172,915 research outputs found
Interaction of market and credit risk: an analysis of inter-risk correlation and risk aggregation
In this paper we investigate the interaction between a credit portfolio and another risk type, which can be thought of as market risk. Combining Merton-like factor models for credit risk with linear factor models for market risk, we analytically calculate their interrisk correlation and show how inter-risk correlation bounds can be derived. Moreover, we elaborate how our model naturally leads to a Gaussian copula approach for describing dependence between both risk types. In particular, we suggest estimators for the correlation parameter of the Gaussian copula that can be used for general credit portfolios. Finally, we use our findings to calculate aggregated risk capital of a sample portfolio both by numerical and analytical techniques. -- Die Berechnung einer bankweit aggregierten Risikokennzahl (normalerweise ausgedrĂŒckt durch das ökonomische Kapital) ist ein Ă€uĂerst wichtiger Bestandteil eines modernen Risikocontrollings and als solches von besonderer Bedeutung fĂŒr bankinterne als auch regulatorische Zwecke. Eine wichtige Frage dabei betrifft die Behandlung von risikoreduzierenden Diversifikationseffekten, die als Folge der GeschĂ€ftsstrategie einer Bank (z.B. durch Produktdiversifikation oder geografische Diversifikation) auftreten können. Solche Diversifikationseffekte stellen einen Wettbewerbsvorteil dar, den Banken deshalb bei der Bestimmung ihrer KapitaladĂ€quanz mit einbeziehen wollen. Auch die Bankenaufsicht erkennt in ihren AusfĂŒhrungen ĂŒber die bankinternen Kapitalbeurteilungsverfahren nach den GrundsĂ€tzen der zweiten SĂ€ule von Basel II die Existenz von Diversifikationseffekten an. Bei der praktischen Berechnung des Diversifikationseffektes unterscheidet man oft zwischen Intrarisiko- und Interrisikodiversifikation. Letztere behandelt die Diversifikation innerhalb einer Risikoart (z.B. Markt- oder Kreditrisiko), wohingegen Interrisiko-Diversifikation die Diversifikation zwischen verschiedenen Risikoarten beschreibt und meist durch eine Interrisiko-Korrelationsmatrix erfasst wird.Risk aggregation,Inter-risk correlation,economic capital,ICAAP,diversification
Implications of return predictability for consumption dynamics and asset pricing
Two broad classes of consumption dynamicsâlong-run risks and rare disastersâhave proven successful in explaining the equity premium puzzle when used in conjunction with recursive preferences. We show that bounds a-lĂ Gallant, Hansen, and Tauchen that restrict the volatility of the stochastic discount factor by conditioning on a set of return predictors constitute a useful tool to discriminate between these alternative dynamics. In particular, we document that models that rely on rare disasters meet comfortably the bounds independently of the forecasting horizon and the asset returns used to construct the bounds. However, the specific nature of disasters is a relevant characteristic at the 1-year horizon: disasters that unfold over multiple years are more successful in meeting the predictors-based bounds than one-period disasters. Instead, at the 5-year horizon, the sole presence of disastersâeven if one-period and permanentâis sufficient for the model to satisfy the bounds. Finally, the bounds point to multiple volatility components in consumption as a promising dimension for long-run risk models
Implications of return predictability for consumption dynamics and asset pricing
Two broad classes of consumption dynamicsâlong-run risks and rare disastersâhave proven successful in explaining the equity premium puzzle when used in conjunction with recursive preferences. We show that bounds a-lĂ Gallant, Hansen, and Tauchen that restrict the volatility of the stochastic discount factor by conditioning on a set of return predictors constitute a useful tool to discriminate between these alternative dynamics. In particular, we document that models that rely on rare disasters meet comfortably the bounds independently of the forecasting horizon and the asset returns used to construct the bounds. However, the specific nature of disasters is a relevant characteristic at the 1-year horizon: disasters that unfold over multiple years are more successful in meeting the predictors-based bounds than one-period disasters. Instead, at the 5-year horizon, the sole presence of disastersâeven if one-period and permanentâis sufficient for the model to satisfy the bounds. Finally, the bounds point to multiple volatility components in consumption as a promising dimension for long-run risk models
Three Essays in Theoretical and Empirical Derivative Pricing
The first essay investigates the option-implied investor preferences by comparing equilibrium option pricing models under jump-diffusion to option bounds extracted from discrete-time stochastic dominance (SD). We show that the bounds converge to two prices that define an interval comparable to the observed option bid-ask spreads for S&P 500 index options. Further, the bounds' implied distributions exhibit tail risk comparable to that of the return data and thus shed light on the dark matter of the divergence between option-implied and underlying tail risks. Moreover, the bounds can better accommodate reasonable values of the ex-dividend expected excess return than the equilibrium models' prices. We examine the relative risk aversion coefficients compatible with the boundary distributions extracted from index return data. We find that the SD-restricted range of admissible RRA values is consistent with the macro-finance studies of the equity premium puzzle and with several anomalous results that have appeared in earlier option market studies.
