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I'm having trouble understanding DV01 convention. From what I understand it stands for "Dollar value of 1 basis point."

For instance, if I have a bond with DV01 = 0.05, does a portfolio composed of 1,000 of this bond also have DV01 of 0.05? Or would it be proportionally larger, since a 1 basis point increase in this bond would result in a $50 loss for the portfolio?

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If the bond's DV01 is 0.05, then the DV01 of 1000 of this bond will be $0.05\times 1000 = 50$. By contrast, if the modified or effective duration of the bond is 0.05, then the modified duration of 1000 of this bond is still 0.05.

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