Today you purchase a five-year, 5 percent coupon bond (paid annually) that is pr
ID: 2764668 • Letter: T
Question
Today you purchase a five-year, 5 percent coupon bond (paid annually) that is priced to yield 4 percent.
a. Given, the price and duration of the bond:
Five-year Bond: Par value = $1,000 Coupon rate = 5% Annual payments
r = 4% Maturity = 5 years
t CFt DFt CFt x DFt CFt x DFt x t
1
50
0.961538
48.077
48.08
2
50
0.924556
46.228
92.46
3
50
0.888996
44.45
133.35
4
50
0.854804
42.74
170.96
5
1050
0.821927
863.02
4315.12
Price $1,044.52 4759.91
Duration = 4,759.91/1,044.52 = 4.56 years
b. If interest rates rise to 5 percent the day after you purchase the bond and your investment horizon is 4.56 years (Duration of the bond), show that you will still earn a 4 percent yield on your investment. If your investment horizon is equal to the Duration of your investment, you have immunized your portfolio.
Value of bond at end of 4.56 years: PV4.56 years =
Future value of interest payments at end of 4.56 years: = $____________*
*Future value of interest payments at end of 4.56 years: = $________
Future value of all cash flows at n = 4.56 years:
Coupon interest payments over four years $_______
Interest on interest at 5 percent for 4.56 years _______
Value of bond at end of year four $_______
Total future value of investment@ 4.56 years $_______
Yield on purchase of asset at 4.56 years:
$__________ (1+i)4.56 = $__________ r=0.040019 or 4%
1
50
0.961538
48.077
48.08
2
50
0.924556
46.228
92.46
3
50
0.888996
44.45
133.35
4
50
0.854804
42.74
170.96
5
1050
0.821927
863.02
4315.12
Explanation / Answer
Answer
Interest
Cashflows
T
CF
DF
CF x DF
CF x DFxT
FV @ 5%
FV @ 5%
1
50
0.952381
47.62
47.62
62.46
50(1+0.05)^4.56
62.46
2
50
0.9070295
45.35
90.70
57.88
50(1+0.05)^3
57.88
3
50
0.8638376
43.19
129.58
55.13
50(1+0.05)^2
55.13
4
50
0.8227025
41.14
164.54
52.50
50(1+0.05)^1
52.50
5
1050
0.8005285
840.55
4202.77
50.00
50(1+0.05)
1,050.00
1017.85
4635.212738
277.96
1,277.96
t
Cf
invested for yrs
Amount received
DF @ 5%
PV @ 5%
FV @ 5%
1
50
4.56
11.4
(50*5%*4.56)
0.95238095
10.85714
14.24059237
11.4*(1+0.05)^4.56
2
50
3
7.5
(50*5%*3)
0.90702948
6.802721
8.6821875
7.5*(1.05)^3
3
50
2
5
(50*5%*2)
0.8638376
4.319188
5.5125
5*(1.05)^2
4
50
1
2.5
(50*5%*1)
0.82270247
2.056756
2.625
2.5*(1.05)^1
26.4
24.03581
31.06027987
total Inflow
4545.95+21.62
4659.248546
Duration
4567.57
4.577528463
or 4.56 yrs
Amount $
Value of bond at end of 4.56 years: PV4.56 years =
1017.85
Future value of interest payments at end of 4.56 years:
277.96
*Future value of interest payments at end of 4.56 years ( including Interest on interest)
(277.96+31.06)
309.03
Future value of all cash flows at n = 4.56 years:
(1277.96+31.06)
1,309.03
Coupon interest payments over four years
200
Interest on interest at 5 percent for 4.56 years present value
24.03580813
Value of bond at end of year four present value of cash flow a investor get at 5 yr = 1050 PV @ 5% , 1yr
1000
Total future value of investment@ 4.56 years = 1000 FV @ 5% , 4.56 yrs + FV Interest on interest
1249.174769
31.06
1280.234769
Yield on purchase of asset at 4.56 years:
1195.8371
[1000 ( 1+0.04)^ 4.56] =
$1,195.84
Interest
Cashflows
T
CF
DF
CF x DF
CF x DFxT
FV @ 5%
FV @ 5%
1
50
0.952381
47.62
47.62
62.46
50(1+0.05)^4.56
62.46
2
50
0.9070295
45.35
90.70
57.88
50(1+0.05)^3
57.88
3
50
0.8638376
43.19
129.58
55.13
50(1+0.05)^2
55.13
4
50
0.8227025
41.14
164.54
52.50
50(1+0.05)^1
52.50
5
1050
0.8005285
840.55
4202.77
50.00
50(1+0.05)
1,050.00
1017.85
4635.212738
277.96
1,277.96
t
Cf
invested for yrs
Amount received
DF @ 5%
PV @ 5%
FV @ 5%
1
50
4.56
11.4
(50*5%*4.56)
0.95238095
10.85714
14.24059237
11.4*(1+0.05)^4.56
2
50
3
7.5
(50*5%*3)
0.90702948
6.802721
8.6821875
7.5*(1.05)^3
3
50
2
5
(50*5%*2)
0.8638376
4.319188
5.5125
5*(1.05)^2
4
50
1
2.5
(50*5%*1)
0.82270247
2.056756
2.625
2.5*(1.05)^1
26.4
24.03581
31.06027987
total Inflow
4545.95+21.62
4659.248546
Duration
4567.57
4.577528463
or 4.56 yrs
Amount $
Value of bond at end of 4.56 years: PV4.56 years =
1017.85
Future value of interest payments at end of 4.56 years:
277.96
*Future value of interest payments at end of 4.56 years ( including Interest on interest)
(277.96+31.06)
309.03
Future value of all cash flows at n = 4.56 years:
(1277.96+31.06)
1,309.03
Coupon interest payments over four years
200
Interest on interest at 5 percent for 4.56 years present value
24.03580813
Value of bond at end of year four present value of cash flow a investor get at 5 yr = 1050 PV @ 5% , 1yr
1000
Total future value of investment@ 4.56 years = 1000 FV @ 5% , 4.56 yrs + FV Interest on interest
1249.174769
31.06
1280.234769
Yield on purchase of asset at 4.56 years:
1195.8371
[1000 ( 1+0.04)^ 4.56] =
$1,195.84
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