S2000magician : Effective duration = Modified duration for option-free bond ?

[alpha668]

S2000magician,

Sorry, I have to confirm with you if :

Effective duration (ED) = Modified duration (MD) for option-free bond ?

[S2000magician]

As long as the cash flows don’t change, yes.

Modified duration assumes that the cash flows don’t change; effective duration allows that cash flows might change.

So, if you have a bond whose cash flows cannot change, modified duration and effective duration will be the same. Generally, that condition is satisfied for option-free bonds. One case where it wouldn’t be true for an option-free bond would be if it were a floating-rate bond: modified duration would assume that the coupons don’t change, effective duration would allow that they might.

[alpha668]

S2000magaician,

Thank you for your response ! Further questions :

1. ED =(V- - V+) / (2 x Vo x Delta Y) while MD = Macaulay Duration / (1 + Y / k), k : number of payments per year.

Will ED = MD ?

2. I know that for zero coupon bonds, Macaulay Duration = Maturity, but how about the MD ? We know that there is no payment for zero coupon bonds until the entire par value is paid at maturity.

[S2000magician]

@alpha668

S2000magaician,

Thank you for your response!

My pleasure.

@alpha668

Further questions :

1. ED =(V- - V+) / (2 x Vo x Delta Y) while MD = Macaulay Duration / (1 + Y / k), k: number of payments per year.

Will ED = MD?

For a bond whose cash flows cannot change (i.e., fixed coupon, no options), almost.

(There are two slight qualifications:

1. The MD calculation is an instantaneous rate of change, whereas the ED calculation is an average rate of change. The smaller the Δy, the closer the ED calculation will be to MD.)
2. k is not the number of payments per year, it’s the number of compounding periods per year. Usually those will be the same (and generally will be 2), but not always. See below.))

@alpha668

2. I know that for zero coupon bonds, Macaulay Duration = Maturity, but how about the MD? We know that there is no payment for zero coupon bonds until the entire par value is paid at maturity.

Unless YTM = 0, modified duration is less than Macaulay duration: MD = Macaulay Duration / (1 + YTM/k). If YTM is given as BEY (the usual convention), then k = 2 here, even though there are no interim payments made.

[alpha668]

@S2000magician

Unless YTM = 0, modified duration is less than Macaulay duration: MD = Macaulay Duration / (1 + YTM/k). If YTM is given as BEY (the usual convention), then k = 2 here, even though there are no interim payments made.

In some countries, coupon payments are made annually (rather than semiannually) for coupon bonds. In this case, how to calculate the MD of a zero-coupon bond ? k = 1 or k = 2 ?

[S2000magician]

@alpha668

@S2000magician

Unless YTM = 0, modified duration is less than Macaulay duration: MD = Macaulay Duration / (1 + YTM/k). If YTM is given as BEY (the usual convention), then k = 2 here, even though there are no interim payments made.

In some countries, coupon payments are made annually (rather than semiannually) for coupon bonds. In this case, how to calculate the MD of a zero-coupon bond? k = 1 or k = 2?

k = 1. Be careful, however, if the YTM is quoted as a BEY; you’ll need to calculate the annual (EAY) YTM.

[alpha668]

@S2000magician

@alpha668

@S2000magician

Unless YTM = 0, modified duration is less than Macaulay duration: MD = Macaulay Duration / (1 + YTM/k). If YTM is given as BEY (the usual convention), then k = 2 here, even though there are no interim payments made.

In some countries, coupon payments are made annually (rather than semiannually) for coupon bonds. In this case, how to calculate the MD of a zero-coupon bond? k = 1 or k = 2?

k = 1. Be careful, however, if the YTM is quoted as a BEY; you’ll need to calculate the annual (EAY) YTM.

Thank you so much for your clarification !

[S2000magician]

@alpha668

@S2000magician

@alpha668

@S2000magician

Unless YTM = 0, modified duration is less than Macaulay duration: MD = Macaulay Duration / (1 + YTM/k). If YTM is given as BEY (the usual convention), then k = 2 here, even though there are no interim payments made.

In some countries, coupon payments are made annually (rather than semiannually) for coupon bonds. In this case, how to calculate the MD of a zero-coupon bond? k = 1 or k = 2?

k = 1. Be careful, however, if the YTM is quoted as a BEY; you’ll need to calculate the annual (EAY) YTM.

Thank you so much for your clarification!

Happy to be of help.