Alfredo D. Egidio dos Reis

We study one of the world’s largest cattle markets by revisiting and extending previous work on the forecasting of Brazil’s Boi Gordo Index (BGI). Using an updated daily dataset (July 2006–September 2025, inflation-adjusted), we evaluate classical and machine learning (ML) approaches for price prediction. Methods include Exponential Smoothing (Simple, Holt, and Holt–Winters), ARMA/ARIMA/SARIMA, GARMA variants, GARCH, Theta, Prophet, and XGBoost; models are compared under a strictly chronological 90/10 holdout (~476 test days) using RMSE, MAE, and MSE, with the AIC guiding within-family selection. Results show that, for the full out-of-sample window, GARMA delivers the best overall accuracy, with ARMA and Holt–Winters close behind, while Prophet and XGBoost perform comparatively worse in this volatile setting. Performance is horizon-dependent: in the first 180 test days, prior to the late-2024 level shift, Holt attains the lowest RMSE/MSE, and XGBoost achieves the lowest MAE. No method anticipates the October–November 2024 exogenous jump and subsequent correction, highlighting the difficulty of structural breaks and the need for timely re-specification. We conclude that GARMA is a robust default for long, turbulent windows, whereas smoothing and ML methods can be competitive on shorter horizons. These findings inform risk measurement and risk mitigation strategies in Brazil’s cattle futures market.

This study analyses a series of live cattle spot and futures prices from the Boi Gordo Index (BGI) in Brazil. The objective is to develop a model that best portrays this commodity’s behaviour to estimate futures prices more accurately. The database created contains 2010 daily entries in which trade in futures contracts occurs, as well as BGI spot sales in the market, from 1 December 2006 to 30 April 2015. One of the most important reasons why this type of risk needs to be measured is to set loss limits. To identify patterns in price behaviour in order to improve future transaction results, investors must analyse fluctuations in asset values for longer periods. Bibliographic research reveals that no other study has conducted a comprehensive analysis of this commodity using this approach. Cattle ranching is big business in Brazil given that in 2021, this sector moved BRL 913.14 billion (USD 169.29 billion). In that year, agribusiness contributed 26.6% of Brazil’s total gross domestic product. Using the proposed risk modelling technique, economic agents can make the best decision about which options within these investors’ reach produce more effective risk management. The methodology is based on Holt–Winters exponential smoothing algorithm, autoregressive integrated moving-average (ARIMA), ARIMA with exogenous inputs, generalised autoregressive conditionally heteroskedastic and generalised autoregressive moving-average (GARMA) models. More specifically, five different methods are applied that allow a comparison of 12 different models as ways to portray and predict the BGI commodity behaviours. The results show that GARMA with order c(2,1) and without intercept is the best model. Investors equipped with such precise modelling insights stand at an advantageous position in the market, promoting informed investment decisions and optimising returns.

In this manuscript we propose a method for pricing insurance products that cover not only traditional risks, but also unforeseen ones. By considering the Poisson process parameter to be a mixed random variable, we capture the heterogeneity of foreseeable and unforeseeable risks. To illustrate, we estimate the weights for the two risk streams for a real dataset from a Portuguese insurer. To calculate the premium, we set the frequency and severity as distributions that belong to the linear exponential family. Under a Bayesian setup , we show that when working with a finite mixture of conjugate priors, the premium can be estimated by a mixture of posterior means, with updated parameters, depending on claim histories. We emphasise the riskiness of the unforeseeable trend, by choosing heavy-tailed distributions. After estimating distribution parameters involved using the Expectation-Maximization algorithm, we found that Bayesian premiums derived are more reactive to claim trends than traditional ones.

For a large motor insurance portfolio, on an open environment, we study the impact of experience rating in finite and continuous time ruin probabilities. We consider a model for calculating ruin probabilities applicable to large portfolios with a Markovian Bonus‐Malus System (BMS), based on claim counts, for an automobile portfolio using the classical risk framework model. New challenges are brought when an open portfolio scenario is introduced. When compared with a classical BMS approach ruin probabilities may change significantly. By using a BMS of a Portuguese insurer, we illustrate and discuss the impact of the proposed formulation on the initial surplus required to target a given ruin probability. Under an open portfolio setup, we show that we may have a significant impact on capital requirements when compared with the classical BMS, by having a significant reduction on the initial surplus needed to maintain a fixed level of the ruin probability.

This project works with the risk model developed by Li et al. (2015) and quests modelling, estimating and pricing insurance for risks brought in by innovative technologies, or other emerging or latent risks. The model considers two different risk streams that arise together, however not clearly separated or observed. Specifically, we consider a risk surplus process where premia are adjusted according to past claim frequencies, like in a Bonus-Malus (BM) system, when we consider a classical or historical risk stream and an unforeseeable risk one. These are unknown risks which can be of high uncertainty that, when pricing insurance (ratemaking and experience rating), suggest a sensitive premium adjustment strategy. It is not clear for the actuary to observe which claim comes from one or the other stream. When modelling such risks it is crucial to estimate the behaviour of such claims, occurrence and their severity. Premium calculation must fairly reflect the nature of these two kinds of risk streams. We start proposing a model, separating claim counts and severities, then propose a premium calculation method, and finally a parameter estimation procedure. In the modelling we assume a Bayesian approach as used in credibility theory, a credibility approach for premium calculation and the use of the Expectation-Maximization (EM) algorithm in the estimation procedure.

Motor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

The dual risk model assumes that the surplus of a company decreases at a constant rate over time, and grows by means of upward jumps which occur at random times with random sizes. In the present work, we study the dual risk renewal model when the waiting times are phase-type distributed. Using the roots of the fundamental and the generalized Lundberg’s equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Then, we address the calculation of expected discounted future dividends particularly when the individual common gains follow a phase-type distribution. We further show that the optimal dividend barrier does not depend on the initial reserve. As far as the roots of the Lundberg equations and the time of ruin are concerned, we address the existing formulae in the corresponding Sparre-Andersen insurance risk model for the first hitting time, and we generalize them to cover also the situations where we have multiple roots. We do that working a new approach and technique, approach we also use for working the dividends, unlike others, it can be also applied for every situation.