Inflation Indexes and Curves 2 (Quantlib comparison)#

This guide replicates and is a comparison to the Quantlib tutorial page at https://www.quantlibguide.com/Inflation%20indexes%20and%20curves.html

Inflation Indexes#

Historical index fixings in rateslib should be indexed to the 1st of the appropriate inflation month.

[1]:

from rateslib import *
from pandas import Series, MultiIndex

[2]:

inflation_fixings = [
    (dt(2022, 1, 1), 110.70),
    (dt(2022, 2, 1), 111.74),
    (dt(2022, 3, 1), 114.46),
    (dt(2022, 4, 1), 115.11),
    (dt(2022, 5, 1), 116.07),
    (dt(2022, 6, 1), 117.01),
    (dt(2022, 7, 1), 117.14),
    (dt(2022, 8, 1), 117.85),
    (dt(2022, 9, 1), 119.26),
    (dt(2022, 10, 1), 121.03),
    (dt(2022, 11, 1), 120.95),
    (dt(2022, 12, 1), 120.52),
    (dt(2023, 1, 1), 120.27),
    (dt(2023, 2, 1), 121.24),
    (dt(2023, 3, 1), 122.34),
    (dt(2023, 4, 1), 123.12),
    (dt(2023, 5, 1), 123.15),
    (dt(2023, 6, 1), 123.47),
    (dt(2023, 7, 1), 123.36),
    (dt(2023, 8, 1), 124.03),
    (dt(2023, 9, 1), 124.43),
    (dt(2023, 10, 1), 124.54),
    (dt(2023, 11, 1), 123.85),
    (dt(2023, 12, 1), 124.05),
    (dt(2024, 1, 1), 123.60),
    (dt(2024, 2, 1), 124.37),
    (dt(2024, 3, 1), 125.31),
    (dt(2024, 4, 1), 126.05),
]
dates, values = zip(*inflation_fixings)
fixings = Series(values, dates)

Rateslib contains an index_value method that will determine such for a given reference value date and other common parameters.

[3]:

index_value(
    index_lag=0,
    index_method="monthly",
    index_fixings=fixings,
    index_date=dt(2024, 3, 15)
)

[3]:

np.float64(125.31)

For example to replicate the Quantlib example of a lagged reference date we can use:

[4]:

index_value(
    index_lag=3,
    index_method="daily",
    index_fixings=fixings,
    index_date=dt(2024, 5, 15)
)

[4]:

np.float64(124.79451612903226)

Inflation Curves#

Create a nominal discount curve for cashflows. Calibrated to a 3% continuously compounded rate.

[5]:

nominal_curve = Curve(
    nodes={dt(2024, 5, 11): 1.0, dt(2074, 5, 18): 1.0},
    interpolation="log_linear",
    convention="Act365F",
    id="discount"
)
solver1 = Solver(
    curves=[nominal_curve],
    instruments=[Value(dt(2074, 5, 11), metric="cc_zero_rate", curves="discount")],
    s=[3.0],
    id="rates",
    instrument_labels=["nominal"],
)

SUCCESS: `func_tol` reached after 8 iterations (levenberg_marquardt), `f_val`: 4.2178219398368675e-19, `time`: 0.0036s

Now create an inflation curve, based on the last known CPI print, calibrated with zero coupon inflation swaps rates. Notice that an inflation curve starts as of the last known fixing as its index_base. This is similar to Quantlib, not be design, but by necessity since this is the only information we have that can define the start of the curve.

