Duration and convexity are fundamental concepts in fixed income analysis. They measure a bond’s sensitivity to changes in interest rates and are essential tools for risk management and portfolio construction. Duration estimates the percentage change in a bond’s price for a given change in interest rates. Convexity accounts for the curvature in the price-yield relationship, providing a more accurate measure of price sensitivity.
Introduction to Interest Rate Risk
Bond prices and interest rates have an inverse relationship. When interest rates rise, bond prices fall, and vice versa. This inverse relationship is the source of interest rate risk for bondholders. The magnitude of the price change depends on several factors, including the bond’s maturity, coupon rate, and yield. Duration and convexity provide a quantitative framework for assessing this risk.
Duration
Duration is a measure of a bond’s sensitivity to changes in interest rates. It is expressed in years and represents the weighted average time to receive the bond’s cash flows. Duration is a key tool for managing interest rate risk and for constructing immunized portfolios.
Macaulay Duration:
Macaulay duration is the weighted average time to receive the bond’s cash flows, where the weights are the present value of each cash flow divided by the bond’s price. It is named after Frederick Macaulay, who introduced the concept in the 1930s. Macaulay duration is expressed in years. It measures the time it takes for the bond’s cash flows to repay the investor’s initial investment. Macaulay Duration = Σ [t × PV(CFt)] / Price, where t is the time to each cash flow and PV(CFt) is the present value of the cash flow. Macaulay duration increases with maturity and decreases with coupon rate.
Modified Duration:
Modified duration is a measure of the price sensitivity of a bond to changes in yield. It is calculated by dividing Macaulay duration by one plus the yield per period. Modified duration estimates the percentage change in a bond’s price for a 100 basis point change in yield. Modified Duration = Macaulay Duration / (1 + y), where y is the yield per period. For example, a modified duration of 5 means that a bond’s price will change by approximately 5% for a 1% change in yield.
Effective Duration:
Effective duration is a measure of a bond’s price sensitivity to changes in interest rates, taking into account embedded options, such as call or put features. It is calculated using a scenario analysis, estimating the bond’s price under two different yield scenarios. Effective duration is more accurate for bonds with embedded options. Effective Duration = (P− − P+) / (2 × P0 × Δy), where P− is the price when yields decrease, P+ is the price when yields increase, P0 is the current price, and Δy is the change in yield.
Key Rate Duration:
Key rate duration measures the sensitivity of a bond’s price to changes in specific points on the yield curve. It is used to manage the risk of non-parallel shifts in the yield curve. Key rate duration is important for portfolio managers seeking to immunize against specific yield curve movements.
Properties of Duration:
Duration has several important properties. Duration is lower for bonds with higher coupon rates. Duration is higher for bonds with longer maturities. Duration is higher for bonds with lower yields. Duration is higher for zero-coupon bonds. Duration is affected by the presence of embedded options.
Convexity
Convexity measures the curvature of the price-yield relationship. It accounts for the fact that the relationship between bond prices and yields is not linear. Convexity is a second-order measure of interest rate risk.
The Price-Yield Relationship:
The price-yield relationship is convex, meaning that the price increases at a decreasing rate as yields fall and decreases at an increasing rate as yields rise. Duration provides a linear approximation of this relationship, which is accurate only for small changes in yield. For larger changes, convexity must be considered.
Convexity Calculation:
Convexity is calculated as the second derivative of the price-yield relationship. Convexity = (P− + P+ − 2P0) / (P0 × (Δy)^2), where P− is the price when yields decrease, P+ is the price when yields increase, P0 is the current price, and Δy is the change in yield. Convexity is expressed in years squared. Higher convexity is beneficial for bondholders, as it means that prices rise more when yields fall and fall less when yields rise.
Positive and Negative Convexity:
Most bonds have positive convexity. Positive convexity means that the price-yield relationship is convex, providing a benefit to bondholders. Callable bonds and mortgage-backed securities can have negative convexity. Negative convexity means that the price-yield relationship is concave, reducing the benefit to bondholders.
Applications of Duration and Convexity
Duration and convexity are widely used in fixed income portfolio management. They are used for immunization, which is a strategy for protecting a portfolio from interest rate changes. They are used for hedging interest rate risk using derivatives. They are used for performance measurement and attribution. They are used for risk management and asset-liability management.
Immunization:
Immunization is a strategy designed to protect a portfolio from interest rate changes. The portfolio is structured so that its duration matches the investment horizon. Immunization ensures that the portfolio’s value is relatively insensitive to interest rate changes.
Limitations of Duration and Convexity:
Duration and convexity are useful tools but have limitations. Duration assumes a parallel shift in the yield curve. Convexity provides a more accurate measure of price sensitivity but is also based on assumptions about the yield curve. Duration and convexity are less accurate for bonds with embedded options.