Variance Portfolio Research Articles

High-Frequency Covariance Matrix Estimation Using Price Durations

We propose a price duration based covariance matrix estimator using high frequency transactions data. The effect of the last-tick time-synchronisation methodology, together with effects of important market microstructure components is analysed through a comprehensive Monte Carlo study. To decrease the number of negative eigenvalues produced by the non positive-semi-definite (psd) covariance matrix, we devise an average covariance estimator by taking an average of a wide range of duration based covariance matrix estimators. Empirically, candidate covariance estimators are implemented on 19 stocks from the DJIA. The duration based covariance estimator is shown to provide comparably accurate estimates with smaller variation compared with competing estimators. An out-of-sample GMV portfolio allocation problem is studied. A simple shrinkage technique is introduced to make the sample matrices psd and well-conditioned. Compared to competing high-frequency covariance matrix estimators, the duration based estimator is shown to give more stable portfolio weights and higher Sharpe ratios while maintaining comparably low portfolio variances.

Dynamic hedging with futures: a copula-based GARCH model with high-frequency data

Modeling the joint distribution of spot and futures returns is crucial for establishing optimal hedging strategies. This paper proposes a new class of dynamic copula-GARCH models that exploits information from high-frequency data for hedge ratio estimation. The copula theory facilitates constructing a flexible distribution; the inclusion of realized volatility measures constructed from high-frequency data enables copula forecasts to swiftly adapt to changing markets. By using data concerning equity index returns, the estimation results show that the inclusion of realized measures of volatility and correlation greatly enhances the explanatory power in the modeling. Moreover, the out-of-sample forecasting results show that the hedged portfolios constructed from the proposed model are superior to those constructed from the prevailing models in reducing the (estimated) conditional hedged portfolio variance. Finally, the economic gains from exploiting high-frequency data for estimating the hedge ratios are examined. It is found that hedgers obtain additional benefits by including high-frequency data in their hedging decisions; more risk-averse hedgers generate greater benefits.

CoAnomaly: Correlation Risk in Stock Market Anomalies

I propose a simple time-series risk measure in trading stock market anomalies, CoAnomaly, the time-varying average pairwise correlation among 34 anomalies, which helps to explain both the time-series and the cross-sectional anomaly return patterns. Since correlations among underlying assets determine the portfolio variance, CoAnomaly is an important state variable for arbitrageurs who hold diversified portfolios of anomalies to boost their performance. Empirically, I show that, first, CoAnomaly is persistent and forecasts long-run aggregate volatility of the diversified anomaly portfolio. Second, CoAnomaly positively predicts future average anomaly returns in the time series. Third, in the cross-section of these 34 anomaly portfolios, CoAnomaly carries a negative price of risk. These return patterns suggest that arbitrageurs take the time-varying correlation into account and their intertemporal hedging demand plays an important role in setting asset prices.

Exchange Rate Targeting and Gold Demand by Central Banks: Modeling International Reserves Composition

This paper explores the composition of international reserves under a central bank’s exchange rate policy target. The model allows for numerical estimation of the shadow price – interpreted as the central bank’s sacrifice of policy precision given additional unit of portfolio variance or return – of the target exchange rate. The simulations indicate a two-regime demand for gold. Derived multiple equilibria suggest that the monetary authority could possibly have a wide percentage range of gold demand. Demanding a higher percentage of gold is more appropriate if the central bank prefers to accumulate international reserves for precautionary reason. The central bank would tend to incur greater exchange rate target sacrifice if it wants to achieve higher portfolio returns. The results suggest that ability to target the exchange rate is unaffected by the higher volatility of monthly returns on gold.

An Exact Test of the Improvement of the Minimum Variance Portfolio

AbstractI propose an exact finite sample test of the risk reduction of the global minimum variance (GMV) portfolio. The GMV test statistic is proportional to the reduction in the variance of the GMV portfolio and has a straightforward geometric and portfolio interpretation and complements the celebrated GRS test in Gibbons et al. (1989). In practical applications, the GMV test leads to a rejection of the null hypothesis of no improvement in the GMV portfolio more often than the GRS test rejects the null hypothesis of no improvement in the risk‐return profile of the tangent portfolio. The power of the GMV test increases with the variance reduction of the GMV portfolio. Using test asset returns scaled by predetermined predictive variables is equivalent to increasing the overall number of test assets and leads to substantial power gains.

