Why Year-on-Year Inflation Swaps May Need Convexity Adjustments

# Intuition for convexity adjustment for year on year inflation swaps


# Intuition for convexity adjustment for year on year inflation swaps












I am trying to understand the intuition behind why a convexity adjustment is required when calculating the YoY rate on inflation swaps.

(Assume no lags for simplicity). The current inflation index is 100. The 3y zero coupon inflation rate in the market is 3% The 4y zero coupon inflation rate in the market is 4%.

This means, the projected inflation index in Year 3 = 100 x (1+3%)^3 = 109.2727 The projected inflation index in year 4 = 100 x (1+4%)^4 = 116.985856

Therefore the YoY rate between years 3 and 4 = 116.985856 / 109.2727 - 1 = 7.0586%.

However, literature tells you that you need to add a convexity adjustment to this and I'm trying to understand why. Both the 3y and 4y are based on market projections, so shouldn't it follow the YoY is also based on that?

## Answer by Attack68 (score 7)

https://quant.stackexchange.com/a/81314

Convexity adjustments are required due to the natural instruments that you use to construct a delta hedged portfolio leaving residual gamma risks. These are the instruments that give rise to the so called numeraire.

Another way of phrasing this is by saying that if you take an instrument and delta hedge it with natural instruments, then if you would prefer to be positioned in either direction after the application of the delta hedges there should be a convexity adjustment to reflect that preference.

Think about some of these examples (there are others) and whether a convexity adjustment is required:

- An IRS with a payment lag of 30 days hedged with a natural 2 day lagged IRS. (yes, there is a preference on direction)

- A STIR future hedged by a single period IRS (yes, there is a preference on direction)

- A 3y1y IRS hedged by a 4y IRS and a 3y IRS (no, there is no preference)

The reason why a delta hedged portfolio of this type would have cross-gamma risks is because a delta hedged portfolio will not make (or lose) any PnL if the CPI market changes, however, its cash profile will change, the 3y ZCIS hedge will pay a net cashflow at the 3y maturity, whilst the 3y1y and 4y ZCIS will pay cashflows at the 4y point, when the CPI market moves around these cashflows will change in scale creating discounting risks.

I have not traded ZCIS much myself but if I were to analyse them the following would be my process.

- Setup a market inline with your data:

```
from rateslib import *  # Python 3.12, rateslib 1.6.0

disc_curve = Curve({dt(2024, 12, 2): 1.0, dt(2027, 12, 2): 1.0, dt(2028, 12, 10): 1.0}, calendar="nyc", id="sofr")
index_curve = IndexCurve({dt(2024, 12, 2): 1.0, dt(2027, 12, 2): 1.0, dt(2028, 12, 10): 1.0}, id="us_cpi", index_base=100.0)

solver = Solver(
    curves=[disc_curve, index_curve],
    instruments=[
        IRS(dt(2024, 12, 2), "3y", spec="usd_irs", curves="sofr"),
        IRS(dt(2024, 12, 2), "4y", spec="usd_irs", curves="sofr"),
        ZCIS(dt(2024, 12, 2), "3y", convention="1+", frequency="A", curves=["us_cpi", "sofr"], leg2_index_base=100.00, leg2_index_method="daily"),
        ZCIS(dt(2024, 12, 2), "4y", convention="1+", frequency="A", curves=["us_cpi", "sofr"], leg2_index_base=100.00, leg2_index_method="daily"),
    ],
    s=[3.9, 3.8, 3.0, 4.0],  # market rates including your 3% and 4%
    instrument_labels=["ir3y", "ir4y", "cpi3y", "cpi4y"]
)
```

- Check the value of your suggested 3y1y ZCIS trade:

```
zcis = ZCIS(dt(2027, 12, 2), "1y",  convention="1+", frequency="A", curves=["us_cpi", "sofr"], leg2_index_method="daily", notional=100e6)

zcis.rate(solver=solver)
# 7.058630
```

- Propose delta hedges using the natural 3y and 4y ZCIS instruments, and check the delta risks:

```
hedge1 = ZCIS(dt(2024, 12, 2), "3y", notional=94.625e6, convention="1+", frequency="A", curves=["us_cpi", "sofr"], leg2_index_base=100.00, leg2_index_method="daily")
hedge2 = ZCIS(dt(2024, 12, 2), "4y", notional=-91.5148e6, convention="1+", frequency="A", curves=["us_cpi", "sofr"], leg2_index_base=100.00, leg2_index_method="daily")

pf = Portfolio([zcis, hedge1, hedge2])
pf.delta(solver=solver).style.format(precision=0)
```

- Check the cross-gamma risks and discover if there is a preference

```
pf.gamma(solver=solver).style.format(precision=0)
```

So there are resultant cross-gamma risks. Assuming all instruments (rates and cpi) are correlated with the same vol this all nets to a small value so it is difficult to ascertain which direction is the preference (if any). I'm sure the literature attempts to break this all down to closed form equations (which I have done in the past for convexity adjustments on zero coupon swaps vs IRS).