The premium of dynamic trading

It is well established that, in a market with inclusion of a risk-free asset, the single-period mean–variance efficient frontier is a straight line tangent to the risky region, a fact that is the very foundation of the classical CAPM. In this paper, it is shown that, in a continuous-time market where the risky prices are described by Itô processes and the investment opportunity set is deterministic (albeit time-varying), any efficient portfolio must involve allocation to the risk-free asset at any time. As a result, the dynamic mean–variance efficient frontier, although still a straight line, is strictly above the entire risky region. This in turn suggests a positive premium, in terms of the Sharpe ratio of the efficient frontier, arising from dynamic trading. Another implication is that the inclusion of a risk-free asset boosts the Sharpe ratio of the efficient frontier, which again contrasts sharply with the single-period case.     

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z _α and target return r _α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.     

Modeling a Dynamic Portfolio for Pension Plans in Emerging Markets With Myopic and Nonmyopic Behavior

ABSTRACT

We introduce a dynamic formulation for the problem of portfolio selection of pension funds in the absence of a risk-free asset. In emerging markets, a risk-free asset might be unavailable, and the approaches commonly used may no longer be suitable. We use a parametric approach to combine dynamic programming and Monte Carlo simulation to gain additional flexibility. This approach is general in the sense that optimal asset allocation is tractable for all HARA utility functions in the absence of a risk-free asset. The traditional case composed of several risky assets and one risk-free asset is compared to a case in which the risk-free asset is unavailable.     

Which demands affect optimal international portfolio choices?

This study analyzes the asset allocations of simple international portfolios that include domestic risky assets, foreign risky assets, and domestic risk-free bonds, through a theoretical analysis. A close-form solution for the optimal holding rates is derived, and can be further sub-divided into three categories of demand: speculative demand, diversified demand, and hedging demands. We carefully explore the essential problem of identifying the underlying reasons for asset allocations, which in turn allows us to answer the question of which of these demands are critical in influencing holding changes.     

Dynamic portfolio choice without cash

We consider the dynamic mean–variance portfolio choice without cash under a game theoretic framework. The mean–variance criterion is investigated in the situation where an investor allocates the wealth among risky assets while keeping no cash in a bank account. The problem is solved explicitly up to solutions of ordinary differential equations by applying the extended Hamilton–Jacobi–Bellman equation system. Given a constant risk aversion coefficient, the optimal allocation without a risk-free asset depends linearly on the current wealth, while that with a risk-free asset turns out to be independent of the current wealth. We also study the minimum-variance criterion, which can be viewed as an extension of the mean–variance model when the risk aversion coefficient tends to infinity. Calibration exercises demonstrate that for large investments, the mean–variance model without cash yields the highest certainty equivalent return for both short-term and long-term investments. Furthermore, the mean–variance portfolio choices with and without cash yield almost the same Sharpe ratio for an investment with large initial wealth.     

Dynamic nonmyopic portfolio behavior

The dynamic nonmyopic portfolio behavior of an investor whotrades a risk-free and risky asset is derived for all HARA utilityfunctions and a stochastic risk premium. Conditions are foundfor when the investor holds more or less than the myopic amountof the risky assets; hedges against or speculates the risk-premiumuncertainty; is long or short on the risky asset; and holdsmore or less of the risky asset at longer horizons. The analyticalsolutions derived take multiple mathematical forms and includeextreme cases in which investors with long but finite horizonscan attain nirvana.     

The implied Sharpe ratio

In an incomplete market, including liquidly traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept—the implied Sharpe ratio—which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.     

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。     

Mean-Variance Asset Liability Management with State-Dependent Risk Aversion

This article investigates the asset liability management problem with state-dependent risk aversion under the mean-variance criterion. The investor allocates the wealth among multiple assets including a risk-free asset and multiple risky assets governed by a system of geometric Brownian motion stochastic differential equations, and the investor faces the risk of paying uncontrollable random liabilities. The state-dependent risk aversion is taken into account in our model, linking the risk aversion to the current wealth held by the investor. An extended Hamilton-Jacobi-Bellman system is established for the optimization of asset liability management, and by solving the extended Hamilton-Jacobi-Bellman system, the analytical closed-form expressions for the time-inconsistent optimal investment strategies and the optimal value function are derived. Finally, numerical examples are presented to illustrate our results.     

