I have questions relating to convexity and duration. Could you explain them conceptually and the practical uses for them? An example of why a certain convexity bids up a bond, or something along those lines, would be extremely helpful.

-- Patrick Golon

Patrick,

I suppose it was inevitable that one of these days I would have to try and explain duration and convexity in simple terms; still, I was hoping that day would never come.

Now that it has, though, I have to say that these concepts are vastly less difficult to explain than I thought they would be.

First, a very brief review of what bonds are, to familiarize everyone with some key terminology.

Many people think bonds are exceedingly complicated, and in some respects they are. But in a way, they couldn't be simpler. When you buy a regular, plain-vanilla bond, you are buying a stream of payments. Assuming the bond has a face value of \$1,000, as most bonds do, the final payment in the stream is the return of principal on the bond's maturity date. And assuming the bond carries a coupon, as most do, the rest of the stream consists of fixed-interest payments, typically paid semiannually. If the coupon is 6%, a \$1,000 bond would pay \$60 a year in two \$30 installments until it matures, when it would make the final \$30 payment and return the \$1,000 of principal.

So what's duration? Sounds like the same thing as maturity, especially because, like maturity, it's measured in years.

It isn't the same thing.

Duration, you may have been told, measures a bond's interest-rate sensitivity. But what exactly is it measuring?

Duration -- which is always shorter than maturity except in the case of zero-coupon bonds, for which duration equals maturity -- measures the amount of time it will take for the investor to receive half of the present value of all future payments from the bond. The discount rate for calculating the present value of the cash flows is the bond's yield.

A 1985 paper by Salomon Brothers' bond portfolio analysis group came up with a beautiful analogy. Imagine a series of buckets lined up on a two-by-four. All of the buckets are the same size, except for the last one on the right-hand end, which is much larger. The small buckets represent all but the last coupon payments from the bond; the final, big bucket represents the principal repayment and the final coupon payment. If it was a 10-year, 6% coupon bond that paid interest semiannually, there would be 20 buckets. The first 19 would represent the \$30 income payments from the bond; the last would represent the final income payment and the principal repayment -- a \$1,030 bucket.

Now imagine that each bucket is filled to a level representing the present value of that payment. The further out the plank you go, the smaller the present value of that payment. This represents the time value of money: \$30 tomorrow is worth more than \$30 five years from now, assuming normal economic conditions.

The plank is essentially a timeline, and duration is the point along the plank where a fulcrum would go to balance the whole system. (The buckets themselves, for argument's sake, don't weigh anything.) Can you see why the duration of a zero-coupon bond would be equal to its maturity? Because the payment stream from a zero-coupon bond consists entirely of a single payment at maturity, that's where the fulcrum would have to go.

So, what's the use of knowing a bond's duration? If you know a bond's duration, you can estimate how much its price will change if its yield changes. Note that there are different types of duration. The one described above is Macaulay duration, defined by Frederick Macaulay in his 1938 study of U.S. financial markets. To easily estimate how much a bond's price will change if its yield changes, you need its modified duration, which you get by dividing the Macaulay duration by one plus half the yield. Once you know the modified duration, you can multiply it by a change in yield to calculate the approximate change in price. For example, a bond with a duration of 6.4 will go up about 6.4% in price if its yield drops by 1% (100 basis points), and down about 6.4% if its yield rises by 100 basis points.

Just a couple more points about duration.

Why it's useful: Duration gives investors a way to compare bonds with different maturities and coupons. A recent Fund Forum talked about the fact that bonds with big coupons are less interest-rate sensitive than bonds with small coupons, all things being equal. Using the Salomon analogy, you can see why. The more that's in those small buckets, relative to the amount in the big bucket, the further to the left the fulcrum will go to balance the system. In other words, the shorter the duration.

Check out this graph: It shows how duration increases with maturity, depending on the size of the coupon. As you can see, the smaller the coupon, the longer the duration.

Got all that? Ready to become a bond trader? Not so fast. As a bond ages, its duration shortens. Every time the investor receives a coupon payment, one bucket gets taken away, and the extra plank gets sawed off. The fulcrum must move to the right from where it was, but because the plank is shorter, so is the duration.

Another complication: As a bond's price and yield fluctuate, so does its duration. As the yield drops, the duration lengthens; as the yield rises, the duration shortens. Here's why: Recall that yield is the rate used to discount the bond's future payments to calculate their present value. As the yield drops, the present value of the payments rises, with the value of the payments furthest in the future rising the most proportionally. So to rebalance the system, the fulcrum would move to the right, indicating a longer duration. Likewise, as the yield rises, the present value of the payments declines, with the value of those furthest in the future declining the most proportionally. To rebalance the system, the fulcrum would move to the left, indicating a shorter duration.

RELATED STORIES:

TSC Fixed-Income Forum aims to provide general bond information. Under no circumstances does the information in this column represent a recommendation to buy or sell bonds, funds or other securities. As originally published, this story contained errors. Please see Corrections and Clarifications.