## Papers in international peer-reviewed journals

### S. Cerreia-Vioglio, F. Ortu, F. Severino, C. Tebaldi (2023)** Multivariate Wold decompositions: a Hilbert ***A*-module approach. *Decisions in Economics and Finance.* Accepted manuscript.

*A*-module approach.

Abstract. Orthogonal decompositions are essential tools for the study of weakly stationary time series. Some examples are given by the Classical Wold Decomposition of Wold (1938) and the Extended Wold Decomposition of Ortu, Severino, Tamoni and Tebaldi (2020), which permits to disentangle shocks with heterogeneous degrees of persistence from a given weakly stationary process. The analysis becomes more involved when dealing with vector processes because of the presence of different simultaneous shocks. In this paper, we recast the standard treatment of multivariate time series in terms of Hilbert A-modules (where matrices replace the field of scalars) and we prove the Abstract Wold Theorem for self-dual pre-Hilbert A-modules with an isometric operator. This theorem allows us to easily retrieve the Multivariate Classical Wold Decomposition and the multivariate version of the Extended Wold Decomposition. The theory helps in handling matrix coefficients and computing orthogonal projections on closed submodules. The orthogonality notion is key to decompose the given vector process into uncorrelated subseries and it implies a variance decomposition.

### F. Severino, M.A. Cremona, É. Dadié (2022) **COVID-19 effects on the Canadian term structure of interest rates.** *Review of Economic Analysis *14(4), 471-502.

Abstract. In Canada, COVID-19 pandemic triggered exceptional monetary policy interventions by the central bank, which in March 2020 made multiple unscheduled cuts to itstarget rate. In this paper we assess the extent to which Bank of Canada interventions affected the determinants of the yield curve. In particular, we apply Functional Principal Component Analysis to the term structure of interest rates. We find that, during the pandemic, the long-run dependence of level and slope components of the yield curve is unchanged with respect to previous months, although the shape of the mean yield curve completely changed after target rate cuts. Bank of Canada was effective in lowering the whole yield curve and correcting the inverted hump of previous months, but it was not able to reduce the exposure to already existing long-run risks.

### S. Cerreia-Vioglio, F. Ortu, F. Rotondi, F. Severino (2022) **On horizon-consistent mean-variance portfolio allocation.** *Annals of Operations Research*. Accepted manuscript.

Abstract. We analyze the problem of constructing multiple buy-and-hold mean-variance portfolios over increasing investment horizons in continuous-time arbitrage-free stochastic interest rate markets. The orthogonal approach to the one-period mean-variance optimization of Hansen and Richard (1987) requires the replication of a risky payoff for each investment horizon. When many maturities are considered, a large number of payoffs must be replicated, with an impact on transaction costs. In this paper, we orthogonally decompose the whole processes defined by asset returns to obtain a mean-variance frontier generated by the same two securities across a multiplicity of horizons. Our risk-adjusted mean-variance frontier rests on the martingale property of the returns discounted by the log-optimal portfolio and features a horizon consistency property. The outcome is that the replication of a single risky payoff is required to implement such frontier at any investment horizon. As a result, when transaction costs are taken into account, our risk-adjusted mean-variance frontier may outperform the traditional mean-variance optimal strategies in terms of Sharpe ratio. Realistic numerical examples show the improvements of our approach in medium- or long-term cashflow management, when a sequence of target returns at increasing investment horizons is considered.

### F. Ortu, F. Severino, A. Tamoni, C. Tebaldi (2020) **A persistence-based Wold-type decomposition for stationary time series**. *Quantitative Economics *11(1), 203-230*. *Paper, supplement and code.

Abstract. This paper shows how to decompose weakly stationary time series into the sum, across time scales, of uncorrelated components associated with different degrees of persistence. In particular, we provide an Extended Wold Decomposition based on an isometric scaling operator that makes averages of process innovations. Thanks to the uncorrelatedness of components, our representation of a time series naturally induces a persistence-based variance decomposition of any weakly stationary process. We provide two applications to show how the tools developed in this paper can shed new light on the determinants of the variability of economic and financial time series.

### M. Marinacci, F. Severino (2018) **Weak time-derivatives and no-arbitrage pricing**.* Finance and Stochastics *22(4), 1007-1036. Accepted manuscript.

Abstract. We prove a risk-neutral pricing formula for a large class of semimartingale processes through a novel notion of weak time-differentiability that permits to differentiate adapted processes. In particular, the weak time-derivative isolates drifts of semimartingales and is null for martingales. Weak time-differentiability enables us to characterize no arbitrage prices as solutions of differential equations, where interest rates play a key role. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (2009) by employing weak time-derivatives.

### F. Severino (2016)** Isometric operators on Hilbert spaces and Wold decomposition of stationary time series**. *Decisions in Economics and Finance* 39(2), 203-234. A preliminary version.

Abstract. The Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.

## Book chapters

### F. Severino, S. Thierry (2022) **Robo-advisors: A big data challenge. **In book: *Big Data in finance: Opportunities and challenges of financial digitalization *by *T. Walker, F. Davis and T. Schwartz, Palgrave-Macmillan.*

Abstract. At the frontier of personal finance and Fintech, robo-advisors aim to provide customized portfolio strategies without human intervention. They typically propose passive strategies that can match the investor’s objectives and risk profile at a low cost. However, digital advisors feature a lack of precision in capturing clients’ attitude towards risk and a (not always suitable) low risk exposure. In this context, leveraging big data and artificial intelligence techniques can improve the main strength of robo-advisors, that is, their ability to automatically provide personalized investment solutions. Text data from dialogue systems, such as chatbots, can be employed to improve the client’s profiling, while recommendation systems can rely on big data from financial social networks to propose targeted investment strategies. Analysis of big data through machine learning methods can also improve the performance of the optimization algorithms employed by digital advisors. The potential for the exploitation of big data and artificial intelligence in automated asset management is still enormous.

