Least-squares Monte-Carlo methods for optimal stopping investment under CEV models

The optimal stopping investment is a kind of mixed expected utility maximization problems with optimal stopping time. The aim of this paper is to develop the least-squares Monte-Carlo methods to solve the optimal stopping investment under the constant elasticity of variance (CEV) model. Such a problem has no closed-form solutions for the value functions, optimal strategies and optimal exercise boundaries due to the early exercised feature. The dual optimal stopping problem is first derived and then the strong duality between the dual and prime problems is established. The least-squares Monte-Carlo methods based on the dual control theory are developed and numerical simulations are provided. Both the power and non-HARA utilities are studied.     

Investment strategies in the long run with proportional transaction costs and a HARA utility function

We consider an agent who invests in a stock and a money market in order to maximize the asymptotic behaviour of expected utility of the portfolio market price in the presence of proportional transaction costs. The assumption that the portfolio market price is a geometric Brownian motion and the restriction to a utility function with hyperbolic absolute risk aversion (HARA) enable us to evaluate interval investment strategies. It is shown that the optimal interval strategy is also optimal among a wide family of strategies and that it is optimal also in a time changed model in the case of logarithmic utility.     

Life insurance decisions under recursive utility

In this paper, we generalize recursive utility to include lifetime uncertainty and utility from bequest. The generalization applies to discrete-time as well as continuous-time recursive utility, and it is an important step forward in the development of recursive utility. We formalize the problem of optimal consumption, investment, and life insurance choice under recursive utility, and we state a verification theorem with a generalized Hamilton-Jacobi-Bellman equation. Our generalization of recursive utility allows us to study optimal consumption, investment, and life insurance choice under separation of (market) risk aversion, elasticity of inter-temporal substitution, and elasticity of substitution between bequest and future utility. The separation gives rise to hump-shaped consumption patterns as observed in realized consumption.     

Correspondence between lifetime minimum wealth and utility of consumption

We establish when the two problems of minimizing a function of lifetime minimum wealth and of maximizing utility of lifetime consumption result in the same optimal investment strategy on a given open interval O in wealth space. To answer this question, we equate the two investment strategies and show that if the individual consumes at the same rate in both problems—the consumption rate is a control in the problem of maximizing utility—then the investment strategies are equal only when the consumption function is linear in wealth on O, a rather surprising result. It then follows that the corresponding investment strategy is also linear in wealth and the implied utility function exhibits hyperbolic absolute risk aversion.      

Consumption–investment optimization with Epstein–Zin utility in incomplete markets

In a market with stochastic investment opportunities, we study an optimal consumption–investment problem for an agent with recursive utility of Epstein–Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The superdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein–Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein–Zin aggregator is neither Lipschitz nor jointly concave in all its variables.     

Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.     

Precautionary Saving and Consumption Smoothing across Time and Possibilities

This paper examines how aversion to risk and aversion to intertemporal substitution determine the strength of the precautionary saving motive in a two-period model with Selden/Kreps–Porteus preferences. For small risks, we derive a measure of the strength of the precautionary saving motive that generalizes the concept of "prudence" introduced by Kimball (1990b) . For large risks, we show that decreasing absolute risk aversion guarantees that the precautionary saving motive is stronger than risk aversion, regardless of the elasticity of intertemporal substitution. Holding risk preferences fixed, the extent to which the precautionary saving motive is stronger than risk aversion increases with the elasticity of intertemporal substitution. We derive sufficient conditions for a change in risk preferences alone to increase the strength of the precautionary saving motive and for the strength of the precautionary saving motive to decline with wealth. Within the class of constant elasticity of intertemporal substitution, constant-relative risk aversion utility functions, these conditions are also necessary.     

