week 27 Fixed-income securities (V) - Duration

Duration

• Duration measures the average maturity of a fixed-income security, reflecting the weighted average timing of a bond’s cash payments.

  • For zero-coupon bonds, duration equals maturity because there is only one payment at the end.

  • For coupon-paying bonds, duration is less than the actual maturity date due to intermediate payments.

Concept of Duration

• Higher coupons result in shorter durations.

• Longer maturities lead to longer durations.

• Higher yields (YTMs) lead to shorter durations.

• Shorter duration means less volatility in bond prices, and vice versa.

Measuring Duration

• Steps in Calculating Duration:

  1. Find the present value of each coupon or principal payment using the prevailing YTM as the discount rate.

  2. Divide the present value by the current market price of the bond to get the "weight."

  3. Multiply this weight by the year in which the cash flow is to be received.

  4. Repeat for each year in the bond's life and sum the results.

• Duration for a Portfolio of Bonds:

  • The duration of a portfolio is the weighted average of the durations of the individual securities in the portfolio.

Bond Duration and Price Volatility

• Bond duration helps investors understand how bond prices will respond to changes in market interest rates, especially for small changes.

• Modified Duration:

  • Modified Duration = \frac{Macaulay\ Duration}{1 + Yield\ to\ Maturity}

• Percentage Change in Bond Prices:

  • Percentage\ Change \approx -1 \cdot Modified\ Duration \cdot Change\ in\ Interest\ Rates

Effective Duration

• Effective Duration (ED) is used for bonds with embedded options like call or conversion features.

• ED = \frac{Price{\downarrow} - Price{\uparrow}}{2 \cdot Price_{initial} \cdot \Delta Yield}

  • Price_{\uparrow} = Bond price if yield increases.

  • Price_{\downarrow} = Bond price if yield decreases.

  • \Delta Yield = Change in yield.