随机通胀风险下基于半鞅理论的最优投资策略

本文在半鞅理论框架下,构建包括可交易风险资产、不可交易风险资产和未定权益的金融投资模型。在考虑随机通胀风险和获取部分市场信息的情形下,研究投资经理人终端真实净财富指数效用最大化问题。运用滤波理论、半鞅和倒向随机微分方程(BSDE)理论,求解带有随机通胀风险的最优投资策略和价值过程精确解。数值分析结果发现,可交易风险资产最优投资额随着预期通胀率的增加而减少,投资价值呈先增后减态势。当通胀波动率无限接近可交易风险资产名义价格波动率时,通胀风险可完全对冲,投资人会不断追加在可交易风险资产的投资额,以期实现终端真实净财富期望指数效用最大化。研究结果为金融市场的投资决策提供更加科学的理论参考。     

The role of index bonds in universal currency hedging

This study examines the demand for index bonds and their role in hedging risky asset returns against currency risks in a complete market where equity is not hedged against inflation risk. Avellaneda's uncertain volatility model with non-constant coefficients to describe equity price variation, forward price variation, index bond price variation and rate of inflation, together with Merton's intertemporal portfolio choice model, are utilized to enable an investor to choose an optimal portfolio consisting of equity, nominal bonds and index bonds when the rate of inflation is uncertain. A hedge ratio is universal if investors in different countries hedge against currency risk to the same extent. Three universal hedge ratios (UHRs) are defined with respect to the investor's total demand for index bonds, hedging risky asset returns (i.e. equity and nominal bonds) against currency risk, which are not held for hedging purposes. These UHRs are hedge positions in foreign index bond portfolios, stated as a fraction of the national market portfolio. At equilibrium all the three UHRs are comparable to Black's corrected equilibrium hedging ratio. The Cameron-Martin-Girsanov theorem is applied to show that the Radon-Nikodym derivative given under a P -martingale, the investor's exchange rate (product of the two currencies) is a martingale. Therefore the investors can agree on a common hedging strategy to trade exchange rate risk irrespective of investor nationality. This makes the choice of the measurement currency irrelevant and the hedge ratio universal without affecting their values.     

通胀风险下基于HARA效用的DC型养老金计划

通货膨胀是养老基金管理过程中最直接最重要的影响因素之一. 假设通胀风险由服从几何布朗运动的物价指数来度量, 且瞬时期望通货膨胀率由Ornstein-Uhlenbeck过程来驱动. 金融市场由n+1种可连续交易的风险资产所构成, 养老基金管理者期望研究和解决通胀风险环境下DC型养老基金在累积阶段的最优投资策略问题, 以最大化终端真实财富过程的期望效用. 双曲绝对风险厌恶(HARA)效用函数具有一般的效用框架, 包含幂效用、指数效用和对数效用作为特例. 假设投资者对风险的偏好程度满足HARA效用, 运用随机最优控制理论和Legendre变换方法得到了最优投资策略的显式表达式.     

Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks

In this paper, we investigate an optimal reinsurance and investment problem for an insurer whose surplus process is approximated by a drifted Brownian motion. Proportional reinsurance is to hedge the risk of insurance. Interest rate risk and inflation risk are considered. We suppose that the instantaneous nominal interest rate follows an Ornstein–Uhlenbeck process, and the inflation index is given by a generalized Fisher equation. To make the market complete, zero-coupon bonds and Treasury Inflation Protected Securities (TIPS) are included in the market. The financial market consists of cash, zero-coupon bond, TIPS and stock. We employ the stochastic dynamic programming to derive the closed-forms of the optimal reinsurance and investment strategies as well as the optimal utility function under the constant relative risk aversion (CRRA) utility maximization. Sensitivity analysis is given to show the economic behavior of the optimal strategies and optimal utility.     

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.     

通货膨胀时期的投资策略分析

目前,各个国家面临着不同程度的通贷膨胀.在此情况下,如何规避通货膨胀所带来的财富稀释是现阶段所有投资人追求的目标.本文从投资产品的价格出发给出了两阶段均值-方差投资问题的最优解.分析了交易费用、风险资产的期望回报率以及波动率对投资策略的影响.最后,我们依据实际算例分析为投资者提供指导,并且给出了政策建议.     

