Risk Factors in USD Libor Market

In this notebook, we perform a classical analysis in risk management. Using 11 years of USD libor data (deposit and swap rates), we:

• Compute a time series of zero-coupon curves, bootstrapped from the deposit and swap rates, using the QuantLib library

• Sample these zero-coupon curves at constant maturities

• Perform a Principal Components Analysis on the change in zero-coupon rates, using the numpy library

The first 3 principal components account for over 95% of the total variance, and can be interpreted as follows:

• The first factor represents an approximate parallel shift

• The second factor represents a twist

• The third factor represents a change in convexity

The notebook demonstrates the use of high-level functions that hide much of the complexity of QuantLib.

import os

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.mlab as ml

import pandas as pd
from quantlib.util.rates import zero_rate, make_term_structure

Bootstrapping Zero-Coupon Yield Curves

A set of zero-coupon yield curves is bootstrapped from deposit and swap rates stored in the pandas DataFrame df_libor.

df_libor = pd.read_pickle(os.path.join('..', 'data', 'df_libor.pkl'))

dtObs = df_libor.index

dtI = dtObs[range(0, len(dtObs) - 1, 60)]
days = [10, 30, 90, 182, 365, 365 * 2, 365 * 3,
            365 * 5, 365 * 10, 365 * 15]

    # maturity in columns, observation days in rows
zc_rate = np.empty((len(dtI), len(days)), dtype='float64')
dt_maturity = np.empty_like(zc_rate, dtype='object')

for i, obs_date in enumerate(dtI):
    rates = df_libor.xs(obs_date) / 100
    # bootstrap a term structure from the libor rates
    ts = make_term_structure(rates, obs_date)
    # compute zero-coupon rates for a range of maturities
    (dt_maturity[i, ], zc_rate[i, ]) = zero_rate(ts, days, obs_date)

Principal Components Analysis and Display

The first three principal components identify the three major risk factors, and account for 95% of the total variance:

• The first factor represents an approximate parallel shift

• The second factor represents a twist

• The third factor represents a change in convexity

from sklearn import decomposition
pca = decomposition.PCA()
# compute rate change
zc_diff = np.diff(zc_rate, axis=0)
# PCA on rate change
zc_pca = pca.fit(zc_diff)

fig = plt.figure()
fig.set_size_inches(15,6)

ax = fig.add_subplot(121)

# compute x-axis limits
dtMin = dt_maturity[0,0]
dtMax = dt_maturity[-1,-1]

ax.set_xlim(dtMin, dtMax)
ax.set_ylim(0.0, 0.1)

# plot a few curves
for i in range(0, len(dtI),3):
    ax.plot(dt_maturity[i,], zc_rate[i,])
plt.title('Zero-Coupon USD Libor: 2000 to 2010')

ax2 = fig.add_subplot(122)
ttm = np.array(days)/365.0
ax2.plot(ttm, zc_pca.components_[0], 'k--', ttm, zc_pca.components_[1], 'k:', ttm, zc_pca.components_[2], 'k')
leg = ax2.legend(('PC 1', 'PC 2', 'PC 3'))
plt.title('First 3 Principal Components of USD Libor')
plt.show()