Mean-Variance Asset Liability Management with State-Dependent Risk Aversion

This article investigates the asset liability management problem with state-dependent risk aversion under the mean-variance criterion. The investor allocates the wealth among multiple assets including a risk-free asset and multiple risky assets governed by a system of geometric Brownian motion stochastic differential equations, and the investor faces the risk of paying uncontrollable random liabilities. The state-dependent risk aversion is taken into account in our model, linking the risk aversion to the current wealth held by the investor. An extended Hamilton-Jacobi-Bellman system is established for the optimization of asset liability management, and by solving the extended Hamilton-Jacobi-Bellman system, the analytical closed-form expressions for the time-inconsistent optimal investment strategies and the optimal value function are derived. Finally, numerical examples are presented to illustrate our results.     

On the Distributional Conditions for a Consumption-oriented Three Moment CAPM

In this paper, we develop sufficient conditions on probability distributions for a three moment (mean, variance, and skewness) consumption-oriented capital asset pricing model (CAPM) to price correctly a subset of assets. The assumptions that individuals in an allocationally efficient capital market have identical probability beliefs and monotone increasing strictly concave utility functions displaying nonincreasing absolute risk aversion imply an aggregate preference function that exhibits preference for expected return, aversion to variance of return, and preference for positive skewness. For otherwise arbitrary preferences, we show that quadratic characteristic lines are sufficient for a subset of assets to be priced according to a three moment consumption-oriented CAPM.     

The Spillover effect of compulsory insurance

The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are not—or are only partially—insurable, while others, such as civil liability or health and workers' injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard—that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing.     

Risk aversion, intertemporal substitution, and the aggregate investment-uncertainty relationship

We analyze the role of risk aversion and intertemporal substitution in a simple dynamic general equilibrium model of investment and savings. Our main finding is that risk aversion cannot by itself explain a negative relationship between aggregate investment and aggregate uncertainty, as the effect of increased uncertainty on investment also depends on the intertemporal elasticity of substitution. In particular, the relationship between aggregate investment and aggregate uncertainty is positive even if agents are very risk averse, as long as the elasticity of intertemporal substitution is low. A negative investment-uncertainty relationship requires that the relative risk aversion and the elasticity of intertemporal substitution are both relatively high or both relatively low. We also show that the implications of our model are consistent with the available empirical evidence.     

模糊厌恶下股指期货风险对冲策略设计及实证分析

投资者对股指期货与现货有着不同的模糊厌恶,本文首先将此假设条件引入带交易成本的Garleanu and Pederson (2013)投资模型中,并以指数基金对冲策略为例,构建了一个股指期货动态对冲的理论模型。与非对冲策略相比,基于上述模型设计的对冲策略投资绩效更好,动态最优成交额占目标交易额的比例更小,目标成交额对收益率预测因子的敏感性更大。借助上述模型,本文选取2010年4月至2021年6月的中国ETF指数基金和股指期货数据,并以2015年9月股指期货管理措施实施为界进行区间划分,实证研究发现:(1)中国A股市场的ETF投资组合进行股指期货对冲显著提升了投资绩效,但股指期货管理会削弱该作用;(2)投资绩效改善主要来源于交易成本的下降与目标成交额因子敏感性的提升,该机制受到股指期货管理的约束;(3)与Garleanu and Pederson (2013)、Zhang et al. (2017)相比,本文对冲策略保留“抗跌”特点的同时增加了“易涨”特性。本文研究结果表明,在当前大力发展机构投资者的背景下应不断丰富股指期货、股指期权产品谱系,降低股指期货交易成本并完善持仓约束。     

How sub-optimal are age-based life-cycle investment products?

We investigate the conditions under which life-cycle investment strategies based on age may be ‘near enough’ to optimal, focusing on the treatment of the pension account balance and assumptions about risk aversion. We show that dynamically adjusting the strategy in response to fluctuations in balance as well as age can lead to moderate improvements over product designs currently seen in the market; although most of the potential gains might be captured by specifying the glide path with reference to measures reflecting the projected balance over time. The risk aversion assumption emerges as a far more important consideration, with much greater reductions in expected utility arising from mismatches between the risk aversion of the investor and that underpinning the glide path design. Our analysis suggests possibilities for improving life-cycle or target date funds, and highlights the benefit of offering a suite of such funds that cater for members with differing risk aversion.     

