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FOClass: 債券殖利率曲線計算

FOClass: 零息債券殖利率計算 中,我們可算出零息債券的殖利率,但如果要套用在實務上的應用中,我們必須將觀察到的各點作一迴歸函式,讓我們可以找到各天期的殖利率。

運用原理為 Cubic Spline 方法。假設債券的 Discount 因子為一個三次方程式:

http://latex.codecogs.com/gif.latex?D%28t%29%20=%201%20+%20a%5Ctimes%20t%20+%20b%5Ctimes%20t%5E2%20+%20c%5Ctimes%20t%5E3

而每張債券的現金流量再套入下方方程式:

http://latex.codecogs.com/gif.latex?PV%20=%20\sum_{i=1}^{n}C_{i}%20\times%20D_{i}

可得到類似 3.3a + 1.2b + 5.5c = 30 的等式。像是代入 前篇文章 的九張債券可得如下式子:

http://latex.codecogs.com/gif.latex?%5Cbegin%7Bbmatrix%7D%20x_%7B11%7D%20&%20x_%7B12%7D%20&%20x_%7B13%7D%5C%5C%20...%20&%20...&%20...%5C%5C%20x_%7B91%7D%20&%20x_%7B92%7D%20&%20x_%7B93%7D%20%5Cend%7Bbmatrix%7D%20%5Cbegin%7Bbmatrix%7D%20a%5C%5C%20b%5C%5C%20c%20%5Cend%7Bbmatrix%7D%20=%20%5Cbegin%7Bbmatrix%7D%2043.33%5C%5C%20...%5C%5C%203,4%20%5Cend%7Bbmatrix%7D

經過矩陣移位:

http://latex.codecogs.com/gif.latex?%5Cbegin%7Bbmatrix%7D%20a%5C%5C%20b%5C%5C%20c%20%5Cend%7Bbmatrix%7D%20=%20%5Cleft%20%28%5Cbegin%7Bbmatrix%7D%20x_%7B11%7D%20&%20x_%7B12%7D%20&%20x_%7B13%7D%5C%5C%20...%20&%20...&%20...%5C%5C%20x_%7B91%7D%20&%20x_%7B92%7D%20&%20x_%7B93%7D%20%5Cend%7Bbmatrix%7D%5E%7Bt%7D%20%5Ccdot%20%5Cbegin%7Bbmatrix%7D%20x_%7B11%7D%20&%20x_%7B12%7D%20&%20x_%7B13%7D%5C%5C%20...%20&%20...&%20...%5C%5C%20x_%7B91%7D%20&%20x_%7B92%7D%20&%20x_%7B93%7D%20%5Cend%7Bbmatrix%7D%5Cright%20%29%5E%7BINV%7D%20%5Ccdot%20%5Cbegin%7Bbmatrix%7D%20x_%7B11%7D%20&%20x_%7B12%7D%20&%20x_%7B13%7D%5C%5C%20...%20&%20...&%20...%5C%5C%20x_%7B91%7D%20&%20x_%7B92%7D%20&%20x_%7B93%7D%20%5Cend%7Bbmatrix%7D%5E%7Bt%7D%20%5Ccdot%20%5Cbegin%7Bbmatrix%7D%2043.33%5C%5C%20...%5C%5C%203,4%20%5Cend%7Bbmatrix%7D

再透過 OLS(ordinary least square) 方法求出 a, b, c 的適當值後,即可代入:

http://latex.codecogs.com/gif.latex?Y%28t%29%20=%20%5Csqrt%5Bt%5D%7B%5Cfrac%7B1%7D%7BD%28t%29%7D%7D%20-%201

便算出殖利率曲線。如下圖:

../../../_images/zero_bond_yield_curve.png

綠色線為零息債券殖利率曲線、紅色線為附息債券殖利率曲線

我們可以看到 20~30年期的殖利率下降的十分奇怪。原因是我們的觀察值債券的年期最大只有 20 年,所以這一條函式在預測 20~30 年期的數據應該是有問題。

Python 程式如下:

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from numpy import array, append, dot, matrix, linalg
class CubicSpline:
     """ 使用最小平方和原則作三次方方程式的迴歸
     """
     def __init__(self):
         self.PVs = array([])
         self.X = array([])


     def addBondData(self, PV=0, Ci=[], Ni=[]):
         self.PV = PV < 0 and PV or -1*PV
         if Ni[0] == 0:
             self.PVs = append(self.PVs,
                                -1*self.PV-Ci[0])
             self.Ci = array(Ci[1:])
             self.Ni = array([[1, t, t**2, t**3]
                                for t in Ni[1:]])
         else:
             self.PVs = append(self.PVs, -1*self.PV)
             self.Ci = array(Ci)
             self.Ni = array([[1, t, t**2, t**3]
                                for t in Ni])

         self.dt = dot(self.Ci, self.Ni)
         if len(self.X):
             self.X = append(self.X,
                            [self.dt[1:]],
                            axis=0)
         else:
             self.X = array([self.dt[1:]])

         self.PVs[-1] -= self.dt[0]


     def runOLS(self):
         self.X = matrix(self.X)
         self.PVs = matrix(self.PVs).T
         self.b = (linalg.inv(self.X.T
                            * self.X)
                                * self.X.T
                                * self.PVs)
         return self.b

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