Dynamic asset allocation for varied financial markets under regime switching framework

Asset allocation among diverse financial markets is essential for investors especially under situations such as the financial crisis of 2008. Portfolio optimization is the most developed method to examine the optimal decision for asset allocation. We employ the hidden Markov model to identify regimes in varied financial markets; a regime switching model gives multiple distributions and this information can convert the static mean–variance model into an optimization problem under uncertainty, which is the case for unobservable market regimes. We construct a stochastic program to optimize portfolios under the regime switching framework and use scenario generation to mathematically formulate the optimization problem. In addition, we build a simple example for a pension fund and examine the behavior of the optimal solution over time by using a rolling-horizon simulation. We conclude that the regime information helps portfolios avoid risk during left-tail events.     

Almost exact risk budgeting with return forecasts for portfolio allocation

We revisit the portfolio allocation problem with designated risk-budget. We generalize the problem of arbitrary risk budgets with unequal correlations to one that includes return forecasts and transaction costs while keeping the no-shorting constraint. We offer a convex second order cone formulation that scales well with the number of assets and explore solutions to problem variants - on equity-bond asset allocation problems as well as formulating portfolios using index constituents from the NASDAQ100 index, illustrating the benefits of this approach.     

A data-driven neural network approach to optimal asset allocation for target based defined contribution pension plans

A data-driven Neural Network (NN) optimization framework is proposed to determine optimal asset allocation during the accumulation phase of a defined contribution pension scheme. In contrast to parametric model based solutions computed by a partial differential equation approach, the proposed computational framework can scale to high dimensional multi-asset problems. More importantly, the proposed approach can determine the optimal NN control directly from market returns, without assuming a particular parametric model for the return process. We validate the proposed NN learning solution by comparing the NN control to the optimal control determined by solution of the Hamilton–Jacobi–Bellman (HJB) equation. The HJB equation solution is based on a double exponential jump model calibrated to the historical market data. The NN control achieves nearly optimal performance. An alternative data-driven approach (without the need of a parametric model) is based on using the historic bootstrap resampling data sets. Robustness is checked by training with a blocksize different from the test data. In both two and three asset cases, we compare performance of the NN controls directly learned from the market return sample paths and demonstrate that they always significantly outperform constant proportion strategies.     

Solving multistage asset investment problems by the sample average approximation method

The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions we address the question of how tree structures, number of stages, number of outcomes and number of assets affect the solution quality. We also present a new method for evaluating the quality of first stage decisions.     

Quadratic programming for portfolio optimization

Portfolio optimization is a procedure for generating a portfolio composition which yields the highest return for a given level of risk or a minimum risk for given level of return. The problem can be formulated as a quadratic programming problem. We shall present a new and efficient optimization procedure taking advantage of the special structure of the portfolio selection problem. An example of its application to the traditional mean-variance method will be shown. Formulation of the procedure shows that the solution of the problem is vector intensive and fits well with the advanced architecture of recent computers, namely the vector processor.     

Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases

In a financial market with one riskless asset and n risky assets whose prices are lognormal, we solve in a closed form the problem of a pension fund maximizing the expected CRRA utility of its surplus till the (stochastic) death time of a representative agent. We consider a unique asset allocation problem for both accumulation and decumulation phases. The optimal investment in the risky assets must decrease during the first phase and increase during the second one. We accordingly suggest it is not optimal to manage the two phases separately, and outsourcing of allocation decisions should be avoided in both phases. JEL: G23, G11 MSC 2000: 62P05, 91B28, 91B30, 91B70, 93E20     

Scenario optimization asset and liability modelling for individual investors

We develop a scenario optimization model for asset and liability management of individual investors. The individual has a given level of initial wealth and a target goal to be reached within some time horizon. The individual must determine an asset allocation strategy so that the portfolio growth rate will be sufficient to reach the target. A scenario optimization model is formulated which maximizes the upside potential of the portfolio, with limits on the downside risk. Both upside and downside are measured vis-à-vis the goal. The stochastic behavior of asset returns is captured through bootstrap simulation, and the simulation is embedded in the model to determine the optimal portfolio. Post-optimality analysis using out-of-sample scenarios measures the probability of success of a given portfolio. It also allows us to estimate the required increase in the initial endowment so that the probability of success is improved.     

Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.     

