Robust multivariate portfolio choice with stochastic covariance in the presence of ambiguity

This paper provides the optimal multivariate intertemporal portfolio for an ambiguity averse investor, who has access to stocks and derivative markets, in closed form. The stock prices follow stochastic covariance processes and the investor can have different levels of uncertainty about the diffusion parts of the stocks and the covariance structure. We find strong evidence that the optimal exposures to stock and covariance risks are significantly affected by ambiguity aversion. Welfare analyses show that investors who ignore model uncertainty incur large losses, larger than those suffered under the embedded one-dimensional cases. We further confirm large welfare losses from not trading in derivatives as well as ignoring intertemporal hedging, we study the impact of ambiguity in that regard and justify the importance of including these factors in the scope of portfolio optimization. Conditions are provided for a well-behaved solution in general, together with verification theorems for the incomplete market case.     

HEDGING INTEREST RATE RISK UNDER TERM STRUCTURE EFFECTS: AN APPLICATION TO FINANCIAL INSTITUTIONS

In this paper we investigate the use of the structured full rank model for hedging the balance sheet of a financial institution. Simulation results suggest that the optimal hedge is insensitive to changes in parameter estimates. In addition, we hedge a portfolio of Treasury bills using both the full information covariance matrix and the structured covariance matrix. We then contrast these results with those obtained from a duration-based model. Empirical results suggest that the structured full rank model is generally more effective in hedging applications than either the full information model or duration-based model.     

Is stochastic volatility relevant for dynamic portfolio choice under ambiguity?

Literature on dynamic portfolio choice has been finding that volatility risk has low impact on portfolio choice. For example, using long-run US data, Chacko and Viceira [2005. “Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets.” The Review of Financial Studies 18 (4): 1369–1402] found that intertemporal hedging demand (required by investors for protection against adverse changes in volatility) is empirically small even for highly risk-averse investors. We want to assess if this continues to be true in the presence of ambiguity. Adopting robust control and perturbation theory techniques, we study the problem of a long-horizon investor with recursive preferences that faces ambiguity about the stochastic processes that generate the investment opportunity set. We find that ambiguity impacts portfolio choice, with the relevant channel being the return process. Ambiguity about the volatility process is only relevant if, through a specific correlation structure, it also induces ambiguity about the return process. Using the same long-run US data, we find that ambiguity about the return process may be empirically relevant, much more than ambiguity about the volatility process. Anyway, intertemporal hedging demand is still very low: investors are essentially focused on the short-term risk–return characteristics of the risky asset.     

A moment computation algorithm for the error in discrete dynamic hedging

This paper develops a computational approach to determining the moments of the distribution of the error in a dynamic hedging or payoff replication strategy under discrete trading. In particular, an algorithm is developed for portfolio affine trading strategies, which lead to portfolio dynamics that are affine in the portfolio variable. This structure can be exploited in the computation of moments of the hedging error of such a strategy, leading to a lattice based backward recursion similar in nature to lattice based pricing techniques, but not requiring the portfolio variable. We use this algorithm to analyze the performance of portfolio affine hedging strategies under discrete trading through the moments of the hedging error.     

Investment decisions when utility depends on wealth and other attributes

The problem of optimal investment under a multivariate utility function allows for an investor to obtain utility not only from wealth, but other (possibly correlated) attributes. In this paper we implement multivariate mixtures of exponential (mixex) utility to address this problem. These utility functions allow for stochastic risk aversions to differing states of the world. We derive some new results for certainty equivalence in this context. By specifying different distributions for stochastic risk aversions, we are able to derive many known, plus several new utility functions, including models of conditional certainty equivalence and multivariate generalisations of HARA utility, which we call dependent HARA utility. Focusing on the case of asset returns and attributes being multivariate normal, we optimise the asset portfolio, and find that the optimal portfolio consists of the Markowitz portfolio and hedging portfolios. We provide an empirical illustration for an investor with a mixex utility function of wealth and sentiment.     

