Modified Duration for Different Bonds

16.4.1 Modified Duration for Different Bonds

Understanding modified duration is crucial for anyone involved in bond investing or fixed income securities analysis. Modified duration is a measure of a bond’s price sensitivity to changes in interest rates, and it plays a vital role in managing interest rate risk. In this section, we will explore the concept of modified duration and how it is calculated for different types of bonds, including zero-coupon bonds, fixed-rate coupon bonds, and floating-rate notes.

What is Modified Duration?

Modified duration is an extension of the Macaulay duration, which measures the weighted average time to receive the bond’s cash flows. Modified duration, on the other hand, adjusts the Macaulay duration to account for changes in interest rates. It provides an estimate of how much the price of a bond will change in response to a 1% change in interest rates. This measure is particularly useful for investors and portfolio managers to assess the interest rate risk of their bond portfolios.

Importance of Modified Duration

Modified duration is a key tool for managing interest rate risk. It helps investors understand how sensitive a bond’s price is to changes in interest rates. By knowing the modified duration, investors can make informed decisions about which bonds to hold, buy, or sell in anticipation of interest rate movements. It also aids in constructing bond portfolios with specific risk and return characteristics.

Calculating Modified Duration

The formula for modified duration is derived from the Macaulay duration and the bond’s yield to maturity (YTM):

Where:

• Macaulay Duration is the weighted average time until cash flows are received.
• YTM is the yield to maturity of the bond.
• n is the number of compounding periods per year.

Let’s delve into how modified duration is calculated for different types of bonds.

Modified Duration for Zero-Coupon Bonds

Zero-coupon bonds do not pay periodic interest. Instead, they are issued at a discount to their face value and mature at par. The modified duration of a zero-coupon bond is equal to its Macaulay duration, which is simply the bond’s time to maturity, since there are no interim cash flows.

Example Calculation

Consider a zero-coupon bond with a face value of $1,000, a maturity of 5 years, and a YTM of 4%.

1. Macaulay Duration: Since it’s a zero-coupon bond, the Macaulay duration is the same as its maturity, which is 5 years.

2. Modified Duration: Using the formula:

   $$ \text{Modified Duration} = \frac{5}{1 + \frac{0.04}{1}} = \frac{5}{1.04} = 4.81 $$

This means that for a 1% increase in interest rates, the price of the bond will decrease by approximately 4.81%.

Modified Duration for Fixed-Rate Coupon Bonds

Fixed-rate coupon bonds pay periodic interest at a fixed rate. The calculation of modified duration for these bonds involves discounting each cash flow by the bond’s YTM and calculating the weighted average time to receive these cash flows.

Example Calculation

Consider a fixed-rate bond with a face value of $1,000, a coupon rate of 5%, a maturity of 5 years, and a YTM of 4%.

1. Calculate Cash Flows: The bond pays $50 annually for 5 years and $1,000 at maturity.

2. Calculate Present Value of Cash Flows: Discount each cash flow by the YTM.

3. Calculate Macaulay Duration: Sum the present values of cash flows, weighted by time, and divide by the bond price.

4. Calculate Modified Duration: Adjust the Macaulay duration using the YTM.

   $$ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{0.04}{1}} $$

The exact calculation involves detailed steps of discounting each cash flow and computing the weighted average, which can be done using spreadsheet software for precision.

Modified Duration for Floating-Rate Notes

Floating-rate notes (FRNs) have variable interest payments linked to a benchmark rate, such as LIBOR. The modified duration for FRNs is typically lower than that of fixed-rate bonds because the interest payments adjust with market rates, reducing interest rate risk.

Example Calculation

Consider a floating-rate note with a face value of $1,000, a benchmark rate of LIBOR + 2%, and a maturity of 5 years.

1. Estimate Cash Flows: Cash flows vary with changes in the benchmark rate.

2. Calculate Present Value of Cash Flows: Use the current benchmark rate to discount expected cash flows.

3. Calculate Macaulay Duration: Since interest payments adjust, the duration is shorter.

4. Calculate Modified Duration: Adjust the Macaulay duration using the current yield.

   The modified duration for FRNs is often close to zero, reflecting their low interest rate sensitivity.

Practical Applications of Modified Duration

Modified duration is used extensively in bond portfolio management to assess and manage interest rate risk. Here are some practical applications:

• Portfolio Immunization: By matching the modified duration of a bond portfolio to the investment horizon, investors can immunize the portfolio against interest rate changes.

• Interest Rate Hedging: Investors can use modified duration to determine the appropriate amount of hedging needed to protect against interest rate fluctuations.

• Risk Assessment: Modified duration provides a clear measure of a bond’s interest rate risk, allowing investors to compare bonds with different maturities and coupon structures.

Conclusion

Modified duration is a vital concept in fixed income analysis, providing insights into a bond’s price sensitivity to interest rate changes. By understanding and calculating modified duration for different bonds, investors can better manage interest rate risk and optimize their bond portfolios. Whether dealing with zero-coupon bonds, fixed-rate coupon bonds, or floating-rate notes, mastering modified duration is essential for successful bond investing.

References

• CFA Program Curriculum - Fixed Income Analysis
• Investopedia - Calculating Duration

Bonds and Fixed Income Securities Quiz: Modified Duration for Different Bonds