Question

2. Suppose that the initial value of an unlevered portfolio of Treasury securities is $200 million and the duration is 7. Suppose further that the manager can borrow $800 million and invest it in the identical Treasury securities so that the levered portfolio has a value of $1 billion. What is the duration of the levered portfolio? What is the duration of the unlevered portfolio (i.e., the duration of the equity)?

Answer 1

The duration of a portfolio measures its sensitivity to changes in interest rates. It is a weighted average of the durations of the individual securities in the portfolio, where the weights are the proportions of the portfolio's value invested in each security.

To find the duration of the levered portfolio, we can use the formula:

Duration of levered portfolio = (Duration of unlevered portfolio * Value of unlevered portfolio + Duration of borrowed funds * Value of borrowed funds) / Value of levered portfolio

Plugging in the given values, we get:

Duration of levered portfolio = (7 * $200 million + 0 * $800 million) / $1 billion

Duration of levered portfolio = 1.4

Therefore, the duration of the levered portfolio is 1.4.

To find the duration of the unlevered portfolio (i.e., the duration of the equity), we can use the fact that the duration of the equity is equal to the duration of the portfolio minus the duration of the borrowed funds. Since the borrowed funds have a duration of 0 (since they are Treasury securities with no price sensitivity to interest rate changes), the duration of the unlevered portfolio is simply equal to its own duration, which is given as 7.

Therefore, the duration of the unlevered portfolio (i.e., the duration of the equity) is 7.