# Bond Duration & Convexity Calculator

> Calculate Macaulay duration, modified duration, and convexity for bonds to understand interest rate sensitivity and price risk.

**Category:** Finance
**Keywords:** bond, duration, convexity, macaulay, modified duration, interest rate risk, fixed income, price sensitivity, finance
**URL:** https://complete.tools/bond-duration-calculator

## How it calculates

The calculator prices the bond by discounting each future cash flow (coupon payments and final principal repayment) at the yield to maturity. For a bond with coupon payment C per period, face value F, yield per period y, and n total periods:

Bond Price = sum of C / (1 + y)^t for t = 1 to n-1, plus (C + F) / (1 + y)^n.

Macaulay Duration is the weighted average time to receive cash flows, where each weight is the present value of the cash flow divided by the bond price: MacD = sum(t x PV(CF_t)) / Price, measured in periods and then divided by the payment frequency to convert to years.

Modified Duration adjusts Macaulay duration for the compounding frequency: ModD = MacD / (1 + y/m), where m is the number of coupon payments per year. Modified duration approximates the percentage price change for a 1% parallel shift in yield.

Convexity captures the curvature of the price-yield relationship that duration alone misses: Convexity = sum(t(t+1) x CF_t / (1+y)^(t+2)) / (Price x (1+y)^2), annualized by dividing by the frequency squared. Convexity is always positive for option-free bonds, meaning prices rise more when yields fall than they drop when yields rise by the same amount.

The estimated price change combines both measures: percentage change is approximately equal to negative modified duration times the yield change plus one-half times convexity times the yield change squared.

## Who should use this

Fixed-income portfolio managers who need to measure and manage interest rate exposure across bond portfolios. Investment analysts comparing the risk profiles of different bonds with varying maturities and coupon structures. Institutional investors performing asset-liability matching or portfolio immunization, where matching the duration of assets to liabilities is critical. Finance students and CFA candidates studying fixed-income valuation and risk measures. Individual investors who hold bonds or bond funds and want to understand how rate changes would affect their holdings. Risk managers calculating Value at Risk or stress testing bond portfolios under different rate scenarios.

## Worked examples

Example 1: A 10-year Treasury bond with a $1,000 face value, 5% coupon rate, semiannual payments, and a 4% yield to maturity. The bond price is approximately $1,081.11 (trading at a premium since coupon exceeds yield). Macaulay duration is approximately 8.08 years, modified duration is approximately 7.92, and convexity is approximately 73.68. If yields rise by 1% to 5%, the estimated price drop is about 7.88%, bringing the price to roughly $995.87. If yields fall by 1% to 3%, the estimated price increase is about 8.29%, or about $1,170.72.

Example 2: A 3-year corporate bond with a $1,000 face value, 8% coupon rate, quarterly payments, and a 6% yield to maturity. The bond price comes to approximately $1,054.17. Macaulay duration is about 2.72 years, and modified duration is about 2.68. With only 3 years to maturity and high coupons, this bond has much lower duration (less interest rate risk) than the 10-year bond above. A 1% yield increase would decrease the price by only about 2.67%.

Example 3: A 30-year zero-coupon-like bond (0.01% coupon to avoid divide-by-zero) with a $1,000 face value and 5% yield. This bond has a Macaulay duration very close to 30 years and extremely high convexity. A 1% yield increase would cause a price decline of roughly 25%, illustrating why long-duration bonds are highly sensitive to rate changes.

## Limitations

This calculator assumes a standard fixed-rate bullet bond with no embedded options. Callable, putable, or convertible bonds have different duration characteristics because their cash flows change depending on rate levels. The calculator uses clean price and does not account for accrued interest, so the actual settlement price (dirty price) will differ between coupon dates. It assumes a flat yield curve and parallel shifts, which rarely occurs in practice; real yield curve movements involve shifts, twists, and butterfly changes that affect different maturities differently. The convexity approximation is second-order and becomes less accurate for very large yield changes (beyond 200-300 basis points). Tax effects, credit risk, and liquidity premiums are not considered. For floating-rate bonds or inflation-linked securities, standard duration measures do not apply in the same way.

## FAQs

**Q:** What is the difference between Macaulay duration and modified duration?


**A:** Macaulay duration is the weighted average time (in years) until you receive the bond's cash flows, where weights are the present values of each cash flow. Modified duration adjusts this figure to directly estimate price sensitivity: it tells you the approximate percentage price change for a 1% change in yield. For example, a modified duration of 7 means the bond price will drop approximately 7% if yields rise by 1%.


**Q:** Why does convexity matter if I already have duration?


**A:** Duration provides a linear (first-order) approximation of price changes, which works well for small yield movements. For larger yield shifts, the actual price-yield relationship is curved, not straight. Convexity captures this curvature. Without the convexity adjustment, duration alone overestimates price declines when yields rise and underestimates price gains when yields fall. The convexity correction becomes increasingly important for longer-maturity bonds and for yield changes exceeding 50 basis points.


**Q:** How does coupon rate affect duration?


**A:** Higher coupon rates reduce duration because a larger proportion of the bond's total value is received earlier through coupon payments, pulling the weighted average time forward. A zero-coupon bond has the maximum duration for its maturity because 100% of its cash flow arrives at maturity. This is why zero-coupon bonds are the most sensitive to interest rate changes among bonds of the same maturity.


**Q:** Is higher or lower duration better?


**A:** Neither is inherently better; it depends on your view and objectives. If you expect interest rates to fall, higher duration bonds will gain more in price. If you expect rates to rise, lower duration bonds will lose less. For liability matching (immunization), you want the duration of your bond portfolio to match the duration of your liabilities, regardless of whether that duration is high or low.


**Q:** What does the dollar duration (DV01) represent?


**A:** Dollar duration, closely related to DV01 (dollar value of a basis point), represents the approximate dollar change in bond price for a 1% change in yield. It is calculated as modified duration multiplied by the bond price, divided by 100. This measure is particularly useful for hedging because it converts the abstract percentage measure into actual dollar exposure.

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