关于长期债券是长期投者无风险资产的研究

假设短期利率服从C IR模型,通过研究投资者如何选择长期债券和短期债券进行资产配置,发现保守的长期投资者在长期债券上投资的比例更大;投资者的投资期限越长,投资的长期债券越多;在进行债券投资时不存在市场时机的选择。对于长期投资者而言,可以为其提供长期稳定收益率的长期债券才是无风险的。     

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??In this paper, we investigate a robust optimal portfolio and reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries about uncertainty in model parameters. We assume that the AAI is allowed to purchase proportional reinsurance and invest his/her wealth in a financial market which consists of a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal expected power utility of terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategies are obtained.     

CIR框架下的投资组合效用微分博弈

建立了Cox-Ingersoll—Ross随机利率下的关于两个投资者的投资组合效用微分博弈模型．市场利率具有CIR动力,博弈双方存在唯一的损益函数,损益函数取决于投资者的投资组合财富．一方选择动态投资组合策略以最大化损益函数,而另一方则最小化损益函数．运用随机控制理论,在一般的效用函数下得到了基于效用的博弈双方的最优策略．特别考虑了常数相对风险厌恶情形,获得了显示的最优投资组合策略和博弈值．最后给出了数值例子和仿真结果以说明本文的结论．     

Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk

In this paper we investigate an optimal investment strategy for a defined-contribution (DC) pension plan member who is loss averse, pays close attention to inflation and longevity risks and requires a minimum performance at retirement. The member aims to maximize the expected S-shaped utility from the terminal wealth exceeding the minimum performance by investing her wealth in a financial market consisting of an indexed bond, a stock and a risk-free asset. We derive the optimal investment strategy in closed-form using the martingale approach. Our theoretical and numerical results reveal that the wealth proportion invested in each risky asset has a V-shaped pattern in the reference point level, while it always increases in the rising lifespan; with a positive correlation between salary and inflation risks, the presence of salary decreases the member’s investment in risky assets; the minimum performance helps to hedge the longevity risk by increasing her investment in risky assets.     

带有通胀风险的退休后最优投资-年金化时刻决策

研究了带通货膨胀的确定缴费养老计划退休后最优投资-年金化决策。假设通货膨胀过程是一个随机过程,建立了真实财富的波动过程。先相对固定年金化时刻,采取目标定位型模型,预设未来各时期的投资目标,利用贝尔曼优化原理,得到从退休时刻到相对固定年金化时刻之间的最优投资策略。接着建立了最优年金化时刻的评估标准,最优的年金化时刻使得年金化前后的累加消费折现均值得到最大。证明了在随机通货膨胀的假设下,传统的自然投资目标不存在;当随机通胀过程退化到确定过程时,求出了自然投资目标的显式表达式,并且在这两种情况下,分析了通胀情况对最优投资策略的影响。最后利用数值分析手段, 研究了通货膨胀、风险偏好、折现率对最优年金化时刻的影响。     

The role of longevity bonds in optimal portfolios

We study the optimal consumption and portfolio for an agent maximizing the expected utility of his intertemporal consumption in a financial market with: (i) a riskless asset, (ii) a stock, (iii) a bond as a derivative on the stochastic interest rate, and (iv) a longevity bond whose coupons are proportional to the population (stochastic) survival rate. With a force of mortality instantaneously uncorrelated with the interest rate (but not necessarily independent), we demonstrate that the wealth invested in the longevity bond must be taken from the ordinary bond and the riskless asset proportionally to the duration of the two bonds. This result is valid for both a complete and an incomplete financial market.     

CIR框架下的投资组合效用微分博弈(英文)

建立了Cox-Ingersoll-Ross随机利率下的关于两个投资者的投资组合效用微分博弈模型.市场利率具有CIR动力,博弈双方存在唯一的损益函数,损益函数取决于投资者的投资组合财富.一方选择动态投资组合策略以最大化损益函数,而另一方则最小化损益函数.运用随机控制理论,在一般的效用函数下得到了基于效用的博弈双方的最优策略.特别考虑了常数相对风险厌恶情形,获得了显示的最优投资组合策略和博弈值.最后给出了数值例子和仿真结果以说明本文的结论.     

