def test_unstable(self, dtype):
"""Demonstrates the instability of Hagan West for extreme cases."""
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Computed with discount rates of [5.0, 4.75, 4.53333333, 4.775]
# which are 100 times the values in the previous test case.
pvs = np.array([
11.561316110080888, 2.6491572753698067, 3.4340789041846866,
1.28732090544209
],
dtype=dtype)
true_discount_rates = np.array([5.0, 4.75, 4.53333333, 4.775],
dtype=dtype)
# Check failure with default initial rates.
results_default = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
maximum_iterations=100,
validate_args=True,
dtype=dtype))
self.assertFalse(results_default.converged)
self.assertTrue(results_default.failed)
self.assertFalse(np.isnan(results_default.discount_rates[0]))
self.assertTrue(np.isnan(results_default.discount_rates[1]))
# It even fails if we underestimate the result even marginally.
# However the behaviour is different if we start above the true values.
# See next test.
results_close = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
initial_discount_rates=true_discount_rates *
0.9999,
maximum_iterations=100))
with self.subTest('Converged'):
self.assertFalse(results_close.converged)
with self.subTest('Failed'):
self.assertTrue(results_close.failed)
with self.subTest('DiscountRates'):
self.assertFalse(np.isnan(results_close.discount_rates[0]))
self.assertFalse(np.isnan(results_close.discount_rates[1]))
self.assertTrue(np.isnan(results_close.discount_rates[2]))
def test_zero_coupon_raises(self):
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with no coupons. This should cause an error.
np.array([1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
pvs = np.array([
999.68155223943393, 1022.322872470043, 1093.9894418810143,
934.20885689015677
],
dtype=dtype)
with self.assertRaises(tf.errors.InvalidArgumentError):
self.evaluate(
bond_curve.bond_curve(cashflows, cashflow_times, pvs))
def test_negative_rates(self):
"""Checks that method works even if the actual rates are negative."""
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
pvs = np.array(
[1029.54933442, 1097.95320227, 1268.65376174, 1249.84175959],
dtype=dtype)
true_discount_rates = np.array([0.02, 0.01, -0.01, -0.03],
dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows, cashflow_times, pvs))
self.assertTrue(results.converged)
self.assertFalse(results.failed)
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-4)
def test_flat_curve(self):
"""Checks that flat curves work."""
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Computed with a flat curve of 15%.
pvs = np.array(
[906.27355957, 840.6517334, 823.73626709, 635.7076416],
dtype=dtype)
true_discount_rates = np.array([0.15] * 4, dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows, cashflow_times, pvs))
self.assertTrue(results.converged)
self.assertFalse(results.failed)
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
def test_non_convex(self):
"""Demonstrates the nonconvexity of Hagan West for extreme cases."""
# This is the same example as the previous one but with different starting
# point.
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Computed with discount rates of [5.0, 4.75, 4.53333333, 4.775]
# which are 100 times the values in the previous test case.
pvs = np.array([
11.561316110080888, 2.6491572753698067, 3.4340789041846866,
1.28732090544209
],
dtype=dtype)
true_discount_rates = np.array([5.0, 4.75, 4.53333333, 4.775],
dtype=dtype)
initial_rates = true_discount_rates * 1.01
# Check failure with default initial rates.
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
initial_discount_rates=initial_rates,
maximum_iterations=100,
validate_args=True,
dtype=dtype))
self.assertTrue(results.converged)
self.assertFalse(results.failed)
# It converges to a different set of rates.
np.testing.assert_allclose(
results.discount_rates,
[4.9610328, 4.17662715, 2.84038942, 2.38737021],
atol=1e-6)
# However, the actual bond prices with the returned rates are indeed
# correct.
implied_pvs = self.evaluate(
_compute_pv(cashflows, cashflow_times, results.discount_rates,
np.array([1.0, 2.0, 3.0, 4.0], dtype=dtype)))
np.testing.assert_allclose(implied_pvs, pvs, rtol=1e-5)
def test_zero_coupon_bond(self):
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 6 months bond with no coupons.