The second essay examines theoretically and empirically a two-factor stochastic volatility model. We adopt an affine two-factor stochastic volatility model, where aggregate market volatility is decomposed into two independent factors; a persistent factor and a transient factor. We introduce a pricing kernel that links the physical and risk neutral distributions, where investor's equity risk preference is distinguished from her variance risk preference. Using simultaneous data from the S&P 500 index and options markets, we find a consistent set of parameters that characterizes the index dynamics under physical and risk-neutral distributions. We show that the proposed decomposition of variance factors can be characterized by a different persistence and different sensitivity of the variance factors to the volatility shocks. We obtain negative prices for both variance factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if those increases have little persistence. We also obtain negative correlations between shocks to the market returns and each volatility factor, where correlation is less significant in transient factor and therefore has a less significant effect on the index skewness. Our empirical results indicate that unlike stochastic volatility model, join restrictions do not lead to the poor performance of two-factor SV model, measured by Vega-weighted root mean squared errors.
In the third essay, we develop a closed-form equity option valuation model where equity returns are related to market returns with two distinct systematic components; one of which captures transient variations in returns and the other one captures persistent variations in returns. Our proposed factor structure and closed-form option pricing equations yield separate expressions for the exposure of equity options to both volatility components and overall market returns. These expressions allow a portfolio manager to hedge her portfolio's exposure to the underlying risk factors. In cross-sectional analysis our model predicts that firms with higher transient beta have a steeper term structure of implied volatility and a steeper implied volatility moneyness slope. Our model also predicts that variances risk premiums have more significant effect on the equity option skew when the transient beta is higher. On the empirical front, for the firms listed on the Dow Jones index, our model provides a good fit to the observed equity option prices
Federated Learning You May Communicate Less Often!
We investigate the generalization error of statistical learning models in a
Federated Learning (FL) setting. Specifically, we study the evolution of the
generalization error with the number of communication rounds between the
clients and the parameter server, i.e., the effect on the generalization error
of how often the local models as computed by the clients are aggregated at the
parameter server. We establish PAC-Bayes and rate-distortion theoretic bounds
on the generalization error that account explicitly for the effect of the
number of rounds, say , in addition to the number of
participating devices and individual datasets size . The bounds, which
apply in their generality for a large class of loss functions and learning
algorithms, appear to be the first of their kind for the FL setting.
Furthermore, we apply our bounds to FL-type Support Vector Machines (FSVM); and
we derive (more) explicit bounds on the generalization error in this case. In
particular, we show that the generalization error of FSVM increases with ,
suggesting that more frequent communication with the parameter server
diminishes the generalization power of such learning algorithms. Combined with
that the empirical risk generally decreases for larger values of , this
indicates that might be a parameter to optimize in order to minimize the
population risk of FL algorithms. Moreover, specialized to the case
(sometimes referred to as "one-shot" FL or distributed learning) our bounds
suggest that the generalization error of the FL setting decreases faster than
that of centralized learning by a factor of ,
thereby generalizing recent findings in this direction to arbitrary loss
functions and algorithms. The results of this paper are also validated on some
experiments
Private Stochastic Optimization With Large Worst-Case Lipschitz Parameter: Optimal Rates for (Non-Smooth) Convex Losses and Extension to Non-Convex Losses
We study differentially private (DP) stochastic optimization (SO) with loss
functions whose worst-case Lipschitz parameter over all data points may be
extremely large. To date, the vast majority of work on DP SO assumes that the
loss is uniformly Lipschitz continuous over data (i.e. stochastic gradients are
uniformly bounded over all data points). While this assumption is convenient,
it often leads to pessimistic excess risk bounds. In many practical problems,
the worst-case (uniform) Lipschitz parameter of the loss over all data points
may be extremely large due to outliers. In such cases, the error bounds for DP
SO, which scale with the worst-case Lipschitz parameter of the loss, are
vacuous. To address these limitations, this work provides near-optimal excess
risk bounds that do not depend on the uniform Lipschitz parameter of the loss.
Building on a recent line of work (Wang et al., 2020; Kamath et al., 2022), we
assume that stochastic gradients have bounded -th order moments for some . Compared with works on uniformly Lipschitz DP SO, our excess risk
scales with the -th moment bound instead of the uniform Lipschitz parameter
of the loss, allowing for significantly faster rates in the presence of
outliers and/or heavy-tailed data. For convex and strongly convex loss
functions, we provide the first asymptotically optimal excess risk bounds (up
to a logarithmic factor). In contrast to (Wang et al., 2020; Kamath et al.,
2022), our bounds do not require the loss function to be differentiable/smooth.
We also devise a linear-time algorithm for smooth losses that has excess risk
that is tight in certain practical parameter regimes. Additionally, our work is
the first to address non-convex non-uniformly Lipschitz loss functions
satisfying the Proximal-PL inequality; this covers some practical machine
learning models. Our Proximal-PL algorithm has near-optimal excess risk.Comment: Appeared in the International Conference on Algorithmic Learning
Theory (ALT) 2023. This version improves the runtime bound in Theorem
Deep Learning Based Residuals in Non-linear Factor Models: Precision Matrix Estimation of Returns with Low Signal-to-Noise Ratio
This paper introduces a consistent estimator and rate of convergence for the
precision matrix of asset returns in large portfolios using a non-linear factor
model within the deep learning framework. Our estimator remains valid even in
low signal-to-noise ratio environments typical for financial markets and is
compatible with weak factors. Our theoretical analysis establishes uniform
bounds on expected estimation risk based on deep neural networks for an
expanding number of assets. Additionally, we provide a new consistent
data-dependent estimator of error covariance in deep neural networks. Our
models demonstrate superior accuracy in extensive simulations and the empirics
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