Using the Black–Derman–Toy interest rate model for portfolio optimization

No-arbitrage interest rate models are designed to be consistent with the current term structure of interest rates. The diffusion of the interest rates is often approximated with a tree, in which the scenario-dependent fair price of any security is calculated as the present value of the risk-neutral expectation by backward induction. To use this tree in a portfolio optimization context it is necessary to account for the so-called “market price of risk”. In this paper we present a method to change the conditional probabilities in the Black–Derman–Toy model to the physical (or real) measure, including the market price of risk, and explore the economic implications for expected spot rates and for expected bond returns.     

Dynamic capital allocation with irreversible investments

Capital allocation models generally assume that the risk portfolio is constructed at a single point in time, when the underwriter has full information about available underwriting opportunities. However, in practice, opportunities are not all known at the beginning but instead arrive over time. Moreover, a commitment to an opportunity is not easy to change as time passes. Thus, to optimize a portfolio, the underwriter must make decisions on opportunities as they arrive while making use of assumptions about what will arrive in the future. This paper studies capital allocation rules in this setting, finding important differences from the static setting. The pricing of an opportunity is based on an expected future marginal cost of risk associated with that opportunity—one that will be fully understood only after the risk portfolio is finalized. The risk charge for today’s opportunity is thus a probability-weighted average of the product of the marginal value of capital in future states of the world and the amount of capital consumed by the opportunity in those future states. Our numerical examples illustrate how the marginal cost of risk for an opportunity is shaped by when it arrives in time, as well as what has arrived before it.     

Investment models based on clustered scenario trees

Stochastic programming is widely applied in financial decision problems. In particular, when we need to carry out the actual calculations for portfolio selection problems, we have to assign a value for each expected return and the associated conditional probability in advance. These estimated random parameters often rely on a scenario tree representing the distribution of the underlying asset returns. One of the drawbacks is that the estimated parameters may be deviated from the actual ones. Therefore, robustness is considered so as to cope with the issue of parameter inaccuracy. In view of this, we propose a clustered scenario-tree approach, which accommodates the parameter inaccuracy problem in the context of a scenario tree.     

Optimal dynamic asset allocation of pension fund in mortality and salary risks framework

In this paper, we consider the optimal dynamic asset allocation of pension fund with mortality risk and salary risk. The managers of the pension fund try to find the optimal investment policy (optimal asset allocation) to maximize the expected utility of terminal wealth. The market is a combination of financial market and insurance market. The financial market consists of three assets: cashes with stochastic interest rate, stocks and rolling bonds, while the insurance market consists of mortality risk and salary risk. These two non-hedging risks cause incompleteness of the market. By martingale method and dynamic programming principle we first derive the approximate optimal investment policy to overcome the difficulty, then investigate the efficiency of the approximation. Finally, we solve an optimal assets liabilities management(ALM) problem with mortality risk and salary risk under CRRA utility, and reveal the influence of these two risks on the optimal investment policy by numerical illustration.     

Optimal dynamic asset allocation for DC plan accumulation/decumulation: Ambition-CVAR

We consider the late accumulation stage, followed by the full decumulation stage, of an investor in a defined contribution (DC) pension plan. The investor’s portfolio consists of a stock index and a bond index. As a measure of risk, we use conditional value at risk (CVAR) at the end of the decumulation stage. This is a measure of the risk of depleting the DC plan, which is primarily driven by sequence of return risk and asset allocation during the decumulation stage. As a measure of reward, we use Ambition, which we define to be the probability that the terminal wealth exceeds a specified level. We develop a method for computing the optimal dynamic asset allocation strategy which generates points on the efficient Ambition-CVAR frontier. By examining the Ambition-CVAR efficient frontier, we can determine points that are Median-CVAR optimal. We carry out numerical tests comparing the Median-CVAR optimal strategy to a benchmark constant proportion strategy. For a fixed median value (from the benchmark strategy) we find that the optimal Median-CVAR control significantly improves the CVAR. In addition, the median allocation to stocks at retirement is considerably smaller than the benchmark allocation to stocks.     

An optimal portfolio model with stochastic volatility and stochastic interest rate

We consider a portfolio optimization problem under stochastic volatility as well as stochastic interest rate on an infinite time horizon. It is assumed that risky asset prices follow geometric Brownian motion and both volatility and interest rate vary according to ergodic Markov diffusion processes and are correlated with risky asset price. We use an asymptotic method to obtain an optimal consumption and investment policy and find some characteristics of the policy depending upon the correlation between the underlying risky asset price and the stochastic interest rate.     

