Question

A bond trader purchased each of the following bonds at a YTM of 8%. Immediately after she purchased the bonds, interest rates fell to 7%. What is the percent change in the price of each bond after the decline in interest rates. What is the duration and modified duration for each bond? Assume Face Value at 1000

I. 4-year, 12% annual coupon

II. 10 year zero coupon

III. 5 year zero coupon

IV. 30 year zero coupon

V. INR 100 perpetual coupon

Answer 1

1. Using BA 2 PLus financial calculator, put 4=N, PMT=120, I/Y= 8, FV= 1000 Then press CPT and PV= 1132.48

when interest rate fell to 7%, follow the same steps and find put 4=N, PMT=120, I/Y= 7, FV= 1000 Then press CPT and PV= 1169.36

wE can see that the price has increased, so percentage increase in price= (1169.36-1132.48)/ 1132.48= 3.25%

Duration calculation=

C1= 120 , C2= 120, C3= 120, C4= 1000+120= 1120

PV1= 120/ 1.08, PV2= 120/(1.08)^2, PV3= 120/(1.08)^3, PV4= 1120/(1.08)^4

therefore, PV1= 111.11, PV2= 102.88, PV3= 95.25, PV4= 823.233

NOW find weghts of each PV

W1= 111.11/1000= 0.111

W2= 0.1028

W3= 0.09525

W4= 0.8232

nOW, w1*1+ w2*2+ w3*3+ w4*4= duration

duration= 0.111 + 0.1028*2+ 0.09525*3 + 0.8232*4= 3.895

Modified duration= Duration/ (1+I/Y)

3.895/(1.08)= 3.606

2. SIMILARLY FOLLOWING ABOVE STEPS ,

Using BA 2 PLus financial calculator, put 4=N, PMT=0, I/Y= 8, FV= 1000 Then press CPT and PV= 735.02

when interest rate fell to 7%, follow the same steps and find put 4=N, PMT=0, I/Y= 7, FV= 1000 Then press CPT and PV= 762.89

wE can see that the price has increased, so percentage increase in price= 3.79%

Duration calculation=

C1= 0 , C2= 0, C3= 0, C4= 1000+0= 1000 (BECAUSE IT IS A ZERO COUPON BOND)

PV1= 0, PV2= 0, PV3= 0 PV4= 1000/(1.08)^4

therefore, PV1= 0, PV2= 0, PV3= 0, PV4= 735.029

NOW find weIghts of each PV

W1= 0

W2= 0

W3= 0

W4= 0.735

nOW, w1*1+ w2*2+ w3*3+ w4*4= duration

duration= 0+ 0+ 0 + 0.735*4=2.9401

Modified duration= Duration/ (1+I/Y)

2.9401/(1.08)= 2.722

2. SIMILARLY FOLLOWING ABOVE STEPS ,

Using BA 2 PLus financial calculator, put 5=N, PMT=0, I/Y= 8, FV= 1000 Then press CPT and PV= 680.58

when interest rate fell to 7%, follow the same steps and find put 5=N, PMT=0, I/Y= 7, FV= 1000 Then press CPT and PV= 712.98

wE can see that the price has increased, so percentage increase in price= 4.7606%

Duration calculation=

C1= 0 , C2= 0, C3= 0, C4= 0, C5= 1000+0= 1000 (BECAUSE IT IS A ZERO COUPON BOND)

PV1= 0, PV2= 0, PV3= 0, PV3= 0, PV5= 1000/(1.08)^5

therefore, PV1= 0, PV2= 0, PV3= 0, PV4= 0, PV5= 680.58

NOW find weIghts of each PV

W1= 0

W2= 0

W3= 0

W4= 0

W5= 0.68

nOW, w1*1+ w2*2+ w3*3+ w4*4 + w5*5= duration

duration= 0+ 0+ 0 + 0+ 0.68*5=3.4029

Modified duration= Duration/ (1+I/Y)

3.4029/(1.08)= 3.1508