透過常態分配累積分配函數之有效率的近似公式分析違約機率

The Analyses of Default Probability by an Efficient Approximate Formula for the Cumulative Distribution Function of Standard Normal Distribution

本文提出一個效率的公式希望能較簡易且快速的計算出標準常態累積分配函數的趨近值，並將此公式結合財務上Kealhofer-McQuown-Vasicek（KMV）違約機率模型及Merton違約機率模型，藉此來分析違約機率在實務分析上的幾個相關議題，諸如：（1）違約機率簡易效率的計算；（2）給定違約機率下，公司資本槓桿與其資產風險之間的關係；（3）當公司的資本結構或資產波動改變時，其違約機率有何變化。為說明公式如何實際應用，文中分別使用標準普爾（Standard & Poor's, S&P）與臺灣企業信用風險指標（Taiwan Corporate Credit Risk Index, TCRI）的信用評等的資料進行數值分析。結果顯示，在違約機率的計算與分析上，與Edous and Eidous（2018）公式相比，本文之公式較具實務上的應用性。文中之數值分析結果亦提供以下幾個有用的訊息供市場參與者參考：（1）當公司為最壞信用評等時，公司的資本槓桿率對其資產風險具最大的影響；（2）資產風險對違約機率的影響比資本槓桿率對違約的機率影響還大；（3）資本結構改變對最壞的信用評等公司的違約機率影響最大。在財務意義上，本文公式有助於市場參與者能易於瞭解違約機率相關變數的變化如何影響違約機率。由於本文所提出的公式易於實務上的應用，相信能協助市場監理者控管金融機構的違約風險，同時，亦能有助市場參與者有效率的管理具違約風險的複雜投資組合與進行最適投資決策。

This study supports an efficiency formula for more simply and more speedily calculating the approximation value of the cumulative distribution function of standard normal distribution. We combine this formula with the probability of default (PD) models, such as the Kealhofer-McQuown-Vasicek (KMV) model and Merton model, to analyze the following issues: (1) the simple and efficient calculations on PD; (2) the relationship between a firm's capital leverage and its asset risk under a given PD; and (3) the changes of PD when the firm changes its capital structure or its asset volatility. Numerical examples using Standard & Poor's (S&P) credit rating reports and Taiwan Corporate Credit Risk Index (TCRI) credit rating data illustrate the application of our formula. The results reveal that our formula owns a better applicability in practice for analyzing the PD compared with the formula shown in Edous and Eidous (2018). Our results also provide market participants the following useful financial information: (1) the influence of firm's capital leverage ratio on its asset risk has the largest effect for the worst credit quality; an increase on debt of asset (or asset volatility) induces a raise in the PD; (2) the influence of the asset risk on PD is larger than the influence of the capital leverage ratio on PD; and (3) if the firm in the worse credit rank, the change of its capital structure has great influence on its PD. On the financial applications, our formula can help market participants to easily understand how sensitive PD is to changes in its relevant variables. This not only can help market supervisors to manage the default risks for financial institutions, but also can help market participants to undertake the optimal investment decisions for the portfolios with default risks.