Question 9 & 10 Question #9: A nine-year, 6,000 par bond with an annual coupon r
ID: 2536981 • Letter: Q
Question
Question 9 & 10
Question #9: A nine-year, 6,000 par bond with an annual coupon rate of 40% paid annually sells for 6,000. Let DA be the Macaulay duration just after the first coupon is paid. Let Ds be the Macaulay duration just before the first coupon is paid. Assume that the yield stays constant at 4.0%. Calculate the ratio: DB/DA Question #10: A bond pays annual coupons, and the next coupon will be paid in one year. The price, par value, and redemption value of the bond are all 100. The Macaulay duration of the bond is 6.88 years. The coupon rate of the bond is 7.4%. Using duration, calculate the estimated price of the bond if the yield of the bond falls to 5.8%.Explanation / Answer
Duration of bond before the first coupon is paid
Year
cash flow
present value of cash flow = cash flow/(1+r)^n r= 4%
present value*time
1
240
230.7692
230.7692
2
240
221.8935
443.787
3
240
213.3591
640.0774
4
240
205.153
820.612
5
240
197.2625
986.3125
6
240
189.6755
1138.053
7
240
182.3803
1276.662
8
240
175.3656
1402.925
9
6240
4384.141
39457.27
value of bond before the first coupon payment
6000
sum of present value of cash flow*time
46396.47
Duration of bond before the first coupon is paid
sum of present value*time/value of bond
46396.47/6000
7.73
Duration of bond after the first coupon is paid
Year
cash flow
present value of cash flow = cash flow/(1+r)^n r= 4%
present value*time
2
240
221.8935
443.787
3
240
213.3591
640.0774
4
240
205.153
820.612
5
240
197.2625
986.3125
6
240
189.6755
1138.053
7
240
182.3803
1276.662
8
240
175.3656
1402.925
9
6240
4384.141
39457.27
value of bond before the first coupon payment
5769.231
sum of present value of cash flow*time
46165.7
Duration of bond after the first coupon is paid
sum of present value*time/value of bond
46396.47/6000
8.00
ratio between duration before the coupon is paid and duration after the coupon is paid
7.7327/8.0020
0.97
% change in bond price if Yield to bond falls to 6.8%
duration*change in YTM rate
4.128
duration
6.88
change in YTM = 6.8-7.4
6.8-7.4
-0.6
Value of bond at 7.4% YTM
100
value of bond at 6.8% YTM
100*1.04128
104.128
As mentioned in the question that market price and par value are same as 100 so it means YTM and coupon rate is same.
Duration of bond before the first coupon is paid
Year
cash flow
present value of cash flow = cash flow/(1+r)^n r= 4%
present value*time
1
240
230.7692
230.7692
2
240
221.8935
443.787
3
240
213.3591
640.0774
4
240
205.153
820.612
5
240
197.2625
986.3125
6
240
189.6755
1138.053
7
240
182.3803
1276.662
8
240
175.3656
1402.925
9
6240
4384.141
39457.27
value of bond before the first coupon payment
6000
sum of present value of cash flow*time
46396.47
Duration of bond before the first coupon is paid
sum of present value*time/value of bond
46396.47/6000
7.73
Duration of bond after the first coupon is paid
Year
cash flow
present value of cash flow = cash flow/(1+r)^n r= 4%
present value*time
2
240
221.8935
443.787
3
240
213.3591
640.0774
4
240
205.153
820.612
5
240
197.2625
986.3125
6
240
189.6755
1138.053
7
240
182.3803
1276.662
8
240
175.3656
1402.925
9
6240
4384.141
39457.27
value of bond before the first coupon payment
5769.231
sum of present value of cash flow*time
46165.7
Duration of bond after the first coupon is paid
sum of present value*time/value of bond
46396.47/6000
8.00
ratio between duration before the coupon is paid and duration after the coupon is paid
7.7327/8.0020
0.97
% change in bond price if Yield to bond falls to 6.8%
duration*change in YTM rate
4.128
duration
6.88
change in YTM = 6.8-7.4
6.8-7.4
-0.6
Value of bond at 7.4% YTM
100
value of bond at 6.8% YTM
100*1.04128
104.128
As mentioned in the question that market price and par value are same as 100 so it means YTM and coupon rate is same.
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