Question

Suppose you own following bond portfolio

	Face Value	Bond Type	Maturity	yield to maturity
Portfolio I	$88 million	Zero Coupon	5 years	4%

You expect interest rate to rise in near future(hence decrease the value of bond portfolio). You decided to sell some of 5-year bond and use that proceed to buy 1.5-year zero coupon bonds with yield to maturity 3%. If you want new duration of the portfolio to be 3 years (that mean after selling 5-year bond and buying 1.5 year bond), what should be the price of 1.5 year bonds?

Answer 1

The duration of a zero-coupon bond is equal to its maturity in years

Hence, the duration of a 5-year zero-coupon bond = 5 years

Duration of a 1.5 year zero coupon bond = 1.5 years

New duration of the portfolio = w(5)*Duration(5) + w(1.5)*Duration(1.5)

w(5) and w(1.5) are the weights of 5 year and 1.5 year zero coupon bond in the portfolio

w(5) +w(1.5) =1

3 = w(5)*5 + (1-w(5))*1.5

w(5) = 0.429

w(1.5) = 1-w(5) = 0.571

Price of 1.5 year bonds = 0.571*$88 million = $50.248 million