Risk-free Asset Research Articles

A Hashing Power Allocation Game with and without Risk-free Asset

Abstract Miners in various blockchain-backed cryptocurrency networks compete to maintain the validity of the underlying distributed ledgers to earn the bootstrapped cryptocurrencies. With limited hashing power, each miner needs to decide how to allocate their resource to different cryptocurrencies so as to achieve the best overall payoff. Together all the miners form a hashing power allocation game. We consider two settings of the game, depending on whether each miner can allocate their fund to a risk-free asset or not. We show that this game admits unique pure Nash equilibrium in closed-form for both settings.

Mean-Variance Portfolio Selection with Dynamic Targets for Expected Terminal Wealth

In a market that consists of multiple stocks and one risk-free asset whose mean return rates and volatility are deterministic, we study a continuous-time mean-variance portfolio selection problem in which an agent is subject to a constraint that the expectation of the agent’s terminal wealth must exceed a target and minimize the variance of the agent’s terminal wealth. The agent can revise the expected terminal wealth target dynamically to adapt to the change of the agent’s current wealth, and we consider the following three targets: (i) the agent’s current wealth multiplied by a target expected gross return rate, (ii) the risk-free payoff of the agent’s current wealth plus a premium, and (iii) a weighted average of the risk-free payoff of the agent’s current wealth and a preset aspiration level. We derive the so-called equilibrium strategy in closed form for each of the three targets and find that the agent effectively minimizes the variance of the instantaneous change of the agent’s wealth subject to a certain constraint on the expectation of the instantaneous change of the agent’s wealth.

Optimal Investment Strategy for DC Pension Plan with Stochastic Income and Inflation Risk under the Ornstein–Uhlenbeck Model

This paper is concerned with the optimal investment strategy for a defined contribution (DC) pension plan. We assumed that the financial market consists of a risk-free asset and a risky asset, where the risky asset is subject to the Ornstein–Uhlenbeck (O-U) process, and stochastic income and inflation risk were also considered in the model. We firstly derived the Hamilton–Jacobi–Bellman (HJB) equation through the stochastic control method. Secondly, under the logarithmic utility function, the closed-form solution of optimal asset allocation was obtained by using the Legendre transform method. Finally, we give several numerical examples and a financial analysis.

A Nash bargaining solution for a multi period competitive portfolio optimization problem: Co-evolutionary approach

This study focuses on proposing a Nash bargaining model to solve a novel multi-period competitive portfolio optimization problem for large investors in the stock market who want to maximize their terminal wealth while taking into account competitors' profits. In this study, a Competitive Portfolio Model (CPM) is developed in accordance with the Cournot competition principle for a static, non-cooperative, and non-zero-sum game with complete information. Transaction costs, risk-free assets and cash are also included to match real-world conditions. Also, three criteria including the average value at risk, the mean absolute semi-deviation, and entropy are considered to control the investment risk in the model. Moreover, due to common constraints between players (free floating shares of risky assets) in stock markets, this study falls into the category of Generalized Nash Equilibrium Problems (GNEP). Therefore, to overcome the problem, a Cooperative Co-evolutionary Algorithm (CCA) based on Particle Swarm Optimization (PSO) is customized and used. A few experimental tests and a numerical example with descriptive analytics (using real data from two large mutual funds who invest in the Iranian Stock Exchange market) are used to evaluate the feasibility of the proposed model and the efficiency of the design algorithm. After solving the model, for each time period, investors' trading strategies (trading signals and stock volume in their portfolio) are determined. The results show that the volume of transactions due to the market power of an investor has a significant effect on the terminal wealth of competitors. Also, the results of sensitivity analysis show that profit-making is inversely related to the degree of risk aversion of investors.

Optimal investment strategies for an insurer with liquid constraint

In the present paper, we investigate the optimal investment strategies for an insurer with the liquid constraint. The insurer’s surplus is modeled by a Lévy process. The insurer can invest in N risky assets and a risk-free asset. However, in reality, the insurer needs to keep partial amount of his wealth as liquid reserves to meet with the management, which is called as the liquid constraint. We assume that the liquid reserve is a proportion of the insurer’s wealth. Using the martingale approach, we derive the explicit optimal investment strategies for both CARA and quadratic utilities.

