Distribution Of Terminal Wealth Research Articles

The benefits of return smoothing in insurer's cover funds – analyzes from a client's perspective

Traditional life insurance typically uses some mechanism that is aimed at smoothing the returns of the (collective) assets in the insurer's so-called cover fund. We consider a generic smoothing mechanism and numerically analyze how it impacts the risk-return characteristics of a traditional life insurance contract distinguishing between pathwise volatility (of the annual returns) and the volatility of terminal wealth. We find that pathwise volatility is significantly reduced while the distribution of terminal wealth is hardly affected. We conclude that using multiple segregated cover funds (that come with different asset allocations) as building blocks for more complex products enables insurers to offer a variety of risk-return profiles of terminal wealth in combination with a rather low pathwise volatility (compared to investments without smoothing mechanism). This increases subjective attractiveness for a typical consumer. We consider a variety of such products (static and dynamic investment products) and compare them to similar purely market-based products that do not use an insurer's cover fund. Analyzing risk-return characteristics, (objective) utility, and (subjective) attractiveness under Cumulative Prospect Theory and extensions of it, we conclude that products that become possible by implementing multiple segregated cover funds can increase both, objective utility and subjective attractiveness.

Across-time risk-aware strategies for outperforming a benchmark

We propose a novel objective function for constructing dynamic investment strategies with the goal of outperforming an investment benchmark at multiple points of evaluation during the investment time horizon. The proposed objective is intuitive, easy to parameterize, and directly targets a favorable tracking difference of the actively managed portfolio relative to the benchmark. Under stylized assumptions, we derive closed-form optimal investment strategies to guide the intuition in more realistic settings. In the case of discrete rebalancing with investment constraints, optimal strategies are obtained using a neural network-based numerical approach that does not rely on dynamic programming techniques. Compared to the targeting of a favorable tracking difference relative to the benchmark only at some fixed time horizon, our results show that the proposed objective offers a number of advantages: (i) The associated optimal strategies exhibit potentially more attractive asset allocation profiles, in that less extreme positions in individual assets are taken early in the investment time horizon, while achieving a similar terminal terminal wealth distribution. (ii) Across-time risk awareness leads to more robust performance and a higher probability of benchmark outperformance during the investment horizon in out-of-sample testing. The resulting strategies therefore exhibit desirable characteristics for active portfolio managers with periodic reporting requirements.

DOLLAR COST AVERAGING RETURNS ESTIMATION

Given a geometric Brownian motion wealth process, a log-Normal lower bound is constructed for the returns of a regular investing schedule. The distribution parameters of this bound are computed recursively. For dollar cost averaging (equal amounts in equal time intervals), parameters are computed in closed form. A lump sum (single amount at time 0) investing schedule is described which achieves a terminal wealth distribution that matches the wealth distribution indicated by the lower bound. Results are applied to annual returns of the S&P Composite Index from the last 150 years. Among data analysis results, the probability of negative returns is less than 2.5% when annual dollar cost averaging lasts over 40 years.

Optimal asset allocation in retirement planning: threshold-based utility maximization

PurposeThe aim of this paper is to modify the shape of utility functions traditionally used in expected utility theory (EUT) to derive optimal retirement saving decisions. Inspired by current reference point based approaches, the authors argue that utility functions with jumps or kinks at certain threshold points might very well be rational.Design/methodology/approachThe authors suggest an alternative to typical utility functions used in EUT, to be applied in the context of retirement saving decisions. The authors argue that certain elements that are used to model biases in behavioral models should–in the context of optimal retirement saving decisions–be considered “rational” and hence be included in a normative setting as well. The authors compare the optimal asset allocation derived under such utility functions with results under traditional power utility.FindingsThe authors find that the considered threshold levels can have a significant impact on the optimal investment decision for some individuals. In particular, the authors show that a much riskier investment than under EUT can become optimal if some level of income is secured by a social security and a significant portion of the distribution of terminal wealth lies below this level.Originality/valueContrary to previous work, this model is especially designed to assess the question of optimal product choice/asset allocation in the specific setting of retirement planning and from a normative point of view. In this regard, the authors first motivate the use of several thresholds and then apply this approach in a capital market model with stochastic stocks and stochastic interest rates to two illustrative investment alternatives.

Portfolio optimization with a prescribed terminal wealth distribution

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck equations with a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We then consider adding consumption during the investment process, to take into account distributions that are either not attainable, or sub-optimal.

Optimal asset allocation for outperforming a stochastic benchmark target

We propose a data-driven Neural Network (NN) optimization framework to determine the optimal multi-period dynamic asset allocation strategy for outperforming a general stochastic target. We formulate the problem as an optimal stochastic control with an asymmetric, distribution shaping, objective function. The proposed framework is illustrated with the asset allocation problem in the accumulation phase of a defined contribution pension plan, with the goal of achieving a higher terminal wealth than a stochastic benchmark. We demonstrate that the data-driven approach is capable of learning an adaptive asset allocation strategy directly from historical market returns, without assuming any parametric model of the financial market dynamics. The optimal adaptive strategy outperforms the benchmark constant proportion strategy, achieving a higher terminal wealth with a 90% probability, a 46% higher median terminal wealth, and a significantly more right-skewed terminal wealth distribution.

