Fabio Trojani - Academia.edu (original) (raw)

Papers by Fabio Trojani

Consider a relaxed multinomial setup, in which there may be mistakes in observing the outcomes of... more Consider a relaxed multinomial setup, in which there may be mistakes in observing the outcomes of the process-this is often the case in real applications. What can we say about the next outcome if we start learning about the process in conditions of prior ignorance? To answer this question we extend the imprecise Dirichlet model to the case of imperfect observations and we focus on posterior predictive probabilities for the next outcome. The results are very surprising: the posterior predictive probabilities are vacuous, irrespectively of the amount of observations we do, and however small is the probability of doing mistakes. In other words, the imprecise Dirichlet model cannot help us to learn from data when the observational mechanism is imperfect. This result seems to rise a serious question about the use of the imprecise Dirichlet model for practical applications, and, more generally, about the possibility to learn from imperfect observations under prior ignorance.

SSRN Electronic Journal, 2000

SSRN Electronic Journal, 2000

ABSTRACT Conventional tests of present-value models tend to over-reject the null of no predictabi... more ABSTRACT Conventional tests of present-value models tend to over-reject the null of no predictability, concluding that price-dividend ratio variations are due to both cash flow and discount rate shocks. We propose a nonparametric Monte Carlo testing method, which does not rely on distributional assumptions to aggregate the information from the time series of price-dividend ratios and dividend growth. We find evidence of return predictability, but no apparent evidence of dividend growth predictability, thus reconciling the diverging conclusions in the literature. Our findings are robust to the specification of the predictive information set and account for the intrinsic probability of detecting predictive relations by chance alone.

SSRN Electronic Journal, 2000

In a present-value model with time-varying risks, we develop a latent variable approach to estima... more In a present-value model with time-varying risks, we develop a latent variable approach to estimate expected market returns and dividend growth consistently with the conditional risk features implied by present-value constraints. We find a timevarying expected dividend growth and expected return, but while the explained fraction of dividend variability is low the predicted portion of return variation is large. Expected dividend growth is more persistent than expected returns and generates large price-dividend ratio components that mask the predictive power for future returns. The model implies (i) predictive regressions consistent with a weak return predictability and a missing dividend predictability by aggregate price-dividend ratios, (ii) predictable market volatilities, (iii) volatile and often counter-cyclical Sharpe ratios, (iv) a time-varying and on average increasing term structure of stock market risk and (v) stocks that can appear as less volatile in the long run using standard testing procedures. These findings show the importance of controlling for time-varying risks and the potential long-run effect of persistent dividend forecasts when studying predictive relations.

SSRN Electronic Journal, 2000

We develop a continuous time general equilibrium yield curve model under ambiguity aversion.

SSRN Electronic Journal, 2000

We solve analytically the Merton's problem of an investor with timeadditive power utility. For ge... more We solve analytically the Merton's problem of an investor with timeadditive power utility. For general state dynamics, we prove existence of two power series representations of the relevant optimal policies and value functions, which hold for all admissible risk aversion parameters. We characterize all terms in the power series by a recursive formula, allowing analytical computations to arbitrary order. Some applications to explicit model settings highlight a very satisfactory accuracy of ¯nite order approximations provided by our power series solution approach.

Journal of Econometrics, 2005

Applied Optimization, 2002

Only a few intertemporal optimal consumption and portfolio problems in partial and general equili... more Only a few intertemporal optimal consumption and portfolio problems in partial and general equilibrium can be solved explicitly. It is illustrated in the paper that perturbation theory is a powerful tool for deriving approximate analytical solutions for the desired optimal policies in problems where general state dynamics are admitted and a preference for robustness is present. Starting from the perturbative approach proposed recently by Kogan and Uppal it is demonstrated how robust equilibria for some formulations of a preference for robustness in the literature can be solved. A crucial requirement for this approach is the existence of a known functional form for the candidate model solutions, a condition which is not satisfied by some models of a preference for robustness. For these cases, £ Fabio Trojani gratefully acknowledges the financial support of the Swiss National Science Foundation (grant 1214-056679).

