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arXiv2013

Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem

We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Ito diffusion controlled by a nondecreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality and we characterize the optimal solution in terms of the base capacity process, i.e. the unique solution of a representation problem in the spirit of Bank and El Karoui (2004). We show that the base capacity is deterministic and it is identified with the free boundary of the associated optimal stopping problem, when the coefficients of the controlled diffusion are deterministic functions of time. This is a novelty in the literature on finite horizon singular stochastic control problems. As a subproduct this result allows us to obtain an integral equation for the free boundary, which we explicitly solve in the infinite horizon case for a Cobb-Douglas production function and constant coefficients in the controlled capacity process.

math.OC↗

arXiv2011

Diversification Return, Portfolio Rebalancing, and the Commodity Return Puzzle

Diversification return is an incremental return earned by a rebalanced portfolio of assets. The diversification return of a rebalanced portfolio is often incorrectly ascribed to a reduction in variance. We argue that the underlying source of the diversification return is the rebalancing, which forces the investor to sell assets that have appreciated in relative value and buy assets that have declined in relative value, as measured by their weights in the portfolio. In contrast, the incremental return of a buy-and-hold portfolio is driven by the fact that the assets that perform the best become a greater fraction of the portfolio. We use these results to resolve two puzzles associated with the Gorton and Rouwenhorst index of commodity futures, and thereby obtain a clear understanding of the source of the return of that index. Diversification return can be a significant source of return for any rebalanced portfolio of volatile assets.

q-fin.PM↗

arXiv2012

Directional Variance Adjustment: improving covariance estimates for high-dimensional portfolio optimization

Robust and reliable covariance estimates play a decisive role in financial and many other applications. An important class of estimators is based on Factor models. Here, we show by extensive Monte Carlo simulations that covariance matrices derived from the statistical Factor Analysis model exhibit a systematic error, which is similar to the well-known systematic error of the spectrum of the sample covariance matrix. Moreover, we introduce the Directional Variance Adjustment (DVA) algorithm, which diminishes the systematic error. In a thorough empirical study for the US, European, and Hong Kong market we show that our proposed method leads to improved portfolio allocation.

q-fin.PM↗

arXiv2012

Using MOEAs To Outperform Stock Benchmarks In The Presence of Typical Investment Constraints

Portfolio managers are typically constrained by turnover limits, minimum and maximum stock positions, cardinality, a target market capitalization and sometimes the need to hew to a style (such as growth or value). In addition, portfolio managers often use multifactor stock models to choose stocks based upon their respective fundamental data. We use multiobjective evolutionary algorithms (MOEAs) to satisfy the above real-world constraints. The portfolios generated consistently outperform typical performance benchmarks and have statistically significant asset selection.

q-fin.PM↗

arXiv2013

Forward Exponential Performances: Pricing and Optimal Risk Sharing

In a Markovian stochastic volatility model, we consider financial agents whose investment criteria are modelled by forward exponential performance processes. The problem of contingent claim indifference valuation is first addressed and a number of properties are proved and discussed. Special attention is given to the comparison between the forward exponential and the backward exponential utility indifference valuation. In addition, we construct the problem of optimal risk sharing in this forward setting and solve it when the agents' forward performance criteria are exponential.

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arXiv2011

Investment Volatility: A Critique of Standard Beta Estimation and a Simple Way Forward

Beta is a widely used quantity in investment analysis. We review the common interpretations that are applied to beta in finance and show that the standard method of estimation - least squares regression - is inconsistent with these interpretations. We present the case for an alternative beta estimator which is more appropriate, as well as being easier to understand and to calculate. Unlike regression, the line fit we propose treats both variables in the same way. Remarkably, it provides a slope that is precisely the ratio of the volatility of the investment's rate of return to the volatility of the market index rate of return (or the equivalent excess rates of returns). Hence, this line fitting method gives an alternative beta, which corresponds exactly to the relative volatility of an investment - which is one of the usual interpretations attached to beta.

