete observations and to develop, (robust), filters for the hidden Markov chain. Then the optimal asset allocation problem of the member is formulated as a two-person, zero-sum stochastic differential game between the member and the market in the economy with complete observations. The BSDE approach is then used to solve the game problem and to characterize the saddle point of the game problem. An explicit expression for the optimal asset mix is obtained in the case of a convex risk measure with quadratic penalty and it can be considered a generalized version of the Merton ratio. An explicit expression for the optimal strategy of the market is also obtained, which leads to a risk-neutral wealth dynamic and may provide some insights into asset pricing in the economy with inflation risk and regime-switching risk. Numerical examples are provided to illustrate financial implications of the BSDE solution.     

The impact of fat tailed returns on asset allocation

This paper analyzes the asset allocation problem of an investor who can invest in equity and cash when there is time variation in expected returns on the equity. The solution methodology is multistage stochastic asset allocation problem with decision rules. The uncertainty is modeled using economic scenarios with Gaussian and stable Paretian non-Gaussian innovations. The optimal allocations under these alternative hypothesis are compared. Our computational results suggest that asset allocation may be up to 20% different depending on the utility function and the risk aversion level of the investor. Certainty equivalent return can be increased up to .13% and utility can be improved up to .72% by switching to the stable Paretian model.     

跳扩散环境下考虑红利支付的动态资产配置问题研究

研究资产价格带跳环境下红利支付对投资者资产配置的影响,投资者将其财富在风险资产和无风险资产中进行分配,在终端财富预期效用最大化标准下,利用动态规划原理建立的HJB方程推导最优配置策略,并得到最优动态资产配置策略的近似解.最后通过数值模拟,分析了跳和红利支付对投资者最优配置策略的影响.结果表明在跳发生的情况下,不管跳的大小和方向如何,投资者都会减少其在风险资产中的配置头寸,同时带有红利支付的资产比不带红利支付的资产对投资者更具吸引力.     

Dynamic asset allocation with event risk,transaction costs and predictable returns

We examine the interplay between event risk, transaction costs and predictability on the dynamic asset allocation of an investor with discrete trading opportunities. The model is calibrated to the U.S. stock market and a Gauss–Hermite quadrature approach is used to solve the investor’s dynamic optimization problem. Numerical scenarios are examined to show the impact of event risk on asset allocations, hedging demands, no-trading regions, and certainty equivalent returns. It is found that event risk shrinks hedging demand. Neglecting event risk can also lead to sizeable certainty equivalent return losses.     

基于Heston随机波动率模型和风险偏好视角的资产负债管理

本文研究基于Heston随机波动率模型的资产负债管理问题。假设金融市场由一个无风险资产和一个风险资产构成,投资者的目标是最大化其终端财富的期望效用。应用随机控制方法,得到了该问题最优资产配置策略的解析表达式和相应值函数的解析解,通过数值算例分析了Heston模型主要参数以及债务对最优资产配置策略的影响。结果表明:配置到风险资产的比例对Heston模型中的参数非常敏感;为了对冲债务风险,负债的引入使得配置到风险资产的比例比无负债情形下的高;在风险厌恶系数变大时,无论投资者是否有负债,其投资到风险资产的比例则越来越低。     

长期投资者的战略资产配置选择

在股票期望收益率服从一个均值回复过程的假设下,推导出具有幂效用函数的投资者的资产配置函数,着重分析了投资期限对投资者资产配置结果的影响,发现长期投资者比短期投资者在股票上配置更大的资产比例.虽然不同投资期限的投资者具有相同的短视配置,但是战略配置随着投资期限的增大而增大.     

跳扩散模型下的非零和随机微分投资组合博弈

现实经济中,当股票价格受到一些重大信息影响而发生突发性的跳跃时,用跳扩散过程来描述股票价格的趋势更符合实际情况。基于这一观察,本文研究跳扩散模型下包含两个投资者的非零和投资组合博弈问题。假设金融市场中包含一种无风险资产和一种风险资产,其中风险资产的价格动态用跳扩散模型来描述。将该非零和博弈问题构造成两个效用最大化问题,每个投资者的目标是最大化终端时刻自身财富与其竞争对手财富差的均值-方差效用。运用随机控制理论,得到了均衡投资策略以及相应值函数的解析表达。最后通过数值仿真算例分析了模型相关参数变动对均衡投资策略的影响。仿真结果显示:当股价发生不连续跳跃,投资者在构造投资策略时考虑跳跃风险可以显著增加其效用水平;同时,随着博弈竞争的加剧,投资者为了在竞争中取得更好的表现,往往会采取更加激进的投资策略,增加对风险资产的投资。     

