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.     

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.     

Efficient fund of hedge funds construction under downside risk measures

We consider portfolio allocation in which the underlying investment instruments are hedge funds. We consider a family of utility functions involving the probability of outperforming a benchmark and expected regret relative to another benchmark. Non-normal return vectors with prescribed marginal distributions and correlation structure are modeled and simulated using the normal-to-anything method. A Monte Carlo procedure is used to obtain, and establish the quality of, a solution to the associated portfolio optimization model. Computational results are presented on a problem in which we construct a fund of 13 CSFB/Tremont hedge-fund indices.     

Asset allocation: How much does model choice matter?

This paper analyzes the optimal portfolio decision of a CRRA investor in models with stochastic volatility and stochastic jumps. The investor follows a buy-and-hold strategy in the stock, the money market account, and one additional derivative. We show that both the type of the model and the structure of the risk premia have a significant impact on the optimal portfolio, on the utility gain from having access to derivatives, and on whether the investor prefers to trade OTM or ATM options. We also show that model mis-specification results in significant utility losses. Omitting jumps in volatility can be devastating, in particular if the investor chooses the seemingly optimal OTM put options. A misestimation of the structure of the risk premia has a less devastating effect, but can still lead to a loss of around 4% in the annual certainty equivalent return.     

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.     

Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility

Given an investor maximizing utility from terminal wealth with respect to a power utility function, we present a verification result for portfolio problems with stochastic volatility. Applying this result, we solve the portfolio problem for Heston's stochastic volatility model. We find that only under a specific condition on the model parameters does the problem possess a unique solution leading to a partial equilibrium. Finally, it is demonstrated that the results critically hinge upon the specification of the market price of risk. We conclude that, in applications, one has to be very careful when exogenously specifying the form of the market price of risk.     

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.     

International asset allocation using the market implied cost of capital

The Black and Litterman (Financ Anal J 48(5):28–43, 1992) (BL) approach to portfolio optimization requires investor views on expected asset returns as an input. I demonstrate that the market implied cost of capital (ICC) is ideal for quantifying those views on a country level. I benchmark this approach against a BL optimization using time-series models as investor views, the equally weighted portfolio, and allocation methods based on stock market capitalization and GDP. I find that the ICC portfolio offers an increase in average return of 2.1 percentage points (yearly) as compared to the value-weighted portfolio, while having a similar standard deviation. The resulting difference in Sharpe ratios is statistically significant and robust to the inclusion of transaction costs, varying BL parameters, and a less strictly defined investment universe.     

Minimizing the lifetime ruin under borrowing and short-selling constraints

In this paper, the optimal investment strategies for minimizing the probability of lifetime ruin under borrowing and short-selling constraints are found. The investment portfolio consists of multiple risky investments and a riskless investment. The investor withdraws money from the portfolio at a constant rate proportional to the portfolio value. In order to find the results, an auxiliary market is constructed, and the techniques of stochastic optimal control are used. Via this method, we show how the application of stochastic optimal control is possible for minimizing the probability of lifetime ruin problem defined under an auxiliary market.     

“KLICing” there and back again: Portfolio selection using the empirical likelihood divergence and Hellinger distance

Stutzer (2000, 2003) proposes the decay-rate maximizing portfolio selection rule wherein the investor selects the asset mix that maximizes the rate at which the probability of shortfall decays to zero. A close examination of this rule reveals that it ranks portfolios by computing the divergence, in the Kullback-Leibler sense, between the unweighted portfolio return distribution and a tilted distribution meaned at the predetermined target or benchmark rate of return selected by or imposed upon the investor. This result implies, in the IID case, that Stutzer's rules can be written as a benchmark constrained Kullback-Leibler-based optimization problem with an endogenous utility interpretation. Here we expand on this idea by introducing two closely related portfolio selection rules based on the empirical likelihood divergence and the Hellinger-Matusita distance. The first of these is the reversed Kullback-Leibler divergence and the second is proportional to the average of the two divergences. The theoretical and in-sample properties of the new criteria suggest them to be competitive with and in some cases better than existing methods, especially in terms of skewness preference.     

Portfolio optimization with stochastic dominance constraints

We consider the problem of constructing a portfolio of finitely many assets whose return rates are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return rate. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.     

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.     

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.     

Optimal payoffs under state-dependent preferences

Most decision theories, including expected utility theory, rank-dependent utility theory and cumulative prospect theory, assume that investors are only interested in the distribution of returns and not in the states of the economy in which income is received. Optimal payoffs have their lowest outcomes when the economy is in a downturn, and this feature is often at odds with the needs of many investors. We introduce a framework for portfolio selection within which state-dependent preferences can be accommodated. Specifically, we assume that investors care about the distribution of final wealth and its interaction with some benchmark. In this context, we are able to characterize optimal payoffs in explicit form. Furthermore, we extend the classical expected utility optimization problem of Merton to the state-dependent situation. Some applications in security design are discussed in detail and we also solve some stochastic extensions of the target probability optimization problem.     

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.     

Smooth investment

In the classical portfolio optimization problem considered by Merton, the resulting constant proportion investment plan requires a diffusive trading strategy. This means that, within any arbitrarily small time interval, the investor must impractically both buy and sell stocks. We study the problems of a mean-square and a power utility investor for whom the trading strategy is constrained to be smooth, i.e. nondiffusive. This means that over sufficiently small time intervals, the investor is either a seller or a buyer of stocks. The mathematical framework is built around quadratic objectives such that trading activity is punished quadratically. Mean-square utility is quadratic, and power utility is covered by quadratic punishment of distance to Merton’s power utility portfolio. We present semi-explicit solutions and, in a series of numerical illustrations, show the impact of trading constraints on the portfolio decision over the investment horizon.     

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.