Independent solution

How to solve this Cash-Flow and Risk Matching question

Setup

Setup

Discount the three equal liabilities and verify that their total present value is 1,000.

PL=402.11(v+v2+v3)=1000,v=1/1.10P_L=402.11(v+v^2+v^3)=1000,\qquad v=1/1.10

Model

Model

Their Macaulay duration is the present-value-weighted average of years one, two, and three.

DL=402.11(v+2v2+3v3)1000=1.93653D_L=\frac{402.11(v+2v^2+3v^3)}{1000}=1.93653

Compute

Compute

For zero-coupon bonds, duration equals maturity; solve the two-point weighted-average equation for the one-year allocation.

1.93653=X+3(1000−X)1000,X=531.7351.93653=\frac{X+3(1000-X)}{1000},\qquad X=531.735

Answer

Answer

The company invests about 532 in one-year zeros, corresponding to choice E.

X≈532(E)\boxed{X\approx532\quad\text{(E)}}