Comonotonic Approximations for the Sum of Log Unified Skew Normal Random Variables: Application in Finance and Actuarial Science

The classical works in finance and insurance for modeling asset returns is the Gaussian model. However, when modeling complex random phenomena, more flexible distributions are needed which are beyond the normal distribution. This is because most of the financial and economic data are skewed and have ＂fat tails＂. Hence symmetric distributions like normal or others may not be good choices while modeling these kinds of data. Flexible distributions like skew normal distribution allow robust modeling of high-dimensional multimodal and asymmetric data. In this paper, we consider a very flexible financial model to construct comonotonic lower convex order bounds in approximating the distribution of the sums of dependent log skew normal random variables. The dependence structure of these random variables is based on a recently developed generalized multivariate skew normal distribution, known as unified skew normal distribution. The approximations are used to calculate the risk measure related to the distribution of terminal wealth. The accurateness of the approximation is investigated numerically. Results obtained from our methods are competitive with a more time consuming method known as Monte Carlo method.

Optimal Asset Allocation for Outperforming a Stochastic Benchmark Target

We propose a data-driven Neural Network (NN) optimization framework to determine the optimal multi-period dynamic asset allocation strategy for outperforming a general stochastic target. We formulate the problem as an optimal stochastic control with an asymmetric, distribution shaping, objective function. The proposed framework is illustrated with the asset allocation problem in the accumulation phase of a defined contribution pension plan, with the goal of achieving a higher terminal wealth than a stochastic benchmark. We demonstrate that the data-driven approach is capable of learning an adaptive asset allocation strategy directly from historical market returns, without assuming any parametric model of the financial market dynamics. Following the optimal adaptive strategy, investors can make allocation decisions simply depending on the current state of the portfolio. The optimal adaptive strategy outperforms the benchmark constant proportion strategy, achieving a higher terminal wealth with a 90% probability, a 46% higher median terminal wealth, and a significantly more right-skewed terminal wealth distribution. We further demonstrate the robustness of the optimal adaptive strategy by testing the performance of the strategy on bootstrap resampled market data, which has different distributions compared to the training data.

Portfolio Optimisation within a Wasserstein Ball

We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy's risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lie within a Wasserstein ball surrounding a benchmark's -- being distributionally close -- and that have a specified dependence/copula -- tying state-by-state outcomes -- to it. The investor then chooses the alternative strategy that minimises a distortion risk measure of terminal wealth. In a general (complete) market model, we prove that an optimal dynamic strategy exists and provide its characterisation through the notion of isotonic projections. We further propose a simulation approach to calculate the optimal strategy's terminal wealth, making our approach applicable to a wide range of market models. Finally, we illustrate how investors with different copula and risk preferences invest and improve upon the benchmark using the Tail Value-at-Risk, inverse S-shaped, and lower- and upper-tail distortion risk measures as examples. We find that investors' optimal terminal wealth distribution has larger probability masses in regions that reduce their risk measure relative to the benchmark while preserving the benchmark's structure.

Portfolio Optimization With a Prescribed Terminal Wealth Distribution

This paper studies a portfolio allocation problem, where the goal is to prescribe the wealth distribution at final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which we solve by a gradient descent algorithm. This involves solving an associated HJB and Fokker–Planck equation by a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We next consider adding consumption during the investment process, to take into account distribution that either not attainable, or sub-optimal.

Dynamic portfolio allocation in goals-based wealth management

We report a dynamic programming algorithm which, given a set of efficient (or even inefficient) portfolios, constructs an optimal portfolio trading strategy that maximizes the probability of attaining an investor’s specified target wealth at the end of a designated time horizon. Our algorithm also accommodates periodic infusions or withdrawals of cash with no degradation to the dynamic portfolio’s performance or runtime. We explore the sensitivity of the terminal wealth distribution to restricting the segment of the efficient frontier available to the investor. Since our algorithm’s optimal strategy can be on the efficient frontier and is driven by an investor’s wealth and goals, it soundly beats the performance of target date funds in attaining investors’ goals. These optimal goals-based wealth management strategies are useful for independent financial advisors to implement behavioral-based FinTech offerings and for robo-advisors.

Dynamic Portfolio Allocation in Goals-Based Wealth Management

Given any set of exogenously provided efficient portfolios, we develop a dynamic programming algorithm that constructs an optimal portfolio trading strategy to maximize the probability of attaining an investor’s specified goal wealth at the end of a designated timeframe. Our algorithm can also accommodate periodic infusions or withdrawals of any size with no degradation in runtime performance. We explore how the terminal wealth distribution is sensitive to restrictions on the segment of the portfolio’s efficient frontier made available to the investor. Because our algorithm’s optimal strategy is on the efficient frontier, allowed to depend on the investor’s wealth, and allowed to depend on the investor’s individual goals and specifications, we show that it soundly beats the performance of target date funds for attaining investors’ goals. These optimal goals-based wealth management strategies are useful for modern day FinTech offerings, both advisor-driven or robo-driven.