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arXiv2011

The Capital Asset Pricing Model as a corollary of the Black-Scholes model

We consider a financial market in which two securities are traded: a stock and an index. Their prices are assumed to satisfy the Black-Scholes model. Besides assuming that the index is a tradable security, we also assume that it is efficient, in the following sense: we do not expect a prespecified self-financing trading strategy whose wealth is almost surely nonnegative at all times to outperform the index greatly. We show that, for a long investment horizon, the appreciation rate of the stock has to be close to the interest rate (assumed constant) plus the covariance between the volatility vectors of the stock and the index. This contains both a version of the Capital Asset Pricing Model and our earlier result that the equity premium is close to the squared volatility of the index.

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arXiv2012

Pricing and Semimartingale Representations of Vulnerable Contingent Claims in Regime-Switching Markets

Using a suitable change of probability measure, we obtain a novel Poisson series representation for the arbitrage- free price process of vulnerable contingent claims in a regime-switching market driven by an underlying continuous- time Markov process. As a result of this representation, along with a short-time asymptotic expansion of the claim's price process, we develop an efficient method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path-dependent claims that we term self-decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the pre-default price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kac representations for the pre-default pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures.

q-fin.CP↗

arXiv2012

Long Horizons, High Risk Aversion, and Endogeneous Spreads

For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk-neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.

q-fin.PM↗

arXiv2012

Optimal investment with intermediate consumption and random endowment

We consider a problem of optimal investment with intermediate consumption and random endowment in an incomplete semimartingale model of a financial market. We establish the key assertions of the utility maximization theory assuming that both primal and dual value functions are finite in the interiors of their domains as well as that random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.

q-fin.PM↗

arXiv2011

Suitability of using technical indicators as potential strategies within intelligent trading systems

The potential of machine learning to automate and control nonlinear, complex systems is well established. These same techniques have always presented potential for use in the investment arena, specifically for the managing of equity portfolios. In this paper, the opportunity for such exploitation is investigated through analysis of potential simple trading strategies that can then be meshed together for the machine learning system to switch between. It is the eligibility of these strategies that is being investigated in this paper, rather than application. In order to accomplish this, the underlying assumptions of each trading system are explored, and data is created in order to evaluate the efficacy of these systems when trading on data with the underlying patterns that they expect. The strategies are tested against a buy-and-hold strategy to determine if the act of trading has actually produced any worthwhile results, or are simply facets of the underlying prices. These results are then used to produce targeted returns based upon either a desired return or a desired risk, as both are required within the portfolio-management industry. Results show a very viable opportunity for exploitation within the aforementioned industry, with the Strategies performing well within their narrow assumptions, and the intelligent system combining them to perform without assumptions.

q-fin.PM↗

arXiv2017

Optimizing expected utility of dividend payments for a Cramér-Lundberg risk proces

We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cramér-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function fulfills the Hamilton-Jacobi-Bellman equation and we identify the optimal dividend strategy.

q-fin.CP↗

arXiv2013

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.

q-fin.PM↗

arXiv2011

An analytical performance comparison of exchanged traded funds with index funds: 2002-2010

Exchange Traded Funds (ETFs) have been gaining increasing popularity in the investment community as is evidenced by the high growth both in the number of ETFs and their net assets since 2000. As ETFs are in nature similar to index mutual funds, in this paper we examined if this growing demand for ETFs can be explained through their outperformance as compared to index mutual funds. We considered the population of all ETFs with inception dates prior to 2002 and then for each ETF found all the passive index mutual funds that had the same investment style as the selected ETF and had inception date prior to 2002. Within each investment style we matched every ETF with all the passive index funds in that investment style and compared the performances of the matched pairs in terms of Sharp Ratios and risk adjusted buy and hold total returns for the period 2002-2010. We then applied the Wilcoxon signed rank test to examine if ETFs had better performances than index mutual funds during the sample period. Out of the 230 paired matches of all the styles, ETFs outperformed index mutual funds in 134 of the times in terms of Sharpe Ratio, however, the test of the hypothesis showed no statistically significant difference between ETFs and index funds performances in terms of Sharpe ratio. Out of the 230 paired matches of all the styles, ETFs outperformed index mutual funds in 125 of the times in terms of risk adjusted buy and hold total return, however, the test of hypothesis showed no statistically significant difference between ETFs and index funds performances in terms of risk adjusted buy and hold total return. These findings indicate there is statistically no significant difference between ETFs and passive index mutual funds performances at the fund level and investors' choice between the two is related to product characteristics and tax advantages.

