# 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?

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.