Original practice · fully worked
Original variant: unknown zero maturity from a duration budget
A liability portfolio has current value 5,000 and Macaulay duration 4. An immunizing asset portfolio holds a time-1 zero-coupon asset worth 1,200 today and one additional zero worth 3,800 today. Find the maturity of the second zero.
- A
4.21 years - B
4.53 years - C
4.95 years - D
5.26 years - E
5.67 years
Answer and calculation
Solving 1,200 + 3,800b = 20,000 gives b = 4.9474 years. The maturity is 4.95 years, so choice C is correct.
Setup
Setup
Present value is already matched, so only the dollar-duration equation is needed to determine the second zero’s maturity.
PVA=1200+3800=5000 Model
Model
For zero-coupon assets, Macaulay duration equals maturity; weight each maturity by its current market value.
1(1200)+b(3800)=4(5000) Compute
Compute
Solving 1,200 + 3,800b = 20,000 gives b = 4.9474 years.
b=380020000−1200=4.947368 Answer
Answer
The maturity is 4.95 years, so choice C is correct.
b=4.95 years(C) 