A Note on the Analytics and Geometry of Limiting Mean—Variance Investment Opportunity Sets

This paper extends the mathematics developed by Merton (1972) to the limiting investment opportunity set as smaller risk assets are added. Investment opportunity sets of risky assets are well-known to be described by hyperbolae in mean-standard deviation space. In practice, the asset classes in portfolios may vary from high risk common stocks to near cash assets. Low variability assets change the appearance of the investment opportunity set to the extent that a unique optimum risky asset portfolio disappears. The limiting result is similar to the investment opportunity set that arises when two assets are perfectly correlated. The location of the IOS is shown to mathematically depend upon the level of the riskless interest rate and one slope parameter. The slope parameter is estimable, using a finite number of assets, and represents a bound on market Sharpe ratios.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

A model of optimal portfolio selection under liquidity risk and price impact

We study a financial model with one risk-free and one risky asset subject to liquidity risk and price impact. In this market, an investor may transfer funds between the two assets at any discrete time. Each purchase or sale policy decision affects the rice of the risky asset and incurs some fixed transaction cost. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to a solvency constraint. This is formulated as an impulse control problem under state constraints and we characterize the value function as the unique constrained viscosity solution to the associated quasi-variational Hamilton–Jacobi–Bellman inequality. We would like to thank Mihail Zervos for useful discussions.     

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.     

Multi-period stochastic optimization models for dynamic asset allocation

Institutional investors manage their strategic asset mix over time to achieve favorable returns subject to various uncertainties, policy and legal constraints, and other requirements. One may use a multi-period portfolio optimization model in order to determine an optimal asset mix. The concept of scenarios is typically employed for modeling random parameters in a multi-period stochastic programming model, and scenarios are constructed via a tree structure. Recently, an alternative stochastic programming model with simulated paths was proposed by Hibiki [Hibiki, N., 2001b. A hybrid simulation/tree multi-period stochastic programming model for optimal asset allocation. In: Takahashi, H. (Ed.), The Japanese Association of Financial Econometrics and Engineering. JAFEE Journal 89–119 (in Japanese); Hibiki, N., 2003. A hybrid simulation/tree stochastic optimization model for dynamic asset allocation. In: Scherer, B. (Ed.), Asset and Liability Management Tools: A Handbook for Best Practice, Risk Books, pp. 269–294], and it is called a hybrid model. The advantage of the simulated path structure compared to the tree structure is to give a better accuracy to describe uncertainties of asset returns. In this paper, we compare the two types of multi-period stochastic optimization models, and clarify that the hybrid model can evaluate and control risk better than the scenario tree model using some numerical tests. According to the numerical results, an efficient frontier of the hybrid model with the fixed-proportion strategy dominates that of the scenario tree model when we evaluate them on simulated paths. Moreover, optimal solutions of the hybrid model are more appropriate than those of the scenario tree model.     

Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment

Abstract

This article considers the compound Poisson insurance risk model perturbed by diffusion with investment. We assume that the insurance company can invest its surplus in both a risky asset and the risk-free asset according to a fixed proportion. If the surplus is negative, a constant debit interest rate is applied. The absolute ruin probability function satisfies a certain integro-differential equation. In various special cases, closed-form solutions are obtained, and numerical illustrations are provided.     

Optimal constrained investment in the Cramer-Lundberg model

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk-free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when short-selling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. For example, for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum, or with the case of no short-selling and no borrowing studied in Azcue and Muler.     

The scaling limit of superreplication prices with small transaction costs in the multivariate case

Kusuoka (Ann. Appl. Probab. 5:198–221, 1995) showed how to obtain non-trivial scaling limits of superreplication prices in discrete-time models of a single risky asset which is traded at properly scaled proportional transaction costs. This article extends the result to a multivariate setup where the investor can trade in several risky assets. The \(G\)-expectation describing the limiting price involves models with a volatility range around the frictionless scaling limit that depends not only on the transaction costs coefficients, but also on the chosen complete discrete-time reference model.     

The two-fund separation theorem revisited

This paper resolves two issues regarding the traditional capital asset pricing model with one risk-free asset which seem to have been overlooked in the literature. First, it provides an elementary and complete proof of the two-fund separation theorem which accounts for the fact that asset demand may become undefined if the limiting slopes of the investor’s indifference curves are finite. Second, it shows that an additional limiting condition on investors’ risk aversions is generally necessary to guarantee existence of an equilibrium. Moreover, a generalized existence result is formulated which includes investors who in equilibrium may not invest in risky assets and a simple condition ensuring positive equilibrium asset prices is given.     

Modern portfolio management with conditioning information

This paper studies models in which active portfolio managers utilize conditioning information unavailable to their clients to optimize performance relative to a benchmark. We derive explicit solutions for the optimal strategies with multiple risky assets, with or without a risk-free asset, and consider various constraints on portfolio risks or weights. The optimal strategies feature a mean–variance efficient component (to minimize portfolio variance), and a hedging demand for the benchmark portfolio (to maximize correlation with the benchmark). A currency portfolio example shows that the optimal strategies improve the measured performance by 53% out of sample, compared with portfolios ignoring conditioning information.