Practical Applications of A Wealth Management Perspective on Factor Premia and the Value of Downside Protection

<h3>Practical Applications Summary</h3> Capital preservation is one of the major concerns of long-term investing, and how best to protect a portfolio against drawdowns is one of its most vexing challenges. <b>Louis Scott</b> and <b>Stefano Cavaglia</b> turned to factor premia, or smart beta, and studied its effects on a 30-year, goals-based investment strategy. They find that, to date, there is limited but very suggestive evidence that supports the downside protection properties of factor premia, and hence their ability to mitigate drawdowns. In their article, <b>A Wealth Management Perspective on Factor Premia and the Value of Downside Protection</b> published in the Spring 2017 issue of <b><i>The Journal of Portfolio Management</i></b>, the authors provide a framework for empirically estimating the value of downside protection offered by a combination of conventional factor premia (value, size, momentum, and quality) overlaid on a portfolio of the developed global equity universe. They find that the premia help reduce the likelihood that investor will lose a significant portion of initial capital by the time of retirement, even if the expected return of the premia were reduced by half. They find also that the premia enhance the distribution of terminal wealth. Finally, their analysis shows that active asset allocation strategies require market-timing skill beyond the reach of most investors in order to outperform a factor premia–based strategy. <b>TOPICS:</b>Analysis of individual factors/risk premia, portfolio construction, emerging, risk management

Hitting Times with Respect to Time-Dependent Barriers

In this short note, we prove that the distribution function of a hitting time of a Brownian Motion with respect to a differentiable, lower threshold solves a Volterra integral equation of second type. Moreover, we give an explicit expression for the kernel of the integral equation in terms of the boundary function and demonstrate how to find a solution of that equation (i.e. the distribution function) using Banach's Fixed-Point Theorem.As a second step, based on the first result, we find an integral representation of the terminal wealth distribution attached to the following, self-financing strategy: Let S_t be standard Geometric Brownian Motion stopped as soon as S_t hits a lower, time-dependent threshold. Then invest all of S_t in the money market account.

Dynamic Portfolio Optimisation with Intermediate Costs: A Least-Squares Monte Carlo Simulation Approach

We propose a dynamic portfolio optimisation strategy that takes liquidity cost into account. Our liquidity cost model is built upon the so-called Marginal Supply-Demand Curve which describes the asset price as a function of the trading volume. We extend the least-squares Monte Carlo algorithm to a stochastic control problem with switching costs and endogenous state variables. This approach is simulation-based, with great flexibility in the choice of investor objective, asset dynamics, portfolio constraints, intermediate consumption and incorporation of different sources of costs, making it easy to implement in practice. We study a portfolio investing in four major sectors in the U.S. market. We benchmark our dynamic strategy against several alternative portfolio strategies in terms of the out-of-sample distribution of terminal wealth and the real market performance. Overall, our dynamic strategy outperforms other standard asset allocation strategies.

Less is more: Increasing retirement gains by using an upside terminal wealth constraint

We solve a portfolio selection problem of an investor with a deterministic savings plan who aims to have a target wealth value at retirement. The investor is an expected power utility-maximizer. The target wealth value is the maximum wealth that the investor can have at retirement.By constraining the investor to have no more than the target wealth at retirement, we find that the lower quantiles of the terminal wealth distribution increase, so the risk of poor financial outcomes is reduced. The drawback of the optimal strategy is that the possibility of gains above the target wealth is eliminated.

Risk-adjusted impact of administrative costs on the distribution of terminal wealth for long-term investment.

The impact of administrative costs on the distribution of terminal wealth is approximated using a simple formula applicable to many investment situations. We show that the reduction in median returns attributable to administrative fees is usually at least twice the amount of the administrative costs charged for most investment funds, when considering a risk-adjustment correction over a reasonably long-term time horizon. The example we present covers a number of standard cases and can be applied to passive investments, mutual funds, and hedge funds. Our results show investors the potential losses they face in performance due to administrative costs.

Expected utility and the non-normal returns of common portfolio rebalancing strategies

This study compares three common strategies: buy-hold, constant mix and constant proportion rebalancing separately for bull, bear and trendless markets using Monte Carlo simulation. These strategies are compared in terms of terminal wealth, risk and expected utility. Our results indicate that rankings of the strategies by general expected utility functions, defined across all moments of the distribution of terminal wealth, often differ from rankings by mean-variance statistics. As rebalancing can produce non-normal payoffs, relying purely on traditional mean-variance analysis may cause investors to select inappropriate rebalancing strategies.

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.

Optimal portfolios when volatility can jump

We consider an asset allocation problem in a continuous-time model with stochastic volatility and jumps in both the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. The demand for jump risk includes a hedging component, which is not present in models without volatility jumps. We further show that the introduction of derivative contracts can have substantial economic value. We also analyze the distribution of terminal wealth for an investor who uses the wrong model, either by ignoring volatility jumps or by falsely including such jumps, or who is subject to estimation risk. Whenever a model different from the true one is used, the terminal wealth distribution exhibits fatter tails and (in some cases) significant default risk.