Nick Roth

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 acquiring financial securities without using 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-front costs per contract:
(1) C1 = 5 - 5Finterest - 3Finflation

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

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

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

First, we should denote the three investments as such:

x1 = # of contracts in investment
x2 = $ in 30-year government bonds (millions)
x3 = $ in stock of other company (millions)

Since the cost of the first investment is $0 and the last two investments are in terms of per million dollars invested, the portfolio 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 interest rate factor. We do this by taking ab interest rate factor from each investments defined in the problem. So, we take the -5 from investment one, the -5 from investment 2, and the 1 from investment three. Thus, the sensitivity to the interest rate factor is:

-5x1 - 5x2 + 1x3

BOOK ERROR WARNING: This can be confusing because the book calls both of the factors "inflation" when defining the factor equations in the problem. The first "inflation" should actually be "interest". I'll make the change bold in the factor equations above so you can see how it's different from the book.

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 as defined in the problem.

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

or

x2 = -1.875

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

Investment 1: Buy $2.75 million worth of swap contracts
Investment 2: Short government bonds by $1.875 million
Investment 3: Purchase $1.875 million in stock of other company

(Grinblatt and Titman 806, 807)

Citation:
Grinblatt, Mark, and Sheridan Titman. Financial Markets and Corporate Strategy. 2nd ed. New York City: McGraw-Hill, 2002.