Final - Interest Rate Sensitivity and Duration in Fixed Income

The percentage change in bond price is inversely related to changes in yield to maturity.
Quantitative Representation: - $ext{Percentage Change in Bond Price} ext{ ∝ } ext{Change in Yield to Maturity}$

Long-term vs. Short-term Bonds: - Long-term bonds tend to exhibit greater sensitivity to interest rate changes compared to short-term bonds.

High-Coupon vs. Low-Coupon Bonds: - Prices of high-coupon bonds are less sensitive to interest rate changes than those of low-coupon bonds.

Example 1: Prices of an 8% coupon bond (with semi-annual payments) affected by YTM. - | YTM | T = 1 | T = 10 | T = 20 | |-------|-------|--------|--------| | 8% | $1,000| $1,000 | $1,000 | | 9% | $990.64| $934.96| $907.99| | Price Change (%) | 0.94% | 6.50% | 9.20% |
Example 2: Prices of zero-coupon bonds (with semi-annual compounding). - | YTM | T = 1 | T = 10 | T = 20 | |-------|--------|---------|---------| | 8% | $924.56| $456.39 | $208.29 | | 9% | $915.73| $414.64 | $171.93 | | Price Change (%) | 0.96% | 9.15% | 17.46% | - Conclusion: The zero-coupon bond demonstrates longer duration and more pronounced price changes when yield shifts from 8% to 9%.

High-YTM vs. Low-YTM Bonds: - Prices of low-YTM bonds show greater sensitivity to interest rate changes than those of high-YTM bonds.

Definition of Duration: - Duration measures the average time to receive the bond's promised cash flows. It provides a metric for how sensitive a bond is to interest rate changes—greater duration equates to higher sensitivity.
Macaulay Duration Equation: - For a bond with cash flows, where $y$ is the yield to maturity: - $P = rac{C}{(1+y)} + rac{C}{(1+y)^2} + … + rac{C+F}{(1+y)^n}$ - Macaulay's Duration is computed as: - $D = rac{ ext{Total Cash Flow Weight}}{P}$ , where: - $w_t = rac{CF_t}{(1+y)^t}$ for cash flows at time $t$ .

Example 1: Determining duration for a 10-year zero-coupon bond priced at $500 with a YTM of 3.526% (semi-annual).
Example 2: Duration for an 18-month bond with an 8% coupon rate, semi-annual payments, BEY of 10%, face value of $1,000.
Example 3: Macaulay’s duration for a 2-year bond with a 4% semiannual coupon and face amount of $2,000, BEY of 2.5%.

Approximation Formula:
- The bond's percentage price change can be estimated using: - $rac{ riangle P}{P} ext{ approx. } -D imes rac{ riangle y}{(1+y)}$ - Modified Duration Definition:
- $D^* = rac{D}{(1+y)}$

Graphical Representation:
- Comparison of Duration Approximation vs. Actual Change: - Actual price changes are not perfectly predicted by duration approximation. - Accuracy diminishes with larger changes in interest rates.

Example 4 Calculation:
- If the YTM changes from 5% to 6% for the 18-month bond (BEY of 10%, coupon rate of 8%, face value of $1,000, initial price of $972.77): - Approximate percentage change in price calculated as: - $rac{ riangle P}{P} = -D imes riangle y = -2.74 ext{ (for } riangle y = 0.01 ext{)}$

The duration of a zero-coupon bond is equivalent to its maturity.
For constant maturity and YTM, a bond's duration increases with a lower coupon rate.
For a constant coupon rate, duration and interest rate sensitivity generally increase with longer maturities.
For consistent factors, a bond's duration is higher with a lower YTM.

If all else being equal, does a higher coupon payment lead to lower duration?
  - A. Yes
  - B. No
  - C. Not enough information

Find the Macaulay’s duration of a 2-year bond with a semiannual coupon rate of 6%, face amount of $1,000, and BEY of 10%:   - Possible options:   - A. 3.82
  - B. 1.21
  - C. 2.74

A bond portfolio has a modified duration of 5 years. If the annual yield for all bonds decreases by 2%, the approximate percentage change in portfolio value can be estimated as:   - A. 2%
  - B. 10%
  - C. 15%