Riskin Nicel Ölçümü ve Bayesçi Kalibrasyon: Piyasa Rejimleri Arasında Kuyruk Riskinin Değerlendirilmesi

Abstract

Küresel finansın kriz sonrası yapısal değişiminde, artan jeopolitik oynaklık ve algoritmik işlem karmaşıklığıyla karakterize edilen statik risk yönetimi ile dinamik nicel ölçüm arasındaki ikilem kritik bir kırılganlık haline gelmiştir. Bu makale, klasik optimizasyonu Bayesçi çıkarımla entegre eden sentezlenmiş bir risk yönetimi çerçevesi önererek bu boşluğu doldurmaktadır. İlk olarak, kredi, piyasa ve operasyonel risklerin teorik bir sınıflandırmasını tanımlıyoruz. İkinci olarak, ortalama-varyans kısıtlamaları altında optimal portföy oluşturma için analitik çözümler elde etmek üzere Lagrangian optimizasyonunu titizlikle uyguluyoruz. Üçüncü olarak, geleneksel Değer-Risk (VaR) yönteminin sınırlamalarını ele alarak, daha tutarlı bir risk ölçüsü olarak Beklenen Açık (ES) kavramının matematiksel bir türetimini sunuyoruz. En önemlisi, Bayesçi Eşlenik Önsel çerçevesini tanıtarak ampirik analizi standart geriye dönük testin ötesine genişletiyoruz. S&P 500'ün tarihsel verilerini kullanarak elde ettiğimiz sonuçlar, önemli bir "volatilite kümelenmesi" (teorik beklenti %1,0 iken ihlal oranı %2,0) ortaya koymakta ve risk modelinin epistemik belirsizliğini nicelleştirmektedir. Bayesçi sonradan analiz, Gauss varsayımları altında düzenleyici başarısızlığın %99,85 olasılığını göstermekte ve 1,93x'lik bir sermaye düzeltme çarpanı gerektirmektedir. Makale, modern finans piyasalarında var olan "parametre belirsizliğini" azaltmak için dinamik, Bayesçi kalibrasyonlu bir iç kontrol sistemi önererek sona ermektedir.

Keywords

• Portföy Optimizasyonu
• Beklenen Kayıp
• Riske Maruz Değer (VaR)
• Bayesçi Kalibrasyon
• Kuyruk Riski

Risk Quantification and Bayesian Calibration: Assessing Tail Risk across Market Regimes

Abstract

Post-pandemic volatility clustering highlights the gap between static governance and quantitative risk measurement. This paper proposes a synthesized framework utilizing complementary components of classical risk measurement and Bayesian inference. First, applying Lagrangian optimization, we illustrate the sensitivity of static mean-variance allocations to regime-specific sample moments. Second, we introduce Expected Shortfall as a complementary theoretical measure and GARCH specifications to account for time-varying conditional variance. Crucially, we complement standard VaR backtesting with a Beta-Binomial conjugate prior framework to calibrate risk exposures. Utilizing S&P 500 daily returns, we demonstrate how this mechanism quantifies the uncertainty surrounding the VaR violation probability. Relative to the 1.0% expectation, the traditional 21-Day Rolling VaR yields a Bayesian breach-rate adjustment factor of 2.40x, whereas the GARCH(1,1) specification lowers this to 1.80x. Ultimately, by bridging the gap between ex-ante predictions and realized shocks, this Bayesian recalibration loop may provide a data-driven approach to ex-post risk calibration.

Keywords

• Portfolio Optimization
• Expected Shortfall (ES)
• Value-at-Risk (VaR)
• Bayesian Calibration
• Tail Risk

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