Question

You have bought 100 bonds with a face value of $1,000, an annual coupon rate of 6 percent, a yield to maturity of 8 percent, and 10 years to maturity. Now you are concerned that rates will rise and the bond value will, therefore, drop. You want to take a short (sell) position on another bond with face value of $1,000, an annual coupon rate of 5 percent and a yield to maturity of 7 percent, and 8 years to maturity. Using a duration hedge how many of these bonds should you short?  

options are

12.1

81.3

94.56

110.82

Answer 1

For the purchased bond

Coupon of the bond = $1000*6% =$60

Price of the purchased bond =60/0.08*(1-1/1.08^10)+1000/1.08^10 = $865.80

Duration of the bond

=(60*1/1.08+60*2/1.08^2+60*3/1.08^3+60*4/1.08^4+60*5/1.08^5+60*6/1.08^6+60*7/1.08^7+60*8/1.08^8+60*9/1.08^9+60*10/1.08^10+1000*10/1.08^10)/865.80

=7.615 years

Modified Duration =7.615 /1.08 =7.051 years

For the bond to be sold

Coupon of the bond = $1000*5% =$50

Price of the purchased bond =50/0.07*(1-1/1.07^8)+1000/1.07^8 = $880.574

Duration of the bond

=(50*1/1.07+50*2/1.07^2+50*3/1.07^3+50*4/1.07^4+50*5/1.07^5+50*6/1.07^6+50*7/1.07^7+50*8/1.07^8+1000*8/1.07^8)/880.574

=6.6935 years

Modified Duration =6.6935 /1.07 =6.2556 years

Using Duration Hedge, if x bonds are shorted

Duration weighted Value of purchased bonds = Duration weighted value of sold bonds

=> 100 bonds * $865.80 * 7.051 years = x* $880.574 * 6.2556 years

=> x = 100*865.80*7.051/(880.574*6.2556)

=110.82 (last option)