Question

A $1,000 face value bond has a coupon of 5% (paid annually) and will mature 20    years from today?

A.   Assume that the yield-to-maturity is 6%. What is the bond’s:

            i. Duration

           ii. Modified Duration

B.    Assume that the bond’s yield-to-maturity immediately changes from 6% to 6.1% (the bond still has 20 years to maturity).

            i. Estimate the % change in the bond’s price using

                                                             modified duration

                                          ii.    What is actual bond price (at YTM = 6.1%), and the % price change (from YTM = 6% to 6.1%)?

C.    Assume that the bond’s yield-to-maturity immediately changes from 6% to 7% (the bond still has 20 years to maturity).

      i.    Estimate the % change in the bond’s price using modified

                duration

                                            ii. What is actual bond price (at YTM = 7%), and the % price    

                                                       change (from YTM = 6%)?

D.   Why is the estimated % price change closer in Part B than it is in Part C? Be precise!

Answer 1

(A)

(i) (ii)

(B) Change in YTM = 0.1% or 10 bps

% Change in Bond Price = (-Modified Duration) x (Change in YTM in BPS/100) = (-11.90) x (10/100) = - 1.19 %

New Bond YTM = 6.1 %

New Bond Price = 50 x (1/0.061) x [1-{1/(1.061)^(20)}] + 1000 / (1.061)^(20) = $ 874.85 approximately.

Actual % Change in Price = (874.85 - 885.30) / 885.30 = - 1.18 % approximately.

(C) Change in YTM = 1 % or 100 bps

% Change in Bond Price = (-11.9) x (100/100) = - 11.9 %

New Bond YTM = 7 %

New Bond Price = 50 x (1/0.07) x [1-{1/(1.07)^(20)}] + 1000/(1.07)^(20) = $ 788.12

Actual % Change in Bond Price = (788.12 - 885.30) / 885.30 = - 0.1098 or - 10.98 % approximately.

(D) The estimation in % price change is more accurate for smaller changes in YTM as compared to large changes owing to the fact that modified duration is a linear approximation of price change for corresponding changes in YTM. However, the actual relationship between YTM and bond price is convex in nature and not linear.