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# Bond Duration and Convexity: Practical Application

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.
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