Retail consumers and risk in centralized energy auctions for indexed long-term contracts in Chile

Centralized energy auctions for long-term contracts are commonly-used mechanisms to ensure supply adequacy, to promote competition, and to protect retail customers from price spikes in Latin America. In Chile, the law mandates that all distribution companies must hold long-term contracts—which are awarded on a competitive centralized auction—to cover 100% of the projected demand from three to fifteen years into the future. These contracts can be indexed to a series of financial parameters, including fossil fuel prices at reference locations. Drawing from portfolio theory, we use a simple example to illustrate the difficulties of selecting, through the current clearing mechanism that focuses on average costs and individual characteristics of the offers, a portfolio of long-term energy contracts that could simultaneously minimize the expected future cost of energy and limit the risk exposure of retail customers. In particular, we show that if the objective of the regulator is to limit the risk to regulated consumers, it could be optimal to include contracts that would not be selected based on individual characteristics of the offers and a least-cost auction objective, but that could significantly reduce the price variance of the overall portfolio due to diversification effects between indexing parameters.

Diversification and the Distribution of Portfolio Variance, Part 1: Sums of IID Variables and Higher-Order Moments

For a weighted sum of asset returns that are independent and identically distributed (IID) up to variance, we derive expressions linking the distribution of variance across assets with higher-order portfolio moments, assuming these quantities are finite. In particular, we show concise relationships for portfolio skewness and kurtosis and, within a general framework, review related metrics in the literature for measuring diversification.

Mean-Reverting Portfolio with Budget Constraint

This paper considers the mean-reverting portfolio (MRP) design problem arising from statistical arbitrage (a.k.a. pairs trading) in the financial markets. It aims at designing a portfolio of underlying assets by optimizing the mean reversion strength of the portfolio, while taking into consideration the portfolio variance and an investment budget constraint. Several specific design problems are considered based on different mean reversion criteria. Efficient algorithms are proposed to solve the problems. Numerical results on both synthetic and market data show that the proposed MRP design methods can generate consistent profits and outperform the traditional design methods and the benchmark methods in the literature.

A Modified Risk Parity Method for Asset Allocation

We propose a return based modification of the portfolio variance matrix for asset allocation using risk parity. The modification is based upon a single scalar parameter which can be tuned to tailor the allocation for desired expected risk and/or return. The present work contributes a new twist on risk parity. While classical risk parity methods are based exclusively on volatility, the new solution (Modified Risk Parity) considers both historical returns and their variance in the construction of an optimal, diversified investment portfolio. We present two examples for periods including the recent financial market crises. The results suggest that the modification may lead to significantly improved risk adjusted returns over those realized by the conventional risk parity method.

Analytic solution to variance optimization with no short positions

We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T of the time series of returns used to estimate risk. The no-short-selling constraint acts as an asymmetric regularizer, setting some of the portfolio weights to zero and keeping the out-of-sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the ban on short positions does not prevent the phase transition in the optimization problem, only shifts the critical point from its non-regularized value of to 2, and changes its character: at the out-of-sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes, while another critical parameter, related to the estimated portfolio weights and the condensate density, diverges at the critical value . We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one’s natural expectation.

An Exact Test of the Improvement of the Minimum Variance Portfolio

I propose an exact finite sample test of the risk reduction of the global minimum variance (GMV) portfolio. The GMV test statistic is proportional to the reduction in the variance of the GMV portfolio and has a straightforward geometric and portfolio interpretation and complements the celebrated GRS test in Gibbons et al. (1989). In practical applications, the GMV test leads to a rejection of the null hypothesis of no improvement in the GMV portfolio more often than the GRS test rejects the null hypothesis of no improvement in the risk‐return profile of the tangent portfolio. The power of the GMV test increases with the variance reduction of the GMV portfolio. Using test asset returns scaled by predetermined predictive variables is equivalent to increasing the overall number of test assets and leads to substantial power gains.

