Nick Roth: Eliminating Factor Loadings w/ Portfolios

Consider a two-factor model where the two factors are interest rate movements and changes in inflation. Assume a company has a future cash flow with factor betas of 2.5 on the interest rate factor and 4.5 on the inflation factor. How could this company eliminate its sensitivity to both factors by acquriing financial securites without usings its own cash?

Suppose the company can (1) enter into a five-year interest rate swap contract, (2) purchase 30-year government bonds, and (3) acquire a sizable chunk of shares in another company. Assume the following factor equations:

Future value of a swap contact with no up-ront costs per contract:
(1) C1 = 5 - 5Finflation - 3Finflation

Future cash flow of 30-year governemtn bond per million dollars invested:
(2) C2 = 10 - 5Finflation - 1Finflation

End-of-period value of stock in another company per million dollars invested:
(3) C3 = 0 + 1Finflation + 1Finflation

What is the proper hedge against inflation and interst rate movements given this information?

First, we should denote the three investments as such:
x1 = the number of contracts in the costless swap investment
x2 = millions of dollars in the 30-year governemtn bonds
x3 = millinos of dollars in stock of other company

Since the cost of the first investment is $0 and the last two investments are in terms of per million dollars invested, the portolfio of these three investments has a cost of:

0x1 + 1,000,000x2 + 1,000,000x3

Next, we want to establish an equation that describes the portfolios sensitivity to the interst rate factor. We do this by taking the interest rate factor of each investment. Thus, the sensitivity to the itnerst rate factor is:

-5x1 - 5x2 + 1x3

Then we do the same thing for the inflation factor. This gives us:

-3x1 - 1x2 + 1x3

With these three equations, we are now able to hedge the portfolio by simultaneously solving them. Notice that we set investment 1 = 0, 2 = -2.5, and 3 = -4.5. These figures are drawn from the betas of the assumed future cash flow.

0x1 + 1,000,000x2 + 1,000,000x3 = 0
-5x1 - 5x2 + 1x3 = -2.5
-3x1 - 1x2 + 1x3 = -4.5

From the first equation, we know that x3 = -x2. For the other two equations, we can imply that:

-5x1 - 6x2 = -2.5
-3x1 - 2x2 = -4.5

By solving for x1 in the first equation (of the two immediately above), we can plug this into the second equation to solve for x2. Thus, we find that

-3((2.5 - 6x2)/5) - 2x2 = -4.5


x2 = -1.875

Now that we know the value of x2, we know that the value of x1 = 2.75. Thus, the solution is

1. Buy 2.75 swap contracts ($2.75 million in notional amount)
2. Short $1.875 million in 30-year government bonds
3. Buy $1.875 million in stock of the other company