Bond Duration Convexity Calculator
Free Bond duration convexity Calculator for bonds. Enter your numbers to see returns, costs, and optimized scenarios instantly.
Bond Duration Convexity Calculator
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Formula: Mac. Duration = Σ(t × PV(CFt)) / Price | Mod. Duration = Mac. Duration / (1 + y/n)
Worked example — Price = $1,077.95 | Mac. Dur. = 7.66 yrs | Mod. Dur. = 7.47 | Convexity = 66.73
Formula
Mac. Duration = Σ(t × PV(CFt)) / Price | Mod. Duration = Mac. Duration / (1 + y/n)
Macaulay duration is the weighted average time of cash flows, where weights are present values. Modified duration adjusts for yield compounding and directly measures price sensitivity to yield changes. Convexity adds a second-order correction for large rate movements.
Worked Examples
Example 1: 10-Year Corporate Bond
Problem:A corporate bond has a face value of $1,000, 6% annual coupon rate (semi-annual), 10 years to maturity, and YTM of 5%. Calculate duration and convexity.
Solution:Face Value = $1,000 | Coupon = 6% (semi-annual = $30 every 6 months) YTM = 5% (2.5% per period) | Periods = 20 Bond Price ≈ $1,077.95 Macaulay Duration ≈ 7.66 years Modified Duration ≈ 7.47 Convexity ≈ 66.73
Result:Price = $1,077.95 | Mac. Dur. = 7.66 yrs | Mod. Dur. = 7.47 | Convexity = 66.73
Example 2: Rate Change Impact Analysis
Problem:Using the bond above, estimate the price change if yields increase by 1%.
Solution:Modified Duration = 7.47 | Convexity = 66.73 | Price = $1,077.95 Duration effect: -7.47 × 0.01 × $1,077.95 = -$80.52 Convexity effect: 0.5 × 66.73 × 0.01² × $1,077.95 = +$3.60 Total change ≈ -$76.92 (-7.14%) New price ≈ $1,001.03
Result:Price drops ~$76.92 (-7.14%) | Duration alone overestimates decline by $3.60
Frequently Asked Questions
What is bond duration and why does it matter?
Bond duration is a measure of the sensitivity of a bond's price to changes in interest rates. Macaulay duration, developed by Frederick Macaulay in 1938, represents the weighted average time until a bond's cash flows are received, measured in years. Modified duration adjusts the Macaulay duration for the yield per period and directly estimates the percentage price change for a one percent change in yield. For example, a bond with a modified duration of 5 would decrease approximately 5% in price for every 1% increase in yield. Duration is critical for portfolio managers who need to manage interest rate risk, match asset and liability durations (immunization), and compare bonds with different maturities and coupon rates.
What is convexity and how does it relate to duration?
Convexity is the second-order measure of a bond's price sensitivity to interest rate changes. While duration provides a linear approximation of price change, convexity captures the curvature in the price-yield relationship. Duration alone underestimates the price increase when yields fall and overestimates the price decrease when yields rise. Convexity corrects this by adding a quadratic term to the price change estimate. Higher convexity means the bond price will rise more when yields fall and decline less when yields rise, making it a desirable property. The total estimated price change combines both: ΔP ≈ -Modified Duration × Δy × P + 0.5 × Convexity × (Δy)² × P. Convexity is particularly important for large yield changes where the linear duration approximation becomes inaccurate.
How do coupon rate and maturity affect duration?
Duration is influenced by three main factors: coupon rate, maturity, and yield to maturity. A higher coupon rate reduces duration because more cash flow is received earlier, reducing the weighted average time. A longer maturity generally increases duration because cash flows extend further into the future. However, for very long-term premium bonds, duration can plateau. A higher yield to maturity reduces duration because the present value of distant cash flows decreases more steeply. A zero-coupon bond has the simplest case: its Macaulay duration equals its maturity since there is only one cash flow at maturity. For all coupon-paying bonds, the Macaulay duration is always less than the maturity. Understanding these relationships is essential for constructing bond portfolios with target duration profiles.
What is DV01 and how is it used in trading?
DV01 (Dollar Value of a Basis Point), also called PV01, measures the dollar change in a bond's price for a one basis point (0.01%) change in yield. It is calculated as DV01 = Modified Duration × Price × 0.0001. For example, a bond with a modified duration of 7 and a price of $1,000 has a DV01 of $0.70, meaning each basis point change in yield changes the price by approximately $0.70 per bond. DV01 is widely used by bond traders for hedging interest rate risk. To hedge a portfolio, traders calculate the total DV01 exposure and take offsetting positions. It allows comparison of interest rate sensitivity across bonds with different face values and prices. Market makers and risk managers rely on DV01 to quantify their exposure to rate movements in real dollar terms.
What is the difference between Macaulay and modified duration?
Macaulay duration and modified duration are related but serve different purposes. Macaulay duration is measured in years and represents the weighted average time to receive all cash flows from a bond, where weights are the present values of each cash flow as a fraction of the total bond price. It answers the question of when, on average, you receive your money back. Modified duration is derived from Macaulay duration by dividing by (1 + y/n), where y is yield and n is compounding frequency. It is a pure sensitivity measure without time units, indicating the approximate percentage change in price for a 1% yield change. For zero-coupon bonds, Macaulay duration equals maturity, while modified duration is slightly less. Modified duration is more commonly used in practice for risk management because it directly translates yield changes into price changes.
What is effective duration and when should it replace modified duration?
Modified duration assumes the cash flows are fixed, which fails for bonds with embedded options such as callable and putable issues and mortgage-backed securities, where the cash flows themselves change as rates move. Effective duration is measured empirically by repricing the security under a small parallel shift up and down and dividing the price difference by twice the shift times the price. Whenever optionality is present, effective duration is the number a risk manager should use.
How is duration used to immunise a future liability?
Immunisation sets the duration of a bond portfolio equal to the horizon of the liability it must fund. At that point a rise in rates cuts the portfolio price but raises the return on reinvested coupons by an offsetting amount, and a fall in rates does the reverse, so the terminal value is largely insulated from parallel rate moves. Because duration drifts as time passes and yields change, an immunised portfolio has to be rebalanced periodically, and it does not protect against non-parallel curve shifts.
How do I combine duration and convexity to estimate a price change?
Use the two-term expansion: the percentage price change is approximately minus modified duration times the yield change, plus one half times convexity times the yield change squared. For a bond with modified duration 6 and convexity 80, a 1% rise in yield gives minus 6 times 0.01, or minus 6%, plus 0.5 times 80 times 0.0001, or plus 0.4%, for a net estimate near minus 5.6%. The convexity term is positive for option-free bonds and grows with the size of the yield move.
Why can a callable bond show negative convexity?
When yields fall far enough that the issuer is likely to call the bond, its price stops rising toward the level an option-free bond would reach and compresses toward the call price. The price-yield curve then bends the wrong way, so further rate declines produce shrinking gains while rate rises still produce full losses. Mortgage-backed securities behave similarly because homeowners prepay when rates drop. Investors expect extra yield to accept this asymmetric profile.
What does key rate duration add to a single duration number?
Modified and effective duration assume the whole yield curve shifts in parallel, which real markets rarely do. Key rate duration measures price sensitivity to a shift at one specific maturity point on the curve while holding the others fixed, giving a profile across the two-year, five-year, ten-year and thirty-year nodes. The key rate durations sum approximately to the overall effective duration, and they reveal exposure to steepening or flattening that a single number hides.
References
Reviewed for accuracy by Sahil, Senior Finance & Tax Editor · Editorial policy
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