Asset pricing in production economies with extrapolative expectations

Introducing extrapolative bias into a standard production-based model with recursive preferences reconciles salient stylized facts about business cycles (low consumption volatility, high investment volatility relative to output) and financial markets (high equity premium, volatile stock returns, low and smooth risk-free rate) with plausible levels of risk aversion and intertemporal elasticity of substitution. Furthermore, the model captures return predictability based upon dividend yield, Q, and investment. Intuitively, extrapolative bias increases the variation in the wealth–consumption ratio, which is heavily priced under recursive preferences; adjustment costs decrease the covariance between marginal utility and asset returns. Empirical support for key implications of the model is also provided.     

Pension Fund Design under Long-term Fairness Constraints

We consider optimal portfolio insurance for a mutually owned with-profits pension fund. First, intergenerational fairness is imposed by requiring that the pension fund is driven towards a steady state. Subject to this condition the optimal asset allocation is identified among the class of constant proportion portfolio insurance strategies by maximising expected power utility of the benefit. For a simple contract approximate analytical results are available and accurate, whereas for a more involved contract Monte Carlo methods must be applied to pick out the best design. The main insights are (i) aggressive investment strategies are disastrous for the clients; (ii) in most cases it is optimal to gear the bonus reserve; and (iii) the results are far less sensitive to the agent's risk aversion than in the classical case of Merton (1969), and as opposed to Merton horizon matters even with constant investment opportunities (because of the serial dependence between bonuses).     

Optimal Portfolios with Credit Default Swaps

Using a continuous-time, stochastic, and dynamic framework, this study derives a closed-form solution for the optimal investment problem for an agent with hyperbolic absolute risk aversion preferences for maximising the expected utility of his or her final wealth. The agent invests in a frictionless, complete market in which a riskless asset, a (defaultable) bond, and a credit default swap written on the bond are listed. The model is calibrated to market data of six European countries and assesses the behaviour of an investor exposed to different levels of sovereign risk. A numerical analysis shows that it is optimal to issue credit default swaps in a larger quantity than that of bonds, which are optimally purchased. This speculative strategy is more aggressive in countries characterised by higher sovereign risk. This result is confirmed when the investor is endowed with a different level of risk aversion. Finally, we solve a static version of the optimisation problem and show that the speculative/hedging strategy is definitely different with respect to the dynamic one.     

Robust portfolio choice with ambiguity and learning about return predictability

We analyze the optimal stock-bond portfolio under both learning and ambiguity aversion. Stock returns are predictable by an observable and an unobservable predictor, and the investor has to learn about the latter. Furthermore, the investor is ambiguity-averse and has a preference for investment strategies that are robust to model misspecifications. We derive a closed-form solution for the optimal robust investment strategy. We find that both learning and ambiguity aversion impact the level and structure of the optimal stock investment. Suboptimal strategies resulting either from not learning or from not considering ambiguity can lead to economically significant losses.     

Optimal growth portfolios reconciling theory and practice

Modern portfolio theory dictates diversification among assets that are not perfectly correlated (that is, asset diversification). Professional investors, on the other hand, contend that one can simply diversify across time (that is, time diversification). The controversy of time diversification versus asset diversification is examined in this article by empirically analyzing the optimum investment strategies for a myopic utility function (the extreme case that supports across asset diversification) under varying degrees of relative risk aversion. While Merton and Samuelson (1974) and Samuelson (1990) show that with a myopic utility function the investment diversification strategy does not change with an increase in the investment horizon, the recommendation of professional investors is also found to hold since for a wide range of relative risk-aversion measures, the optimum portfolio is shown to consist almost entirely of equities.     

Dynamic Asset Allocation under Inflation

We develop a simple framework for analyzing a finite-horizon investor's asset allocation problem under inflation when only nominal assets are available. The investor's optimal investment strategy and indirect utility are given in simple closed form. Hedge demands depend on the investor's horizon and risk aversion and on the maturities of the bonds included in the portfolio. When short positions are precluded, the optimal strategy consists of investments in cash, equity, and a single nominal bond with optimally chosen maturity. Both the optimal stock–bond mix and the optimal bond maturity depend on the investor's horizon and risk aversion.