Duration, convexity, and rate risk

In short

You have probably looked at a bond price and wondered why it moves at all. The issuer promised a fixed payment, so why does the price change? The answer lives in two ideas, duration and convexity. Duration tells you how far a bond's price may fall or rise when interest rates move. Convexity tells you how much that first estimate needs fixing. Look up the duration of any bond or bond fund you own, and write down what a 1 percent rise in rates would do to it. Do that math once and the next rate headline will feel a lot calmer.

The whole of it

What it is

A friend of mine once bought a long bond and felt very smart about the yield. Then rates rose and the price fell hard. He had never heard the word duration. Duration is a number, measured in years, that shows how sensitive a bond's price is to a change in rates. In its pure form it is the weighted average time you wait to get your money back, counting every coupon and the final payment. A bond that pays you back fast has a low duration. A bond that makes you wait has a high one.

Convexity is the next layer. If you are holding a bond, you may have noticed that its price does not move in a perfectly straight line as rates change. Convexity measures that bend. It tells you how much the duration number itself shifts as rates move. Low duration means less price pain when rates rise. High duration means more.

How it works

I once watched a seesaw on a school playground for a long while. One side goes up and the other goes down. Bonds work the same way with rates and prices. When rates rise, bond prices fall. When rates fall, bond prices rise. Duration tells you how far the seesaw tips for each 1 percent move in rates.

The U.S. Securities and Exchange Commission explains on its Investor.gov site that a bond fund with a duration of 5 years would be expected to lose about 5 percent in value if interest rates rose by 1 percentage point. That is the plain, usable idea this whole section rests on. Take the duration, multiply by the rate change, and you have a rough guess at the price change.

Convexity then corrects that guess. Duration alone draws a straight line, but real bond prices curve. A bond with high convexity loses less than duration predicts when rates rise. It also gains more than duration predicts when rates fall. That is a nice trait to own.

You have probably seen the words modified duration on a fund fact sheet. Modified duration is the version built for price math. It takes the raw number and adjusts it for the bond's yield, so you can use it straight in the seesaw math above. Macaulay duration is the older form. It is the weighted average time to receive each cash flow. The two numbers sit close together for most bonds, and they come from the same family.

The numbers, and where to find yours

If you are holding a single bond, your broker will often show the duration right on the detail page. For a bond fund, look at the fund's fact sheet or prospectus. The fund company must keep that document current and available to you. The SEC runs EDGAR, a free public database where you can pull prospectuses and annual reports for registered funds. That is a good first stop.

For Treasury bonds, TreasuryDirect.gov is the primary source. It is run by the U.S. Department of the Treasury. Coupon rate, maturity, and yield are the three inputs that drive duration. You do not need a formula tattooed on your arm. You need to know where to find the number and what it means once you have it.

No law sets a limit on duration the way it sets limits on retirement accounts. Duration is a math output, not a legal rule. If you hold a bond fund inside a retirement plan, the plan's own documents will say how the fund is run. Read them. Every word.

A worked example

Maria is 41 years old, and she is looking at two bonds sitting side by side in her brokerage account. The first bond matures in 3 years and pays a coupon of 4 percent a year. Its modified duration is 2.8. The second bond matures in 20 years and also pays 4 percent a year. Its modified duration is 12.5. Both bonds cost $1,000 today.

Now suppose rates rise by 1 percent across the board. For the short bond, Maria multiplies 2.8 by 1 percent and gets a price drop of about 2.8 percent. That is $1,000 times 0.028, or $28. She ends up near $972. For the long bond, she multiplies 12.5 by 1 percent and gets about 12.5 percent. That is $1,000 times 0.125, or $125. She ends up near $875. Same coupon. Same starting price. Very different damage.

Now Maria adds convexity to the picture. Her long bond has a convexity of 185, and her short bond has a convexity of 10. The usual fix is to add one half times convexity times the rate change squared. For the long bond, that is 0.5 times 185 times 0.01 times 0.01, which comes to 0.00925, or about 0.9 percent. On $1,000, that is about $9. So her loss is not $125. It is closer to $116, and she ends up near $884. For the short bond, the same math gives 0.5 times 10 times 0.0001, which is 0.0005, or about 5 cents on $1,000. That is tiny. She did not gain money from convexity. She lost less than the straight line said she would. That is what convexity does. It softens the blow on the way down and adds a little extra on the way up.

Where it goes wrong

You have probably heard someone say a short term bond is safe. That is only half true. Duration risk is lower on short bonds, but credit risk is its own thing. A bond from a shaky borrower can lose value even with a duration of 1. Duration measures rate sensitivity. It says nothing about whether the borrower pays you back. I have seen people mix up those two risks at the kitchen table with their statements spread out in front of them, and it is an easy mistake to make.

Callable bonds are another place things go sideways. A callable bond lets the issuer pay you back early, and issuers tend to do that when rates fall. That cuts off your upside. Duration math gets tangled here because the payment schedule is not fixed. Your broker or the fund prospectus will note whether a bond is callable. Read that line with care. The standard formula assumes a fixed schedule, and a callable bond breaks that.

Inflation is a third quiet troublemaker. A long bond locks in a fixed payment, so the number on the check never changes. Inflation can still eat what that check buys, year after year. Duration does not account for it. Treasury Inflation Protected Securities, which the Treasury calls TIPS, adjust their principal for inflation. If that worry keeps you up at night, the TIPS pages on TreasuryDirect.gov are written plainly and worth an hour of your time.

Questions to answer before you leave this page

Before you close this page, sit with a few questions. Do you know the modified duration of each bond or bond fund you own right now, and if not, do you know where to look, starting with EDGAR or your fund's own fact sheet? If rates rose by 1 percent tomorrow, could you estimate how much your bond holdings might drop in price, and does that number sit comfortably with the rest of your finances? Are any of your bonds callable, and have you read what that means for your return if rates fall? Do you own long bonds for a reason tied to your timeline, or did they just land in your account because the yield looked good that day? Do you keep credit quality separate from rate sensitivity in your mind, or have you been treating low duration as a stand in for safety of every kind? And if a friend asked you tonight why bond prices fall when rates rise, could you explain it in plain words, the way you would describe a seesaw to a child?