[6]:

inflation_curve = Curve(
    nodes={
        dt(2024, 4, 1): 1.0,  # <- last known inflation print.
        dt(2025, 5, 11): 1.0,  # 1y
        dt(2026, 5, 11): 1.0,  # 2y
        dt(2027, 5, 11): 1.0,  # 3y
        dt(2028, 5, 11): 1.0,  # 4y
        dt(2029, 5, 11): 1.0,  # 5y
        dt(2031, 5, 11): 1.0,  # 7y
        dt(2034, 5, 11): 1.0,  # 10y
        dt(2036, 5, 11): 1.0,  # 12y
        dt(2039, 5, 11): 1.0,  # 15y
        dt(2044, 5, 11): 1.0,  # 20y
        dt(2049, 5, 11): 1.0,  # 25y
        dt(2054, 5, 11): 1.0,  # 30y
        dt(2064, 5, 11): 1.0,  # 40y
        dt(2074, 5, 11): 1.0,  # 50y
    },
    interpolation="log_linear",
    convention="Act365F",
    index_base=126.05,
    index_lag=0,
    id="inflation"
)
solver = Solver(
    pre_solvers=[solver1],
    curves=[inflation_curve],
    instruments=[
        ZCIS(dt(2024, 5, 11), "1y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "2y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "3y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "4y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "5y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "7y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "10y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "12y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "15y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "20y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "25y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "30y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "40y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "50y", spec="eur_zcis", curves=["inflation", "discount"], leg2_index_fixings=fixings),
    ],
    s=[2.93, 2.95, 2.965, 2.98, 3.0, 3.06, 3.175, 3.243, 3.293, 3.338, 3.348, 3.348, 3.308, 3.228],
    instrument_labels=["1y", "2y", "3y", "4y", "5y", "7y", "10y", "12y", "15", "20y", "25y", "30y", "40y", "50y"],
    id="zcis",
)

SUCCESS: `func_tol` reached after 8 iterations (levenberg_marquardt), `f_val`: 1.4302416694642844e-17, `time`: 0.0373s

The data can be output to a table or plotted as below.

[7]:

inflation_curve.plot("1m")

[7]:

(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x111005310>])
_images/z_inflation_indexes_13_1.png 
[8]:

inflation_curve.plot_index(right=dt(2030, 6, 1))

[8]:

(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x11224e710>])
_images/z_inflation_indexes_14_1.png

Some of the forecast values from the curve can be obtained directly from Curve methods.

[9]:

inflation_curve.index_value(dt(2027, 4, 1), index_lag=3, interpolation="monthly")

[9]:

<Dual: 135.440324, (inflation0, inflation1, inflation2, ...), [-0.0, -0.0, -50.9, ...]>

[10]:

inflation_curve.index_value(dt(2027, 4, 1), index_lag=0, interpolation="monthly")

[10]:

<Dual: 136.382058, (inflation0, inflation1, inflation2, ...), [-0.0, -0.0, -15.8, ...]>

[11]:

inflation_curve.index_value(dt(2027, 5, 15), index_lag=3, interpolation="daily")

[11]:

<Dual: 135.896278, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, -33.9, ...]>

Seasonality#

The way rateslib handles seasonality is to replicate it via its CompositeCurve framework. With this approach, adding seasonality can be done in a multitude of ways, but since Quantlib has used seasonaility factors we will synthesize that approach within rateslib’s framework. We create a new curve with node dates reflecting the key dates used by Quantlib.

The theory here for rateslib is as follows.

A discount factor on a rateslib CompositeCurve is very approximately the product of its contained curves (c1 and c2):

\[v_{cc} \approx v_{c1} v_{c2}\]

The CompositeCurve index value takes its index base from the first (main) curve in its composition, therefore:

\[I_{value(cc)} = \frac{I_{base(c1)}}{v_{cc}} \approx = \frac{I_{base(c1)}}{v_{c1}v_{c2}} = I_{value(c1)} \frac{1}{v_{c2}}\]

If we set the index base on the seasonality curve (c2) exactly to 1.0 this equation reduces to:

\[I_{value(cc)} \approx I_{value(c1)} I_{value(c2)}\]

So the index value on the CompositeCurve equals the product of the two index values.

Thus, if we set the set the index values on the seasonality curve (c2) directly they can act directly as multipliers to the underlying inflation values on the inflation curve. This seasonality curve can be calibrated using a Solver, and in this particular case (to match Quantlib) they are calibrated directly with index values.

