HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.     

Optimal design of equity-linked products with a probabilistic constraint

There is a rich variety of tailored investment products available to the retail investor. These products combine upside participation in bull markets with downside protection in bear markets. Examples include the equity-linked products sold by insurance companies and the structured products marketed by banks. This paper examines a particular contract design for products of this nature. The paper finds the optimal design from the investor's viewpoint. It is assumed that the investor wishes to maximize expected utility of the terminal wealth subject to certain constraints. These constraints include a guaranteed rate of return as well as the opportunity to outperform a benchmark portfolio with a given probability. We derive the explicit form of the optimal design assuming both constraints apply and we illustrate the nature of the solution using some specific examples.     

A bootstrap-based comparison of portfolio insurance strategies

This study presents a systematic comparison of portfolio insurance strategies. We implement a bootstrap-based hypothesis test to assess statistical significance of the differences in a variety of downside-oriented risk and performance measures for pairs of portfolio insurance strategies. Our comparison of different strategies considers the following distinguishing characteristics: static versus dynamic protection; initial wealth versus cumulated wealth protection; model-based versus model-free protection; and strong floor compliance versus probabilistic floor compliance. Our results indicate that the classical portfolio insurance strategies synthetic put and constant proportion portfolio insurance (CPPI) provide superior downside protection compared to a simple stop-loss trading rule and also exhibit a higher risk-adjusted performance in many cases (dependent on the applied performance measure). Analyzing recently developed strategies, neither the TIPP strategy (as an ‘improved’ CPPI strategy) nor the dynamic VaR-strategy provides significant improvements over the more traditional portfolio insurance strategies.     

The 4% strategy revisited: a pre-commitment mean-variance optimal approach to wealth management

In contrast to single-period mean-variance (MV) portfolio allocation, multi-period MV optimal portfolio allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider multi-period MV optimal portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated between a risk-free investment and a risky asset, the price of which is assumed to follow a jump diffusion process. We consider two wealth management applications: optimal de-accumulation rates for a defined contribution pension plan and sustainable withdrawal rates for an endowment. Several numerical illustrations are provided, with some interesting implications. In the pension de-accumulation context, Bengen (1994)’s [J. Financial Planning, 1994, 7, 171–180], historical analysis indicated that a retiree could safely withdraw 4% of her initial retirement savings annually (in real terms), provided that her portfolio maintained an even balance between diversified equities and U.S. Treasury bonds. Our analysis does support 4% as a sustainable withdrawal rate in the pension de-accumulation context (and a somewhat lower rate for an endowment), but only if the investor follows an MV optimal portfolio allocation, not a fixed proportion strategy. Compared with a constant proportion strategy, the MV optimal policy achieves the same expected wealth at the end of the investment horizon, while significantly reducing the standard deviation of wealth and the probability of shortfall. We also explore the effects of suppressing jumps so as to have a pure diffusion process, but assuming a correspondingly larger volatility for the latter process. Surprisingly, it turns out that the MV optimal strategy is more effective when there are large downward jumps compared to having a high volatility diffusion process. Finally, tests based on historical data demonstrate that the MV optimal policy is quite robust to uncertainty about parameter estimates.     

Optimal Bond Portfolio for Investors with Long Time Horizons

We study the optimal bond portfolio for an investor with long time horizonusing Japanese interest rate data. A simple one-factor term structure modelis used for our numerical example. The optimal portfolio is computed using thetechnique of stochastic flows and Monte Carlo simulation. The hedgingportfolio is not negligible and the mean variance portfolio is very sensitiveto parameter values. The optimal portfolio is highly leveraged for a typicalparameter value. The investor holds a zero-coupon bond because of the lowerbound restriction on investor's wealth. The lower bound constraint may makethe optimal portfolio more realistic.     

When do jumps matter for portfolio optimization?

We consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on quantities known only in the true model. We apply our method to a calibrated affine model. Our findings are threefold: Jumps matter more, i.e. our approximation is less accurate, if (i) the expected jump size or (ii) the jump intensity is large. Fixing the average impact of jumps, we find that (iii) rare, but severe jumps matter more than frequent, but small jumps.     

Portfolio management with stochastic interest rates and inflation ambiguity

We solve, in closed form, a stock-bond-cash portfolio problem of a risk- and ambiguity-averse investor when interest rates and the inflation rate are stochastic. The expected inflation rate is unobservable, but the investor can learn about it from observing realized inflation and stock and bond prices. The investor is ambiguous about the inflation model and prefers a portfolio strategy which is robust to model misspecification. Ambiguity about the inflation dynamics is shown to affect the optimal portfolio fundamentally different than ambiguity about the price dynamics of traded assets, for example the optimal portfolio weights can be increasing in the degree of ambiguity aversion. In a numerical example, the optimal portfolio is significantly affected by the learning about expected inflation and somewhat affected by ambiguity aversion. The welfare loss from ignoring learning or ambiguity can be considerable.     

