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.     

Lifetime asset allocation with idiosyncratic and systematic mortality risks

This paper considers a lifetime asset allocation problem with both idiosyncratic and systematic mortality risks. The novelty of the paper is to integrate stochastic mortality, stochastic interest rate and stochastic income into a unified framework. An investor, who is a wage earner receiving a stochastic income, can invest in a financial market, consume part of his wealth and purchase life insurance or annuity so as to maximize the expected utility from consumption, terminal wealth and bequest. The problem is solved via the dynamic programming principle and the Hamilton–Jacobi–Bellman equation. Analytical solutions to the problem are derived, and numerical examples are provided to illustrate our results. It is shown that idiosyncratic mortality risk has significant impacts on the investor’s investment, consumption, life insurance/annuity purchase and bequest decisions regardless of the length of the decision-making horizon. The systematic mortality risk is largely alleviated by trading the longevity bond. However, its impacts on consumption, purchase of life insurance/annuity and bequest as well as the value function are still pronounced, when the decision-making horizon is sufficiently long.     

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.     

Portfolio Choice and Life Insurance: The CRRA Case

We solve a portfolio choice problem that includes life insurance and labor income under constant relative risk aversion (CRRA) preferences. We focus on the correlation between the dynamics of human capital and financial capital and model the utility of the family as opposed to separating consumption and bequest. We simplify the underlying Hamilton–Jacobi–Bellman equation using a similarity reduction technique that leads to an efficient numerical solution. Households for whom shocks to human capital are negatively correlated with shocks to financial capital should own more life insurance with greater equity/stock exposure. Life insurance hedges human capital and is insensitive to the family's risk aversion, consistent with practitioner guidance.     

One-Period Model of Individual Consumption, Life Insurance, and Investment Decisions

This article studies individuals' optimal decisions on consumption, life insurance, and stock purchases in a one‐period framework. With exponential utility functions, individuals' life insurance and stock purchases are independent of each other; life insurance purchases are affected only by individuals' future income, bequest intensity, risk attitude, survival probability, and the insurance risk premium; stock purchases are affected only by individuals' risk attitude, the risk‐free rate of return, the stock return, and stock volatility. With power utility functions, life insurance and stock purchases are positively related with each other and are affected by all the factors.     

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.     

Long-Term Care Insurance and Life Insurance Demand

This article investigates the interaction between life insurance and long-term care insurance markets on the demand side. In the model utility depends on both consumption and bequest, and utility from consumption is contingent on the state of health. While the demand for life insurance increases both with decreasing income and with a rising degree of altruism, the influences of these two parameters on the demand for long-term care insurance are ambiguous. If the utility shock arising from disability declines, both insurance demands will rise.     

Can the Life Insurance Market Provide Evidence for a Bequest Motive?

Using U.K. microeconomic data, we analyze the empirical determinants of participation in the life insurance market. We find that term insurance demand is positively correlated with measures of bequest motives like being married, having children, and/or subjective measures of strong bequest motives. We then show that a life‐cycle model of life insurance demand, saving, and portfolio choice can rationalize quantitatively the data in the presence of a bequest motive. These findings provide evidence supporting the presence of a bequest motive.     

Endogenous Life‐Cycle Housing Investment and Portfolio Allocation

This paper develops a life‐cycle portfolio allocation model to address the effects of housing investment on the portfolio allocation of households. The model employs a comprehensive housing investment structure, Epstein–Zin recursive preferences, and a stock market entry cost. Furthermore, rather than resorting to calibration we estimate the value of the relative risk aversion and elasticity of intertemporal substitution. The model shows that housing investment has a strong crowding out effect on investment in risky assets throughout the life‐cycle. We further find that the effect of the presence of housing investment on households portfolio allocation is larger than the effect of having EZ recursive preferences.     

Optimal life insurance with no-borrowing constraints: duality approach and example

We solve an optimal portfolio choice problem under a no-borrowing assumption. A duality approach is applied to study a family’s optimal consumption, optimal portfolio choice, and optimal life insurance purchase when the family receives labor income that may be terminated due to the wage earner’s premature death or retirement. We establish the existence of an optimal solution to the optimization problem theoretically by the duality approach and we provide an explicitly solved example with numerical illustration. Our results illustrate that the no-borrowing constraints do not always impact the family’s optimal decisions on consumption, portfolio choice, and life insurance. When the constraints are binding, there must exist a wealth depletion time (WDT) prior to the retirement date, and the constraints indeed reduce the optimal consumption and the life insurance purchase at the beginning of time. However, the optimal consumption under the constraints will become larger than that without the constraints at some time later than the WDT.     

“Skewness and Stock Option Prices”, Hans Gerber and Bruno Landry,July 1997

We determine the optimal amount of life insurance for a household of two wage earners. We consider the simple case of exponential utility, thereby removing wealth as a factor in buying life insurance, while retaining the relationship among life insurance, income, and the probability of dying and thus losing that income. For insurance purchased via a single premium or premium payable continuously, we explicitly determine the optimal death benefit. We show that if the premium is determined to target a specific probability of loss per policy, then the rates of consumption are identical under single premium or continuously payable premium. Thus, not only is equivalence of consumption achieved for the households under the two premium schemes, it is also obtained for the insurance company in the sense of equivalence of loss probabilities.     

Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin

Abstract

I study the problem of how individuals should invest their wealth in a risky financial market to minimize the probability that they outlive their wealth, also known as the probability of lifetime ruin. Specifically, I determine the optimal investment strategy of an individual who targets a given rate of consumption and seeks to minimize the probability of lifetime ruin. Two forms of the consumption function are considered: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of his or her wealth. The first is arguably more realistic, but the second has a close connection with optimal consumption in Merton’s model of optimal consumption and investment under power utility.

For constant force of mortality, I determine (a) the probability that individuals outlive their wealth if they follow the optimal investment strategy; (b) the corresponding optimal investment rule that tells individuals how much money to invest in the risky asset for a given wealth level; (c) comparative statics for the functions in (a) and (b); (d) the distribution of the time of lifetime ruin, given that ruin occurs; and (e) the distribution of bequest, given that ruin does not occur. I also include numerical examples to illustrate how the formulas developed in this paper might be applied.     

Target-Bequest Investment and Insurance Fund

We propose a new type of investment fund, a Target-Bequest Fund (TBF), for which the manager of the fund invests to maximize the probability of reaching a bequest goal, specified by the investor. We assume the fund pays dividends at a rate proportional to the value of the fund, with the proportion also specified by the investor. In addition to considering this basic fund, we propose two extensions. The first extension is to impose a no-borrowing constraint. Indeed, unless the investment fund is a hedge fund, it will likely not allow the manager to invest more in the risky asset than is available in the fund, so this constraint is a realistic one. The second extension is to allow the fund manager to buy life insurance to help reach the bequest goal. We consider both extensions in the special case for which the proportional dividend rate is less than or equal to the riskless rate of return. Our focus is to obtain explicit solutions in a simplified market and insurance setting to give actuaries implementing this fund design some rules-of-thumb for investing in a financial market and buying life insurance in more realistic settings. We fully expect these rules-of-thumb to be generally valid in those more realistic settings.     

Optimal consumption,portfolio and life insurance rules for an uncertain lived individual in a continuous time model

A continuous time model for optimal consumption, portfolio and life insurance rules, for an investor with an arbitrary but known distribution of lifetime, is derived as a generalization of the model by Merton (1971). The classic Tobin-Markowitz separation theorem obtains with the mutual funds being identical to those obtained under the assumption of certain lifetime. The investor is found to have a ‘human capital’ component of wealth, which is independent of his preferences and risky market opportunities and represents the certainty equivalent of his future net (wage) earnings. Explicit solutions, which are linear in wealth, are found for the investor with constant relative risk aversion.     

Optimal Gradual Annuitization: Quantifying the Costs of Switching to Annuities

We compute the optimal dynamic annuitization and asset allocation policy for a retiree with Epstein–Zin preferences, uncertain investment horizon, potential bequest motives, and pre‐existing pension income. In our setting the retiree can decide each year how much he consumes and how much he invests in stocks, bonds, and life annuities, while the prior literature mostly considered restricted so‐called deterministic or stochastic switching strategies. We show that postponing the annuity purchase is no longer optimal in the gradual annuitization (GA) case since investors are able to attain the optimal mix between liquid assets (stocks and bonds) and illiquid life annuities each year. In order to assess potential utility losses, we benchmark various restricted annuitization strategies against the unrestricted GA strategy.     

Inventories and optimal monetary policy in a small open economy

We study how inventory investment affects the design of optimal monetary policy in a New Keynesian small open economy model. We find that under producer currency pricing, when the intratemporal elasticity of substitution is smaller than 1, optimal monetary policy in our model with inventories is similar to a standard model without inventories. However, when the intratemporal elasticity of substitution is larger than 1, inventory investment increases the importance of nominal exchange rate stabilization relative to a standard model without inventories. The importance of nominal exchange rate stabilization increases with the intratemporal elasticity of substitution.     

Consumption and Portfolio Choice over the Life Cycle

This article solves a realistically calibrated life cycle modelof consumption and portfolio choice with non-tradable laborincome and borrowing constraints. Since labor income substitutesfor riskless asset holdings, the optimal share invested in equitiesis roughly decreasing over life. We compute a measure of theimportance of human capital for investment behavior. We findthat ignoring labor income generates large utility costs, whilethe cost of ignoring only its risk is an order of magnitudesmaller, except when we allow for a disastrous labor incomeshock. Moreover, we study the implications of introducing endogenousborrowing constraints in this incomplete-markets setting.     

Optimal consumption and investment with Epstein–Zin recursive utility

We study continuous-time optimal consumption and investment with Epstein–Zin recursive preferences in incomplete markets. We develop a novel approach that rigorously constructs the solution of the associated Hamilton–Jacobi–Bellman equation by a fixed point argument and makes it possible to compute both the indirect utility and, more importantly, optimal strategies. Based on these results, we also establish a fast and accurate method for numerical computations. Our setting is not restricted to affine asset price dynamics; we only require boundedness of the underlying model coefficients.     

Multivariate utility maximization with proportional transaction costs

We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor’s preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for this include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.     

Optimal investment strategies for general utilities under dynamic elasticity of variance models

This paper studies the optimal investment strategies under the dynamic elasticity of variance (DEV) model which maximize the expected utility of terminal wealth. The DEV model is an extension of the constant elasticity of variance model, in which the volatility term is a power function of stock prices with the power being a nonparametric time function. It is not possible to find the explicit solution to the utility maximization problem under the DEV model. In this paper, a dual-control Monte-Carlo method is developed to compute the optimal investment strategies for a variety of utility functions, including power, non-hyperbolic absolute risk aversion and symmetric asymptotic hyperbolic absolute risk aversion utilities. Numerical examples show that this dual-control Monte-Carlo method is quite efficient.