np.array([1020], dtype=dtype),
# 1 year bond with 5% semi-annual coupon.
np.array([25, 1025], dtype=dtype),
# 2 year bond with 8% annual coupon.
np.array([80, 1080], dtype=dtype),
# 3 year bond with 3% annual coupon.
np.array([30, 30, 1030], dtype=dtype)
]
cashflow_times = [
np.array([0.5], dtype=dtype),
np.array([0.5, 1.0], dtype=dtype),
np.array([1.0, 2.0], dtype=dtype),
np.array([1.0, 2.0, 3.0], dtype=dtype)
]
pvs = np.array([1000.0, 1000.0, 1000.0, 1000.0], dtype=dtype)
# We can calculate discount rates going step-by-step.
r1 = -math.log(pvs[0] / cashflows[0][0]) / cashflow_times[0]
r2 = -(math.log(
(pvs[1] - cashflows[1][0] *
math.exp(-r1 * cashflow_times[1][0])) / cashflows[1][1]) /
cashflow_times[1][1])
r3 = -(math.log(
(pvs[2] - cashflows[2][0] *
math.exp(-r2 * cashflow_times[2][0])) / cashflows[2][1]) /
cashflow_times[2][1])
r4 = -(math.log(
(pvs[3] -
cashflows[3][0] * math.exp(-r2 * cashflow_times[3][0]) -
cashflows[3][1] * math.exp(-r3 * cashflow_times[3][1])) /
cashflows[3][2]) / cashflow_times[3][2])
true_discount_rates = np.array([r1, r2, r3, r4], dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
validate_args=True,
dtype=dtype))
self.assertTrue(results.converged)
self.assertFalse(results.failed)
self.assertEqual(results.iterations, 4)
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
np.testing.assert_allclose(results.times, [0.5, 1.0, 2.0, 3.0],
atol=1e-6)
def test_final_cashflow_is_the_largest_error(self, dtype):
with self.assertRaises(tf.errors.InvalidArgumentError):
self.evaluate(
bond_curve.bond_curve(bond_cashflows=[
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
np.array([30.0, 30.0, 30.0, 3.0], dtype=dtype)
],
bond_cashflow_times=[
np.array([0.25, 0.5, 0.75, 1.0],
dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0],
dtype=dtype)
],
present_values=np.array([999.0, 1022.0],
dtype=dtype),
validate_args=True,
dtype=dtype))
def test_correctness(self, dtype):
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
pvs = np.array([
999.68155223943393, 1022.322872470043, 1093.9894418810143,
934.20885689015677
],
dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
validate_args=True,
dtype=dtype))
with self.subTest('Times'):
np.testing.assert_allclose(results.times, [1.0, 2.0, 3.0, 4.0])
with self.subTest('Converged'):
self.assertTrue(results.converged)
with self.subTest('NotFailed'):
self.assertFalse(results.failed)
expected_discount_rates = np.array([5.0, 4.75, 4.53333333, 4.775],
dtype=dtype) / 100
expected_discount_factors = np.exp(-expected_discount_rates *
[1.0, 2.0, 3.0, 4.0])
with self.subTest('DiscountRates'):
np.testing.assert_allclose(results.discount_rates,
expected_discount_rates,
atol=1e-6)
with self.subTest('DiscountFactors'):
np.testing.assert_allclose(results.discount_factors,
expected_discount_factors,
atol=1e-6)
def test_negative_forwards(self, dtype):
"""Checks that method works if the rates are positive by fwds are not."""
true_discount_rates = np.array([0.12, 0.09, 0.02, 0.01, 0.01318182],
dtype=dtype)
# Note the implied forward rates for this rate curve are:
# [0.12, 0.06, -0.05, -0.01, 0.02]
cashflows = [
np.array([1.2, 10.], dtype=dtype),
np.array([1.1, 2.2, 1.4, 15.5], dtype=dtype),
np.array([1.22, 0.45, 2.83, 96.0], dtype=dtype),
np.array([12.33, 9.84, 1.15, 11.87, 0.66, 104.55], dtype=dtype),
np.array([5.84, 0.23, 5.23, 114.95], dtype=dtype)
]
cashflow_times = [
np.array([0.15, 0.25], dtype=dtype),
np.array([0.1, 0.2, 0.4, 0.5], dtype=dtype),
np.array([0.22, 0.45, 0.93, 1.0], dtype=dtype),
np.array([0.33, 0.84, 0.92, 1.22, 1.45, 1.5], dtype=dtype),
np.array([0.43, 0.77, 1.3, 2.2], dtype=dtype)
]
pvs = np.array([
10.88135262, 19.39268844, 98.48426722, 137.91938533, 122.63546542
],
dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
validate_args=True,
dtype=dtype))
with self.subTest('Converged'):
self.assertTrue(results.converged)
with self.subTest('NotFailed'):
self.assertFalse(results.failed)
with self.subTest('NumIterations'):
self.assertEqual(results.iterations, 6)
with self.subTest('DiscountRates'):
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
with self.subTest('Times'):
np.testing.assert_allclose(results.times,
[0.25, 0.5, 1., 1.5, 2.2],
atol=1e-6)
def test_cashflow_times_are_strongly_ordered_error(self):
dtypes = [np.float64, np.float32]
for dtype in dtypes:
with self.assertRaises(tf.errors.InvalidArgumentError):
self.evaluate(
bond_curve.bond_curve(bond_cashflows=[
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
np.array([30.0, 30.0, 30.0, 1030.0], dtype=dtype)
],
bond_cashflow_times=[
np.array([0.25, 0.5, 0.75, 1.0],
dtype=dtype),
np.array([0.5, 1.0, 1.5, 1.5],
dtype=dtype)
],
present_values=np.array(
[999.0, 1022.0], dtype=dtype),
validate_args=True,
dtype=dtype))
def test_only_zero_coupon_bonds(self, dtype):