QUADRATIC INVESTMENT PORTFOLIO BASED ON VALUE-AT-RISK WITH RISK-FREE ASSETS: FOR STOCKS OF THE MINING AND ENERGY SECTOR

The mining and energy sector is still the driving force for economic development and community empowerment, especially around mining and energy activities. Therefore, increased investment in the mining and energy sectors needs to be increased and balanced with stricter safety and environmental policies. This paper aims to formulate a quadratic investment portfolio optimization model, and apply it to several stocks in the mining and energy sectors. In this paper, it is assumed that risk is measured using Value-at-Risk (VaR), so that the optimization modeling is carried out using the quadratic investment portfolio approach to the Mean-VaR model with risk-free assets. Furthermore, the model is used to determine the efficient portfolio surface based on several values of risk aversion levels. Based on the results of the analysis, it is found that an efficient portfolio surface has a minimum portfolio return value with an average of 0.766522 and a VaR risk of 0.038687. In addition, the results of the analysis can be concluded that the greater the level of risk aversion, the smaller the VaR value, which is followed by the smaller the portfolio average value.Keywords: Mining and energy sector, risk free assets, investment, Value-at-Risk, portfolio optimization.JEL Classifications: A12, C61, G11, Q48DOI: https://doi.org/10.32479/ijeep.11165

Rule-based strategies for dynamic life cycle investment

In this work, we consider rule-based investment strategies for managing a defined contribution pension savings scheme, under the Dutch pension fund testing model. We find that dynamic, rule-based investment strategies can outperform traditional static strategies, by which we mean that the investor may achieve the target retirement income with a higher probability or limit the shortfall when the target is not met. In comparison with dynamic programming-based strategies, the rule-based strategies have more stable asset allocations throughout time and avoid excessive transactions that may be hard to explain to an investor. We also study a combined strategy of a rule-based target with dynamic programming. A key feature of our setting is that there is no risk-free asset, instead, a matching portfolio is introduced for the investor to avoid unnecessary risk.

Optimal Bitcoin trading with inverse futures

We consider an optimal trading problem for an investor who trades Bitcoin spot and Bitcoin inverse futures, plus a risk-free asset. The investor seeks an optimal strategy to maximize her expected utility of terminal wealth. We obtain explicit solutions to the investor’s optimal strategies under both exponential and power utility functions. Empirical studies confirm that optimal strategies perform well in terms of Sharpe ratio and Sortino ratio and beat the long-only strategy in Bitcoin spot.

A hybrid stochastic differential reinsurance and investment game with bounded memory

This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest her wealth in a financial market that contains a risk-free asset and a risky asset which is described by the constant elasticity of variance (CEV) model. The two insurers, as the followers of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. The competitive relationship between two insurers is modeled by the non-zero-sum game, and their decision-making will consider the relative performance measured by the difference in their terminal wealth. The bounded memory feature is characterized by the wealth process with delay. The purpose of the reinsurer is to maximize the expected utility of her own terminal wealth with delay. The two insurers aim to maximize the expected utility of the combination of her terminal wealth and the relative performance with delay. By using the idea of backward induction and the dynamic programming approach, we derive the equilibrium strategy and value functions explicitly. Then, we provide the corresponding verification theorem. Finally, some numerical examples and sensitivity analysis are presented to demonstrate the effects of model parameters on the equilibrium strategy. We find the delay factor discourages or stimulates investment depending on the length of delay. Moreover, competitive factors between two insurers make their optimal reinsurance-investment strategy interact, and reduce reinsurance demand and reinsurance premium price.

Coherent portfolio performance ratios

In Quantitative Finance 2016, Chen, Hu and Lin (CHL) claimed the following: ‘ … there is yet no coherent risk measure related to investment performance.’ (p. 682). Our paper suggests and analyzes four coherence axioms that portfolio performance ratios should satisfy. Our Portfolio Riskless Translation Invariance axiom must be satisfied to assure separation of the objective decision to optimize a portfolio’s risky composition from the subjective decision to optimize the weight of the portfolio's level of risk-free asset. Performance ratios with fixed thresholds other than the risk-free rate do not satisfy this axiom, allowing portfolio managers to affect an ex-ante performance ratio merely by changing the proportion of the risk-free asset in the portfolio rather than by improving the composition of the portfolio’s risky components. The magnitude of this potential drawback is examined using S&P-500 stock index data. Replacing the fixed threshold, T, with a threshold that equals γ times the portfolio’s risk premium plus (1-γ) times the risk-free rate, eliminates the above shortcoming for any selected γ. In addition, using performance ratios with threshold rather than fixed T, assures consistency of performance ratios of effective stochastic dominance and risk-free asset rules.

Optimal asset allocation under search frictions and stochastic interest rate

In this paper, we investigate an optimal asset allocation problem in a financial market consisting of one risk-free asset, one liquid risky asset and one illiquid risky asset. The liquidity risk stems from the asset that cannot be traded continuously, and the trading opportunities are captured by a Poisson process with constant intensity. Also, it is assumed that the interest rate is stochastically varying and follows the Cox–Ingersoll–Ross model. The performance functional of the decision maker is selected as the expected logarithmic utility of the total wealth at terminal time. The dynamic programming principle coupled with the Hamilton–Jacobi–Bellman equation has been adopted to solve this stochastic optimal control problem. In order to reduce the dimension of the problem, we introduce the proportion of the wealth invested in the illiquid risky asset and derive the semi-analytical form of the value function using a separation principle. A finite difference method is employed to solve the controlled partial differential equation satisfied by a function which is an important component of the value function. The numerical examples and their economic interpretations are then discussed.

Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay

This paper investigates the asset-liability management problem for an ordinary robust insurance system between a reinsurer and an insurer under the Heston model with delay. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the Heston model. The insurer, as the follower of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. Under the consideration of the performance-related capital inflow/outflow, the wealth processes of the insurer and reinsurer are modeled by stochastic differential delay equations (SDDEs). This paper aims to find the equilibrium strategies for the reinsurer and insurer by maximizing the expected utility of the player’s terminal wealth with delay under the worst-case scenario of the alternative measures. By using the idea of backward induction and dynamic programming approach, the explicit expressions of the robust equilibrium strategies and value functions are derived. Finally, some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the robust optimal reinsurance and investment strategies are performed.

DYNAMIC ASSET ALLOCATION FOR TARGET DATE FUNDS UNDER THE BENCHMARK APPROACH

ABSTRACTTarget date funds (TDFs) are becoming increasingly popular investment choices among investors with long-term prospects. Examples include members of superannuation funds seeking to save for retirement at a given age. TDFs provide efficient risk exposures to a diversified range of asset classes that dynamically match the risk profile of the investment payoff as the investors age. This is often achieved by making increasingly conservative asset allocations over time as the retirement date approaches. Such dynamically evolving allocation strategies for TDFs are often referred to as glide paths. We propose a systematic approach to the design of optimal TDF glide paths implied by retirement dates and risk preferences and construct the corresponding dynamic asset allocation strategy that delivers the optimal payoffs at minimal costs. The TDF strategies we propose are dynamic portfolios consisting of units of the growth-optimal portfolio (GP) and the risk-free asset. Here, the GP is often approximated by a well-diversified index of multiple risky assets. We backtest the TDF strategies with the historical returns of the S&P500 total return index serving as the GP approximation.

Modern Approaches to Dynamic Portfolio Optimization

Although appealing from a theoretical point of view, empirical assessments of dynamic portfolio optimizations in a mean-variance framework often fail to reach the high expectations set forth by analytical evaluations. A major reason for this shortfall is the imprecise estimation of asset moments and in particular, the expected return. This work levers recent advancements in the field of machine learning and employs three types of artificial neural networks in an attempt to improve the accuracy of the asset return estimation and the expected associated portfolio performances. After an introduction of the dynamic portfolio optimization framework and the artificial neural networks, their suitability for the considered application is analyzed in a two asset universe of a market and a risk-free asset. A comparison of the corresponding risk-return characteristics and those achieved using a more traditional exponentially weighted moving average estimator is subsequently drawn. While outperformance of the artificial neural networks is found for daily and monthly estimated returns, significance can only be established in the latter case, especially in light of trading costs. Multiple robustness checks are performed before an outlook for subsequent research opportunities is given. Keywords: Portfolio optimization; machine learning; multilayer perceptron; convolutional neural network; long short-term memory.

Continuous-time stochastic mutual fund management game between active and passive funds

This paper investigates a Stackelberg game between a mutual fund manager and an investor. Throughout the paper, we follow the literature (see, e.g. Li and Sethi. A review of dynamic stackelberg game models. Discrete Contin. Dyn. Syst. B, 2016, 22(1), 125) on Stackelberg games, i.e. leader-follower games, and call the leader and follower, respectively, her and him. In our model, the leader (she) refers to the mutual fund manager, while the follower (he) is the investor. The individual investor (he), manages an active mutual fund and can only allocate his wealth among a risk-free asset, the active mutual fund, and a passive index fund. The passive index fund is composed of an exogenously given portfolio of all stocks in the market. The investor aims to maximize the expected constant relative risk aversion utility of his terminal wealth, while the mutual fund manager's objective is to maximize the expected value of the accumulative discounted management fees from the investor. Assume that the mutual fund manager faces portfolio constraints, which reflects the investment objective and style of the fund. We first focus on a special case in which the mutual fund is a sector fund, investing only in a small subset of available stocks in the market, namely a specific sector or industry. Then, this special case is extended to the general case in which the portfolio constraint is given by a nonempty, closed convex set. By applying the dynamic programming principle approach, we solve two associated Hamilton-Jacobi-Bellman (HJB) equations and obtain Stackelberg equilibrium strategies for both the mutual fund manager and the investor in closed-form expressions. Finally, in the special case we provide numerical examples to analyze the effects of some parameters on the equilibrium strategies.

