Question

Do you agree or not agree with this statement made by a finance student? Explain why. lf two bonds with the same term to maturity and the same yield have dierent coupon rates, the one with the higher coupon rate will have the higher (longer) duration

Answer 1

I do not agree with the student's statement.
Higher the coupon rate higher the duration is a wrong statement because higher the coupon rate lower the duration.This is because which  coupons are high then the price of bond is recovered lot earlier than expected maturity hence the duration will be less. Lowe coupon rate bonds recovery is slightly later , hence duration is higher. Zero coupon bond has duration equal to maturity because of no coupons.

Maculay Duration =(PV of Coupon_t*time+Par Value * Maturity)/Price of Bond
Coupon_t means coupon at time period t
modified Duration =Maculay Duration/(1+YTM)

As per this formula when coupon rate increases percentage increase in the numerator is less as compared to percentage increase in price of bond . Hence the duration decreases with increase in coupon.

Example
Let par Value of bond = 1000
Number of years = 2
YTM = 10%
Price of zero coupon bond = 1000/(1+10%)^2= 826.45
a zero coupon bond duration as per formula = (1000*2/(1+10%)^2)/826.45 = 2 years

Let us take another coupon bearing bond
Let coupon rate = 10%
coupon = 100
Since YTM and Coupon rate are same price of bond = Par Value
Duration using formula =(100*1/(1+10%)+100*2/(1+10%)^2+1000*2/(1+10%)^2)/1000 =1.91 years

from example it is clear if all other condition are constant then duration of bond is higher when coupon rate is lower.