再探永續應急可轉債之結構式評價

本論文係延伸修改李宇昂 (2021) 永續應急可轉債之評價模型與評價做法，該模型納入利率、普通股權益第一類資本比率、股價、槓桿比率等四項變數，設定觸發損失條件為發行人之普通股權益第一類資本比率低於一特定門檻值，並考量無到期日與發行人可贖回之發行條件，是目前較為完整且符合實務的應急可轉債評價模型。在評價做法上，李宇昂 (2021) 根據永續應急可轉債的週期性性質，引用積分法及迭代演算法，透過對原始變數進行正交化程序、對正交變數進行維度切割，採逆推法以迭代方式計算贖回日之持有價值向量，直到兩個贖回日持有價值的被贖回位置是相同的為止，從而以收斂持有價值向量逆推回評價日，對應評價日各變數實際資料點，得出債券價格；又輔以蒙地卡羅模擬，以降低離散化誤差。李宇昂 (2021) 對BACR 7.750% Perp的評價結果明顯高於市場價格，我們推測最可能有兩個原因，第一、受限於電腦運算效能，正交變數切割數不足導致之離散化誤差，第二、以兩個贖回日持有價值大於等於贖回價格的節點個數相同作為判斷持有價值之被贖回位置相同的條件，可能造成評價結果未完全收斂。為此，我們修改評價方法：第一、對正交變數的切割節點索引值進行線性內插，進而計算內插後的持有價值向量，並將李宇昂 (2021) 直接以矩陣公式進行迭代計算的方式，改為個別計算每個時間點持有價值向量每個節點的持有價值，以逐期內插、逐期逆推的方式，求算贖回日之持有價值向量；第二、新增判定收斂持有價值向量的收斂條件，除了採李宇昂 (2021) 的判斷標準外，另要求兩個贖回日之持有價值向量中，超過99%之節點的贖回行為是一致的，贖回行為一致係指兩個持有價值向量同一個座標值之持有價值同時大於等於或同時小於贖回價格。考量利率在模型設定的重要特殊性，我們在策略上採取增加利率變數的分割數，對其他三項正交變數進行線性內插之敏感度分析，以決定需內插之變數及內插分割的組數。敏感度分析結果顯示僅普通股權益第一類資本比率具內插效果，據此，重行計算債券價格。評價結果顯示：透過增加利率變數之切割組數及較嚴格之收斂持有價值向量的判定條件，無需輔以蒙地卡羅模擬，我們未內插的債券價格與李宇昂 (2021) 之評價結果相近；對普通股權益第一類資本比率之正交變數進行內插後之債券價格明顯低於未内插之價格，亦較接近市場價格。

關鍵字

永續應急可轉債；結構式評價模型；週期性性質；積分法；迭代演算法；線性內插

並列摘要

This thesis modifies the pricing method based on the structural model for pricing perpetual CoCo bonds in Lee (2021). Lee’s (2021) model, which includes the stochastic variables of interest rate, common equity Tier 1 ratio, stock price and leverage ratio and brings into the covenants of trigger events, optional redemption and no scheduled maturity, is relatively thorough and in line with reality. The pricing steps of the quadrature method from Lee (2021) are as follows: estimating the parameters of the model, orthogonalizing the four stochastic variables, dividing the orthogonalized variables by setting the number of divisions, and calculating the holding value vector by backward induction in an iterative manner. Then, by the periodic property and the iteration algorithm, the convergent holding value vector is determined in accordance with the pre-specified convergent condition. After deriving the convergent holding value vector, CoCo bond price can be obtained through combining with the Monte Carlo Simulation. The pricing results from Lee (2021) showed that the price of BACR 7.750% Perp, the Coco bond issued by Barclays PLC, is much higher than its market price. There are two possible reasons for the differences. The first reason is that Lee (2021) could not set sufficient numbers of divisions for the orthogonalized variables due to the limited computer computing capacity. It caused the discretization errors. The second reason is that the convergent condition to determine the convergent holding value vector is not robust enough. In order to determine the convergent holding vector, it is necessary to theoretically judge whether the index positons of redemption for the holding value vectors on two adjacent call dates are same. The judging standard in Lee (2021) is that the number of the abscissas that holding values are greater or equal to the redemption price in the holding value vectors on two adjacent call dates are the same. Therefore, I try to make improvements to Lee’s (2021) pricing method. Firstly, I interpolate the index values of the division position for the orthogonalized variables. With the interpolated index values, the interpolated holding value vector can be derived. Then, using backward induction period by period, I can complete an iteration and judge if the holding value vector is convergent. Secondly, I modify the convergent conditions to determine the convergent holding value vector. In addition to the criterion adopted by Lee (2021), another criterion is considered to be met simultaneously. The additional criterion requires that the percentage of nodes with the same redemption behavior in the holding values vectors on the two adjacent call dates is over 99%. The same redemption behavior means the holding values at the same abscissas in different holding value vectors are both above the redemption price or both lower than the redemption price. Through the sensitivity analysis of the interpolation, I find that the interpolation makes the transition probability distribution of the common equity Tier 1 ratio more diffusive, but make less diffusion effect to the variables such as stock price and leverage ratio. Using the modified pricing method, I evaluate the pricing of BACR 7.750% Perp CoCo bond. The pricing results show the bond prices are lower than the prices of Lee (2021) and closer to the market price.

並列關鍵字

Perpetual CoCo Bond ； Structural Pricing Model ； Periodic Property ； Quadrature Method ； Iteration Algorithm ； Linear Interpolation

參考文獻

李宇昂 (2021)。永續應急可轉債之結構式評價模型。國立台灣大學管理學院國際企業研究所碩士論文。

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再探永續應急可轉債之結構式評價

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