Risk Parity in the Brazilian Market

Using sectorial indices of the Brazilian market, we compare the portfolio optimization approach known as risk parity with minimum variance and equally weighted approaches. We apply various estimators for the covariance matrix to each portfolio strategy, since portfolio variance is considered as risk measure. Empirical results demonstrate that the risk parity approach provides more diversified portfolios and stable weights in the out-of-sample than the other two approaches, thereby avoiding the dangers of excessive concentration and reducing transaction costs. Furthermore, the results demonstrate that different estimators of the covariance matrix had little influence on the results obtained through the risk parity approach.

Hybrid bi-objective portfolio optimization with pre-selection strategy

Classical Markowitz mean-variance model is widely used for portfolio assets selection and allocation, which aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio variance. Many numerical approaches and metaheuristic algorithms have been proposed to effectively solve this portfolio optimization problem under an ideal condition. However, introducing various realistic constraints inadvertently leads to a non-convex search space, which has hindered the application of many classic, exact algorithms such as quadratic programming (QP). The increasing size of available assets and complex constraints has made the effectiveness of metaheuristic algorithms deteriorated. This paper proposes a hybrid bi-objective algorithm combining with the respective advantages of local search algorithm, evolutionary algorithm and QP with a pre-selection strategy. The algorithm first down select the assets that have greater contribution to the Pareto frontier by applying the pre-selection strategy. Then local search and evolutionary algorithm combined with QP are employed to fully exploit the useful assets combination modes to lead the search process toward the frontier direction quickly. The experimental study demonstrates that the proposed hybrid approach can obtain faster and better convergence compared with eight state-of the-art multi-objective evolutionary algorithms. The results also show that the proposed method with the pre-selection strategy always displays a closer proximity to the Pareto frontier compared with k-means strategy.

Focused Shrinkage Estimators for the Global Minimum Variance Portfolio

We propose a shrinkage estimator for covariance matrices designed to minimize estimation error of the Global Minimum Variance (GMV) portfolio. Implementing the GMV portfolio requires estimating the asset covariance matrix and using this to obtain variance-minimizing portfolio weights. Standard estimation approaches for this application utilize shrinkage. These estimators use shrinkage weights that are not designed to directly minimize estimation error of the final object of interest: GMV portfolio weights. We develop a focused shrinkage approach to the problem. This method utilizes the form of the trading rule to derive a shrinkage estimator that directly controls estimation error of GMV portfolio weights. Extensive simulations are conducted to compare performance with nine standard competitors. Our estimator uniformly outperformed all competitors across portfolios of various sizes. The methods are applied to several standard portfolios of US and international assets. Similar improvements are found. Our estimator achieves the smallest out-of-sample portfolio variance in 25 of 28 data sets considered.

Knowing What to Worry About: A Goals-BasedApplication of Fragility Detection Theory

Drawing from the theory of nonlinear risk and fragility detection, we analyze which portfolio management attributes should be of most concern to the practitioner in a goal-based setting. We begin by constructing a risk measurement mechanism using a Gaussian stochastic Monte Carlo process. We then analyze which portfolio inputs are most sensitive to errors. Counterintuitively, we find that portfolio variance is of least concern, whereas factors affecting the future required minimum wealth level are of greatest concern. We show how goal-based practitioners get the most bang for their buck by focusing on the accuracy of their inflation projections (not often a first focus) and portfolio variance last (often the first focus). These results can help both the practitioner and researcher dedicate resources to answering the questions that are of most importance when a future goal is at stake. <b>TOPICS:</b>Portfolio construction, simulations, tail risks, wealth management

Multi objective mean–variance–skewness model with Burg’s entropy and fuzzy return for portfolio optimization

A new non-Shannon fuzzy Mean–Variance–Skewness-entropy model is proposed with stock returns are considered as triangular fuzzy numbers. The fuzzy stock portfolio selection models are presented with credibility theory that maximizes mean and skewness and minimizes portfolio variance and cross-entropy in terms of Burg. With addition of Burg’s entropy in the multi objective non linear models, focus is the generation of well diversified portfolios within the optimal allocation. For an imprecise capital market, this study facilitates a more reasonable investment decisions with four objective decision criteria including Burg’s entropy. Numerical examples with case studies are used to illustrate the entire method which can be efficiently used in practical purposes like national stock exchanges.

