[Quantlib-users] Implied inflation vs results from ZeroCouponInflationSwapHelper and PieceWiseZeroI

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@Luigi Ballabio <lui...@gm...> posted this very
informative piece,
https://www.implementingquantlib.com/2024/05/inflation-curves.html
on Inflation indexes and curves which has been of great help.

What I am struggling with, is that when I price a CPI bond, I would expect
the value of the bond priced off the real curve to equal the price of the
CPI bond priced off the nominal curve with coupons increased by the
forecasted inflation rate as defined by implied inflation being the
difference between the nominal and real zero curves.

I have investigated this here by adapting the Python code from
https://www.implementingquantlib.com/2024/05/inflation-curves.html, by
defining a Nominal and Real curve, with implied inflation being the
geometric diff of the two curves.

The problem is that I was expecting ql.CPI.laggedFixing(hicp, some_date,
observation_lag, ql.CPI.Linear))/ ValueBase  to be very close to 1.0/
implied_infl_curve.discount(some_date)) however this does not seem to be
the case:

node nom real imp_inf base_val inf diff
May 11th, 2024    1.0000000  1.0000000  1.0000000  124.6732258
1.0000000
-
May 12th, 2025    1.0620111  1.0298161  1.0312628  124.6732258
1.0308708 -0.0003802

May 11th, 2026    1.1274969  1.0607752  1.0628989  124.6732258
1.0620177 -0.0008291

May 11th, 2027    1.1972174  1.0930260  1.0953238  124.6732258
1.0939309 -0.0012716

May 11th, 2028    1.2714581  1.1266872  1.1284926  124.6732258
1.1264751 -0.0017878

May 11th, 2029    1.3500807  1.1619297  1.1619297  124.6732258
1.1594619 -0.0021240

May 12th, 2031    1.5224620  1.2390781  1.2287054  124.6732258
1.2251594 -0.0028860

May 11th, 2034    1.8227180  1.3739282  1.3266471  124.6732258
1.3215592 -0.0038352

May 12th, 2036    2.0557845  1.4762652  1.3925577  124.6732258
1.3863294 -0.0044726

May 11th, 2039    2.4608164  1.6392202  1.5012116  124.6732258
1.4932337 -0.0053143

May 11th, 2044    3.3228469  1.9504444  1.7036358  124.6732258
1.6918066 -0.0069435

May 11th, 2049    4.4861115  2.3106996  1.9414516  124.6732258
1.9248788 -0.0085363

May 11th, 2054    6.0566127  2.7320222  2.2168973  124.6732258
2.1941767 -0.0102488

May 12th, 2064  11.0431267  3.7591653  2.9376540  124.6732258
2.8960907 -0.0141484

May 11th, 2074  20.1251968  5.0281959  4.0024687  124.6732258
3.9279265 -0.0186241

To replicate please see the Python below:

import QuantLib as ql
import pandas as pd
today = ql.Date(11, ql.May, 2024)
ql.Settings.instance().evaluationDate = today

index = ql.EUHICP()

inflation_fixings = [
    ((2022, ql.January), 110.70),
    ((2022, ql.February), 111.74),
    ((2022, ql.March), 114.46),
    ((2022, ql.April), 115.11),
    ((2022, ql.May), 116.07),
    ((2022, ql.June), 117.01),
    ((2022, ql.July), 117.14),
    ((2022, ql.August), 117.85),
    ((2022, ql.September), 119.26),
    ((2022, ql.October), 121.03),
    ((2022, ql.November), 120.95),
    ((2022, ql.December), 120.52),
    ((2023, ql.January), 120.27),
    ((2023, ql.February), 121.24),
    ((2023, ql.March), 122.34),
    ((2023, ql.April), 123.12),
    ((2023, ql.May), 123.15),
    ((2023, ql.June), 123.47),
    ((2023, ql.July), 123.36),
    ((2023, ql.August), 124.03),
    ((2023, ql.September), 124.43),
    ((2023, ql.October), 124.54),
    ((2023, ql.November), 123.85),
    ((2023, ql.December), 124.05),
    ((2024, ql.January), 123.60),
    ((2024, ql.February), 124.37),
    ((2024, ql.March), 125.31),
    ((2024, ql.April), 126.05),
]