Stochastic models for strategic resource allocation in nonprofit foreclosed housing acquisitions

Increased rates of mortgage foreclosures in the U.S. have had devastating social and economic impacts during and after the 2008 financial crisis. As part of the response to this problem, nonprofit organizations such as community development corporations (CDCs) have been trying to mitigate the negative impacts of mortgage foreclosures by acquiring and redeveloping foreclosed properties. We consider the strategic resource allocation decisions for these organizations which involve budget allocations to different neighborhoods under cost and return uncertainty. Based on interactions with a CDC, we develop stochastic integer programming based frameworks for this decision problem, and assess the practical value of the models by using real-world data. Both policy-related and computational analyses are performed, and several insights such as the trade-offs between different objectives, and the efficiency of different solution approaches are presented.     

Optimal asset allocation aid system: From “one-size” vs “tailor-made” performance ratio

Optimal asset allocation well-fitting investors’ goals is a pressing challenge in risk management. Making a step forward to the Sharpe ratio, the parameter-dependent Sortino–Satchell, Generalized Rachev and Farinelli–Tibiletti performance ratios are suggested for personalizing asset allocation. Tailor-made optimal asset paths for five different investor risk profiles are traced over a rolling 12 month investing horizon. Our simulations show a satisfactorily good match between asset allocation and correspondent risk profile. Specifically, Generalized Rachev ratios outperform in personalized allocation for “extreme” risk profiles, i.e. conservative and aggressive investors, whereas Sortino–Satchell and Farinelli–Tibiletti ratios for those that are more moderate. Sharpe ratio confirms its ability in constructing steady-diversified portfolios, although underperformed.     

Optimal allocation to Deferred Income Annuities

In this paper we employ a lifecycle model that uses utility of consumption and bequest to determine an optimal Deferred Income Annuity (DIA) purchase policy. We lay out a mathematical framework to formalize the optimization process. The method and implementation of the optimization is explained, and the results are then analyzed. We extend our model to control for asset allocation and show how the purchase policy changes when one is allowed to vary asset allocation. Our results indicate that(i) refundable DIAs are less appealing than non-refundable DIAs because of the loss of mortality credits; (ii) the DIA allocation region is larger under the fixed asset allocation strategy due to it becoming a proxy for fixed-income allocation; and (iii) when the investor is allowed to change asset-allocation, DIA allocation becomes less appealing. However, a case for higher DIA allocation can be made for those individuals who perceive their longevity to be higher than the population.     

A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem

In this paper, we consider an extension of the Markovitz model, in which the variance has been replaced with the Value-at-Risk. So a new portfolio optimization problem is formulated. We showed that the model leads to an NP-hard problem, but if the number of past observation T or the number of assets K are low, e.g. fixed to a constant, polynomial time algorithms exist. Furthermore, we showed that the problem can be formulated as an integer programming instance. When K and T are large and α^VaR is small—as common in financial practice—the computational results show that the problem can be solved in a reasonable amount of time.     

Comments on “A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem”

Benati and Rizzi [S. Benati, R. Rizzi, A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem, European Journal of Operational Research 176 (2007) 423–434], in a recent proposal of two linear integer programming models for portfolio optimization using Value-at-Risk as the measure of risk, claimed that the two counterpart models are equivalent. This note shows that this claim is only partly true. The second model attempts to minimize the probability of the portfolio return falling below a certain threshold instead of minimizing the Value-at-Risk. However, the discontinuity of real-world probability values makes the second model impractical. An alternative model with Value-at-Risk as the objective is thus proposed.     

A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset

This paper focuses on the computation issue of portfolio optimization with scenario-based CVaR. According to the semismoothness of the studied models, a smoothing technology is considered, and a smoothing SQP algorithm then is presented. The global convergence of the algorithm is established. Numerical examples arising from the allocation of generation assets in power markets are done. The computation efficiency between the proposed method and the linear programming (LP) method is compared. Numerical results show that the performance of the new approach is very good. The remarkable characteristic of the new method is threefold. First, the dimension of smoothing models for portfolio optimization with scenario-based CVaR is low and is independent of the number of samples. Second, the smoothing models retain the convexity of original portfolio optimization problems. Third, the complicated smoothing model that maximizes the profit under the CVaR constraint can be reduced to an ordinary optimization model equivalently. All of these show the advantage of the new method to improve the computation efficiency for solving portfolio optimization problems with CVaR measure.