Fabio Trojani

Correlation Risk and Optimal Portfolio Choice

In this paper we solve an intertemporal portfolio problem with correlation risk, using a new approach for simultaneously modeling stochastic correlation and volatility. The solutions of the model are in closed form and include an optimal portfolio demand for hedging correlation risk. We calibrate the model and find that the optimal demand to hedge correlation risk is a non negligible fraction of the myopic portfolio, which often dominates the pure volatility hedging demand. The hedging demand for correlation risk is larger in settings with high average correlations and correlation variances. Moreover, it is increasing in the number of assets available for investment as the dimension of uncertainty with regard to the correlation structure becomes proportionally more important. JEL classification: D9, E3, E4, G12

Robust inference with GMM estimators

The local robustness properties of generalized method of moments (GMM) estimators and of a broad class of GMM based tests are investigated in a unified framework. GMM statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded. Since in many applications this function is unbounded, it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a GMM estimator or test. We show how this can be obtained when a reference model for the data distribution can be assumed. We develop a flexible algorithm for constructing a robust GMM (RGMM) estimator leading to stable GMM test statistics. The amount of robustness can be controlled by an appropriate tuning constant. We relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or (and) the power of a GMM test and the amount of contamination that one can reasonably assume given some information on the data. Finally, we illustrate the RGMM methodology with some simulations of an application to RGMM testing for conditional heteroscedasticity in a simple linear autoregressive model. In this example we find a significant instability of the size and the power of a classical GMM testing procedure under a non-normal conditional error distribution. On the other side, the RGMM testing procedures can control the size and the power of the test under non-standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution.

A Geometric Approach To Multiperiod Mean Variance Optimization of Assets and Liabilities

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization that largely simplifies the mathematical analysis and the economic interpretation of such model settings. We show that multiperiod mean variance optimal policies can be decomposed in an orthogonal set of basis strategies, each having a clear economic interpretation. This implies that the corresponding multi period mean variance frontiers are spanned by an orthogonal basis of dynamic returns. Specifically, in a k-period model the optimal strategy is a linear combination of a single k-period global minimum second moment strategy and a sequence of k local excess return strategies which expose the dynamic portfolio optimally to each single-period asset excess return. This decomposition is a multi period version of Hansen and Richard (1987) orthogonal representation of single-period mean variance frontiers and naturally extends the basic economic intuition of the static Markowitz model to the multiperiod context. Using the geometric approach to dynamic mean variance optimization we obtain closed form solutions in the i.i.d. setting for portfolios consisting of both assets and liabilities (AL), each modelled by a distinct state variable. As a special case, the solution of the mean variance problem for the asset only case in Li and Ng (2000) follows directly and can be represented in terms of simple products of some single period orthogonal returns. We illustrate the usefulness of our geometric representation of multiperiods optimal policies and mean variance frontiers by discussing specific issued related to AL portfolios: The impact of taking liabilities into account on the implied mean variance frontiers, the quantification of the impact of the investment horizon and the determination of the optimal initial funding ratio.

Ambiguity Aversion and the Term Structure of Interest Rates

This paper studies the term structure implications of a simple structural economy in which the representative agent displays ambiguity aversion, modeled by Multiple Priors Recursive Utility. Bond excess returns reflect a premium for ambiguity, which is observationally distinct from the risk premium of affine yield curve models. The ambiguity premium can be large even in the simplest logutility model and is non zero also for stochastic factors that have a zero risk premium. A calibrated low-dimensional two-factor economy with ambiguity is able to reproduce the deviations from the expectations hypothesis documented in the literature, without modifying in a substantial way the nonlinear mean reversion dynamics of the short interest rate. In this economy, we do not find any apparent tradeoffs between fitting the first and second moments of the yield curve and the large equity premium.