for (year, month), fixing in inflation_fixings:
    index.addFixing(ql.Date(1, month, year), fixing)

index.fixing(ql.Date(15, ql.March, 2024))

observation_lag = ql.Period(3, ql.Months)

ql.CPI.laggedFixing(
    index, ql.Date(15, ql.May, 2024), observation_lag, ql.CPI.Linear
)

real_quotes = [
    (ql.Period(1, ql.Years), 2.93),
    (ql.Period(2, ql.Years), 2.95),
    (ql.Period(3, ql.Years), 2.965),
    (ql.Period(4, ql.Years), 2.98),
    (ql.Period(5, ql.Years), 3.0),
    (ql.Period(7, ql.Years), 3.06),
    (ql.Period(10, ql.Years), 3.175),
    (ql.Period(12, ql.Years), 3.243),
    (ql.Period(15, ql.Years), 3.293),
    (ql.Period(20, ql.Years), 3.338),
    (ql.Period(25, ql.Years), 3.348),
    (ql.Period(30, ql.Years), 3.348),
    (ql.Period(40, ql.Years), 3.308),
    (ql.Period(50, ql.Years), 3.228),
]

calendar = ql.TARGET()
observation_lag = ql.Period(3, ql.Months)
day_counter = ql.Actual365Fixed()
interpolation = ql.CPI.Linear

nominal_curve = ql.YieldTermStructureHandle(
    ql.FlatForward(today, 0.06, ql.Actual365Fixed()) # Continuous, Annual
)

rates,dates = [],[]
for tenor, quote in real_quotes:
    maturity = calendar.advance(today, tenor)
    rates.append(quote/100.0)
    dates.append(maturity)

real_curve = ql.ZeroCurve([today] + dates,[0.0] + rates,day_counter) #
Continuous, Annual
implied_infl_curve = ql.DiscountCurve([today] + dates,[1.0]+[nominal_curve.
discount(d)/real_curve.discount(d) for d in dates],ql.Actual365Fixed())

helpers = []

for tenor, quote in real_quotes:
    maturity = calendar.advance(today, tenor)
    helpers.append(
        ql.ZeroCouponInflationSwapHelper(
            ql.makeQuoteHandle(implied_infl_curve.zeroRate(maturity,ql.
Actual365Fixed(),ql.Continuous).rate()), # Continuous, Annual
            observation_lag,
            maturity,
            calendar,
            ql.Following,
            day_counter,
            index,
            interpolation,
            nominal_curve,
        )
    )

ql.Date(1,1,2024)
fixing_frequency = ql.Monthly

inflation_curve = ql.PiecewiseZeroInflation(
    today,index.lastFixingDate(), fixing_frequency, ql.Actual365Fixed(),
helpers
)
hicp = ql.EUHICP(ql.ZeroInflationTermStructureHandle(inflation_curve))
hicp.fixing(ql.Date(2, ql.February, 2074))
hicp.fixing(ql.Date(11, ql.May, 2024))

pd.DataFrame(inflation_curve.nodes(), columns=["node", "rate"]).style.format
(
    {"rate": "{:.4%}"}
)

df = pd.DataFrame(real_curve.nodes(), columns=["node", "yreal"])
df['nom'] = df.node.map(lambda x: 1.0/nominal_curve.discount(x))
df['real'] = df.node.map(lambda x: 1.0/real_curve.discount(x))
df['imp_inf'] = df.node.map(lambda x: 1.0/implied_infl_curve.discount(x))
df['base_val'] = df.node.map(lambda x: ql.CPI.laggedFixing(index, df.loc[0,
'node'], observation_lag, ql.CPI.Linear))
df['inf'] = df.node.map(lambda x: ql.CPI.laggedFixing(hicp, x,
observation_lag, ql.CPI.Linear))
df['inf'] = df.inf/df.base_val
df['diff'] = df.inf/df.imp_inf-1
df

Am I on the right track or am I missing something? Any feedback would be
much appreciated.

[Quantlib-users] Implied inflation vs results from ZeroCouponInflationSwapHelper and PieceWiseZeroI

[Quantlib-users] Implied inflation vs results from ZeroCouponInflationSwapHelper and PieceWiseZeroInflation