Duration and Convexity: How to Measure Bond Price Sensitivity to Interest Rates

Macaulay Duration: The Weighted-Average Time to Receive Cash Flows

Macaulay duration, named after economist Frederick Macaulay, is the weighted-average time to receive a bond's cash flows, measured in years. Each cash flow (coupon payments and principal repayment) is weighted by its present value as a percentage of the bond's total price. A zero-coupon bond's Macaulay duration equals its maturity because there is only one cash flow at maturity. A coupon bond's Macaulay duration is less than its maturity because some cash flows (coupon payments) arrive before maturity. For example, a 10-year bond with a 4% coupon might have a Macaulay duration of 8.2 years -- meaning the average time to receive cash flows is 8.2 years, shorter than the 10-year maturity because coupons are received earlier. Macaulay duration is measured in years and is used primarily to understand the timing of cash flows. It is the foundation for modified duration, which is more directly useful for risk measurement. Longer Macaulay duration means greater interest rate sensitivity. All else equal, bonds with lower coupons and longer maturities have higher Macaulay durations. Calculate bond yields and understand how they affect duration →

Modified Duration: The Percentage Price Sensitivity

Modified duration is derived from Macaulay duration and directly estimates the percentage price change for a 1% change in yield. The formula is: Modified Duration = Macaulay Duration / (1 + Yield/n), where n is the number of compounding periods per year (typically 2 for semiannual bonds). If a bond has a Macaulay duration of 8.2 years and a yield of 4% (semiannual), the modified duration is 8.2 / (1 + 0.04/2) = 8.04 years. This means the bond's price is expected to change by approximately 8.04% for each 1% change in yield. Modified duration is the most commonly cited duration measure for risk management. Portfolio managers use modified duration to estimate how much a bond portfolio will gain or lose for a given change in interest rates. A portfolio with a modified duration of 5 years will change approximately 5% in value for a 1% parallel shift in the yield curve. Modified duration is valid for small yield changes (up to about 50-100 basis points) but becomes increasingly inaccurate for larger changes, which is where convexity becomes important.

Dollar Duration: The Dollar Change in Price

Dollar duration (also called DV01 -- dollar value of 01, or the price value of a basis point) measures the actual dollar change in a bond's price for a 1 basis point (0.01%) change in yield. For a bond with a modified duration of 8.04 years and a price of $1,000, the dollar duration is approximately: $1,000 x 0.0804 x 0.0001 = $0.0804 per $1,000 face value. A bond's DV01 is approximately $0.08 per $1,000 of face value, meaning a 1 basis point yield increase causes the bond's price to fall by about $0.08. Dollar duration is essential for hedging. If a portfolio has a dollar duration of $1 million per basis point, a 10 basis point rise in yields causes a $10 million loss. The hedger then takes an offsetting position (such as Treasury futures or interest rate swaps) with an equal and opposite dollar duration. DV01 is also additive across positions -- you can sum the DV01 of individual bonds to get the portfolio's total interest rate exposure. This makes it the most practical measure for risk managers and traders. Hedging interest rate risk with duration-based strategies →

Convexity: The Adjustment for Large Rate Moves

Convexity measures the curvature of the price-yield relationship. The price-yield curve of a standard bond is convex (curving upward), meaning that as yields fall, prices rise at an increasing rate; as yields rise, prices fall at a decreasing rate. This is positive convexity, and it benefits bondholders. Duration is the slope of the price-yield curve at a single point. Convexity is the rate of change of that slope. For small yield changes, the slope (duration) is a reasonable approximation. For large changes, the curvature (convexity) becomes significant. The convexity adjustment to the duration estimate is: 0.5 x Convexity x (Change in Yield)^2. Convexity is expressed as a positive number for standard bonds (positive convexity). Callable bonds, mortgage-backed securities, and some other fixed-income instruments can have negative convexity, meaning the price-yield curve bends downward. Negative convexity hurts investors: prices fall more when yields rise and rise less when yields fall, compared to what duration predicts. Investors should understand convexity when holding bonds or bond funds in a volatile rate environment. Negative convexity in mortgage-backed securities →