q-fin.PM↗

arXiv2011

Time-Inconsistent Stochastic Linear--Quadratic Control

In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we find an explicit equilibrium control. As an application, we then consider a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic.

math.OC↗

arXiv2011

Copula-based Hierarchical Aggregation of Correlated Risks. The behaviour of the diversification benefit in Gaussian and Lognormal Trees

The benefits of diversifying risks are difficult to estimate quantitatively because of the uncertainties in the dependence structure between the risks. Also, the modelling of multidimensional dependencies is a non-trivial task. This paper focuses on one such technique for portfolio aggregation, namely the aggregation of risks within trees, where dependencies are set at each step of the aggregation with the help of some copulas. We define rigorously this procedure and then study extensively the Gaussian Tree of quite arbitrary size and shape, where individual risks are normal, and where the Gaussian copula is used. We derive exact analytical results for the diversification benefit of the Gaussian tree as a function of its shape and of the dependency parameters. Such a "toy-model" of an aggregation tree enables one to understand the basic phenomena's at play while aggregating risks in this way. In particular, it is shown that, for a fixed number of individual risks, "thin" trees diversify better than "fat" trees. Related to this, it is shown that hierarchical trees have the natural tendency to lower the overall dependency with respect to the dependency parameter chosen at each step of the aggregation. We also show that these results hold in more general cases outside the gaussian world, and apply notably to more realistic portfolios (LogNormal trees). We believe that any insurer or reinsurer using such a tool should be aware of these systematic effects, and that this awareness should strongly call for designing trees that adequately fit the business. We finally address the issue of specifying the full joint distribution between the risks. We show that the hierarchical mechanism does not require nor specify the joint distribution, but that the latter can be determined exactly (in the Gaussian case) by adding conditional independence hypotheses between the risks and their sums.

q-fin.RM↗

arXiv2013

Recovering Model Structures from Large Low Rank and Sparse Covariance Matrix Estimation

Many popular statistical models, such as factor and random effects models, give arise a certain type of covariance structures that is a summation of low rank and sparse matrices. This paper introduces a penalized approximation framework to recover such model structures from large covariance matrix estimation. We propose an estimator based on minimizing a non-likelihood loss with separable non-smooth penalty functions. This estimator is shown to recover exactly the rank and sparsity patterns of these two components, and thus partially recovers the model structures. Convergence rates under various matrix norms are also presented. To compute this estimator, we further develop a first-order iterative algorithm to solve a convex optimization problem that contains separa- ble non-smooth functions, and the algorithm is shown to produce a solution within O(1/t^2) of the optimal, after any finite t iterations. Numerical performance is illustrated using simulated data and stock portfolio selection on S&P 100.

stat.ME↗

arXiv2012

Performance-based regularization in mean-CVaR portfolio optimization

We introduce performance-based regularization (PBR), a new approach to addressing estimation risk in data-driven optimization, to mean-CVaR portfolio optimization. We assume the available log-return data is iid, and detail the approach for two cases: nonparametric and parametric (the log-return distribution belongs in the elliptical family). The nonparametric PBR method penalizes portfolios with large variability in mean and CVaR estimations. The parametric PBR method solves the empirical Markowitz problem instead of the empirical mean-CVaR problem, as the solutions of the Markowitz and mean-CVaR problems are equivalent when the log-return distribution is elliptical. We derive the asymptotic behavior of the nonparametric PBR solution, which leads to insight into the effect of penalization, and justification of the parametric PBR method. We also show via simulations that the PBR methods produce efficient frontiers that are, on average, closer to the population efficient frontier than the empirical approach to the mean-CVaR problem, with less variability.

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