Optimal Investment and Consumption with Default Risk: HARA Utility

In this paper, we consider a portfolio optimization problem in a defaultable market. The representative investor dynamically allocates his or her wealth among the following securities: a perpetual defaultable bond, a money market account and a default-free risky asset. The optimal investment and consumption policies that maximize the infinite horizon expected discounted HARA utility of the consumption are explicitly derived. Moreover, numerical illustrations are also presented.     

Robust portfolio choice with stochastic interest rates

We determine the optimal investment strategy for an ambiguity-averse investor in a setting with stochastic interest rates. The investor has access to stocks, bonds, and a bank account and he is ambiguous about the expected rate of return of both bonds and stocks. The investor can have different levels of ambiguity aversion about the two types of risky assets. We find that it is more important to take model uncertainty about the stock dynamics than model uncertainty about the bond dynamics into account. Furthermore, the investor’s ambiguity increases his hedging demand. Consequently, the bond/stock ratio increases with his ambiguity and implies less extreme positions in the bank account. Altogether, our model yields portfolio allocations which are more in line with what is implementable in practice. Finally, we demonstrate that neglecting model uncertainty implies significant losses for the investor.     

Multiperiod Optimal Investment-Consumption Strategies with Mortality Risk and Environment Uncertainty

Abstract

In this article we investigate three related investment-consumption problems for a risk-averse investor: (1) an investment-only problem that involves utility from only terminal wealth, (2) an investment-consumption problem that involves utility from only consumption, and (3) an extended investment-consumption problem that involves utility from both consumption and terminal wealth. Although these problems have been studied quite extensively in continuous-time frameworks, we focus on discrete time. Our contributions are (1) to model these investmentconsumption problems using a discrete model that incorporates the environment risk and mortality risk, in addition to the market risk that is typically considered, and (2) to derive explicit expressions of the optimal investment-consumption strategies to these modeled problems. Furthermore, economic implications of our results are presented. It is reassuring that many of our findings are consistent with the well-known results from the continuous-time models, even though our models have the additional features of modeling the environment uncertainty and the uncertain exit time.     

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.     

How do investors react under uncertainty?

It has long been accepted that risk plays an important role in determining valuation where risk reflects that investors are unsure of future returns but are able to express their prior expectations by a probability distribution of these returns. Knight (1921) introduced the concept of uncertainty where investors possess incomplete knowledge about this distribution and so are unable to formulate priors over all possible outcomes. One common approach for making uncertainty tractable is to assume that investors faced with uncertainty will base their decisions on the worst case scenario (i.e. follow maxmin expected utility). As a consequence it is postulated that investors will become more pessimistic as uncertainty increases, upgrading bad news and downgrading good news. Using Australian data, we find evidence that investors react to bad news at times of high market uncertainty but largely ignore good news which is consistent with them taking on a pessimistic bias. However, we also find evidence of the reverse when market uncertainty is low with investors taking on an optimistic stance by ignoring bad news but reacting to good news. We also find that the impact that market uncertainty has on the reaction of investors to new information is modified by the prevailing market sentiment at the time of the announcement. Besides throwing light on the question of how uncertainty impacts on investor behaviour, our findings seriously challenge the common assumption made that investors consistently deal with uncertainty by applying maxmin expected utility.     

Uncertainty,Time‐Varying Fear,and Asset Prices

I construct an equilibrium model that captures salient properties of index option prices, equity returns, variance, and the risk‐free rate. A representative investor makes consumption and portfolio choice decisions that are robust to his uncertainty about the true economic model. He pays a large premium for index options because they hedge important model misspecification concerns, particularly concerning jump shocks to cash flow growth and volatility. A calibration shows that empirically consistent fundamentals and reasonable model uncertainty explain option prices and the variance premium. Time variation in uncertainty generates variance premium fluctuations, helping explain their power to predict stock returns.     