SSRN Electronic Journal, 2000

We analyze and compare analytically continuous-time financial equilibria where heterogeneous risk... more We analyze and compare analytically continuous-time financial equilibria where heterogeneous risk averse investors care about model misspecification through some preference for robustness and in the presence of a stochastic opportunity set. This incorporates a concern for model misspecification into equilibrium asset prices. Since no exact equilibrium computations are possible in this model setting, perturbation theory is used to provide first order asymptotics for the implied equilibria. We find that to first order robustness enhances effective risk aversion while keeping constant the preference for intertemporal substitution. Therefore, equilibrium consumption, equilibrium capital stock dynamics (in production economies) and equilibrium stock price processes (in exchange economies) are not directly modified by a preference for robustness. By contrast, robustness affects directly optimal portfolios, causing lower equilibrium interest rates -and thereby enhanced risk premia -when the speculative investment motive dominates the intertemporal hedging demand. Finally, at variance with other robustness specifications, definitions of robustness that mimic Knightian uncertainty produce state dependent effective risk aversions that generate first order risk aversion effects on optimal portfolios, equilibrium interest rates and equity premia. This yields functional forms for some key equilibrium variables like equity premia which are structurally different from those implied by standard risk aversion or other robustness definitions, which reflect all second order risk aversion. Moreover, under Knightian uncertainty the structure of an equilibrium depends strongly on the completeness of the underlying economy. For instance, within complete production economies we find that Knightian uncertainty can generate an endogenous stock market participation, a feature that cannot be obtained by the other robustness definitions. The richness of the equilibrium effects generated in our heterogenous economies suggests that definitions of robustness which mimic Knightian uncertainty can generate the largest variety of robust economic behaviours in the presence of model uncertainty.

SSRN Electronic Journal, 2000

SSRN Electronic Journal, 2000

We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance op... more We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance optimization of assets and liabilities (AL). This makes the analysis more challenging than in a setting based on purely exogenous liabilities. We show that under general conditions for the joint AL dynamics the arising optimal policies and MVF can be decomposed into an orthogonal set of basis returns using exterior algebra. The geometric representation in Leippold, for the exogenous liabilities case follows as a special case. Using a numerical example, we illustrate our methodology by studying the impact of the rebalancing frequency on the MVF and by highlighting the main differences between exogenous and endogenous liabilities.

Review of Financial Studies, 2006

We propose a new continuous-time framework for studying asset prices under learning and ambiguity... more We propose a new continuous-time framework for studying asset prices under learning and ambiguity aversion. In a Lucas economy with time-additive power utility, a discount for ambiguity arises for a relative risk aversion below one or, equivalently, an intertemporal elasticity of substitution above one. The joint presence of learning and ambiguity enforces large equity premia and model predictions consistent with well-known asset pricing puzzles.

Review of Finance, 2000

We analyze the empirical predictions arising from settings of ambiguity aversion in intertemporal... more We analyze the empirical predictions arising from settings of ambiguity aversion in intertemporal heterogenous agents economies. We study equilibria for two tractable wealth-homothetic settings of ambiguity aversion in continuous time. Such settings are motivated by a different robust control optimization problem. We show that ambiguity aversion affects optimal portfolio exposures in a way that is similar to an increase in risk aversion. A distinct property of the second of our settings of ambiguity aversion is that such increase is state-dependent and highly pronounced at moderate portfolio exposures. This feature causes quite prudent levels of equity market participation over a nontrivial set of states of the economy. In general equilibrium, ambiguity aversion tends to induce a higher equilibrium equity premium and lower interest rates. A distinct feature of the second of our settings of ambiguity aversion is that the equity premium part due to ambiguity aversion dominates when the exogenous random factors in the economy have low volatility. Thus, such setting can account for some distinct empirical predictions -like a limited equity market participation and ambiguity equity premia that dominate equity premia for small equity volatilities -which are unavailable under the first of our settings of ambiguity aversion.

Quantitative Finance, 2011

Journal of Economic Dynamics and Control, 2006

Journal of Economic Dynamics and Control, 2004

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization... more 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 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 multi-periods 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.

Journal of Economic Dynamics and Control, 2002

The paper presents a robust version of a simple two-assets Merton's 1969 model where the optimal ... more The paper presents a robust version of a simple two-assets Merton's 1969 model where the optimal choices and the implied shadow market prices of risk for a representative robust decision maker RDM can be easily described.

SSRN Electronic Journal, 2002

European Financial Management, 2002

A typical problem arising in the financial planning for private investors consists in the fact th... more A typical problem arising in the financial planning for private investors consists in the fact that the initial investor's portfolio, the one determined by the consulting process of the financial institution and the universe of instruments made available to the investor have to be matched/optimized when determining the relevant portfolio choice. We call this problem the three-portfolios matching problem. Clearly, the resulting portfolio selection should be as close as possible to the optimal asset allocation determined by the consulting process of the financial institution. However, the transition from the investor's initial portfolio to the final one is complicated by the presence of transaction costs and some further more specific constraints. Indeed, usually the portfolios under consideration are structured at different aggregation levels, making portfolios comparison and matching more difficult. Further, several investment restrictions have to be satisfied by the final portfolio choice. Finally, the arising portfolio selection process should be sufficiently transparent in order to incorporate the subjective investor's trade-off between the objectives 'optimal portfolio matching' and 'minimal portfolio transition costs'. In this paper, we solve the three-portfolios matching problem analytically for a simplified setting that illustrates the main features of the arising solutions and numerically for the more general situation.

