Week 1 - Matrices & portfolio analysis, Linear independence

Question 1

3 conditions for an arbitrage portfolio

Answer 1

1. y1 + y2 +…+ ym = 0
   - costs nothing
2. for all j, (YR)j >=0
   - never makes a loss
3. for at least one j, (YR)j >0
   - can make a profit

Question 2

How to decide if an arbitrage portfolio exists? Theorem

Answer 2

Suppose that R is a returns matrix.
- If R has an SPV, there is NO arbitrage portfolio for R.
- If R has NO SPV, there is an arbitrage portfolio for R.
*SPV = state price vector

Rp = (1 1…1) and for ALL i, pi > 0
> must be ALL positive solutions to be SPV

Question 3

Linear dependence vs Linear independence

Answer 3

Linear dependence
- must find at least 1 NON-TRIVIAL SOLUTION, ie. where at least one of the xi is NOT zero
- at least 1 of the vectors in the set can be written as a LINEAR COMBINATION of the others

Linear independence
- ONLY has the trivial solution, ALL xi = 0
- no linear combination

Question 4

How to use the Master theorem (first pass)?

Answer 4

If A is a nxn matrix (SQUARE MATRIX), use master theorem to determine linear in/dependence,
ie. look for ROW OF ZEROES for LI

*If not square matrix, solve using row operations & see what solutions we get. If not square, row of zeroes doesn’t necessarily mean anything.

Question 5

The master theorem - important 2nd & 3rd pass from Week 2

Answer 5

2. matrix A has rank n
3. |A| =/= 0