# DV01 of Interest rate swap

I am a beginner in financial risk management and recently I have been studying the plain vanilla interest rate swap.

I came across several articles talking about DV01 of interest rate swap as follow.

$$DV01(t) = \frac{\partial V_{swap}(t)}{\partial R_{fix}} = \sum_{j=1}^N \alpha_j Z_t(t_j)$$

My question is: In the equation, $R_{fix}$ should be in decimals. Then for a 1 basis point (i.e. $1/10000$) change in $R_{fix}$, shouldn't it lead to a $(1/10000) \sum_{j=1}^N \alpha_j Z_t(t_j)$ change in the price of the interest rate swap?

Thank you.

• The units in which DV01 is expressed are not always consistent, different software or books use different conventions. Sometimes "dollars per 1 bp", sometimes "dollars per 100 bp", etc. In your example they are apparently using dollars per 100 percentage points. This post gives the derivation quant.stackexchange.com/questions/31548/… Jun 4 '17 at 12:35
• When you read these equations, you should take the first equal sign as the definition of DV01 in this paper, and then remember that it is 10000 times larger than the definition you are used to. Jun 4 '17 at 12:45

## 1 Answer

Since DV01 and PV01 are short for 'dollar value of a basis point' and 'present value of a basis point' (for currencies not in USD) respectively then you are right that the value should be scaled to give the right value inline with the definition.

As a commenter highlighted definitions are not necessarily consistent across sources. I have written a number of articles myself and am guilty of this inconsistency, unfortunately. Since it generally obfuscates delta and gamma formulae to have $10^{-4}$ and $10^{-8}$ scalars I tend to leave them out with a footnote. The results and terminology often follow from context.

One other important item you should be aware that is not consistent is direction. Traders of interest rates can reference long/short differently and have plus/minus sign conventions for the risk of the same interest rate move.