Consider a relaxed multinomial setup, in which there may be mistakes in observing the outcomes of... more Consider a relaxed multinomial setup, in which there may be mistakes in observing the outcomes of the process-this is often the case in real applications. What can we say about the next outcome if we start learning about the process in conditions of prior ignorance? To answer this question we extend the imprecise Dirichlet model to the case of imperfect observations and we focus on posterior predictive probabilities for the next outcome. The results are very surprising: the posterior predictive probabilities are vacuous, irrespectively of the amount of observations we do, and however small is the probability of doing mistakes. In other words, the imprecise Dirichlet model cannot help us to learn from data when the observational mechanism is imperfect. This result seems to rise a serious question about the use of the imprecise Dirichlet model for practical applications, and, more generally, about the possibility to learn from imperfect observations under prior ignorance.

SSRN Electronic Journal, 2000

SSRN Electronic Journal, 2000

ABSTRACT Conventional tests of present-value models tend to over-reject the null of no predictabi... more ABSTRACT Conventional tests of present-value models tend to over-reject the null of no predictability, concluding that price-dividend ratio variations are due to both cash flow and discount rate shocks. We propose a nonparametric Monte Carlo testing method, which does not rely on distributional assumptions to aggregate the information from the time series of price-dividend ratios and dividend growth. We find evidence of return predictability, but no apparent evidence of dividend growth predictability, thus reconciling the diverging conclusions in the literature. Our findings are robust to the specification of the predictive information set and account for the intrinsic probability of detecting predictive relations by chance alone.

SSRN Electronic Journal, 2000

In a present-value model with time-varying risks, we develop a latent variable approach to estima... more In a present-value model with time-varying risks, we develop a latent variable approach to estimate expected market returns and dividend growth consistently with the conditional risk features implied by present-value constraints. We find a timevarying expected dividend growth and expected return, but while the explained fraction of dividend variability is low the predicted portion of return variation is large. Expected dividend growth is more persistent than expected returns and generates large price-dividend ratio components that mask the predictive power for future returns. The model implies (i) predictive regressions consistent with a weak return predictability and a missing dividend predictability by aggregate price-dividend ratios, (ii) predictable market volatilities, (iii) volatile and often counter-cyclical Sharpe ratios, (iv) a time-varying and on average increasing term structure of stock market risk and (v) stocks that can appear as less volatile in the long run using standard testing procedures. These findings show the importance of controlling for time-varying risks and the potential long-run effect of persistent dividend forecasts when studying predictive relations.

SSRN Electronic Journal, 2000

We develop a continuous time general equilibrium yield curve model under ambiguity aversion.

SSRN Electronic Journal, 2000

We solve analytically the Merton's problem of an investor with timeadditive power utility. For ge... more We solve analytically the Merton's problem of an investor with timeadditive power utility. For general state dynamics, we prove existence of two power series representations of the relevant optimal policies and value functions, which hold for all admissible risk aversion parameters. We characterize all terms in the power series by a recursive formula, allowing analytical computations to arbitrary order. Some applications to explicit model settings highlight a very satisfactory accuracy of ¯nite order approximations provided by our power series solution approach.

Journal of Econometrics, 2005

Applied Optimization, 2002

Only a few intertemporal optimal consumption and portfolio problems in partial and general equili... more Only a few intertemporal optimal consumption and portfolio problems in partial and general equilibrium can be solved explicitly. It is illustrated in the paper that perturbation theory is a powerful tool for deriving approximate analytical solutions for the desired optimal policies in problems where general state dynamics are admitted and a preference for robustness is present. Starting from the perturbative approach proposed recently by Kogan and Uppal it is demonstrated how robust equilibria for some formulations of a preference for robustness in the literature can be solved. A crucial requirement for this approach is the existence of a known functional form for the candidate model solutions, a condition which is not satisfied by some models of a preference for robustness. For these cases, £ Fabio Trojani gratefully acknowledges the financial support of the Swiss National Science Foundation (grant 1214-056679).