Welfare effects of information and rationality in portfolio decisions under parameter uncertainty

We analyse and quantify, in a financial market with parameter uncertainty and for a Constant Relative Risk Aversion investor, the utility effects of two different boundedly rational (i.e. sub-optimal) investment strategies (namely, myopic and unconditional strategies) and compare them with each other and with the utility effect of full information. We show that effects are mainly caused by full information and predictability, being the effect of learning marginal. We also investigate the saver's decision regarding whether to manage her/his portfolio personally (DIY investor) or hire, against the payment of a management fee, a professional investor and find that delegation is mainly motivated by the belief that professional advisors are, depending on investment horizon and risk aversion, either better informed (‘insiders’) or more capable of gathering and processing information, rather than possessing the ability to learn from financial data. In particular, for very short investment horizons, delegation is primarily, if not exclusively, motivated by the beliefs that professional investors are better informed.     

Investor sentiment and management earnings forecast bias

This study investigates whether investor sentiment is associated with behavioral bias in managers’ annual earnings forecasts that are generally issued early in the year when uncertainty is relatively high. I provide evidence that management earnings forecast optimism increases with investor sentiment. Furthermore, I find that managers’ annual earnings forecasts are more pessimistic during low‐sentiment periods than during normal‐sentiment periods. Since managers lack incentives to further deflate stock prices during a low‐sentiment period, this evidence indicates that sentiment‐related management earnings forecast bias is likely to be unintentional. In addition, I find that the relationship between management earnings forecast bias and investor sentiment is stronger for firms with higher uncertainty, consistent with investor sentiment having a greater influence on management earnings forecasts when uncertainty is higher.     

Portfolio implications of systemic crises

Systemic crises can have grave consequences for investors in international equity markets, because they cause the risk-return trade-off to deteriorate severely for a longer period. We propose a novel approach to include the possibility of systemic crises in asset allocation decisions. By combining regime switching models with Merton [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247–257]-style portfolio construction, our approach captures persistence of crises much better than existing models. Our analysis shows that incorporating systemic crises greatly affects asset allocation decisions, while the costs of ignoring them is substantial. For an expected utility maximizing US investor, who can invest globally these costs range from 1.13% per year of his initial wealth when he has no prior information on the likelihood of a crisis, to over 3% per month if a crisis occurs with almost certainty. If a crisis is taken into account, the investor allocates less to risky assets, and particularly less to the crisis prone emerging markets.     

Robust Portfolio Rules and Asset Pricing

I present a new approach to the dynamic portfolio and consumptionproblem of an investor who worries about model uncertainty (inaddition to market risk) and seeks robust decisions along thelines of Anderson, Hansen, and Sargent (2002). In accordancewith max-min expected utility, a robust investor insures againstsome endogenous worst case. I first show that robustness dramaticallydecreases the demand for equities and is observationally equivalentto recursive preferences when removing wealth effects. Unlikestandard recursive preferences, however, robustness leads toenvironment-specific "effective" risk aversion. As an extension,I present a closed-form solution for the portfolio problem ofa robust Duffie-Epstein-Zin investor. Finally, robustness increasesthe equilibrium equity premium and lowers the risk-free rate.Reasonable parameters generate a 4% to 6% equity premium.     

A dynamic equilibrium model for U-shaped pricing kernels

This paper proposes a dynamic equilibrium model that can provide a unified explanation for the stylized facts observed in stock index markets such as the fat tails of the risk-neutral return distribution relative to the physical distribution, negative expected returns on deep OTM call options and negative realized variance risk premiums. In particular, we focus on the U-shaped pricing kernel against the stock index return, which is closely related to the negative call returns. We assume that the stock index return follows a time-changed Lévy process and that a representative investor has power utility over the aggregate consumption that forms a linear regression of the stock index return and its stochastic activity rate. This model offers a macroeconomic interpretation of the stylized facts from the perspective of the sensitivity of the activity rate and stock index return on aggregate consumption as well as the investor’s risk aversion.     

