Question

You need to create a portfolio with a duration of 8 years. You can use a 5 year zero-coupon bond and a perpetuity which pays $100 each and every year forever and has yield of 10%. How much of the portfolio value in percentage you would have to invest in the zero-coupon bond, and how much in the perpetuity?

Answer 1

Duration of a 5 year zero-coupon bond = 5 years

Duration of a perpetuity = (1 + y) / y = (1 + 10%) / 10% = 11 years

Let's assume w be the portfolio value in %age invested in zero coupon bond then proportion invested in perpetuity = 1 - w

Hence, duration of portfolio = 8 years = w x 5 years + (1 - w) x 11 years

Hence, 8 = 11 - 6w

Hence, w = 3 / 6 = 0.50 = 50%

The portfolio value in percentage we would have to invest in the zero-coupon bond = w = 50%, and percentage portfolio value in the perpetuity = 1- w = 1 - 50% = 50%