SSRN Electronic Journal, 2000

We analyze and compare analytically continuous-time financial equilibria where heterogeneous risk... more We analyze and compare analytically continuous-time financial equilibria where heterogeneous risk averse investors care about model misspecification through some preference for robustness and in the presence of a stochastic opportunity set. This incorporates a concern for model misspecification into equilibrium asset prices. Since no exact equilibrium computations are possible in this model setting, perturbation theory is used to provide first order asymptotics for the implied equilibria. We find that to first order robustness enhances effective risk aversion while keeping constant the preference for intertemporal substitution. Therefore, equilibrium consumption, equilibrium capital stock dynamics (in production economies) and equilibrium stock price processes (in exchange economies) are not directly modified by a preference for robustness. By contrast, robustness affects directly optimal portfolios, causing lower equilibrium interest rates -and thereby enhanced risk premia -when the speculative investment motive dominates the intertemporal hedging demand. Finally, at variance with other robustness specifications, definitions of robustness that mimic Knightian uncertainty produce state dependent effective risk aversions that generate first order risk aversion effects on optimal portfolios, equilibrium interest rates and equity premia. This yields functional forms for some key equilibrium variables like equity premia which are structurally different from those implied by standard risk aversion or other robustness definitions, which reflect all second order risk aversion. Moreover, under Knightian uncertainty the structure of an equilibrium depends strongly on the completeness of the underlying economy. For instance, within complete production economies we find that Knightian uncertainty can generate an endogenous stock market participation, a feature that cannot be obtained by the other robustness definitions. The richness of the equilibrium effects generated in our heterogenous economies suggests that definitions of robustness which mimic Knightian uncertainty can generate the largest variety of robust economic behaviours in the presence of model uncertainty.

SSRN Electronic Journal, 2000

SSRN Electronic Journal, 2000

We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance op... more We study the optimal policies and mean-variance frontiers (MVF) of a multiperiod mean-variance optimization of assets and liabilities (AL). This makes the analysis more challenging than in a setting based on purely exogenous liabilities. We show that under general conditions for the joint AL dynamics the arising optimal policies and MVF can be decomposed into an orthogonal set of basis returns using exterior algebra. The geometric representation in Leippold, for the exogenous liabilities case follows as a special case. Using a numerical example, we illustrate our methodology by studying the impact of the rebalancing frequency on the MVF and by highlighting the main differences between exogenous and endogenous liabilities.

Review of Financial Studies, 2006

We propose a new continuous-time framework for studying asset prices under learning and ambiguity... more We propose a new continuous-time framework for studying asset prices under learning and ambiguity aversion. In a Lucas economy with time-additive power utility, a discount for ambiguity arises for a relative risk aversion below one or, equivalently, an intertemporal elasticity of substitution above one. The joint presence of learning and ambiguity enforces large equity premia and model predictions consistent with well-known asset pricing puzzles.

Review of Finance, 2000

We analyze the empirical predictions arising from settings of ambiguity aversion in intertemporal... more We analyze the empirical predictions arising from settings of ambiguity aversion in intertemporal heterogenous agents economies. We study equilibria for two tractable wealth-homothetic settings of ambiguity aversion in continuous time. Such settings are motivated by a different robust control optimization problem. We show that ambiguity aversion affects optimal portfolio exposures in a way that is similar to an increase in risk aversion. A distinct property of the second of our settings of ambiguity aversion is that such increase is state-dependent and highly pronounced at moderate portfolio exposures. This feature causes quite prudent levels of equity market participation over a nontrivial set of states of the economy. In general equilibrium, ambiguity aversion tends to induce a higher equilibrium equity premium and lower interest rates. A distinct feature of the second of our settings of ambiguity aversion is that the equity premium part due to ambiguity aversion dominates when the exogenous random factors in the economy have low volatility. Thus, such setting can account for some distinct empirical predictions -like a limited equity market participation and ambiguity equity premia that dominate equity premia for small equity volatilities -which are unavailable under the first of our settings of ambiguity aversion.

Quantitative Finance, 2011

Journal of Economic Dynamics and Control, 2006

Journal of Economic Dynamics and Control, 2004

We present a geometric approach to discrete time multiperiod mean variance portfolio optimization... more 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 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 multi-periods 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.

Journal of Economic Dynamics and Control, 2002

The paper presents a robust version of a simple two-assets Merton's 1969 model where the optimal ... more The paper presents a robust version of a simple two-assets Merton's 1969 model where the optimal choices and the implied shadow market prices of risk for a representative robust decision maker RDM can be easily described.

SSRN Electronic Journal, 2002

European Financial Management, 2002

A typical problem arising in the financial planning for private investors consists in the fact th... more A typical problem arising in the financial planning for private investors consists in the fact that the initial investor's portfolio, the one determined by the consulting process of the financial institution and the universe of instruments made available to the investor have to be matched/optimized when determining the relevant portfolio choice. We call this problem the three-portfolios matching problem. Clearly, the resulting portfolio selection should be as close as possible to the optimal asset allocation determined by the consulting process of the financial institution. However, the transition from the investor's initial portfolio to the final one is complicated by the presence of transaction costs and some further more specific constraints. Indeed, usually the portfolios under consideration are structured at different aggregation levels, making portfolios comparison and matching more difficult. Further, several investment restrictions have to be satisfied by the final portfolio choice. Finally, the arising portfolio selection process should be sufficiently transparent in order to incorporate the subjective investor's trade-off between the objectives 'optimal portfolio matching' and 'minimal portfolio transition costs'. In this paper, we solve the three-portfolios matching problem analytically for a simplified setting that illustrates the main features of the arising solutions and numerically for the more general situation.