Robust minimum variance portfolio optimization modelling under scenario uncertainty

Our purpose in this article is to develop a robust optimization model which minimizes portfolio variance for a finite set of covariance matrices scenarios. The proposed approach aims at the proper selection of portfolios, in a way that for every covariance matrix estimate included in the analysis, the calculated portfolio variance remains as close to the corresponding individual minimum value, as possible. To accomplish this, we formulate a mixed-integer non-linear program with quadratic constraints. With respect to practical underlying concerns, investment policy constraints regarding the portfolio structure are also taken into consideration. The validity of the proposed approach is verified through extensive out-of-sample empirical testing in the EuroStoxx 50, the S&P 100, the S&P 500, as well as a well-diversified investment universe of ETFs. We report consistent generation of stable out-of-sample returns, which are in most cases superior to those of the worst-case scenario. Moreover, we provide strong evidence that the proposed robust model assists in selective asset picking and systematic avoidance of excessive losses.

RISKY RISK MEASURES: A NOTE ON UNDERESTIMATING FINANCIAL RISK UNDER THE NORMAL ASSUMPTION

This note compares three different risk measures based on the same stock return data: (1) the portfolio variance as in Markowitz (1952), (2) the value at risk based on the historical return distribution, and (3) the value at risk based on a t copula. Unless return series follow a Normal distribution, Normal-based risk measures underestimate risk, particularly so during periods of market stress, when accurate risk measurement is essential. Based on these insights, we recommend that supervisors discontinue to accept Normal-based value at risk estimations. We are happy to share our commented R-code with practitioners who wish to implement our methodology. Risk measurement is the foundation of risk management and hence of vital importance in any financial institution. Supervisory capital requirements according to Basel III or Solvency II are also derived from risk measures. Investors are interested in ratings which are based on risk assessments. This note is therefore relevant to practitioners and supervisors alike.

Portfolio Selection and Dynamic Behavior in Heston's Stochastic Volatility Model Using a Contingent Claim

In the existing literature, little is known about the dynamic behavior of the optimal portfolio in terms of market inputs and arbitrary stochastic factor dynamics in an incomplete market with a stochastic volatility. In this paper, to study the optimal portfolio behavior, we compute and analyze the mean and the variance of the optimal portfolio and of its adjustment speed in terms of market inputs in an incomplete market. The incompleteness arises from the additional source of uncertainty of the volatility in Heston’s stochastic volatility model. We discuss a novel way of completing the incomplete market by introducing an artificial European option, which is then made completely uninteresting to the investor. We then apply the martingale method to the completed market to obtain the optimal portfolio in a closed form. Conducting sensitivity analysis for the mean and the variance of the optimal portfolio process as well as its adjustment speed to the market parameters, we find several interesting investor’s behavioral patterns toward asset price and its volatility shocks. Our results are robust and convergent by the agreement from two simulation methods for different time step increments and the numbers of Monte Carlo simulation paths.

Application of the phase‐type mortality law to life contingencies and risk management

The class of phase‐type distributions has recently gained much popularity in insurance applications due to its mathematical tractability and denseness in the class of distributions defined on positive real line. In this paper, we show how to use the phase‐type mortality law as an efficient risk management tool for various life insurance applications. In particular, pure premiums, benefit reserves, and risk‐loaded premiums using CTE for standard life insurance products are shown to be available in analytic forms, leading to efficient computation and straightforward implementation. A way to explicitly determine provisions for adverse deviation for interest rate and mortality is also proposed. Furthermore, we show how the interest rate risk embedded in life insurance portfolios can be analyzed via interest rate sensitivity index and diversification index which are constructed based on the decomposition of portfolio variance. We also consider the applicability of phase‐type mortality law under a few non‐flat term structures of interest rate. Lastly, we explore how other properties of phase‐type distributions may be applied to joint‐life products as well as subgroup risk ordering and pricing within a given pool of insureds. Copyright © 2017 John Wiley &amp; Sons, Ltd.