The market trades in three fixed-income
securities: A, B, and C. Security A is a 2-year bond paying 40 at date 1
and 80 at date 2. Security B is a 1-year bond paying 50 at date 1. Security
C pays 30 at date 1 and 40 at date 2. Prices are pa=100, pb=45 and pc=62.
Determine the rank of this market. Is this market complete?
To determine the rank of the market, we need to consider the number of linearly independent portfolios that can be constructed using the available securities. If there are more linearly independent portfolios than available securities, the market is said to have full rank and is complete.
Let's check the rank of this market:
The available securities are:
- Security A: 2-year bond paying 40 at date 1 and 80 at date 2.
- Security B: 1-year bond paying 50 at date 1.
- Security C: 2-year bond paying 30 at date 1 and 40 at date 2.
We can represent the securities as vectors:
Security A: [40, 80]
Security B: [50, 0]
Security C: [30, 40]
Now, let's consider the linear combinations of these securities:
x * Security A + y * Security B + z * Security C = [0, 0]
x * [40, 80] + y * [50, 0] + z * [30, 40] = [0, 0]
Simplifying the equation, we get the following system of equations:
40x + 50y + 30z = 0
80x + 40z = 0
Solving these equations, we find that there is one linearly independent portfolio that satisfies the equation: x = -2, y = 1, z = 1.
Portfolio P1: -2 * Security A + Security B + Security C = [0, 0]
Since we have found at least one linearly independent combination, the rank of the market is 1.
Now, let's check if the market is complete. If the rank of the market is equal to the number of available securities, then the market is complete. In this case, we have 1 rank and 3 available securities.
Since the rank is less than the number of securities, the market is not complete.
In summary:
- The rank of this market is 1.
- The market is not complete.