Optimal asset allocation aid system: From “one-size” vs “tailor-made” performance ratio

Optimal asset allocation well-fitting investors’ goals is a pressing challenge in risk management. Making a step forward to the Sharpe ratio, the parameter-dependent Sortino–Satchell, Generalized Rachev and Farinelli–Tibiletti performance ratios are suggested for personalizing asset allocation. Tailor-made optimal asset paths for five different investor risk profiles are traced over a rolling 12 month investing horizon. Our simulations show a satisfactorily good match between asset allocation and correspondent risk profile. Specifically, Generalized Rachev ratios outperform in personalized allocation for “extreme” risk profiles, i.e. conservative and aggressive investors, whereas Sortino–Satchell and Farinelli–Tibiletti ratios for those that are more moderate. Sharpe ratio confirms its ability in constructing steady-diversified portfolios, although underperformed.     

基于DEA博弈交叉效率和投资者心理的模糊投资组合研究

考虑到投资者并不是完全理性的,本文结合DEA博弈交叉效率方法研究了带有投资者心理因素的多目标模糊投资组合决策问题。首先,为了充分描绘投资者的心理因素和风险感知,本文基于可能性理论推导了带有风险态度的可能性均值和半绝对偏差。其次,将候选的风险资产视为互相竞争的博弈者,采用基于熵权法的DEA博弈交叉效率模型衡量它们的综合表现,从而得到每项资产的博弈交叉效率和奇异指数,并将其分别作为额外的收益和风险决策准则。基于此,提出了更加综合的可能性均值—半绝对偏差—博弈交叉效率—奇异指数模型。最后,通过一个应用实例验证了所提出的模型的合理性和有效性,从而为不同类型的投资者提供具有个性化的投资策略。     

Optimal dynamic asset allocation for DC plan accumulation/decumulation: Ambition-CVAR

We consider the late accumulation stage, followed by the full decumulation stage, of an investor in a defined contribution (DC) pension plan. The investor’s portfolio consists of a stock index and a bond index. As a measure of risk, we use conditional value at risk (CVAR) at the end of the decumulation stage. This is a measure of the risk of depleting the DC plan, which is primarily driven by sequence of return risk and asset allocation during the decumulation stage. As a measure of reward, we use Ambition, which we define to be the probability that the terminal wealth exceeds a specified level. We develop a method for computing the optimal dynamic asset allocation strategy which generates points on the efficient Ambition-CVAR frontier. By examining the Ambition-CVAR efficient frontier, we can determine points that are Median-CVAR optimal. We carry out numerical tests comparing the Median-CVAR optimal strategy to a benchmark constant proportion strategy. For a fixed median value (from the benchmark strategy) we find that the optimal Median-CVAR control significantly improves the CVAR. In addition, the median allocation to stocks at retirement is considerably smaller than the benchmark allocation to stocks.     

动态非短视资产负债管理

用均值-回复过程刻画股票价格变化,本文研究了股票收益可预测金融市场中的连续时间资产负债管理问题。运用动态规划方法,求得了最优资产负债管理策略的闭合解。结果表明,最优策略是风险溢价的线性函数,随着投资期限的缩短,股票上的投资金额不断降低。数值分析表明,投资期限、股票风险溢价和债务对于最优资产配置策略和股票风险溢价不确定性跨期对冲需求都存在显著影响。     

Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management

ABSTRACT

Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these two objectives in a unified framework. We look to maximize the expected growth rate differential between the wealth of the investor’s portfolio and that of a performance benchmark while penalizing risk-weighted deviations from a given tracking portfolio. Using stochastic control techniques, we provide explicit closed-form expressions for the optimal allocation and we show how the optimal strategy can be related to the growth optimal portfolio. The admissible benchmarks encompass the class of functionally generated portfolios (FGPs), which include the market portfolio, as the only requirement is that they depend only on the prevailing asset values. Finally, some numerical experiments are presented to illustrate the risk–reward profile of the optimal allocation.     