Here we add the same seasonality factors in years 2025, 2026 and 2027.

[12]:

season = Curve(
    nodes={
        dt(2024, 4, 1): 1.0,
        dt(2024, 12, 1): 1.0,
        **{dt(2025, _, 1): 1.0 for _ in range(1, 13)},
        **{dt(2026, _, 1): 1.0 for _ in range(1, 13)},
        **{dt(2027, _, 1): 1.0 for _ in range(1, 13)},
        dt(2074, 5, 11): 1.0,
    },
    interpolation="log_linear",
    convention="Act365F",
    index_base=1.0,  #  <- as per theory above
    index_lag=0,  # <- matches the main inflation curve
    id="seasonality"
)

The seasonality factors will be added by directly inserting index values at specific dates.

[13]:

multipliers = [
    1.003245,
    1.001994,
    0.999715,
    1.000495,
    1.000929,
    0.998687,
    0.995949,
    0.994682,
    0.995949,
    1.000519,
    1.003705,
    1.004186,
]
season_solver = Solver(
    curves=[season],
    instruments=\
        [Value(dt(2024, 12, 1), curves=[season], metric="index_value")] +\
        [Value(dt(2025, _, 1), curves=[season], metric="index_value") for _ in range(1, 13)] +\
        [Value(dt(2026, _, 1), curves=[season], metric="index_value") for _ in range(1, 13)] +\
        [Value(dt(2027, _, 1), curves=[season], metric="index_value") for _ in range(1, 13)],
    s=[1.0] + multipliers*3
)

SUCCESS: `func_tol` reached after 8 iterations (levenberg_marquardt), `f_val`: 1.290941550558499e-12, `time`: 0.0474s

[14]:

season.plot("1b", left=dt(2024, 4, 1), right=dt(2028, 4, 1))

[14]:

(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x1125f2850>])
_images/z_inflation_indexes_23_1.png

Once the seasonality is designed it must be composited with an underlying inflation curve and the instruments re-solved

[15]:

inflation_with_season = CompositeCurve([inflation_curve, season], id="inflation_s")

[16]:

solver = Solver(
    pre_solvers=[solver1],  # nominal discount curve
    curves=[inflation_with_season, inflation_curve],
    instruments=[
        ZCIS(dt(2024, 5, 11), "1y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "2y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "3y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "4y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "5y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "7y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "10y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "12y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "15y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "20y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "25y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "30y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "40y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
        ZCIS(dt(2024, 5, 11), "50y", spec="eur_zcis", curves=["inflation_s", "discount"], leg2_index_fixings=fixings),
    ],
    s=[2.93, 2.95, 2.965, 2.98, 3.0, 3.06, 3.175, 3.243, 3.293, 3.338, 3.348, 3.348, 3.308, 3.228],
    instrument_labels=["1y", "2y", "3y", "4y", "5y", "7y", "10y", "12y", "15y", "20y", "25y", "30y", "40y", "50y"],
    id="zcis",
)

SUCCESS: `func_tol` reached after 5 iterations (levenberg_marquardt), `f_val`: 1.7223655493350175e-13, `time`: 0.0377s

[17]:

inflation_with_season.plot("1b", comparators=[inflation_curve], left=dt(2024, 9, 1), right=dt(2030, 9, 1))

[17]:

(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x1127f5590>,
  <matplotlib.lines.Line2D at 0x1127f56d0>])
_images/z_inflation_indexes_27_1.png 
[18]:

inflation_with_season.plot_index(comparators=[inflation_curve], left=dt(2024, 9, 1), right=dt(2030, 9, 1))

[18]:

(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x112a1c410>,
  <matplotlib.lines.Line2D at 0x112a1c550>])
_images/z_inflation_indexes_28_1.png

New sampled values#

[19]:

inflation_with_season.index_value(dt(2027, 4, 1), index_lag=3, interpolation="monthly")

[19]:

<Dual: 135.613192, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, -50.9, ...]>

[20]:

inflation_with_season.index_value(dt(2027, 4, 1), index_lag=0, interpolation="monthly")

[20]:

<Dual: 136.170810, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, -15.7, ...]>

[21]:

inflation_with_season.index_value(dt(2027, 5, 15), index_lag=3, interpolation="daily")

[21]:

<Dual: 135.754993, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, -33.8, ...]>

The trick here is obviously to find a representation of a seasonaility curve that matches one’s expectation of seasonality adjustments. Here, a Solver calibration was used to separately solve the seasonality curve to inflation rate adjustments.

