Wednesday, June 17, 2009

More on Duration

We're probably going a little overboard here, but let's take a quick look at the actual formula used to calculate duration. Not that I think you'll have to calculate it on the exam, but the concept becomes clearer by fussing with the formula a little. Remember that duration is a weighted average of a bond's cash flows--the longer you have to wait to receive a significant part of your bond investment back, the higher the duration. Let's say you buy a 2-year T-note with a 5% yield. If so, you would receive exactly four interest payments and one principal payment. The total amount of money received would be $1,100. The formula would look like this:

.5($25/$1,100) + 1($25/$1,100) + 1.5($25/$1,100) + 2($25/$1,100) + 2($1,000/$1,100)

That might look crazy at first, but if we break it down, it makes perfect sense. The little ".5," "1," "1.5" and so on are representing the income payments received at the first half-year, the first year, the first year-and-a-half, etc. In parentheses, we see that the income received is a percentage of the total $1,100 that will be returned to the investor. At the very end, $1,000 of principal is returned, along with the last income payment of $25. The duration turns out to be just a little less than 2 (1.92 approximately), and, of course, if the number were higher than 2, we messed up. As the test question might say, the duration of a bond paying interest is always lower than the term to maturity. But, a zero coupon bond's duration equals the maturity. Looking at the formula above, we see that it would have to. You would only have one entry on a zero coupon bond, since it only makes one payment.

For fun, run the calculation with a bond paying 10%, and you'll see that the duration is lower with a higher coupon rate. The bond would be paying ($50/$1,200) with each interest payment and returning ($1,000/$1,200) at maturity, making the duration on this bond/note approximately 1.86.

Ah, there. Now I can get some sleep.

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