Economic Uncertainty, Disagreement, and Credit Markets

We study how the equilibrium risk sharing of agents with heterogeneous perceptions of aggregate consumption growth affects bond and stock returns. Although credit spreads and their volatilities increase with the degree of heterogeneity, the decreasing risk premium on moderately levered equity can produce a violation of basic capital structure no-arbitrage relations. Using bottom-up proxies of aggregate belief dispersion, we give empirical support to the model predictions and show that risk premia on corporate bond and stock returns are systematically explained by their exposures to aggregate disagreement shocks. This paper was accepted by Jerome Detemple, finance.

A note on robustness in Merton's model of intertemporal consumption and portfolio choice

The paper presents a robust version of a simple two-assets Merton (1969, Review of Economics and Statistics 51, 247-57) model where the optimal choices and the implied shadow market prices of risk for a representative robust decision maker (RDM) can be easily described. With the exception of the log-utility case, precautionary behaviour is induced in the optimal consumption-investment rules through a substitution of investment in risky assets with both current consumption and riskless saving. For the log-utility case, precautionary behaviour arises only through a substitution between risky and riskless assets. On the financial side, the decomposition of the market price of risk in a standard consumption based component and a further price for model uncertainty risk (which is positively related to the robustness parameter) is independent of the underlying risk aversion parameter.

Robust GMM Tests for Structural Breaks

We propose a class of new robust GMM tests for endogenous structural breaks. The tests are based on supremum, average and exponential functionals derived from robust GMM estimators with bounded influence function. We study the theoretical local robustness properties of the new tests and show that they imply a uniformly bounded asymptotic sensitivity of size and power under general local deviations from a reference model. We then analyze the finite sample performance of the new robust tests in some Monte Carlo simulations, and compare it with that of classical GMM tests for structural breaks. In large samples, we find that the performance of classical asymptotic GMM tests can be quite unstable already under slight departures from some given reference distribution. In particular, the loss in power can be substantial in some models. Robust asymptotic tests for structural breaks yield important power improvements already under slight local departures from the reference model. This holds both in exactly identified and overidentified model settings. In small samples, bootstrapped versions of both the classical and the robust GMM tests provide a very accurate and very stable empirical size also for quite small sample sizes. However, bootstrapped robust GMM tests are found to provide again a higher finite sample efficiency.

Robust GMM analysis of models for the short rate process

We re-examine the empirical evidence concerning a well-known class of one-factor models for the short rate process (cf. Chan et al. [Journal of Finance 47 (1992) 1209] (CKLS)) and some recent extensions allowing for a nonlinear drift and for changing parameters with a new statistical methodology based on robust statistics, the Robust Generalized Method of Moments (RGMM). We find that standard GMM model selection procedures are highly unstable in these applications. When testing the CKLS models with the RGMM we find that they are all clearly misspecified and we identify a clustering of influential observations in the 1979–1982 subperiod, a time span that is well known to coincide with a temporary change in the monetary policy of the Federal Reserve. This clustering of influential observations does not disappear when we introduce a non-linearity in the drift and allow for a parameter shift during the 1979–1982 period. Moreover, a Cox–Ingersoll–Ross model (selected by the RGMM) might offer a satisfactory data description for the period after 1982, since there only a few isolated outliers are found. Comparable results are obtained for the Euro-mark case.

Limits of learning about a categorical latent variable under prior near-ignorance

It is well known that complete prior ignorance is not compatible with learning, at least in a coherent theory of (epistemic) uncertainty. What is less widely known, is that there is a state similar to full ignorance, that Walley calls near-ignorance, that permits learning to take place. In this paper we provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of startingstatistical inferencein conditionsof very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is latent. We argue that such a setting is by far the most common case in practice, and we show, for the case of categorical latent variables (and general manifest variables) that there is a sufficient condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied in the most common statistical problems.

A General Multivariate Threshold GARCH Model With Dynamic Conditional Correlations

We introduce a new multivariate GARCH model with multivariate thresholds in conditional correlations and develop a two-step estimation procedure that is feasible in large dimensional applications. Optimal threshold functions are estimated endogenously from the data and the model conditional covariance matrix is ensured to be positive definite. We study the empirical performance of our model in two applications using U.S. stock and bond market data. In both applications our model has, in terms of statistical and economic significance, higher forecasting power than several other multivariate GARCH models for conditional correlations.