Minimum option prices under decreasing absolute risk aversion

We establish bounds on option prices in an economy where the representative investor has an unknown utility function that is constrained to belong to the family of nonincreasing absolute risk averse functions. For any distribution of terminal consumption, we identify a procedure that establishes the lower bound of option prices. We prove that the lower bound derives from a particular negative exponential utility function. We also identify lower bounds of option prices in a decreasing relative risk averse economy. For this case, we find that the lower bound is determined by a power utility function. Similar to other recent findings, for the latter case, we confirm that under lognormality of consumption, the Black Scholes price is a lower bound. The main advantage of our bounding methodology is that it can be applied to any arbitrary marginal distribution for consumption. This revised version was published online in June 2006 with corrections to the Cover Date.     

Long‐Run Consumption Risk and Asset Allocation under Recursive Utility and Rational Inattention

We study the portfolio decision of a household with limited information‐processing capacity (rational inattention [RI]) in a setting with recursive utility. We find that RI combined with a preference for early resolution of uncertainty could lead to a significant drop in the share of portfolios held in risky assets, even when the departure from the standard expected utility setting with full‐information rational expectations is small. In addition, we show that the equilibrium equity premium increases with the degree of inattention because inattentive investors with recursive utility face greater long‐run risk and thus require higher compensation in equilibrium.     

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 investment and consumption under a continuous-time cointegration model with exponential utility

In this paper, we study the effects of cointegration on optimal investment and consumption strategies for an investor with exponential utility. A Hamilton-Jacobi-Bellman (HJB) equation is derived first and then solved analytically. Both the optimal investment and consumption strategies are expressed in closed form. A verification theorem is also established to demonstrate that the solution of the HJB equation is indeed the solution of the original optimization problem under an integrability condition. In addition, a simple and sufficient condition is proposed to ensure that the integrability condition is satisfied. Financially, the optimal investment and consumption strategies are decomposed into two parts: the myopic part and the hedging demand caused by cointegration. Discussions on the hedging demand are carried out first, based on analytical formulae. Then numerical results show that ignoring the information about cointegration results in a utility loss.     

Inflation Uncertainty, Real-Interest-Rate Uncertainty, and the Liquidity Premium on Government Bonds

This paper shows that the components of uncertainty about nominal interest rates, real-rate uncertainty and inflation uncertainty, have different effects on the liquidity premium. An increase in inflation uncertainty should increase the equilibrium liquidity premium because investors reduce the effect of inflation uncertainty on the riskiness of their portfolios by holding more short-term bonds. In contrast, an investor can reduce the effects of uncertainty about future ex-ante real rates on portfolio return by matching more closely the maturity dates of the bonds held with the date on which the portfolio is to be liquidated for consumption purposes. Thus, the effect of an increase in real-rate uncertainty on the equilibrium liquidity premium is ambiguous, depending on the relative magnitudes of long-term and short-term saving and the proportions of short-term and long-term bonds issued by the government.     

Optimal dynamic hedging in incomplete futures markets

This article derives optimal hedging demands for futures contracts from an investor who cannot freely trade his portfolio of primitive assets in the context of either a CARA or a logarithmic utility function. Existing futures contracts are not numerous enough to complete the market. In addition, in the case of CARA, the nonnegativity constraint on wealth is binding, and the optimal hedging demands are not identical to those that would be derived if the constraint were ignored. Fictitiously completing the market, we can characterize the optimal hedging demands for futures contracts. Closed-form solutions exist in the logarithmic case but not in the CARA case, since then a put (insurance) written on his wealth is implicitly bought by the investor. Although solutions are formally similar to those that obtain under complete markets, incompleteness leads in fact to second-best optima.     

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.     

Staging of Venture Financing, Investor Opportunism and Patent Law

Abstract:  Stage financing provides a real option that is valuable when facing external uncertainty. However, it may also induce investor hold-up, if the property rights on an invention are not sufficiently protected. As a consequence, the entrepreneur may not work hard. Investor opportunism is less likely to occur, if investors' residual cash-flow-rights are contingent on verifiable 'milestones' in the previous stage. Equity-linked financing also provides high-powered incentives to the investor not to 'steal the idea' because his payoff becomes sensitive to the project value. The paper provides a new explanation for both types of contractual provisions.