[22]:

for i, date in enumerate([dt(2025, _, 1) for _ in range(1, 13)]+[dt(2026, _, 1) for _ in range(1, 13)]):
    a = inflation_curve.index_value(date, index_lag=0, interpolation="monthly")
    b = inflation_with_season.index_value(date, index_lag=0, interpolation="monthly")
    print(f"Date: {date}, Inflation curve: {float(a):.5f}, With seasonality: {float(b):.5f}, Multiplier: {float(b/a):.7f}, Intended Multiplier: {multipliers[i%12]:.7f}")

Date: 2025-01-01 00:00:00, Inflation curve: 127.58510, With seasonality: 127.99907, Multiplier: 1.0032447, Intended Multiplier: 1.0032450
Date: 2025-02-01 00:00:00, Inflation curve: 127.75932, With seasonality: 128.01404, Multiplier: 1.0019938, Intended Multiplier: 1.0019940
Date: 2025-03-01 00:00:00, Inflation curve: 127.91688, With seasonality: 127.88042, Multiplier: 0.9997150, Intended Multiplier: 0.9997150
Date: 2025-04-01 00:00:00, Inflation curve: 128.09155, With seasonality: 128.15494, Multiplier: 1.0004949, Intended Multiplier: 1.0004950
Date: 2025-05-01 00:00:00, Inflation curve: 128.26081, With seasonality: 128.37995, Multiplier: 1.0009289, Intended Multiplier: 1.0009290
Date: 2025-06-01 00:00:00, Inflation curve: 128.56987, With seasonality: 128.40108, Multiplier: 0.9986871, Intended Multiplier: 0.9986870
Date: 2025-07-01 00:00:00, Inflation curve: 128.93158, With seasonality: 128.40934, Multiplier: 0.9959495, Intended Multiplier: 0.9959490
Date: 2025-08-01 00:00:00, Inflation curve: 129.30641, With seasonality: 128.61884, Multiplier: 0.9946826, Intended Multiplier: 0.9946820
Date: 2025-09-01 00:00:00, Inflation curve: 129.68234, With seasonality: 129.15706, Multiplier: 0.9959495, Intended Multiplier: 0.9959490
Date: 2025-10-01 00:00:00, Inflation curve: 130.04718, With seasonality: 130.11466, Multiplier: 1.0005189, Intended Multiplier: 1.0005190
Date: 2025-11-01 00:00:00, Inflation curve: 130.42525, With seasonality: 130.90842, Multiplier: 1.0037046, Intended Multiplier: 1.0037050
Date: 2025-12-01 00:00:00, Inflation curve: 130.79218, With seasonality: 131.33961, Multiplier: 1.0041855, Intended Multiplier: 1.0041860
Date: 2026-01-01 00:00:00, Inflation curve: 131.17243, With seasonality: 131.59803, Multiplier: 1.0032446, Intended Multiplier: 1.0032450
Date: 2026-02-01 00:00:00, Inflation curve: 131.55378, With seasonality: 131.81606, Multiplier: 1.0019938, Intended Multiplier: 1.0019940
Date: 2026-03-01 00:00:00, Inflation curve: 131.89917, With seasonality: 131.86159, Multiplier: 0.9997150, Intended Multiplier: 0.9997150
Date: 2026-04-01 00:00:00, Inflation curve: 132.28264, With seasonality: 132.34811, Multiplier: 1.0004949, Intended Multiplier: 1.0004950
Date: 2026-05-01 00:00:00, Inflation curve: 132.65479, With seasonality: 132.77801, Multiplier: 1.0009289, Intended Multiplier: 1.0009290
Date: 2026-06-01 00:00:00, Inflation curve: 132.99141, With seasonality: 132.81681, Multiplier: 0.9986872, Intended Multiplier: 0.9986870
Date: 2026-07-01 00:00:00, Inflation curve: 133.29532, With seasonality: 132.75542, Multiplier: 0.9959495, Intended Multiplier: 0.9959490
Date: 2026-08-01 00:00:00, Inflation curve: 133.61010, With seasonality: 132.89966, Multiplier: 0.9946827, Intended Multiplier: 0.9946820
Date: 2026-09-01 00:00:00, Inflation curve: 133.92562, With seasonality: 133.38316, Multiplier: 0.9959495, Intended Multiplier: 0.9959490
Date: 2026-10-01 00:00:00, Inflation curve: 134.23168, With seasonality: 134.30133, Multiplier: 1.0005189, Intended Multiplier: 1.0005190
Date: 2026-11-01 00:00:00, Inflation curve: 134.54867, With seasonality: 135.04711, Multiplier: 1.0037045, Intended Multiplier: 1.0037050
Date: 2026-12-01 00:00:00, Inflation curve: 134.85614, With seasonality: 135.42058, Multiplier: 1.0041855, Intended Multiplier: 1.0041860

