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Calculating the 5-Year Forward Inflation Swap Rate

Article Quant Q&A · Author: filifunk
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Summary

The document explains why a common Bloomberg approximation for the five-year inflation rate beginning five years ahead uses twice the 10-year zero-coupon inflation swap rate minus the 5-year rate. It contrasts that linear calculation with a compounded forward-rate calculation: under annual compounding, the forward inflation rate is obtained from the ratio of the 10-year accumulated inflation factor to the 5-year factor, then adjusted for the five-year forward period.

The answer says the linear and compounded methods usually produce similar values in market data, while differences can grow when inflation is higher. It characterizes the linear convention as simpler and the compounded calculation as more theoretically consistent with zero-coupon swaps. The document gives no market observations or derivation, so it does not quantify the difference or discuss details such as compounding conventions, day counts, or swap-market adjustments.

Key ideas

  • The linear 5-year forward approximation is twice the 10-year swap rate minus the 5-year swap rate.
  • A compounded forward calculation uses the ratio of accumulated inflation factors across the two maturities.
  • The two approaches are described as producing similar market values in ordinary conditions.
  • The compounded method can differ more when inflation rates are higher.
  • The answer favors compounding as more consistent with zero-coupon swap definitions.

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Fixed incomeDerivatives pricingStatistics
Full text
# Why is the formula for the 5y5y inflation swap forward on bloomberg 2*USSWIT10 Curncy-USSWIT5 Curncy


# Why is the formula for the 5y5y inflation swap forward on bloomberg 2*USSWIT10 Curncy-USSWIT5 Curncy












If USSWIT10 Curncy is the expected average inflation for ten years and the USSWIT5 Curncy is the expected average inflation for five years, I don't know why that formula would tell me what the expected 5 year inflation rate would be 5 years from now. Shouldn't there be some sort of compounding taken into account?

I almost feel like you should have to take ((1+USSWIT10)^10 - (1+USSWIT5)^5)/5. Something like that.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/70497

The standard method is 2*USSWIT10 - USSWIT5 which as you say, has no compounding. The compounded version that you attempted should be (1+USSWIT10)^2/((1+USSWIT5). If you experiment with actual market values you will find the two are very similar. The higher inflation rates get, the more potential difference there could be. I assume that the non compounded method has been used for simplicity but you are right, the compounded would be more correct since the USSWIT values are zero coupon swaps by definition.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.