Duration of a Bond Portfolio

The duration of a bond portfolio is the weighted average of the durations of the individual bonds, weighted by market value. If a portfolio has $10M of bonds with duration 5 years and $10M of bonds with duration 15 years, the portfolio duration is (0.5 x 5) + (0.5 x 15) = 10 years. However, this simple weighted average is only an approximation. For precise risk measurement, portfolio managers use key rate durations, which measure sensitivity to yield changes at specific maturities along the yield curve (e.g., 2-year, 5-year, 10-year, 30-year). Key rate duration captures that a steepening of the yield curve (short rates rise more than long rates, or vice versa) has different effects than a parallel shift. Portfolio duration is the starting point for asset-liability management, where institutions match the duration of their assets to the duration of their liabilities to minimize interest rate risk. Pension funds and insurance companies are the primary practitioners of duration matching. How institutions use duration matching for risk management →

What is the difference between Macaulay duration and modified duration?

Macaulay duration measures the weighted-average time to receive a bond's cash flows, expressed in years. Modified duration measures the percentage price sensitivity of a bond to yield changes, expressed as a number (not years). Modified duration = Macaulay duration / (1 + yield/n). For a bond with 8.2 years Macaulay duration and 4% yield, modified duration is approximately 8.04. Macaulay duration helps investors understand the timing of cash flows and is used in immunization strategies. Modified duration is the practical measure for estimating price changes. When investors refer to duration without qualification, they typically mean modified duration or its close approximation. Both measures increase with longer maturity and decrease with higher coupon rates. A zero-coupon bond's Macaulay duration equals its maturity, giving it the highest duration for a given maturity.

How does duration affect bond ETFs and mutual funds?

A bond ETF or mutual fund has a portfolio duration that reflects the weighted average duration of its holdings. A fund with a 7-year duration will decline approximately 7% for a 1% rise in interest rates. This is critical for understanding bond fund risk: in 2022, when the Federal Reserve raised rates aggressively, the Bloomberg US Aggregate Bond Index (duration ~6.5 years) fell approximately 13% -- the worst year in its history. Long-term bond ETFs with 15+ year durations fell over 30%. Short-term bond ETFs with 1-3 year durations fell only 3-5%. Duration explains these differences. When choosing a bond fund, the duration tells you how sensitive the fund is to rate changes. In a rising rate environment, choose short-duration funds to minimize losses. In a falling rate environment, choose long-duration funds to maximize price appreciation. The duration of a bond fund is not static -- it changes as the fund's portfolio composition changes and as yields change. Bond ETFs vs individual bonds: duration and interest rate risk →

What is negative convexity and why does it matter?

Negative convexity occurs when the price-yield curve bends downward rather than upward. This means that as yields fall, the bond's price rises less than duration predicts (or in extreme cases, may even fall). As yields rise, the price falls more than duration predicts. Negative convexity is most commonly associated with callable bonds and mortgage-backed securities (MBS). For callable bonds, when yields fall, the likelihood that the issuer will call the bond increases, capping the price appreciation (the price approach the call price rather than rising further). For MBS, when yields fall, homeowners refinance their mortgages, causing prepayments that shorten the security's life. For MBS investors, this means they receive principal back early when they would rather keep the high-yielding security. Negative convexity is a disadvantage for the investor. Securities with negative convexity must offer higher yields to compensate. Investors can hedge negative convexity using options, interest rate swaps, or by holding positively convex securities to offset the exposure. Understanding prepayment risk and negative convexity in MBS →

How do I use duration to build a bond ladder?

A bond ladder is a portfolio of bonds with staggered maturities. Duration helps you design the ladder: shorter rungs have low duration (low sensitivity, stable returns); longer rungs have high duration (higher sensitivity, more potential return). A typical ladder has rungs at 1, 2, 3, 4, 5, 7, and 10 years. The portfolio duration is the weighted average of each rung. By matching the ladder's duration to your investment horizon, you can immunize the portfolio against interest rate changes. If you need the money in 5 years, build a ladder with a portfolio duration of approximately 5 years. As each bond matures, reinvest the proceeds in the longest rung to maintain the ladder. Duration-based immunization is the foundation of liability-driven investing. For individual investors, a 10-year bond ladder with annual rungs provides a natural duration-matching strategy for long-term goals like retirement. How to build and manage bond ladders →

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