# Duration + Jensen's Inequality

#### forty two

I know that duration decreases as spot rates increase. This is evident from taking the partial derivative of duration w.r.t to the yield. But how can that relationship be expressed in the form of Jensen's inequality: (E[X^{2}] \leq E[X]^{2})

Any help will be appreciated!!!

#### Sanket Patel

##### i do stuff
Take a look at the attached PDF.

It's a tad sloppy, but it should get the point across.

#### Attachments

• jensen.pdf
44.8 KB · Views: 31

#### Bastian Gross

##### German Mathquant
This estimation isn't tough, but vital to understand.

#### forty two

Take a look at the attached PDF.

It's a tad sloppy, but it should get the point across.

Thank you for your reply - it is actually quite simple. I didn't think to look at duration as an expectation.

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