cashflows = [
# 1 year bond with no coupons.
np.array([1010], dtype=dtype),
# 2 year bond with no coupons.
np.array([1030], dtype=dtype),
# 3 year bond with no coupons.
np.array([1020], dtype=dtype),
# 4 year bond with no coupons.
np.array([1040], dtype=dtype)
]
cashflow_times = [
np.array([1.0], dtype=dtype),
np.array([2.0], dtype=dtype),
np.array([3.0], dtype=dtype),
np.array([4.0], dtype=dtype)
]
true_discount_rates = np.array([0.001, 0.2, 0.03, 0.0], dtype=dtype)
pvs = np.array(
[(cashflows[i][0] * math.exp(-rate * cashflow_times[i][0]))
for i, rate in enumerate(true_discount_rates)],
dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
discount_tolerance=1e-6,
validate_args=True,
dtype=dtype))
with self.subTest('Converged'):
self.assertTrue(results.converged)
with self.subTest('NotFailed'):
self.assertFalse(results.failed)
with self.subTest('NumIterations'):
self.assertEqual(results.iterations, 1)
with self.subTest('DiscountRates'):
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
with self.subTest('ResultTimes'):
np.testing.assert_allclose(results.times, [1.0, 2.0, 3.0, 4.0],
atol=1e-6)
def test_only_zero_coupon_bonds(self):
dtypes = [np.float64, np.float32]
for dtype in dtypes:
cashflows = [
# 1 year bond with no coupons.
np.array([1010], dtype=dtype),
# 2 year bond with no coupons.
np.array([1030], dtype=dtype),
# 3 year bond with no coupons.
np.array([1020], dtype=dtype),
# 4 year bond with no coupons.
np.array([1040], dtype=dtype)
]
cashflow_times = [
np.array([1.0], dtype=dtype),
np.array([2.0], dtype=dtype),
np.array([3.0], dtype=dtype),
np.array([4.0], dtype=dtype)
]
true_discount_rates = np.array([0.001, 0.2, 0.03, 0.0],
dtype=dtype)
pvs = np.array(
[(cashflows[i][0] * math.exp(-rate * cashflow_times[i][0]))
for i, rate in enumerate(true_discount_rates)],
dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
validate_args=True,
dtype=dtype))
self.assertTrue(results.converged)
self.assertFalse(results.failed)
self.assertEqual(results.iterations, 1)
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
np.testing.assert_allclose(results.times, [1.0, 2.0, 3.0, 4.0],
atol=1e-6)
def test_flat_curve(self, dtype):
"""Checks that flat curves work."""
cashflows = [
# 1 year bond with 5% three monthly coupon.
np.array([12.5, 12.5, 12.5, 1012.5], dtype=dtype),
# 2 year bond with 6% semi-annual coupon.
np.array([30, 30, 30, 1030], dtype=dtype),
# 3 year bond with 8% semi-annual coupon.
np.array([40, 40, 40, 40, 40, 1040], dtype=dtype),
# 4 year bond with 3% semi-annual coupon.
np.array([15, 15, 15, 15, 15, 15, 15, 1015], dtype=dtype)
]
cashflow_times = [
np.array([0.25, 0.5, 0.75, 1.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Computed with a flat curve of 15%.
pvs = np.array([906.27355957, 840.6517334, 823.73626709, 635.7076416],
dtype=dtype)
true_discount_rates = np.array([0.15] * 4, dtype=dtype)
results = self.evaluate(
bond_curve.bond_curve(cashflows,
cashflow_times,
pvs,
discount_tolerance=1e-6,
validate_args=True,
dtype=dtype))
with self.subTest('Converged'):
self.assertTrue(results.converged)
with self.subTest('NotFailed'):
self.assertFalse(results.failed)
with self.subTest('DiscountRates'):
np.testing.assert_allclose(results.discount_rates,
true_discount_rates,
atol=1e-6)