Dynamic preferences for popular investment strategies in pension funds

In this paper, we characterize dynamic investment strategies that are consistent with the expected utility setting and more generally with the forward utility setting. Two popular dynamic strategies in the pension funds industry are used to illustrate our results: a constant proportion portfolio insurance (CPPI) strategy and a life-cycle strategy. For the CPPI strategy, we are able to infer preferences of the pension fund’s manager from her investment strategy, and to exhibit the specific expected utility maximization that makes this strategy optimal at any given time horizon. In the Black–Scholes market with deterministic parameters, we are able to show that traditional life-cycle funds are not optimal to any expected utility maximizers. We also prove that a CPPI strategy is optimal for a fund manager with HARA utility function, while an investor with a SAHARA utility function will choose a time-decreasing allocation to risky assets in the same spirit as the life-cycle funds strategy. Finally, we suggest how to modify these strategies if the financial market follows a more general diffusion process than in the Black–Scholes market.     

Optimal asset allocation for participating contracts under the VaR and PI constraint

ABSTRACT

Participating contracts provide a maturity guarantee for the policyholder. However, the terminal payoff to the policyholder should be related to financial risks of participating insurance contracts. We investigate an optimal investment problem under a joint value-at-risk and portfolio insurance constraint faced by the insurer who offers participating contracts. The insurer aims to maximize the expected utility of the terminal payoff to the insurer. We adopt a concavification technique and a Lagrange dual method to solve the problem and derive the representations of the optimal wealth process and trading strategies. We also carry out some numerical analysis to show how the joint value-at-risk and the portfolio insurance constraint impacts the optimal terminal wealth.     

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.     

Dynamic Asset Allocation with Event Risk

Major events often trigger abrupt changes in stock prices and volatility. We study the implications of jumps in prices and volatility on investment strategies. Using the event-risk framework of Duffie, Pan, and Singleton (2000), we provide analytical solutions to the optimal portfolio problem. Event risk dramatically affects the optimal strategy. An investor facing event risk is less willing to take leveraged or short positions. The investor acts as if some portion of his wealth may become illiquid and the optimal strategy blends both dynamic and buy-and-hold strategies. Jumps in prices and volatility both have important effects.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

On the Optimality of Portfolio Insurance

This paper examines the optimality of an insurance strategy in which an investor buys a risky asset and a put on that asset. The put's striking price serves as the insurance level. In complete markets, it is highly unlikely that an investor would utilize such a strategy. However, in some types of less complete markets, an investor may wish to purchase a put on the risky asset. Given only a risky asset, a put, and noncontinuous trading, an investor would purchase a put as a way of introducing a risk-free asset into the portfolio. If, in addition, there is a risk-free asset and the investor's utility function displays constant proportional risk-aversion, then the investor would buy the risk-free asset directly and not buy a put. In sum, only under the most incomplete markets would an investor find an insurance strategy optimal.     

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.     

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.     

Asset allocation and liquidity breakdowns: what if your broker does not answer the phone?

This paper analyzes the portfolio decision of an investor facing the threat of illiquidity. In a continuous-time setting, the efficiency loss due to illiquidity is addressed and quantified. For a logarithmic investor, we solve the portfolio problem explicitly. We show that the efficiency loss for a logarithmic investor with 30 years until the investment horizon is a significant 22.7% of current wealth if the illiquidity part of the model is calibrated to the Japanese data of the aftermath of WWII. For general utility functions, an explicit solution does not seem to be available. However, under a mild growth condition on the utility function, we show that the value function of a model in which only finitely many liquidity breakdowns can occur converges uniformly to the value function of a model with infinitely many breakdowns if the number of possible breakdowns goes to infinity. Furthermore, we show how the optimal security demands of the model with finitely many breakdowns can be used to approximate the solution of the model with infinitely many breakdowns. These results are illustrated for an investor with a power utility function.     

Portfolio selection with mental accounts and delegation

Das et al. (2010) develop an elegant framework where an investor selects portfolios within mental accounts but ends up holding an aggregate portfolio on the mean-variance frontier. This investor directly allocates the wealth in each account among available assets. In practice, however, investors often delegate the task of allocating wealth among assets to portfolio managers who seek to beat certain benchmarks. Accordingly, we extend their framework to the case where the investor allocates the wealth in each account among portfolio managers. Our contribution is threefold. First, we provide an analytical characterization of the existence and composition of the optimal portfolios within accounts and the aggregate portfolio. Second, we present conditions under which such portfolios are not on the mean-variance frontier, and conditions under which they are. Third, we show that the aforementioned analytical characterization is also applicable within the framework of Das et al. and thus improves upon their numerical approach.     

Computing optimal rebalance frequency for log-optimal portfolios

Log-optimal investment portfolio is deemed to be impractical and cost-prohibitive due to inherent need for continuous rebalancing and significant overhead of trading cost. We study the question of how often a log-optimal portfolio should be rebalanced for any given finite investment horizon. We develop an analytical framework to compute the expected log of portfolio growth when a given discrete-time periodic rebalance frequency is used. For a certain class of portfolio assets, we compute the optimal rebalance frequency. We show that it is possible to improve investor log utility using this quasi-passive or hybrid rebalancing strategy. Simulation studies show that an investor shall gain significantly by rebalancing periodically in discrete time, overcoming the limitations of continuous rebalancing.     

Optimal portfolio allocation with higher moments

We model the risky asset as driven by a pure jump process, with non-trivial and tractable higher moments. We compute the optimal portfolio strategy of an investor with CRRA utility and study the sensitivity of the investment in the risky asset to the higher moments, as well as the resulting wealth loss from ignoring higher moments. We find that ignoring higher moments can lead to significant overinvestment in risky securities, especially when volatility is high.      

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.