Question

In: Finance

Suppose a semiannual coupon bond with a 7% annualized coupon rate has an annual yield of...

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?

Solutions

Expert Solution

(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

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.   


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