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Estimating Bond Time P&L Through Carry and Roll-Down

Article Quant Q&A · Author: Sohail
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Summary

The note considers how to estimate a bond’s price change from the passage of time, in a context where PV01 is used to approximate P&L from yield-curve movements. The central point is that time passage alone does not define a unique scenario: the calculation must specify what is assumed to happen to the curve. Possible assumptions include an unchanged term-structure shape, realized forward rates, or another stated curve scenario.

With an assumption selected, the bond can be repriced at a future date and compared with its current price to estimate the time-related effect. The answer cautions that “theta” is uncommon terminology for bonds and portfolios; carry, roll-down, or roll-yield may better describe the relevant return components. A second response offers the simpler idea of moving the valuation forward by one day, but does not address the curve assumptions needed to interpret that result. No worked example or empirical evidence is provided, so the output depends on the chosen scenario.

Key ideas

  • A bond’s time-related P&L estimate requires a defined assumption about how the yield curve evolves.
  • Possible scenarios include an unchanged curve shape, realized forwards, or another hypothesized curve path.
  • Repricing at a future date under the chosen scenario gives an estimate of the passage-of-time effect.
  • Carry, roll-down, and roll-yield are often more suitable terms than theta for bond returns.

Tags

Fixed incomeCarryDerivatives pricing
Full text
# How to calculate the theta of a bond?


# How to calculate the theta of a bond?












For calculating P&L from interest rate risk, we often use PV01 to estimate the day over day P&L by multiplying PV01 with a change in curve.

Is there any approach to calculate theta P&L in a similar way?

## Answer by Quantifeye (score 2)

https://quant.stackexchange.com/a/30261

To answer that question you first have to define what "no change other than the passage of time" means. So you could make one of the following "no change" assumptions.

- the shape of the term structure will remain unchanged.

- assumption of realized forwards.

- assumption that some other hypothesized scenario will realize.

Based on one of those assumptions you can then reprice the bond based on what you think the price of the bond will be at some point in the future in order to gauge the effect of time.

It is somewhat unusual to use the term theta in the context of a bond or a bond portfolio however. It would be more appropriate to focus such a discussion around the components of return. I think what you are referring to is carry-roll-down or roll-yield.

For an overview of the subject see Fixed Income Securities: Tools for Today's Markets 3rd Edition.

## Answer by César (score 0)

https://quant.stackexchange.com/a/30224

Theta is a position's sensitivity to a small change in time to maturity.

You just simply would see what happens to the bond when you go one day ahead.

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.