具非線性相依利率-違約率-回收率之可違約債券評價模型—評價演算法與風險分析

Abstract

本研究將提出一個新的建模方法，用以評價可違約債券的價格與風險值估計。此方法的特點在於考量可違約債中利率、違約率、回收率間的非線性相依性，並允許他們的相關性與動態過程可為任意線性或非線性的設定形式，增加評價模型的設定彈性。本議題的研究意義在於他們間的相關性在實證上發現存在非線性的特徵，且此特徵將可能隨著日後國際間非預期的危機數量增加，而使得過往對債券持有期間內設定相關性為定值或線性關係的假設可信度下降。 本研究蒐集美國市場於2018年5月至2023年5月的股匯債市與主權違約交換報價等資料，證實非線性相依性的存在。首先是利率、違約率與回收率的有界性。第二是利率與違約率存在正斜率且凹口向上的非線性關係。同時，本研究發現除了利率，市場景氣狀況亦為影響違約率的重要因子。第三是違約率與回收率間具負斜率且凹口向上的函數關係。至今，有鑑於2023年全球風險報告書揭露不斷攀升的多領域脆弱度，各國政府恐將更頻繁的運用貨幣工具維持國內金融穩定，因此考量非線性相依性的必要性更顯重要。 本研究根據實證結果進行模型與參數的設定，如假使利率、違約率、回收率為兩個具均值回歸特性之狀態變量的仿射函數。基於可解析性的考量，本研究先透過動差與共變異數擬合的方法將狀態變量的隨機過程離散化，將其轉換為連續時間的馬可夫鏈，並藉以求得相應馬可夫鏈間狀態的轉移率。而由於本研究標的為可違約債，故再加入吸收態去捕捉進入違約狀態的可能。自此，本研究遂可運用馬可夫調整泊松過程去估計考量非線性相依性之可違約債券價格，並進一步預測未來的風險值。最後，數值分析發現，本研究所提出的非線性模型相較過往的線性模型傾向估計出較高的債券價格與較保守的風險估計。

Our research intends to propose a novel methodology to evaluate the price and measure the risks of defaultable bonds. The distinctive feature of this method is taking the nonlinear dependence among interest rates, default rates, and recovery rates into account, allowing their correlations and dynamics to be specified as arbitrary linear or nonlinear forms, thereby increasing the model’s flexibility. The significance of this research topic lies in the nonlinear characteristics among these rates found in this paper. With increasing unexpected shocks or crises, it leads to a decline in the credibility of previous fixed or linear relationship assumptions during the bond's holding period.

We obtain several insights concerning their relations via US market from May 2018 to May 2023. Three key findings are listed below. The first feature is the boundedness of rates. Second, there is a positive-slope, concave-upward nonlinear relation between interest and default rates. In addition, we found that not only interest rate, but the US market condition also influences default rate. Third, the relationship between default rate and recovery rate exhibits the negatively sloping with upward concavity patterns. Based on the Global Risks Report 2023, it cites pieces of evidence to show the increasing vulnerabilities across multiple domains and sound a warning of poly-crises. Hence, to cope with more shocks in the future, governments will utilize monetary tools more frequently, highlighting the importance of considering the nonlinear dependency.

Our model setup and parameter settings are inspired by our empirical research and past literature. For instance, we assume that these three rates are the affine functions of two state variables with mean-reverting features. To maintain analytical tractability, we will discretise the stochastic processes of the state variables through moment and covariance matching approach, transforming them into two Continuous-Time Markov Chains with absorbing state. Finally, we employ the Markovian Modulated Poisson Process to estimate the price of defaultable bond and its VaR which consider the nonlinear dependencies. At last, the numerical analyses show that compared with the benchmark model, our nonlinear model with truncated dynamic of interest rates will predict a relatively higher ZCB price and more negative VaR.