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Fixed Income Volatility: Modified Duration

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Understanding Bond Price Sensitivity

Introduction

Modified duration helps us figure out how sensitive a bond is to changes in interest rates. It tells us how much a bond's price will change when interest rates go up or down by 1%. Let's break it down in simpler terms.

How We Calculate Modified Duration

To find modified duration, we use a simple formula: We take something called the Macaulay duration (which is like the average time it takes to get back the money you invested) and divide it by a number that includes the interest rate.

  • Modified Duration = Macaulay Duration / (1 + Interest Rate)
  • Macaulay Duration = Average time to get back money
  • Interest Rate = How much the bond pays you (also known as yield)

Example

Imagine you have a bond with a face value of $1,000, a 5% coupon rate (meaning it pays you 5% of $1,000 every year), and it takes 5 years to mature. If the interest rate is 4%, the modified duration is about 4.73 years.

Comparing Modified Duration and Macaulay Duration

The big difference is this: Macaulay duration tells us when we get our money back on average, while modified duration tells us how the bond's price changes when interest rates move.

Why It Matters

  1. Price Sensitivity: Modified duration helps us see how bond prices react to interest rate changes. This is important for understanding the risks.
  2. Comparing Bonds: It lets us compare different bonds, even if they have different coupon rates or maturity times. It's like a standard way of looking at things.

Things to Watch Out For

  1. Convexity Consideration: Sometimes, the relationship between bond price and interest rate changes isn't straightforward. Ignoring this can lead to mistakes.
  2. Sensitivity to Big Changes: Modified duration works well for small changes in interest rates but may not be as accurate for big changes.

When It's Useful

Use modified duration when you want to:

  • Check how interest rate changes might affect bond prices.
  • Compare different bonds, even if they have different features.

Common Mistakes

  1. Absolute Comparison: Don't compare modified duration directly between bonds with different durations. It's more about how each bond reacts on its own.
  2. Ignoring Convexity: Remember, modified duration doesn't cover all the details. Ignoring the curve in the bond's price-yield relationship can lead to errors.

In Conclusion

Modified duration is a handy tool for understanding how bonds react to changes in interest rates. It helps us compare different bonds and manage risks. But to get the whole picture, it's good to know about convexity too. Knowing the upsides, downsides, and possible mistakes with modified duration helps us make better decisions when dealing with bonds.

This article takes inspiration from a lesson found in FIN 4243 at the University of Florida.