In: Finance
4. You are managing a bond portfolio of $1 million. Your target duration is 10 years, and you can invest in two bonds, a zero-coupon bond with maturity of five years and a zero-coupon bond with maturity of 15 years, each currently yielding 5%.
(1) What are your weights in these two zero-coupon bonds to have the target duration?
(2) How will these factions change next year if the target duration is still ten years? .
We will first calculate the duration of each of the 2 bonds as below - we know that the duration for zero coupon bond is equal to its time to maturity. Hence the duration will be as below:
Duration ?5 year zero coupon? = 5 years and Duration ?15 year zero coupon? = 15 years
Now to construct a portfolio with 10 year duration we will construct an equal weighted portfolio (50% weight for each zero coupon bond) to have a portfolio duration of 10 years.
Next year, since the residual maturity for each bond will be less by 1 year and hence the duration will also change accordingly - the duration 1 year hence will be as below:
Duration ?5 year zero coupon? = 4 years and Duration ?15 year zero coupon? = 14 years
Now lets assume the weight x% for 5 year bond, hence (1-x%) weight for 15 year bond. The duration will be
x% * 4 + (1-x%) * 14 = 10 ; solving this equation we get 4x% + 14 - 14x% = 10 or x% = 40% (or x = 0.4).
Thus the portfolio weight for 1 year hence will be 5 year bond 40% and 15 year bond at 60%.