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Separating Carry and Rolldown for Inflation-Linked Bonds

Article Quant Q&A · Author: sciencemonk
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Summary

The document asks how to decompose nominal returns on inflation-protected bonds into carry and rolldown. Its proposed starting point is to project inflation index levels and hold the nominal discount curve fixed, then compare the bond’s present value now with its value one year later. The response offers a currency-based framework: treat inflation-adjusted dollars as a separate currency, so an inflation-linked bond becomes a fixed-coupon bond denominated in that unit.

Under this view, carry is the predetermined coupon accrual in inflation-adjusted dollars. Rolldown is the remaining change in value as the valuation date advances while the real yield curve is unchanged; more generally, it can be derived from the bond’s theta after subtracting carry. The explanation applies the same idea to TIPS and other inflation-linked bonds. It is conceptual rather than a worked numerical derivation, and it explicitly sets aside embedded deflation floors, which can affect actual bond values and attribution.

Key ideas

  • Treat inflation-adjusted dollars as a separate currency for analyzing inflation-linked bonds.
  • In that currency, a bond with fixed real cash flows resembles a fixed-coupon bond.
  • Carry is the predetermined coupon accrual expressed in inflation-adjusted units.
  • Rolldown can be estimated from the value change as time advances with the yield curve held fixed, less carry.
  • Embedded floors are excluded from the proposed simplification.

Tags

Fixed incomeCarryDerivatives pricingRisk management
Full text
# What is the most convincing method/formula for carry and rolldown (in nominal terms) of inflation protected bonds


# What is the most convincing method/formula for carry and rolldown (in nominal terms) of inflation protected bonds












It is interesting that there is no thorough discussion and clear derivation on this per my search. I know TIPS are complex (compared to nominal bonds). The naive use of simple spot/forward yield difference seems to be the "carry/rolldown" in real terms but not nominal.

My best hunch is that: assuming we ignore floor protection, and assuming we have built a breakeven inflation curve (seasonally adjusted) using whatever method. We fix 1) nominal market discount (treasury curve) and 2) all breakeven price indexes in the future, then we calculate the fair present value of TIPS as of today, then we recalculate the present value one year later, in which we keep nominal term structure the same, and breakeven prices indexes realized. The difference is then the total expected nominal value change of TIPS.

Any better idea?

## Answer by Dimitri Vulis (score 3)

https://quant.stackexchange.com/a/55386

Long ago, I built a good (IMHO) P&L-explain for Latin American inflation-linked bonds, which included usable C&RD. I hope the below ideas might help.

In markets like Mexico, Chile, Colombia an "inflation-adjusted currency" is treated as a separate currency. This is extremely convenient! You can say that you will have a fixed cash flow of $N$ "unidades de fomento" on a future date. You can then project how much that might turn out to be in later in the nominal currency or in USD. You should take the same approach with U.S. TIPS (or Brazil NTN-Bs, Japanese, British, etc bonds) - create an "inflation-adjusted USD" currency. Then (ignoring any embedded floors!) TIPS are very simple fixed-coupon bonds denominated in this currency.

Their carry is just the predetermined accrued expressed in "inflation-adjusted USD".

There are many ways of calculating bond "theta", but they all predict the P&L if the valiation date changes, but TIPS yield curve does not change. Subtracting the carry gives the rolldown.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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