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.

Duration, Interest Rate Risk

A customer just emailed me for clarification on a very tough practice question:

Which of the following securities would react the most to a change in interest rates?
A. 10-year corporate subordinated debenture
B. 11-year AAA-rated municipal bond
C. 20-year US Treasury bond
D. 20-year US Treasury STRIP

EXPLANATION: this question is about "duration," which is a measure of a bond's interest-rate risk. The textbook definition is "a weighted average of a bond's cash flows." Sounds tough at first, but it really isn't. A "weighted average" means that we give more points to certain items than others, as in school, when homework might = 10% of your grade, 40% for the midterm, and 50% for the final exam. But, rather than focus on the "weighted average" part, just remember the "cash flow" part, which is key to understanding duration. You don't have to calculate duration, but if you did, you would give more weight to certain income payments than others. What's important here is a general understanding that it is "safer" to hold bonds that pay out big income streams--it calms people down when they're getting $120 a year on a 12% bond. But, if they're getting $40 a year on a 4% bond (after paying $1,000 for the bond), that could make them a little nervous. So, bonds with high coupon rates have lower durations (less interest rate risk) than bonds with chinsy little coupon rates. Also, the longer the term on the bond, the more nervous investors are about holding it.
So, in this question we look for the longest term to maturity, which is 20 years. One of these bonds has the highest duration--is the most sensitive to interest rate changes. If we have a 20-year T-bond and a 20-year STRIP, we have to remember that zero coupons pay NO cash flow--so they have to have a higher duration than bonds of equal maturities that DO pay cash flow. Duration is founded on the concept that investors will receive part or all of the principal they paid for the bond through the interest payments received--the faster that happens, the safer it is to hold the bond. Even if it's only a 3% nominal yield on a U S Treasury bond, after 20 years, you'd collect $600 on a 20-year bond. On a 20-year zero coupon (strip), you'd still be waitin' and a hopin' you receive the par value upon maturity. So, the test wants you to know that a 20-year bond paying interest will always have a lower duration than a 20-year zero coupon. Similarly, if you loaned $1,000 to a friend, would you rather have him pay back $100 a month or give him 5 years to pay it all back in a lump sum?
Me, I'd be a lot more laid back receiving payments regularly.

ANSWER: D