Optimal consumption and asset allocation with unknown income growth

Recent empirical evidence supports the view that the income process has an individual-specific growth rate component [Baker, M., 1997. Growth-rate heterogeneity and the covariance structure of life-cycle earnings. Journal of Labor Economics 15, 338-375; Guvenen, F., 2007b. Learning your earning: Are labor income shocks really very persistent? American Economic Review 97, 687-712; Huggett, M., Ventura, G., Yaron, A., 2007. Sources of life-cycle inequality. Working paper, University of Pennsylvania]. Moreover, the individual-specific growth component may be stochastic. Motivated by these empirical observations, I study an individual's optimal consumption-saving and portfolio choice problem when he does not observe his income growth. As in standard income fluctuation problems, the individual cannot fully insure himself against income shocks. In addition to the standard income-risk-induced precautionary saving demand, the individual also has learning-induced precautionary saving demand, which is greater when belief is more uncertain. With constant unobserved income growth, changes in belief are not predictable. However, with stationary stochastic income growth, belief is no longer a martingale. Mean reversion of belief reduces hedging demand on average and in turn mitigates the impact of estimation risk on consumption-saving and portfolio decisions.     

Optimizing international portfolios with options and forwards

We develop a stochastic programming model to address in a unified manner a number of interrelated decisions in international portfolio management: optimal portfolio diversification and mitigation of market and currency risks. The goal is to control the portfolio’s total risk exposure and attain an effective balance between risk and expected return. By incorporating options and forward contracts in the portfolio optimization model we are able to numerically assess the performance of alternative tactics for mitigating exposure to the primary risks. We find that control of market risk with options has more significant impact on portfolio performance than currency hedging. We demonstrate through extensive empirical tests that incremental benefits, in terms of reducing risk and generating profits, are gained when both the market and currency risks are jointly controlled through appropriate means.     

Deep hedging

We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in Section 4 that the set of constrained trading strategies used by our algorithm is large enough to ε-approximate any optimal solution. Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available. We illustrate our approach by an experiment on the S&P500 index and by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard ‘complete-market’ solution.     

A mixed C-vine copula model for hedging price and volumetric risk in wind power trading

When energy trading companies enter into long-term agreements with wind power producers, where a fixed price is paid for the fluctuating production, they are facing a joint price and volumetric risk. Since the pay-off of such agreements is non-linear, a hedging portfolio would ideally consist of not only forwards, but also a basket of e.g. call and put options. Illiquidity and an almost non-existent market for options challenge however the optimal hedging of joint price and volumetric risk in many market places. Here, we consider the case of the Danish power market, and exploit its strong positive correlation with the much more liquid German market to construct a proxy hedge. We propose a three-dimensional mixed vine copula to model the evolution of the Danish and German spot electricity prices and the Danish wind power production. We construct a realistic hedging portfolio by identifying various instruments available in the market, such as real options in the form of the right to transfer electricity across the border and the right to convert electricity to heat. Using the proposed vine copula to determine optimal hedging decisions, we show that significant benefits are to be drawn by extending the hedging portfolio with the proposed instruments.     

Modern portfolio management with conditioning information

This paper studies models in which active portfolio managers utilize conditioning information unavailable to their clients to optimize performance relative to a benchmark. We derive explicit solutions for the optimal strategies with multiple risky assets, with or without a risk-free asset, and consider various constraints on portfolio risks or weights. The optimal strategies feature a mean–variance efficient component (to minimize portfolio variance), and a hedging demand for the benchmark portfolio (to maximize correlation with the benchmark). A currency portfolio example shows that the optimal strategies improve the measured performance by 53% out of sample, compared with portfolios ignoring conditioning information.     

Optimal investment under multi-factor stochastic volatility

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with a multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that investors who invest myopically, ignore derivative assets, model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.     

Dynamic correlations and volatility effects in the Balkan equity markets

This paper investigates the presence of time-varying comovements, volatility implications and dynamic correlations in major Balkan and leading mature equity markets, in order to provide quantified responses to international asset allocation decisions. Since asset returns and correlation dynamics are critical inputs in asset pricing, portfolio management and risk hedging, emphasis is placed on the respective (constant and dynamic) equity market correlations produced by alternative multivariate GARCH forms, the Constant Conditional Correlation and the Asymmetric Dynamic Conditional Correlation models. The Balkan stock markets are seen to exhibit time-varying correlations as a peer group, although correlations with the mature markets remain relatively modest. In conjunction with sensitivity analysis on the asymmetric variance–covariance matrix, active portfolio diversification to the Balkan equity markets indicates to potentially improve investors’ risk-return trade-off.     

HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.     