### D. Di Virgilio, F. Ortu, F. Severino, C. Tebaldi (2019) **Optimal asset allocation with heterogeneous persistent shocks and myopic and intertemporal hedging demand**. In book: *Behavioral finance: the coming of age* by *I. Venezia, World Scientific*. A preliminary version.

## Working papers

### F. Severino: **Long-term risk with stochastic interest rates.**

Abstract. Investors with heterogeneous trading horizons require compensation for the exposure to different risks. The no-arbitrage valuation over increasing horizons is described by the evolution of stochastic discount factors (SDFs). Each of them exhibits a multiplicative decomposition into deterministic growth term, permanent and transient component, provided by Hansen and Scheinkman (2009). In particular, the growth rate captures the deterministic discounting for risks that are relevant in the long term. When interest rates in the market are constant, the SDF growth rate coincides with the instantaneous rate. On the contrary, when rates of interest are stochastic, the SDF growth rate is given by the long-term yield of zero-coupon bonds, which is unsuitable for instantaneous no-arbitrage valuation.

We show how to reconcile the long-run properties of the SDF with the instantaneous rela- tions between returns and rates in stochastic-rate markets. In particular, we introduce a rate adjustment in pricing that isolates the short-term variability of rates. No-arbitrage prices are then factorized into rate-adjusted prices and a rate adjustment that is absent when interest rates are constant. Rate-adjusted prices employ constant yields to maturity for discounting future payoffs over time. The rate-adjusted SDF features the same long-term growth rate of the SDF in the market but has no transient component in its Hansen-Scheinkman decomposition. Therefore, rate-adjusted prices provide the proper valuation for long-term interest rate risk. Moreover, we show how this novel notion is fruitful for managing the interest rate risk related to fixed-income derivatives, life insurances and annuities.

### M. Madotto, F. Severino:** Heterogeneous awareness in financial markets.**

Abstract. The overlook of some economic scenarios may result in unforeseen negative outcomes for investors. In this paper, we consider an order-driven financial market in which a fraction of the traders is only partially aware of the possible payoffs of a risky asset, but is aware of the possibility of facing unknown contingencies. Investors decide whether to acquire a costly signal about the payoff of a risky asset and whether to buy such asset given their awareness level and their perceived relations among signals, order flows, and prices. We show that as unawareness becomes more severe, the value of the signal to the partially aware traders diminishes. In turn, through its impact on the price, the reduced number of partially aware informed investors increases the incentives of the fully aware to acquire the signal. This may partly or fully offset the previous effect, so that the overall proportion of informed investors in the market is (weakly) decreasing in the unawareness level. As for the equilibrium price, a lower amount of informed traders makes it more difficult for market makers to distinguish between good and bad signals, and this brings the conditional expectations of the price closer to the unconditional one and reduces the price variance.

## Work in progress

### F. Ortu, P. Reggiani, F. Severino:** Persistence-based portfolio choice along the FOMC cycle.**

Abstract. The Federal Reserve holds two main sets of monetary policy meetings, the ‘Federal Open Market Committee’ (FOMC) and the ‘Board Meetings’, which gather with 6-week and 2-week cadence respectively. Cieslak, Morse, and Vissing-Jorgensen (2019) show that the cadence of these meetings is associated with cycles of corresponding frequency in stock markets. These can be fruitfully exploited through a portfolio strategy that invests in the whole market at alternate weeks (‘even-week strategy’). This simple investment rule is based on the cycles identified empirically but, so far, lacks a theoretical foundation. In this paper, we provide a rigorous framework to detect cycles in the stock market, and to determine optimal portfolio choices which profit from such cycles. We use the filtering approach for stationary time series of Ortu, Severino, Tamoni and Tebaldi (2020) to isolate uncorrelated components of stock returns that are precisely associated with two- and six-week cycles. Then, we replicate these components using tradeable assets from the U.S. market, and design an optimal portfolio strategy that maximizes the investor’s wealth and outperforms the even-week strategy.

### F. Ortu, F. Severino, C. Tebaldi:** Persistence-based Beveridge-Nelson decomposition**.

Abstract. Given an integrated process, the Beveridge-Nelson decomposition allows to isolate a permanent component (or random walk) from a cyclical one. We formalize the nature of the random walk in a functional analytical setting in which this component is a functional reached by taking a weak limit over time. Such functional shares many of the features of the Wiener process in a discrete-time setting. Hence, we provide a characterization of integrated process, based on their asymptotic behaviour in discrete time, as an alternative to the Functional Central Limit Theorem approach. In this framework we show that the convergence to the random walk can be obtained by taking the limit over increasingly wider time scales, by exploiting the persistence-based decomposition of Ortu, Severino, Tamoni and Tebaldi. After providing an Extended Beveridge-Nelson Decomposition, we describe several ways to distinguish weakly stationary from integrated processes by analysing the heterogeneous layers of persistence.