# Duration and Coupon Relationships

Hi Everyone

I seem to battle with this scenario:

3 bonds
Each have 5 year maturities
YTM = 9%
Coupons = 7%, 9% and 11% respectively

Question - How do you plot the relationship between the Duration and coupon int. that is paid?

Are the frequencies the same; e.g., all three bonds pay semiannually?

Hi Magician

All are annual.

Thanks for the prompt response.

T

In that case, the 7% coupon will have the longest duration and the 11% coupon will have the shortest duration.

This is most easily seen by computing their Macaulay durations. The higher the coupon, the greater the early payments, so the greater the percentage of present value that comes early, so the shorter the Macaulay duration. And as Macaulay duration goes, so goes modified duration.

one way is to use the R plot function, where x is the coupon rate and y is the duration. as you’d expect, the duration decreases as the coupon rate increases. you need to define the functions (just turn the textbook formula into code). assuming i didn’t make an embarrassing mistake, the graph should look like this:

I used Excel’s MDURATION function and got 4.00 years, 3.89 years, and 3.80 years, respectively. I verified those with my own calculations.

i’m guessing excel uses modified duration, while i used Macaulay duration.
note that:
r = .07 gives
Macaulay Duration = 4.358789
Modified Duration = 3.998889

r = .09
Macaulay Duration = 4.23972
Modified Duration = 3.889651

r = .11
Macaulay Duration = 4.137839
Modified Duration = 3.796182

edit: fixed. i meant coupon rate r, not i.

Yes: MDURATION is modified duration, not Macaulay.

You are a star - makes perfect sense - i did them manually and they are correct
Thanks Champ!

Thanks a Mil @S2000magician @not_a_CFA much appreciated

you are very welcome, and good luck in your studies

My pleasure.

there’s a fun website here Duration

just above the bottom us “Modified Duration Calculation”
you can use the sliders on that page to change YTM, coupon rate, and tenor to see the effect on the (modified) duration
it rounds off the numbers given by not_a_CFA
r=0.07 modified duration 4
r=0.09 modified duration 3.89
r=0.11 modified duration 3.8