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This study uses generalized additive models to identify and analyze nonlinear relationships between accounting-based and market-based independent variables and how these affect bankruptcy predictions. Specifically, it examines the independent variables that Altman (J Financ 23:589–609, 1968; Predicting financial distress of companies. Revisiting the Z-score and ZETA® models. Working paper, 2000) and Campbell et al. (J Financ 63:2899–2939, 2008) used and analyzes what specific form these nonlinear relationships take. Drawing on comprehensive data on listed U.S. companies, we show empirically that the bankruptcy prediction is influenced by statistically and economically relevant nonlinear relationships. Our results indicate that taking into account these nonlinear relationships improves significantly several statistical validity measures. We also use a validity measure that is based on the profitability of the bankruptcy prediction models in the context of credit scoring. The findings demonstrate that taking into account nonlinear relationships can substantially increase the discriminatory power of bankruptcy prediction models.

Handling risk factors in the context of a multi-asset risk parity portfolio allocation has created increased interest in recent literature. When allocating along risk factors through principal components, one major problem that persists is the potential existence of leverage or short positions for the optimal portfolio. It is possible, however, to avoid these positions if investors accept equal risk contributions for the first few risk factors, which explain most of the portfolio variation, and allow residual risk contributions to "float". Convex polytopes describe the entire set of solutions if the restrictions hold. A model to handle these kind of factor risk parity allocations with portfolio weight constraints is the subject of this paper. Focusing on equality in the first two risk contributions and requiring a sufficiently high explanation level for them is the main challenge. Different numerical allocation strategies for backtesting and performing the empirical analysis along two multi-asset data sets with different period lengths and a different number of assets are used.