# Modified duration and Money duration

Hi!

I am having troubles studying about Modified duration and Money duration, and hoping to get some helps.

1. Modified duration can be used to measure the percentage price change upon absolute change in YTM. I am confused about YTM here. Modified duration can be shown as

Modified duration = (dP/P)/d(1+r)

And I think the r here represents periodic interest rate.

So，suppose there is a semiannually paid bond has a modified duration of 5, and the YTM declines 0.25%. In calculating the price percentage change, why use 5*0.25%, instead of 5*0.25%/2 as the periodic interest rate?

2. If there is no any statement, is money duration per bps, per 1% or per 1 YTM? For example, if the money duration is \$10,000, and the YTM increases 25 bps, the value of the bond would change 10000*25, 10000*0.25, or 10000*0.0025?

Thanks!

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waaaaaz wrote:
Hi!

Howdy.

waaaaaz wrote:
I am having troubles studying about Modified duration and Money duration, and hoping to get some helps.

1. Modified duration can be used to measure the percentage price change upon absolute change in YTM. I am confused about YTM here. Modified duration can be shown as

Modified duration = (dP/P)/d(1+r)

And I think the r here represents periodic interest rate.

Assuming that you’re using “d” to represent the differential (derivative) operator, this is correct (I think), but it’s overly complicated.

Modified duration measures the (negative of the) percentage change in a portfolio’s value for a change in its YTM.  You could write it this way:

−(dP/dy) / P

The way you wrote it is a bit confusing: you separated out the differential of the price from the differential of the YTM, and you added 1 to the YTM which doesn’t change the differential.

Also, the r in your formula is the annual rate, not the periodic rate.

waaaaaz wrote:
So，suppose there is a semiannually paid bond has a modified duration of 5 …

Note that this is, properly, 5 years.

waaaaaz wrote:
… and the YTM declines 0.25%. In calculating the price percentage change, why use 5*0.25%, instead of 5*0.25%/2 as the periodic interest rate?

Because modified duration is based on the annual YTM, not the semiannual TYM.

waaaaaz wrote:
2. If there is no any statement, is money duration per bps, per 1% or per 1 YTM?

Thanks!

Money duration is (modified or effective) duration multiplied by the value of the portfolio, so it’s the (negative of the) change in portfolio value for a 1% change in YTM.

waaaaaz wrote:
For example, if the money duration is \$10,000, and the YTM increases 25 bps, the value of the bond would change 10000*25, 10000*0.25, or 10000*0.0025?

First, note that the units on money duration (for a portfolio denominated in \$) are \$years, not \$.

The value of the bond would decrease by approximately \$10,000 × 0.25 = \$2,500

waaaaaz wrote:
Thanks!

You’re welcome.

Simplify the complicated side; don't complify the simplicated side.

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