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Foreign. Hi, I'm Andy Tempte and welcome to Money Lessons. Join me every Saturday morning for bite sized lessons that are designed to improve financial literacy around the world. Today is February 28, 2026. Last week we explored the yield curve and what its shape tells us about economic expectations. Today we tackle a concept that I promised at the end of that episod concept known as duration. If you're going to invest in bonds, duration is essential knowledge. And as a bonus, today we'll discover how advanced mathematics quietly powers the financial world. Don't worry, we're not going to get too deep in advanced math. But let's start today with the question that duration answers. We know from our January 17th lesson that when interest rates rise, bond prices fall, and vice versa. This inverse relationship is fundamental to bond investing. But here's what we didn't answer. We didn't answer by how much? If interest rates jump by 1%, does your bond lose 5% of its value? 10%? 20%? The answer depends on the bond's interest rate sensitivity or its duration. Duration measures how sensitive a bond's price is to changes in interest rates, and economist Frederick Macaulay developed the concept way back in 1938 while studying interest rates, bond yields and stock prices for the National Bureau of Economic Research. Macaulay realized that a bond's maturity date alone doesn't tell you how the bond will behave when rates change. A ten year bond that pays annual coupons behaves differently than a ten year zero coupon bond, which is a bond that pays no interest while you own it and simply returns face value at maturity. So even though bond both bonds mature on the same date, they're going to have different interest rate sensitivity and duration captures that difference. Here's the practical rule. Duration tells you approximately how much a bond's price will change for every 1% change in interest rates. If a bond has a duration of seven years, a 1% increase in interest rates will cause its price to drop by approximately 7%. Conversely, if rates fall by 1%, that same bond rises by about 7%. Duration is your interest rate sensitivity measure and I want you to keep track in the back of your mind this word approximate. Because duration does not give exact or precise estimates, it gives approximations. Now let's discover or explore why longer term bonds move more in price than short term bonds in most cases. So think about what you're buying when you purchase a bond. It is a series of future cash flows. Some arrive sooner as coupon payments, while the big payment, the return of your principal Arrives at maturity. Duration calculates when on average, you receive your money back. And that average is weighted by the present value of each payment. Consider two bonds, each maturing in 10 years. The first pays a 5% coupon, sending you interest payments every six months while you own it. The second bond is a zero coupon bond, which means no, no interest rate payments at all. And then at maturity, you get your principal back. Even though both bonds mature on the same date, they have different durations. The coupon bond might have a duration of around eight years because you receive meaningful cash flows along the way. The zero coupon bond has a duration of exactly 10 years because all of your future cash flows arrive at year 10. This difference matters enormously for bond investors. For a 30 year zero coupon bond, you receive nothing until the very end. So duration equals the full 30 years. That's three decades of exposure to interest rate movements. Which explains why long term zero coupon bonds are among the most volatile fixed income securities. I'm going to say that again. Long term zero coupon bonds are among the most volatile fixed income securities as measured by changes in price. Now let me show you why this matters. I'm going to take you back to 2022. Back in 2022, inflation was rising dramatically. And the Federal Reserve responded by raising interest rates from near zero to over 4%. One of the most aggressive rate H campaigns in history. The results were brutal for bond investors. The broad U.S. bond market fell more than 13% that year. 10 year treasury notes lost over 16% of their value. And 30 year zero coupon bonds, well, they collapsed nearly 40%. That's 4,0% the worst performance for long dated U.S. bonds since records began in 1754. Why such carnage? Well, this is duration or interest rate risk in action. Those long term bonds had duration stretching well beyond 20 years. When rates jumped by 4 percentage points, duration predicted massive losses. And losses indeed occurred. Investors who understood duration weren't really surprised. Those who didn't learned an expensive lesson about interest rate risk. Now here's a caveat, and this is where it gets interesting. For anyone who thinks that advanced mathematics has no real world application. Duration provides a linear estimate of what is actually a curvilinear relationship between bond prices and interest rates. The relationship between prices and interest rates is more like an arc than a straight line. Duration works well for small interest rate changes, but for larger movements, the estimate becomes less and less precise. Remember when we said that a 4% rise in rates created catastrophic losses back in 2022? That's true, but using duration alone would have overestimated those losses. A 30 year zero coupon bond with a DUR of 30 years mathematically predicts a 120% loss for a 4% rise in interest rates, which is impossible. The actual loss of 40%, while devastating, was far less. To get more accurate predictions, financial professionals use something called convexity, which accounts for the curvature of the price yield relationship duration and convexity together provide a much better estimate estimate of actual losses. Mathematically, duration is the first derivative of bond price with respect to interest rates, while convexity is the second derivative. And if that sounds like calculus, well, it is. This is calculus in action, working quietly behind the scenes every time a portfolio manager makes a decision. So the next time someone asks why advanced math matters, you have an answer. It helps manage trillion of dollars in the global bond market. But what does all this mean for you and your investing decisions? Duration is a measure that helps you match your bond portfolio to your investment horizon and risk tolerance. If you're investing for the long term and can stomach volatility, longer duration bonds typically offer higher yields as compensation for their greater interest rate risk. But if you need stability and have a shorter time horizon, shorter duration bonds provide much more protection against interest rate movements. Many investors reduce portfolio duration as they approach retirement because their investment horizon is shrinking. So the yield curve discussion from last week connects right in with right here. When the yield curve is steep and upward sloping, you're being paid more to accept more interest rate risk or more duration risk. When the yield curve is flat or even inverted, that extra compensation shrinks or even disappears. Understanding duration helps you evaluate whether the trade off this risk return trade off makes sense. Next week we'll wrap up our discussion of debt securities by putting everything together how to build a bond portfolio that serves your financial goals. We're going to talk about laddering strategies, how bonds fit alongside stocks, and practical steps for implementing what you've learned in this series. So until next week, I wish you grace, dignity and compassion. My name is Andy Tempte. This is Money Lessons. You can find the show on all the major streaming services as well as out on YouTube. YouTube. Please like subscribe rate and most importantly, share this public good with your friends, your family, your neighbors, and maybe a colleague if you like them. The show is produced by Nick Tempte and we'll see you next week on Money Lessons.