Modified duration is a measure of the sensitivity of a bond's price to changes in interest rates. It represents the approximate percentage change in a bond's price for a 1% change in yield, taking into account the time value of money and the bond's coupon payments.
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Modified duration is a more accurate measure of interest rate risk than Macaulay duration, as it takes into account the time value of money and the bond's coupon payments.
A bond with a higher modified duration is more sensitive to changes in interest rates, meaning its price will fluctuate more for a given change in yield.
Modified duration can be used to estimate the percentage change in a bond's price for a small change in yield, assuming the yield curve remains constant.
The formula for modified duration is: $\text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{\text{Yield}}{\text{Coupon Frequency}}}$
Modified duration is an important metric for fixed-income investors to consider when managing interest rate risk in their portfolios.
Review Questions
Explain how modified duration differs from Macaulay duration and why it is a more accurate measure of interest rate risk.
Modified duration is a more accurate measure of interest rate risk compared to Macaulay duration because it takes into account the time value of money and the bond's coupon payments. Macaulay duration only considers the average time it takes to receive the present value of a bond's cash flows, while modified duration adjusts for the impact of the bond's yield and coupon frequency on the bond's price sensitivity to interest rate changes. This makes modified duration a more precise tool for estimating the percentage change in a bond's price for a given change in yield.
Describe how the yield curve and convexity relate to modified duration and their impact on a bond's interest rate risk.
The shape of the yield curve can affect a bond's modified duration, as the relationship between a bond's price and yield is not linear but convex. Convexity measures the curvature of this price-yield relationship, and it can be used to estimate the change in a bond's duration as yields change. A bond's modified duration will be lower when yields are higher, and higher when yields are lower, due to the convex nature of the price-yield curve. Additionally, the slope and shape of the yield curve can influence a bond's modified duration, as the term structure of interest rates affects the bond's cash flow patterns and sensitivity to rate changes.
Analyze how an investor can use modified duration to manage interest rate risk in a fixed-income portfolio, particularly in the context of the 'Risks of Interest Rates and Default' topic.
Investors can use modified duration as a tool to manage interest rate risk in their fixed-income portfolios. By understanding a bond's modified duration, they can estimate the potential price change of the bond for a given change in yield, which is especially important in the context of the 'Risks of Interest Rates and Default' topic. If an investor expects interest rates to rise, they can reduce the modified duration of their portfolio by shifting investments towards bonds with lower modified durations, which will be less sensitive to the rising rates. Conversely, if rates are expected to fall, the investor can increase the modified duration of their portfolio to take advantage of the potential price appreciation. This active management of modified duration can help mitigate the interest rate risk in a fixed-income portfolio.
Macaulay duration is a measure of the average time it takes to receive the present value of a bond's cash flows, weighted by the present value of each cash flow.
The yield curve is a graphical representation of the relationship between the yield (interest rate) and the time to maturity of a set of similar bonds, typically U.S. Treasury securities.
Convexity is a measure of the curvature of a bond's price-yield relationship, which can be used to estimate the change in a bond's duration as yields change.