Question

9. Consider a bullet portfolio comprising a 20 year zero coupon bond with a face value of
100,000 and a barbell portfolio comprising a 10 year zero coupon bond with a face value of
25,174 and a 30 year zero coupon bond with a face value of 91,898. The 10-year rate is 6.0%
p.a. nominal, the 20-year rate is 6.5% p.a. nominal and the 30-year rate is 6.4% p.a. nominal.
These portfolios have the same market value today. Assuming semi-annual compounding and
that the yield curve shifts upwards by 100 basis points at all maturities then:
A. The barbell outperforms the bullet by $106.97.
B. The bullet outperforms the barbell by $106.97.
C. The barbell outperforms the bullet by $159.18.
D. The bullet outperforms the barbell by $159.18.
E. The barbell outperforms the bullet by $29.47.

Answer 1

Nominal 10-year rate = 6%

Nominal 20-year rate = 6.5%

Nominal 30-year rate = 6.4%

Yield curve shifts up by 100 basis points or 1%

New nominal yields are

Nominal 10-year rate = 6% +1% = 7%

Nominal 20-year rate = 6.5% +1% = 7.5%

Nominal 30-year rate = 6.4% +1% = 7.4%

Assuming semi-annual compounding

Effective annual rate = (1+nominal annual rate/2)² - 1

Effective 10-year rate = (1+7%/2)² - 1 = 1.071225 - 1 = 0.071225 = 7.1225%

Effective 20-year rate = (1+7.5%/2)² - 1 = 1.07640625 - 1 = 0.07640625 = 7.6406%

Effective 30-year rate = (1+7.4%/2)² - 1 = 1.075369 - 1 = 0.075369 = 7.5369%

Value of a zero coupon bond = Face Value /(1+r)ⁿ

Where r is the yield rate

n is the maturity of the bond

Face Value of 10-year bond = $25,174

Face Value of 20-year bond = $100,000

Face Value of 30-year bond = $91,898

Value of 10-year bond = $25,174 / (1+7.1225%)¹⁰ = $25,174 / 1.989789 = $12,651.59

Value of 20-year bond = $100,000 / (1+7.6406%)²⁰ = $100,000 / 4.360379 = $22,933.79

Value of 30-year bond = $91,898/ (1+7.5369%)³⁰ = $91,898/ 8.845561 = $10,389.17

Value of Bullet Portfolio = Value of 20 year zero coupon bond = $22,933.79

Value of Barbell Portfolio = Value of 10 year zero coupon bond + Value of 30 year zero coupon bond

= $12,651.59 + $10,389.17 = $23,040.76

As you can see barbell portfolio has a higher value

Difference in value = Value of Barbell Portfolio - Value of Bullet Portfolio = $23,040.76 - $22,933.79 = $106.97

Hence the barbell outperforms the bullet by $106.97

Answer is Option A