Lesson Objective: To understand the measurement and management of interest rate risk using duration and convexity.

In-Depth Notes:

1. The Concept of Interest Rate Risk:
Interest rate risk is the risk that the value of a bond will decline due to a rise in market interest rates. This is the most significant risk faced by bond investors, and understanding its measurement and management is crucial for fixed-income portfolio management.

2. Duration:
Duration is the primary metric used to measure a bond’s sensitivity to interest rate changes. It represents the weighted average time to receive the bond’s cash flows (coupons and principal), where the weights are the present value of each cash flow as a percentage of the bond’s price.

• Macaulay Duration: The weighted average time to receive the bond’s cash flows, measured in years. The formula for Macaulay duration is:
  Macaulay Duration = Σ [t × (PV of Cash Flow t) / Bond Price]
  Where t is the time period and PV is the present value of the cash flow.

• Modified Duration: A measure of the bond’s price sensitivity to a 1% (100 basis point) change in yield, expressed as a percentage price change. Modified duration is calculated as:
  Modified Duration = Macaulay Duration / (1 + YTM / m)
  Where m is the number of coupon payments per year.

• Interpretation: For a bond with a modified duration of 5 years, a 1% increase in yield is expected to result in a price decline of approximately 5%. Conversely, a 1% decrease in yield is expected to result in a price increase of approximately 5%.

3. Determinants of Duration:

• Maturity: Generally, the longer the maturity, the higher the duration (all else being equal). Longer-term bonds are more sensitive to interest rate changes.

• Coupon Rate: The higher the coupon rate, the lower the duration. A higher coupon means that a larger portion of the bond’s value is received earlier, reducing the weighted average time to cash flow.

• Yield to Maturity: The higher the yield, the lower the duration. Higher yields mean that future cash flows are discounted more heavily, reducing their present value and their weight in the duration calculation.

4. Convexity:
Duration is a linear approximation of the price-yield relationship. Convexity captures the curvature of this relationship.

• Positive Convexity: Most bonds exhibit positive convexity, meaning that the bond’s price increases more when yields fall than it decreases when yields rise by the same amount. Positive convexity is a desirable feature for bond investors, as it provides additional price protection.

• The Convexity Adjustment: The bond price change due to a yield change can be more accurately estimated by adding a convexity adjustment to the duration estimate:
  Price Change (%) ≈ -Modified Duration × ΔYield + 0.5 × Convexity × (ΔYield)^2

• Negative Convexity: Some bonds, such as callable bonds, exhibit negative convexity at certain yield levels. This means that the bond’s price increases less when yields fall than it decreases when yields rise.