Bond prices vs. bond yields

Bond Valuation and Analysis in Python

Joshua Mayhew

Options Trader

Historical government bond yields

Plotting bond prices vs. yields

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

bond_yields = np.arange(0, 20, 0.1)
print(bond_yields)

[0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
...
18.5 18.6 18.7 18.8 18.9 19.0 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9]

Plotting bond prices vs. yields

bond = pd.DataFrame(bond_yields, columns=['bond_yield'])
print(bond)

     bond_yield
0           0.0
1           0.1
..          ...
198        19.8
199        19.9
[200 rows x 1 columns]

Plotting bond prices vs. yields

bond['bond_price'] = -npf.pv(rate=bond['bond_yield'] / 100, nper=10, pmt=5, fv=100)
print(bond)

     bond_yield  bond_price
0           0.0  150.000000
1           0.1  148.731575
..          ...         ...
198        19.8   37.527719
199        19.9   37.319493
[200 rows x 2 columns]

Plotting bond prices vs. yields

plt.plot(bond['bond_yield'], bond['bond_price'])

plt.xlabel('Yield (%)')

plt.ylabel('Bond Price (USD)')

plt.title("10 Year Bond 5% Annual Coupon")

plt.show()

The relationship between price and yield

Prices move inversely to yields

Higher yield = higher discount rate = lower PV

Higher price lowers the return on investment (yield)

The relationship between price and yield

Bond premium vs. bond discount

Premium: Price > 100, Yield < Coupon

Discount: Price < 100, Yield > Coupon

Par: Price = 100, Yield = Coupon

The relationship between price and yield

Price/yield relationship is non-linear

The line we have plotted is not a straight line

This is due to something called convexity

The relationship between price and yield

Summary of key points:

Prices and yields move inversely
Premium: Price > 100, Yield < Coupon
Discount: Price < 100, Yield > Coupon
Par: Price = 100, Yield = Coupon
The price/yield relationship is non-linear

Let's practice!

Bond Valuation and Analysis in Python