Computing optimal rebalance frequency for log-optimal portfolios in linear time

The pure form of log-optimal investment strategies are often considered to be impractical due to the inherent need for continuous rebalancing. It is however possible to improve investor log utility by adopting a discrete-time periodic rebalancing strategy. Under the assumptions of geometric Brownian motion for assets and approximate log-normality for a sum of log-normal random variables, we find that the optimum rebalance frequency is a piecewise continuous function of investment horizon. One can construct this rebalance strategy function, called the optimal rebalance frequency function, up to a specified investment horizon given a limited trajectory of the expected log of portfolio growth when the initial portfolio is never rebalanced. We develop the analytical framework to compute the optimal rebalance strategy in linear time, a significant improvement from the previously proposed search-based quadratic time algorithm.     

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.     

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 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.     

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.     

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.     

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.     

Time Diversification: Empirical Tests

This paper investigates the relationship between the performance of equity and the length of the investment horizon used by investors. We examine optimal portfolio time diversification and two definitions of ex ante time diversification. Using almost two centuries of US and UK data we find some support for the hypothesis that equity represents a significantly better investment over long investment horizons than over short investment horizons. Where this result holds, the likely explanation is mean-aversion in fixed-income asset returns. However, these results are sensitive to changes in investor risk preference, changes in utility function specification, changes in the sample period used, changes in investor constraints, and the definition of time diversification adopted. They also differ between the US and UK markets.     

Optimal Hedging and Valuation of Nontraded Assets

This paper examines a number of valuation problems faced by an expected-utility-maximizing investor who, over a given time horizon, is constrained to hold an asset which cannot be replicated by dynamic trading and which therefore does not have a unique no-arbitrage price. We first derive the private valuation which the investor assigns to the nontradedasset in order to determine his optimal investment in the traded assets. We thereby show that, as part of this portfolio, the investor hedges the private valuation process of the nontraded asset, rather than its market price process. We also study the price at which the investor would be willing to sell the nontraded asset if he were subsequently prohibited from trading in it, as well as the amount the investor would be willing to pay to removethe trading restriction. All three values are shown to depend in an intuitive manner on the investor's risk aversion, the residual risk of the nontraded asset unhedged by the traded assets, the difference between the constrained holding and optimal unconstrained holding of the asset and the length of the time horizon over which the asset cannot be traded.     

Investment Horizon Effects

Abstract:   Boudry and Gray (2003) have documented that the optimal buy‐and‐hold demand for Australian stocks is not necessarily increasing in the investment horizon when returns are predictable. Such finding is in contrast with Barberis (2000) who shows that positive monotonic horizon effects predominate for US stocks. Using a closed‐form approximation to the asset allocation problem, this paper relates the return dynamics to the investor's portfolio choice for different investment horizons. In the special case of a single risky asset, it is shown that return predictability under stationarity may induce both positive and negative horizon effects in the optimal allocation to the risky asset. The paper extends previous empirical results by solving for the optimal portfolio when two risky assets with predictable returns are available for investment.     

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.     

风险资产配置与货币政策规则——黏性价格均衡下的宏观资产配置模型初探

本文将一个基于动态新凯恩斯理论的连续时间黏性价格一般均衡模型与随机动态资产配置模型相结合,进而研究基于内生宏观经济动态和货币政策规则进行资产配置的问题。在最优配置策略下,投资者相对风险偏好随无风险名义利率的增大而单调减小,而随通胀率的变化呈“U”型,说明投资者在通胀偏离稳态幅度较大时配置风险资产的相对意愿较高。此外,本文也给出了使用该模型讨论投资者最大化跨期效用对经济反作用这一宏观审慎问题的方式。     

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.     

Time to wealth goals in capital accumulation

This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.     

A knowledge‐based framework for incorporating investor's preference into portfolio decision‐making

This paper describes a framework for the integration of a rule‐based system capable of identifying an investor's risk preference into a quantitative portfolio model based on risk and expected return. By inferring rules consisting of an investor's objective and subjective risk preferences, the integrated methodology provides the assets suitable for the preferences. Through investment in the portfolio composed of the assets, the investor is able to obtain the following bene?ts: reduction of costs and time spent to determine target assets, and alleviation of anxiety from ‘out‐of‐favor’ assets. The framework is applied to the development of a knowledge‐based portfolio system for constructing an investor's preference‐oriented portfolio. In the procedure of the system for ?nding an optimal portfolio, the system uses an arti?cial intelligence method of a case‐based reasoning to obtain preference thresholds for an investor when the investor's past investment records are available. Experimental results show that the framework contributes signi?cantly to the construction of a better portfolio from the perspective of an investor's bene?t/cost ratio than that produced by the existing portfolio models. Copyright © 2004 John Wiley & Sons, Ltd.     

Tests of the multiperiod two-parameter model

Although investors face multiperiod decision problems, there are conditions under which the results of the one-period two-parameter model apply period by period. In addition to the assumptions made in the development of the two-parameter model itself (a perfect capital market, investor risk aversion, and normal distributions of one-period portfolio returns), the critical assumption in a multiperiod context is that, for any t, returns on portfolio assets from t?1 to t are independent of stochastic elements of the state-of-the-world at time t that affect investor tastes for given levels of wealth to be obtained at t.One such element of the state-of-the-world is the nature of investment opportunities to be available at t. For example, if the level of expected returns on investment portfolios to be available at time t is uncertain at time t?1, and if the returns from t?1 to t on some investment assets are more strongly related to the level of expected returns at t than returns on other assets, then the former assets are better vehicles for hedging against the level of expected returns at t. This can affect the demands for assets and their prices in such a way that the simple results of the one-period two-parameter model do not hold.The empirical tests of this paper reveal no evidence of measurable relationships between the returns on portfolio assets from t?1 to t and the level of expected returns to be available at t. Indeed, in our opinion there is no reliable evidence that the level of expected returns changed during the 1953–1972 period.     

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.     

Islamic and conventional portfolios optimization under investor sentiment states: Bayesian vs Markowitz portfolio analysis

This paper investigates the portfolio optimization under investor’s sentiment states of Hidden Markov model and over a different time horizon during the period 2004–2016. To compare the efficient portfolios of the Islamic and the conventional stock indexes, we have employed two approaches: the Bayesian and Markowitz mean-variance. Our findings reveal that the Bayesian efficient frontier of Islamic and conventional stock portfolios is affected by the investor’s sentiment state and the time horizon. Our findings also indicate that the investor’s sentiment regimes change the Islamic and the conventional optimal diversified portfolios.Moreover, the results show that the potential diversification benefits seem to be more important when using the Bayesian approach than when applying the Markowitz approach. This finding is valid for the bearish, depressed, bullish and calm states in Islamic stock markets. However, the diversification of potential portfolios is significant only for the bullish and the bubble states in the conventional financial markets.The findings of the study provided additional evidence for investors to exploit googling investor sentiment states to evaluate the portfolio performance and make an optimal portfolio allocation.     

Portfolio allocation and the investment horizon: a multiscaling approach

A new approach is proposed for analysing portfolio allocation over various time scales. This new approach is based on wavelet analysis, which decomposes a given time series on a scale-by-scale basis. Empirical results indicate that, as the investment horizon lengthens, a greater weighting should be allocated to stocks. An explanation for this result is that the mean-reverting property of stock returns causes investors to perceive that stocks are less risky than bonds and T-bills at longer time scales compared with shorter time scales. When we include the effect of risk aversion, it is found that the higher the risk aversion, the less the Sharpe ratio, indicating that a more conservative investor prefers a smoother consumption stream.