Hedging and competition

We consider firms that, all else equal, wish to minimize variability in their internal capital (due to convex costs of raising external funds). The firms can hedge the cash flow risk of the project, but not that of winning or losing the auction. We characterize optimal hedging and bidding strategies in this competition framework. We show that access to financial markets makes firms bid more aggressively, possibly even above their valuation for the project. In addition, hedging increases the variance of bids and makes firm values more dispersed. Further, with hedging, the covariance of internal capital changes with the risk factor is negative, and is more negative, the higher the correlation of the hedging instrument with the risk factor.     

Income hedging and portfolio decisions

We examine whether the decision to participate in the stock market and other related portfolio decisions are influenced by income hedging motives. Economic theory predicts that the market participation propensity should increase as the correlation between income growth and stock market returns decreases. Surprisingly, empirical studies find limited support for the income hedging motive. Using a rich, unique Dutch data set and the National Longitudinal Survey of the Youth (NLSY) from the United States, we show that when the income-return correlation is low, individuals exhibit a greater propensity to participate in the market and allocate a larger proportion of their wealth to risky assets. Even when the income risk is high, individuals exhibit a higher propensity to participate in the market when the hedging potential is high. These findings suggest that income hedging is an important determinant of stock market participation and asset allocation decisions.     

Hedging efficiently under correlation

We show that when a derivative portfolio has different correlated underlyings, hedging using classical greeks (first-order derivatives) is not the best possible choice. We first show how to adjust greeks to take correlation into account and reduce P&L volatility. Then we embed correlation-adjusted greeks in a global hedging strategy that reduces cost of hedging without increasing P&L volatility, by optimization of hedge re-adjustments. The strategy is justified in terms of a balance between transaction costs and risk-aversion, but, unlike more complex proposals from previous literature, it is completely defined by observable parameters, geometrically intuitive, and easy to implement for an arbitrary number of risk factors. We test our findings on a CVA hedging example. We first consider daily re-hedging: in this test, correlation-adjusted greeks allow the reduction of P&L volatility by more than 30% compared to standard deltas. Then we apply our general strategy to a context where a CVA portfolio is exposed to both credit and interest rate risk. The strategy keeps P&L volatility in line with daily standard delta-hedging, but with massive cost-saving: only six rebalances of the illiquid credit hedge are performed, over a period of six months.     

Hedging and Reserving for Single-Premium Segregated Fund Contracts

Abstract

Three methods for determining suitable provision for maturity guarantees for single-premium segregated fund contracts are compared. Actuarial reserving assumes funds are held in risk-free assets, to give a prescribed probability of meeting the guarantee liability. Dynamic hedging uses the Black-Scholes framework to determine the replicating portfolio. Static hedging assumes a counterparty is willing to sell the options required to meet the guarantee. Using a stochastic cash flow projection, we consider how to assess which approach is most profitable. The example given assumes a typical Canadian segregated fund contract.     

Up and down credit risk

This paper discusses the main modeling approaches that have been developed for handling portfolio credit derivatives, with a focus on the question of hedging. In particular, the so-called top, top down and bottom up approaches are considered. We give some mathematical insights regarding the fact that information, namely the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multi-name derivative by single-name derivatives. We then illustrate by means of numerical simulations (semi-static hedging experiments) why and when the portfolio loss process may not be a ‘sufficient statistic’ for the purpose of valuation and hedging of portfolio credit risk.     

An international analysis of REITs and stock portfolio management based on dynamic conditional correlation models

This paper analyzes the volatility spillovers and asymmetry between REITs and stock prices for nine countries (Australia, Belgium, Germany, Italy, Japan, The Netherlands, Singapore, the United Kingdom, and the United States) using eight different multivariate GARCH models. We also analyze the optimal weights, hedging effectiveness, and hedge ratios for REIT-stock portfolio holdings with respect to the results. The empirical results indicate that dynamic conditional correlation (DCC) models provide a better fit than the constant conditional correlation models. The DCC with volatility spillovers and asymmetry (DCC-SA) model provides a better fit than the other multivariate GARCH models. The DCC-SA model also provides the best hedging effectiveness for all pairs of REIT-stock assets. More importantly, this result holds for all cases and for all models that we consider, which means that by taking spillover and asymmetry into consideration, hedging effectiveness can be vastly improved.