Inflation Swap and DV01#

We can easily construct a ZCIS or other type of inflation based instrument and use the native delta and gamma methods associated with a Solver to extract risk sensitivities.

[23]:

zcis = ZCIS(dt(2024, 3, 11), "4y", spec="eur_zcis", curves=["inflation_s", "discount"], fixed_rate=3.0, leg2_index_fixings=fixings)

[24]:

zcis.rate(solver=solver)

[24]:

<Dual: 2.926124, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, 0.0, ...]>

[25]:

zcis.npv(solver=solver)

[25]:

<Dual: -2869.908680, (inflation0, inflation1, inflation2, ...), [0.0, 0.0, 0.0, ...]>

[26]:

df = zcis.delta(solver=solver)
descriptors = df.agg(["sum"])
descriptors.index = MultiIndex.from_tuples([("sum", "sum", "sum")])
df.style.format(precision=0).concat(descriptors.style.format(precision=0))

[26]:
    local_ccy eur
    display_ccy eur
type solver label  
instruments rates nominal 1
zcis 1y 4
2y -23
3y 93
4y 297
5y -0
7y 0
10y 0
12y 0
15y 0
20y 0
25y 0
30y 0
40y -0
50y -0
sum sum sum 373

This risk display shows no exposure to the seasonality configuration becuase the Solver that was used to calibrate the seasonality was not added as a pre_solver to the Solver used in the last step. Therefore it treats the seasonality as a constant and not as variable to which risk sensitivities can be obtained.

To observe the change, simply replace pre_solvers=[solver1] with pre_solvers=[solver1, season_solver], in the above Solver and re run the cells.

[27]:

zcis.gamma(solver=solver).style.format(precision=1)

[27]:
        type instruments
        solver rates zcis
        label nominal 1y 2y 3y 4y 5y 7y 10y 12y 15y 20y 25y 30y 40y 50y
local_ccy display_ccy type solver label                              
eur eur instruments rates nominal -0.0 -0.0 0.0 -0.0 -0.1 -0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 0.0 0.0
zcis 1y -0.0 -0.0 -0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 -0.0 -0.0
2y 0.0 -0.0 0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 0.0 0.0
3y -0.0 0.0 -0.0 -0.0 0.0 0.0 0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 -0.0 -0.0
4y -0.1 0.0 -0.0 0.0 0.1 0.0 -0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 -0.0 -0.0
5y -0.0 0.0 -0.0 0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 0.0 -0.0
7y 0.0 -0.0 0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 -0.0 0.0
10y 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 0.0 -0.0
12y 0.0 -0.0 0.0 -0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 -0.0 0.0
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