Asset liability management for open pension schemes using multistage stochastic programming under Solvency-II-based regulatory constraints

Open private pension schemes are subject to risk-based regulation. In this context, asset and liability management (ALM) frameworks for pension plan operators are increasingly based on multistage stochastic programming (MSP). The significant advances in MSP modeling notwithstanding, previous works ignore risk-based regulatory constraints such as those in the Solvency II Directive. In this work, we propose an ALM model for open pension schemes based on an MSP model with a thorough representation of a risk-based regulation. Our proposal aims to define a dynamic optimal asset allocation including a detailed depiction of bond coupon payments, based on insolvency risk measures over a planning horizon. We present a realistic case study based on the Brazilian market, where the regulator imposes Solvency-II-compatible constraints on credit, underwriting, and operational risks. We develop a computationally tractable MSP model with explicit regulatory constraints, which induce risk aversion for even risk-neutral open pension plan operators.     

模型不确定和极端事件冲击下带通胀的最优投资组合选择问题研究

本文研究了投资者在极端事件冲击下带通胀的最优投资组合选择问题, 其中投资者不仅对损失风险是厌恶的而且对模型不确定也是厌恶的. 投资者在风险资产和无风险资产中进行投资. 首先, 利用Ito公式推导考虑通胀的消费篮子价格动力学方程, 其次由通胀折现的终端财富预期效用最大化, 对含糊厌恶投资者的最优期望效用进行刻画. 利用动态规划原理, 建立最优消费和投资策略所满足的HJB方程. 再次, 利用市场分解的方法解出HJB方程, 获得投资者最优消费和投资策略的显式解. 最后, 通过数值模拟, 分析了含糊厌恶、风险厌恶、跳和通胀因素对投资者最优资产配置策略的影响.     

A note on portfolio selection with restrictions on leverage

This note focuses on a mean–variance asset allocation framework having restrictions on leverage, namely where investors are constrained either to hold funds in a risk-free asset (i.e. to lend) or to hold debt (i.e. to borrow). It is shown that the optimal portfolio in a constrained leverage situation will not have the same composition as the optimal portfolio in an unconstrained situation. We give formal justification for the intuitive notion that the more debt an investor is constrained to hold, the more the investor should tilt the remaining investments towards a portfolio of less risky assets. Conversely, the greater the proportion an investor is constrained to hold in a risk-free asset, the more the investor should tilt remaining investment towards a portfolio of more risky assets.     

Heston随机波动率市场中带VaR约束的最优投资策略

本文研究了Heston随机波动率市场下, 基于VaR约束下的动态最优投资组合问题。
假设Heston随机波动率市场由一个无风险资产和一个风险资产构成,投资者的目标为最大化其终端的期望效用。与此同时, 投资者将动态地评估其待选的投资组合的VaR风险,并将其控制在一个可接受的范围之内。本文在合理的假设下,使用动态规划的方法,来求解该问题的最优投资策略。在特定的参数范围内,利用数值方法计算出近似的最优投资策略和相应值函数, 并对结果进行了分析。     

Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market

This paper study an optimal time-consistent reinsurance-investment strategy selection problem in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process. The aggregate claim process and the price process of risky asset are correlated by a common Poisson process. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. Since this problem is time-inconsistent, we attack it by placing the problem within a game theoretic framework and looking for subgame perfect Nash equilibrium strategy. We investigate the problem using the extended Hamilton–Jacobi–Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained. Numerical examples and economic significance analysis are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary.     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

Optimal consumption and investment problem with random horizon in a BMAP model

In this paper, we consider the consumption and investment problem with random horizon in a Batch Markov Arrival Process (BMAP) model. The investor invests her wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the riskless asset and the risky asset are modulated by a continuous-time Markov chain, which is the phase process of a BMAP. The possible consumption or investment are restricted to a sequence of random discrete time points which are determined by the same BMAP. The investor has only consumption opportunities at some of these random time points, has both consumption and investment opportunities at some other random time points, and can do nothing at the remaining random time points. The object of the investor is to select the consumption–investment strategy that maximizes the expected total discounted utility. The purpose of this paper is to analyze the impact of the consumption–investment opportunity and the economic state on the value functions and consumption–investment strategies. The general solution and the exact solution under the assumption that the consumption and the terminal wealth are evaluated by the power utility are obtained. Finally, a numerical example is presented.