Optimal Reinsurance and Investment Strategy with Delay in Heston’s SV Model

In this paper, we consider an optimal investment and proportional reinsurance problem with delay, in which the insurer’s surplus process is described by a jump-diffusion model. The insurer can buy proportional reinsurance to transfer part of the insurance claims risk. In addition to reinsurance, she also can invests her surplus in a financial market, which is consisted of a risk-free asset and a risky asset described by Heston’s stochastic volatility (SV) model. Considering the performance-related capital flow, the insurer’s wealth process is modeled by a stochastic differential delay equation. The insurer’s target is to find the optimal investment and proportional reinsurance strategy to maximize the expected exponential utility of combined terminal wealth. We explicitly derive the optimal strategy and the value function. Finally, we provide some numerical examples to illustrate our results.

Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

<p style='text-indent:20px;'>This paper studies a robust optimal investment problem under the mean-variance criterion for a defined contribution (DC) pension plan with an ambiguity-averse member (AAM), who worries about model misspecification and aims to find robust optimal strategy. The member has access to a risk-free asset (i.e., cash or bank account) and a risky asset (i.e., the stock) in a financial market. In order to get closer to the actual environment, we assume that both the income level and stock price are driven by Heston's stochastic volatility model. A continuous-time mean-variance model with ambiguity aversion for a DC pension plan is established. By using the Lagrangian multiplier method and stochastic optimal control theory, the closed-form expressions for robust efficient strategy and efficient frontier are derived. In addition, some special cases are derived in detail. Finally, a numerical example is presented to illustrate the effects of model parameters on the robust efficient strategy and the efficient frontier, and some economic implications have been revealed.</p>

Optimal strategies for a defined contribution pension plan with SAHARA utility under DEV model

This paper investigates the optimal investment problems for a defined contribution pension plan in an incomplete financial market that consists of a risk-free asset anda risky asset whose price process obeys the dynamic elasticity of variance model and a correlated inflation risk model. The pension plan participant aims to maximize the expected utility of the terminal wealth under the symmetric asymptotic hyperbolic absolute risk aversion utility. This paper derives the Hamilton-Jacobi-Bellman equations and the verification results, develops lower and upper bounds for the value function, and computes optimal strategies by applying the dual control Monte Carlo algorithm. Numerical examples are presented to verify the accuracy of the approach.

Robust optimal investment strategy of DC pension plans with stochastic salary and a return of premiums clause

In this paper, we investigate a robust optimal investment problem for a defined contribution (DC) pension plan with a return of premiums clause, taking account of the inflation risk and the salary risk. The members in the pension plan continuously contribute a fixed proportion of their stochastic salaries into the pension fund and receive the benefits at retirement time. Due to a return of premiums clause, the members who die in the accumulation phase have the right to withdraw their accumulated premiums. The pension fund manager can invest the wealth of the fund into a financial market consisting of a risk-free asset, an inflation-indexed bond and a stock. The manager is assumed to be ambiguity-averse and attempt to maximize the expected utility of the terminal wealth in each alive member’s pension account under the worst-case scenario. By adopting a new state variable which represents the accumulated premiums and using dynamic programming approach, we derive the robust optimal investment strategy and the corresponding value function. Finally, the effects of the stochastic salary, the return of premiums clause and the attitude about the model uncertainty on the robust optimal investment strategy are illustrated by mathematical and numerical analysis.

Portfolio selection with tail nonlinearly transformed risk measures—a comparison with mean-CVaR analysis

We compare portfolio selection under the tail nonlinearly transformed risk measure (TNT) with mean-CVaR analysis for normally distributed returns. TNT arises as a natural extension to CVaR that additionally distorts the portfolio returns' outcomes by means of a concave transformation function. We first address the portfolio setting where no risk-free asset exists. Here, the global minimum CVaR portfolio, also under fairly realistic asset returns, may not exist. TNT is able to resolve this shortcoming: its additional transformation of the portfolio outcomes ensures that the global minimum TNT portfolio and, accordingly, the ()-efficient frontiers, do always exist. Second, we analyze the setting where the risk-free asset is available. In this case, Tobin's theorem holds both for CVaR and TNT, under the condition that the respective investors are sufficiently risk averse. Under TNT, this condition is less restrictive, Tobin's theorem holds more often. We third address the choice of optimal portfolios under CVaR and TNT. Under CVaR, optimal portfolios exhibit plunging: either the investor invests entirely in the risky tangency portfolio or she invests entirely in the risk-free asset; diversification is never optimal. TNT, in contrast, yields more realistic portfolio structures. Below a certain minimum risk premium, we do not find any risky investment. Once this minimum risk premium is passed, the risky investment is continuous and strictly monotonously increasing in the risk premium. This pattern is in line with empirical evidence on the stock market participation puzzle and the equity premium puzzle.