You purchased another bond with the following characteristics: $1,000 par value

Question

You purchased another bond with the following characteristics:

$1,000 par value                                                                           6.5% coupon, annual payments

25 years to maturity                                                                        Callable in 7 years at $1,065.

You paid $1063.92 for the bond.                                                  Macaulay duration is 13.34 years

Assume market rates drop by 50 basis points.

a. What will be the new bond price?

b. Using modified duration, estimate the value of the bond following the decrease in interest rates.

c. The estimate (from part b), is fairly close to the actual (in part a). What explains the difference in the two values? Be specific.

d. Calculate the effective duration of this bond. Use shifts of 50 basis points.

e. Calculate the yield to call.

f. Of the three duration measures (Macaulay, modified, effective) which is the most appropriate measure for this bond? Why?

Explanation / Answer

Bond face value                1,000 Bond Price        1,063.92 Bond Annual interest @6.5%=                    65 Years to Maturity =                    25 YTM Formula= [Annual Interest+(Par Value-Market Value)/Years to Maturity]/(Par value+Market Price*2)/3   YTM =[65+(1000-1063.92)/25]/(1000+2*1063.92)/3 YTM is 6% Macaulay's Duration =13.34 years Modified duration = Macaulay's duration/(1+YTM)=              12.58 Years Assume YTM drops by 0.5% Bond Price will be : Years Ineterst +Maturity PV factor @5.5% PV of Cash flows Year 1                    65           0.948                 62 Year 2                    65           0.898                 58 Year 3                    65           0.852                 55 Year 4                    65           0.807                 52 Year 5                    65           0.765                 50 Year 6                    65           0.725                 47 Year 7                    65           0.687                 45 Year 8                    65           0.652                 42 Year 9                    65           0.618                 40 Year 10                    65           0.585                 38 Year 11                    65           0.555                 36 Year 12                    65           0.526                 34 Year 13                    65           0.499                 32 Year 14                    65           0.473                 31 Year 15                    65           0.448                 29 Year 16                    65           0.425                 28 Year 17                    65           0.402                 26 Year 18                    65           0.381                 25 Year 19                    65           0.362                 24 Year 20              1,065           0.343               365     1,119.50 a So new Price of the Bond = $    1,119.50 b Modified Duration =              12.58 Yrs Interest cahneg =-0.5% So Price change will be =-Modified duration *Interest change =-12.58*-0.5%=6.29 % increase No Changed price =1063.92*1.0629= $    1,130.84 c There is a difference due to the call feature which may cahnage   in cash dlow and that is not accounted by modofied duration. d When interest goes up by 0.5%, Thn YTM =6.5% Price change =-12.58*0.5%=-6.29% So New Price =1063.92*0.9371= $       997.00 Effective duration : (P1-P2)/[2*P0(Y2-Y1) Here P1=1130.84 P2=997 P0=1063.92 Y2-Y1=0.005 Effcetive Duration =(1130.84-997)/(2*1063.92*0.005) =12.58 Years e Yield to call= YTCFormula= [Annual Interest+( Call Value-Market Value)/Years to Call ]/(Call value+Market Price)/2 Bond face value                1,000 Bond Price        1,063.92 Bond Annual interest @6.5%=                    65 Years to Maturity =                    25 caLL Value =              1,065 Years to call                      7 YTC =[65+(1065-1063.92)/7]/(1065+1063.92)/2 YTC =6.12% So Yield to call is 6.12% f For this bond effective duration is more appropriate as the bond has embeded   call option and possibility of change in   cash flow with change in interest rate.

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