Konveksitas

Penilaian dan Analisis Obligasi dengan Python

Joshua Mayhew

Options Trader

Memplot harga prediksi vs aktual

import numpy as np
import numpy_financial as npf
import pandas as pd
import matplotlib.pyplot as plt

price = -npf.pv(rate=0.05, nper=20, pmt=5, fv=100)
price_up = -npf.pv(rate=0.06, nper=20, pmt=5, fv=100)
price_down = -npf.pv(rate=0.04, nper=20, pmt=5, fv=100)
duration = (price_down - price_up) / (2 * price * 0.01)
dollar_duration = duration * price * 0.01
print("Bond Price (USD): ", price)
print("Dollar Duration (USD): ", dollar_duration)

Bond Price (USD): 100.00
Dollar Duration (USD): 12.53

Memplot harga prediksi vs aktual

bond_yields = np.arange(0, 10, 0.1)
bond = pd.DataFrame(bond_yields, columns=['bond_yield'])
bond['price'] = -npf.pv(rate=bond['bond_yield'] / 100, nper=20, pmt=5, fv=100)

    bond_yield       price
0          0.0  200.000000
1          0.1  196.978503
..         ...         ...
98         9.8   58.570780
99         9.9   57.997210

[100 rows x 2 columns]

Memplot harga prediksi vs aktual

bond['yield_change'] = bond['bond_yield'] - 5

    bond_yield       price  yield_change
0          0.0  200.000000        -5.0
1          0.1  196.978503        -4.9
2          0.2  194.013231        -4.8
..         ...         ...           ...
97         9.7   59.153044         4.7
98         9.8   58.570780         4.8
99         9.9   57.997210         4.9

[100 rows x 3 columns]

Memplot harga prediksi vs aktual

$ \text{Price Change} = -100 \times \text{Dollar Duration} \times \Delta y$

bond['price_change'] = -100 * dollar_duration * bond['yield_change'] / 100

    bond_yield       price  yield_change  price_change
0          0.0  200.000000          -5.0     62.650619
1          0.1  196.978503          -4.9     61.397607
..         ...         ...           ...           ...
98         9.8   58.570780           4.8    -60.144594
99         9.9   57.997210           4.9    -61.397607

[100 rows x 4 columns]

Memplot harga prediksi vs aktual

bond['predicted_price'] = price + bond['price_change']

   bond_yield       price  yield_change  price_change  predicted_price
0          0.0  200.000000          -5.0     62.650619       162.650619
1          0.1  196.978503          -4.9     61.397607       161.397607
2          0.2  194.013231          -4.8     60.144594       160.144594
..         ...         ...           ...           ...              ...
97         9.7   59.153044           4.7    -58.891582        41.108418
98         9.8   58.570780           4.8    -60.144594        39.855406
99         9.9   57.997210           4.9    -61.397607        38.602393

[100 rows x 5 columns]

Memplot harga prediksi vs aktual

plt.plot(bond['bond_yield'], bond['price'])
plt.plot(bond['bond_yield'], bond['predicted_price'])
plt.xlabel('Yield (%)')
plt.ylabel('Price (USD)')
plt.title("Actual Bond Prices vs. 
    Predicted Prices Using Duration")
plt.legend(["Actual Price", "Predicted Price"])
plt.show()

Keterbatasan durasi

Durasi adalah ukuran linear
Harga obligasi bersifat cembung
Durasi akurat hanya untuk perubahan yield kecil

Apa itu konveksitas?

Mengukur kelengkungan harga obligasi
Untuk meningkatkan prediksi harga dan pengukuran risiko
Konveksitas lebih tinggi = hubungan harga/yield lebih melengkung

Rumus konveksitas

Kita akan gunakan rumus konveksitas yang disederhanakan:

${\large Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

$P_{down}$ = Harga obligasi saat yield 1% lebih rendah
$P_{up}$ = Harga obligasi saat yield 1% lebih tinggi
$2 \times P$ = Dua kali harga obligasi pada yield saat ini
$(\Delta y)^2$ = Perubahan yield kuadrat (kita gunakan 1% ^ 2)

Contoh konveksitas

Obligasi 10 tahun, kupon 5% per tahun, yield to maturity 4%, berapa konveksitasnya?

${ Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

price = -npf.pv(rate=0.04, nper=10, pmt=5, fv=100)

price_up = -npf.pv(rate=0.05, nper=10, pmt=5, fv=100)
price_down = -npf.pv(rate=0.03, nper=10, pmt=5, fv=100)

convexity = (price_down + price_up - 2 * price) / (price * 0.01 ** 2)
print("Convexity: ", convexity)

Convexity:  77.56

Ringkasan

Durasi adalah ukuran linear
Harga obligasi melengkung
Durasi akurat hanya untuk perubahan yield kecil
Konveksitas mengukur kelengkungan ini

${ Convexity = \frac{P_{down}\ +\ P_{up} \ - \ 2\times P}{P\ \times\ (\Delta y)^2}}$

Ayo berlatih!

Penilaian dan Analisis Obligasi dengan Python