Porfolio Analysis
Math 272  Prof. Leise

Email completed labs to tleise@amherst.edu by 4pm Monday.

Today we will see how some linear algebra ideas connect to portfolio analysis.  The material for this lab was adapted from various sections of Mathematics for Economists by Simon and Blume (1994).

Investment assets

Suppose an investor has a choice of A investment assets, to be bought at the beginning of an investment period and then sold at the end.

There is uncertainty in the outcomes of these investments, so suppose that S different financial climates are possible during the coming period.  These different possible climates are called states of nature.  Exactly one of these states of nature will occur, but we don't know which one.  An asset will have different returns in different states of nature, and we assume that we can predict the return for each potential state of nature (a big assumption!).

Let v[i] be the current value of one unit of asset i and y[s,i] be the value of one unit of asset i at the end of the investment period if state s occurs.

Let n[i] be the number of units of asset i that the investor buys (can be positive, indicating a "long position" where the investor buys, or negative, indicating a "short position" where the investor borrows).

Let x[i]=n[i]v[i] (amount invested in asset i).

The vector {x[1],x[2],...,x[A]} is called a portfolio.

We will consider various portfolios under a budget constraint:  x[1]+x[2]+...+x[A]  = 1.

Return on investment

The return on asset i in state s is R[s,i]=y[s,i]/v[i] (amount received per dollar invested in asset i).  We will be given a matrix R[s,i] in order to analyze various portfolios.

The total return in state s is defined to be 

R[s] = y[s,1]n[1] + ... + y[x,A]n[A]
        = ((y[s,1])/(v[1]))n[1]v[1] + ... + ((y[s,A])/(v[A]))n[A]v[A]
        = R[s,1]x[1] + ... + R[s,A]x[A]

This looks like matrix multiplication!  Given a return matrix R (whose entries are R[s,i], s gives the row and i the column) and a portfolio vector x, we can compute the return {r[1],...,r[S]} on that portfolio via matrix multiplication Rx.

Suppose there are 2 assets and 3 possible states.  
If state 1 occurs, each unit of asset 1 returns R[1,1]=1, each unit of asset 2 returns R[1,2]=3.
If state 2 occurs, each unit of asset 1 returns R[2,1]=2, each unit of asset 2 returns R[2,2]=2.
If state 3 occurs, each unit of asset 1 returns R[3,1]=3, each unit of asset 2 returns R[3,2]=1.

Exercise 1.  Define the return matrix R.  If a portfolio is x={3/4,1/4}, calculate its return R.x (remember that the period denotes matrix multiplication in Mathematica).  Note that R.x will be a vector listing the returns in the possible states of nature.

Exercise 2. A portfolio is called riskless if it provides the same nonzero return (or payoff) in every state.  Find a riskless portfolio for the matrix R given above.  Hint: assume payoff vector {p,p,p} (same payoff value in every state of nature) and write out the three equations resulting from R.{x1,x2}={p,p,p} plus the budget constraint x1+x2=1.  This gives 4 equations in the 3 unknowns x1, x2, and p.  Solve this new system (be sure to reorganize the system so x1, x2, and p are on the left hand side--treat p here as “x3”).  What is the resulting payoff value p?

Exercise 3. Given any set-up with 2 assets and 3 states, do you expect to always be able to find a riskless portfolio?  What vector(s) must be in the column space of R for a riskless portfolio to exist?

Exercise 4. A portfolio x is called duplicable if there exists a different portfolio w with exactly the same payoff: Rx=Rw.  What does the existence of a duplicable portfolio tell you about the nullspace of the matrix R? Remember that we also want to satisfy the budget constraint (requires an extra free variable).

Exercise 5. Are there any duplicable portfolios for the matrix R given above?  Explain why or why not in linear algebra terms.

Exercise 6. Suppose we now have the following set of returns on 3 assets with 3 possible states:

If state 1 occurs, each unit of asset 1 returns R[1,1]=1, each unit of asset 2 returns R[1,2]=3, and each unit of asset 3 returns R[1,3]=2.
If state 2 occurs, each unit of asset 1 returns R[2,1]=2, each unit of asset 2 returns R[2,2]=2, and each unit of asset 3 returns R[2,3]=2.
If state 3 occurs, each unit of asset 1 returns R[3,1]=1, each unit of asset 2 returns R[3,2]=2, and each unit of asset 3 returns R[3,3]=3.

(a) Find the null space of R.  Are there any duplicable portfolios?

(b) Find the column space of R and note that it contains the vector {1,1,1}, indicating that a riskless portfolio exists.  What is the rank of R? Find the riskless portfolio that satisfies the budget constraint.  What is the payoff for this riskless portfolio?

Exercise 7. Suppose we now have the following set of returns on 4 assets with 3 possible states:

R[1,1]=1, R[1,2]=3, R[1,3]=2, R[1,4]=2.
R[2,1]=2, R[2,2]=3, R[2,3]=4, R[2,4]=1.
R[3,1]=1, R[3,2]=2, R[3,3]=2, R[3,4]=1.

Find the nullspace of R.  Are there any duplicable portfolios?  If so, find all portfolios that yield the same return as {1,0,0,0} and also satisfy the budget constraint.  (Recall that both positive and negative numbers are okay in the portfolio vectors.)