# 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?

• Please, please, please: A bond does not have a Theta. Theta and Gamma are concepts that apply to options, which involve hedging or speculating on future contingent events. Bonds have duration, convexity, rolldown and other things but you will only confuse everyone if you speak of bond theta. Please specify exactly what you wish to calculate for bonds. Jun 19 '16 at 20:40

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.

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.

• That's fine, that's the short term (1 day) return, which in the absence of additional assumptions is a random variable. If you make the assumption that time moves on by one day while the shape of the yield curve (the ZCB curve) remains unchanged then it is the "rolldown" and is a specific computable number. With different assumptions you get different, interesting, numbers; I just don't call them "thetas". Sep 20 '16 at 22:39