Affine model of inflation-indexed derivatives and inflation risk premium

This paper proposes an affine-based approach which jointly captures the nominal interest rate, the real interest rate, and the inflation risk premium to price inflation-indexed derivatives, including zero-coupon inflation-indexed swaps, year-on-year inflation-indexed swaps, inflation-indexed swaptions, and inflation-indexed caps and floors. We provide an example and explain how to use traded zero-coupon inflation-indexed swap rates to estimate inflation risk premiums.     

Robust optimal reinsurance and investment strategies for an AAI with multiple risks

This paper studies the robust optimal reinsurance and investment problem for an ambiguity averse insurer (abbr. AAI). The AAI sells insurance contracts and has access to proportional reinsurance business. The AAI can invest in a financial market consisting of four assets: one risk-free asset, one bond, one inflation protected bond and one stock, and has different levels of ambiguity aversions towards the risks. The goal of the AAI is to seek the robust optimal reinsurance and investment strategies under the worst case scenario. Here, the nominal interest rate is characterized by the Vasicek model; the inflation index is introduced according to the Fisher’s equation; and the stock price is driven by the Heston’s stochastic volatility model. The explicit forms of the robust optimal strategies and value function are derived by introducing an auxiliary robust optimal control problem and stochastic dynamic programming method. In the end of this paper, a detailed sensitivity analysis is presented to show the effects of market parameters on the robust optimal reinsurance policy, the robust optimal investment strategy and the utility loss when ignoring ambiguity.     

Vasicek利率模型下带有零息票债券的投资-消费模型

应用随机最优控制理论研究Vasicek利率模型下的投资-消费问题,其中假设无风险利率是服从Vasicek利率模型的随机过程,且与股票价格过程存在一般相关性.假设金融市场由一种无风险资产、一种风险资产和一种零息票债券所构成,投资者的目标是最大化中期消费与终端财富的期望贴现效用.应用变量替换方法得到了幂效用下最优投资-消费策略的显示表达式,并分析了最优投资-消费策略对市场参数的灵敏度.     

Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives

For many years, the longevity risk of individuals has been underestimated, as survival probabilities have improved across the developed world. The uncertainty and volatility of future longevity has posed significant risk issues for both individuals and product providers of annuities and pensions. This paper investigates the effectiveness of static hedging strategies for longevity risk management using longevity bonds and derivatives (q-forwards) for the retail products: life annuity, deferred life annuity, indexed life annuity, and variable annuity with guaranteed lifetime benefits. Improved market and mortality models are developed for the underlying risks in annuities. The market model is a regime-switching vector error correction model for GDP, inflation, interest rates, and share prices. The mortality model is a discrete-time logit model for mortality rates with age dependence. Models were estimated using Australian data. The basis risk between annuitant portfolios and population mortality was based on UK experience. Results show that static hedging using q-forwards or longevity bonds reduces the longevity risk substantially for life annuities, but significantly less for deferred annuities. For inflation-indexed annuities, static hedging of longevity is less effective because of the inflation risk. Variable annuities provide limited longevity protection compared to life annuities and indexed annuities, and as a result longevity risk hedging adds little value for these products.     

Optimal consumption and portfolio under inflation and Markovian switching

In this paper, a financial market with Markovian switching and inflation is described, and the problem of maximizing expected utility of consumption discounted by inflation is given. Then, by a generalized Itô formula for Markov-modulated processes, the verification theorems of optimal policies are derived. Finally, the optimal consumption and portfolio policies for the constant relative risk aversion utility are given explicitly, and an economic analysis is made by numerical examples.     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Unhedgeable inflation risk within pension schemes

Pension schemes generally aim to protect the purchasing power of their participants, but cannot completely do this when due to market incompleteness inflation risk cannot be fully hedged. Without a market price for inflation risk the value of a pension contract depends on the investor’s risk appetite and inflation risk exposure. We develop a valuation framework to deal with two sources of unhedgeable inflation risk: the absence of instruments to hedge general consumer price inflation risk and differences in group-specific consumption bundles from the economy-wide bundle. We find that the absence of financial instruments to hedge inflation risks may reduce lifetime welfare by up to 6% of certainty-equivalent consumption for commonly assumed degrees of risk aversion. Regulators face a dilemma as young (workers) and old participants (retirees) have different capacities to absorb losses from unhedgeable inflation risks and as a consequence have a different risk appetite.