In: Finance
Suppose a semiannual coupon bond with a 7% annualized coupon rate has an annual yield of 5% compounded semiannually. It matures in 10 years to its face of $10,000. 1)Compute the price. 2) What its modified duration?
(1)-Price of the Bond
The Current Price of the Bond is the Present Value of the Coupon Payments plus the Present Value of the face Value
Face Value of the bond = $10,000
Semi-annual Coupon Amount = $350 [$10,000 x 7% x ½]
Semi-annual Yield to Maturity = 2.50% [5% x ½]
Maturity Period = 20 Years [15 Years x 2]
The Price of the Bond = Present Value of the Coupon Payments + Present Value of the face Value
= $350[PVIFA 2.50%, 20 Years] + $10,000[PVIF 2.50%, 20 Years]
= [$350 x 15.589162] + [$10,000 x 0.610721]
= $5,456.21 + $6,102.71
= $11,558.92
“Therefore, the Price of the Bond = $11,558.92”
(2)-Modified Duration of the Bond
Step-1, Calculation of Macaulay Duration of the Bond
Period (1) |
Cash Flow (2) |
Present Value Factor at 2.50% (3) |
Present Value (4) = (3) x (2) |
Weight (5) |
Duration (6) = (1) x (5) |
0.50 |
350 |
0.97561 |
341.46 |
0.02954 |
0.01 |
1.00 |
350 |
0.95181 |
333.14 |
0.02882 |
0.03 |
1.50 |
350 |
0.92860 |
325.01 |
0.02812 |
0.04 |
2.00 |
350 |
0.90595 |
317.08 |
0.02743 |
0.05 |
2.50 |
350 |
0.88385 |
309.35 |
0.02676 |
0.07 |
3.00 |
350 |
0.86230 |
301.80 |
0.02611 |
0.08 |
3.50 |
350 |
0.84127 |
294.44 |
0.02547 |
0.09 |
4.00 |
350 |
0.82075 |
287.26 |
0.02485 |
0.10 |
4.50 |
350 |
0.80073 |
280.25 |
0.02425 |
0.11 |
5.00 |
350 |
0.78120 |
273.42 |
0.02365 |
0.12 |
5.50 |
350 |
0.76214 |
266.75 |
0.02308 |
0.13 |
6.00 |
350 |
0.74356 |
260.24 |
0.02251 |
0.14 |
6.50 |
350 |
0.72542 |
253.90 |
0.02197 |
0.14 |
7.00 |
350 |
0.70773 |
247.70 |
0.02143 |
0.15 |
7.50 |
350 |
0.69047 |
241.66 |
0.02091 |
0.16 |
8.00 |
350 |
0.67362 |
235.77 |
0.02040 |
0.16 |
8.50 |
350 |
0.65720 |
230.02 |
0.01990 |
0.17 |
9.00 |
350 |
0.64117 |
224.41 |
0.01941 |
0.17 |
9.50 |
350 |
0.62553 |
218.93 |
0.01894 |
0.18 |
10.00 |
10,350 |
0.61027 |
6,316.30 |
0.54644 |
5.46 |
TOTAL |
11,558.92 |
7.56 Years |
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Macaulay Duration = 7.56 Years
Step-2, Calculation of Modified Duration of the Bond
Modified Duration of the Bond = Macaulay Duration / [1 + (YTM / Number of coupon payments per year)]
= 7.56 Years / [1 + (0.05/ 2)]
= 7.56 Years / (1 + 0.025)
= 7.56 Years / 1.025
= 7.38 Years
“Hence, the Modified Duration of the Bond = 7.38 Years”
NOTE
-The formula for calculating the Present Value Annuity Inflow Factor (PVIFA) is [{1 - (1 / (1 + r)n} / r], where “r” is the Yield to Maturity of the Bond and “n” is the number of maturity periods of the Bond.
-The formula for calculating the Present Value Inflow Factor (PVIF) is [1 / (1 + r)n], where “r” is the Yield to Maturity of the Bond and “n” is the number of maturity periods of the Bond.