Original practice · fully worked
Original variant: face-amount ratio after dollar-duration matching
A liability has present value 200,000 and Macaulay duration 6. Asset A is a four-year zero priced at 0.90 per unit of face; Asset B is a nine-year zero priced at 0.80 per unit. Find the ratio of face amount in A to face amount in B that matches value and duration.
- A
0.8889 - B
1.0667 - C
1.2000 - D
1.3333 - E
1.5000
Answer and calculation
The market values are 120,000 and 80,000. Converting each to face amount gives a ratio of 1.3333. The face-amount ratio is 1.3333, corresponding to choice D.
Setup
Setup
First solve for current market values X and Y; duration weights market value, not face amount.
X+Y=200000,4X+9Y=6(200000) Model
Model
Value and dollar-duration matching give a two-equation system with asset durations 4 and 9.
X=120000,Y=80000 Compute
Compute
The market values are 120,000 and 80,000. Converting each to face amount gives a ratio of 1.3333.
FBFA=Y/0.80X/0.90=1.333333 Answer
Answer
The face-amount ratio is 1.3333, corresponding to choice D.
FA/FB=1.3333(D) 