Adrian Pizzinga

A Row-Wise Stacking of the Runoff Triangle: State Space Alternatives for IBNR Reserve Prediction

This work deals with prediction of IBNR reserve under a different data ordering of the noncumulative runoff triangle. The rows of the triangle are stacked, resulting in a univariate time series with several missing values. Under this ordering, two approaches entirely based on state space models and the Kalman filter are developed, implemented with real data, and compared with a well-established IBNR estimation method – the chain ladder. The remarks from the empirical results are: (i) computational feasibility and efficiency; (ii) accuracy improvement for IBNR prediction; and (iii) flexibility regarding IBNR modeling possibilities. Key-words: IBNR, Kalman filter, mean square error, missing values, state space model. JEL codes: C22, C53, G22. IME codes: IM10, IM42. ∗Department of Electrical Engineering of the Pontifical Catholic University of Rio de Janeiro. †Corresponding author. Financial and Actuarial Risk Management Institute of the Pontifical Catholic University of Rio de Janeiro (IAPUC). E-mail: adrianhpster@gmail.com ‡Department of Electrical Engineering of the Pontifical Catholic University of Rio de Janeiro. 1

Diffuse Restricted Kalman Filtering

This article investigates how the use of an initial diffuse state vector affects the use of the Kalman smoother under linear restrictions. The contribution is twofold. First, using elementary linear and matrix algebra, it is established that it is still possible to obtain restricted smoothed state vectors in the “diffuse” period under quite general conditions. Second, using results from general conditional expectation theory, it is proven that the extension of the restricted Kalman smoother also preserves conditional statistical efficiency, given some meaningful information sets.

State space models for the exchange rate pass-through: determinants and null/full pass-through hypotheses

In this article, we formulate linear Gaussian state space models for the estimation of the exchange rate pass-through of the Brazilian Real against the US Dollar, using monthly data from August 1999 to August 2008. The state space/Kalman filtering framework allows the investigation of some empirical aspects previously suggested in the literature, such as time-varying coefficients and null/full pass-through hypotheses. We also test whether some theoretical ‘determinants’ of the pass-through are statistically significant in the period considered. The principal findings are as follows: (1) the data offer strong support to a time-varying pass-through; and (2) the variance of the exchange rate pass-through, the monetary policy and the trade flow have shown to be relevant determinants of the exchange rate pass-through.

A pairs trading strategy based on linear state space models and the Kalman filter

Among many strategies for financial trading, pairs trading has played an important role in practical and academic frameworks. Loosely speaking, it involves a statistical arbitrage tool for identifying and exploiting the inefficiencies of two long-term, related financial assets. When a significant deviation from this equilibrium is observed, a profit might result. In this paper, we propose a pairs trading strategy entirely based on linear state space models designed for modelling the spread formed with a pair of assets. Once an adequate state space model for the spread is estimated, we use the Kalman filter to calculate conditional probabilities that the spread will return to its long-term mean. The strategy is activated upon large values of these conditional probabilities: the spread is bought or sold accordingly. Two applications with real data from the US and Brazilian markets are offered, and even though they probably rely on limited evidence, they already indicate that a very basic portfolio consisting of a sole spread outperforms some of the main market benchmarks.

Modeling and predicting IBNR reserve: extended chain ladder and heteroscedastic regression analysis

This work deals with two methodologies for predicting incurred but not reported (IBNR) actuarial reserves. The first is the traditional chain ladder, which is extended for dealing with the calendar year IBNR reserve. The second is based on heteroscedastic regression models suitable to deal with the tail effect of the runoff triangle – and to forecast calendar year IBNR reserves as well. Theoretical results regarding closed expressions for IBNR predictors and mean squared errors are established – for the case of the second methodology, a Monte Carlo study is designed and implemented for accessing finite sample performances of feasible mean squared error formulae. Finally, the methods are implemented with two real data sets. The main conclusions: (i) considering tail effects does not imply theoretical and/or computational problems; and (ii) both methodologies are interesting to design softwares for IBNR reserve prediction.

Yield Curve Forecasts and the Predictive Power of Macro Variables in a VAR Framework

Recent macro-finance papers have documented the importance of adding information from macro variables in order to improve out-of-sample forecasting performance of bond yields. This paper aims at investigating the reasons for this success. We use Diebold and Li’s dynamic version of the Nelson and Siegel exponential approximation of the yield curve to estimate the factors that govern its dynamics. Factors and macro variables are modeled simultaneously in a VAR framework, which is then used to forecast the factors. Our main conclusions are (i) this framework is useful in forecasting slope and curvature factors, but not the level factor; and (ii) to get good results in forecasting the level factor, one needs a macro model which incorporates variables related to long-run trends and expectations.

Restricted Kalman Filtering: Theoretical Issues

This entire chapter will be devoted to a discussion of several topics concerning the theory of imposing linear restrictions enunciated under a quite general form in (2.5) from Assumption 2.1. In Sect. 3.1, I will present and compare three different derivations of the restricted Kalman updating and smoothing equations under an augmented modeling approach. In Sect. 3.2, the statistical efficiency due to the imposition of restrictions is proved, and this shall be done using a geometrical framework. Moving forward, I try in Sect. 3.3 to establish the equivalence between restricted Kalman filtering and something that could be termed a recursive restricted least squares estimator. Finally, in Sect. 3.4, I investigate how initial diffuse state vectors affect the use of the Kalman smoother under linear restrictions.

Restricted Kalman Filtering: Methodological Issues

This chapter is concerned with some methods for imposing linear restrictions in state space modeling. The plan I will follow is this. In Sect. 4.1, I discuss an alternative restricted Kalman filtering that is indicated for situations where the linear restrictions are time-invariant and the state vector follows a general random walk. This approach was first featured in Pizzinga(2009). In Sect. 4.2, I present another alternative restricted Kalman filtering, this time due to Pizzinga(2010), that is based on a reduced linear state space model; this method will be compared with the previous augmented restricted Kalman filtering from several standpoints. Finally, Sect. 4.3 deals with a method developed in Pizzinga (2010) to impose linear restrictions in the prediction of a state vector.

Linear State Space Models and Kalman Filtering

A linear wide-sense state space model for an observable p-variate stochastic process Y t , defined on an appropriate probability space \((\Omega,\mathcal{F},\mathcal{P})\), is described by the following set of equations: