Bond Duration & Convexity Calculator

There are three main types of duration, and each answers a slightly different question. Understanding when to use each one is essential for fixed-income analysis.

Macaulay Duration is the weighted average time (in years) until you receive the bond's cash flows, where each cash flow is weighted by its present value. It was developed by Frederick Macaulay in 1938.

• Formula: Mac Duration = (1/P) × Σ[t × PV(CF_t)]
• Units: Years
• Use case: Portfolio immunization and liability matching. If you need your assets to mature at the same "weighted average time" as your liabilities, Macaulay duration is your tool.

Modified Duration takes Macaulay duration and adjusts it for the yield level to give a direct measure of price sensitivity.

• Formula: Mod Duration = Macaulay Duration / (1 + y/n), where y is the periodic yield
• Units: Percentage price change per 1% yield change
• Use case: Day-to-day risk management. Modified duration tells you directly how much a bond's price will change for a given yield movement, making it the most practical version for trading and risk management.

Effective Duration is a numerical (brute-force) approximation that works by repricing the bond at slightly higher and lower yields.

• Formula: Eff Duration = (P_down − P_up) / (2 × P × Δy)
• Use case: Bonds with embedded options (callable, putable, MBS). For these bonds, cash flows change when rates change, so Macaulay and modified duration break down. Effective duration captures this optionality because it actually reprices the bond.

For plain-vanilla bonds (no embedded options), all three duration measures will be very close to each other. Modified duration is the most commonly used in practice for option-free bonds.

Convexity measures the curvature in the relationship between a bond's price and its yield. While duration gives you a straight-line (linear) estimate of how price changes with yield, convexity captures the fact that this relationship is actually curved.

Why duration alone isn't enough: Duration assumes that price changes are proportional to yield changes — a 1% yield increase causes the same dollar price drop as a 2% increase would cause twice the drop. But in reality, the price-yield curve is not a straight line. It's convex (bowed toward the origin), which means:

• When yields fall, bond prices rise more than duration alone predicts
• When yields rise, bond prices fall less than duration alone predicts

This asymmetry is good for bondholders. Convexity means you gain more on the upside than you lose on the downside, all else equal.

The convexity adjustment formula:

ΔP/P ≈ −ModDuration × Δy + 0.5 × Convexity × (Δy)²

The first term is the duration effect (linear, can be positive or negative). The second term is always positive, which is the convexity bonus. For small yield changes (under 25bp), convexity barely matters. For large moves (100bp+), ignoring it produces meaningful pricing errors.

Practical implications:

• Positive convexity is desirable — Given two bonds with the same duration and yield, the one with higher convexity is more valuable because it benefits more from rate volatility.
• Callable bonds have negative convexity — When rates fall far enough, the issuer calls the bond, capping your upside. The price-yield curve bends backward, which is bad for bondholders.
• Zero-coupon bonds have the highest convexity for a given maturity and duration because all the cash flow sits at one single point in the future.

Duration provides a quick approximation of how much a bond's price will change when yields shift. The formula is straightforward:

Duration-only estimate:

ΔP/P ≈ −Modified Duration × Δy

Where Δy is the change in yield expressed as a decimal (e.g., 50 basis points = 0.005). The negative sign reflects the inverse relationship between prices and yields.

Example: A bond with a modified duration of 6.5 and a price of $1,020. If yields rise by 75 basis points (0.0075):

• Duration-only: ΔP/P ≈ −6.5 × 0.0075 = −4.875%
• Dollar change: −4.875% × $1,020 = −$49.73
• Estimated new price: $1,020 − $49.73 = $970.27

Adding the convexity adjustment:

ΔP/P ≈ −ModDuration × Δy + 0.5 × Convexity × (Δy)²

If this bond has a convexity of 55, the convexity adjustment is: 0.5 × 55 × (0.0075)² = 0.00155 = +0.155%

So the adjusted estimate is −4.875% + 0.155% = −4.72%, giving a new price of $971.85 instead of $970.27. The convexity correction reduced the estimated loss by about $1.58.

When does the correction matter most? The convexity term grows with the square of the yield change. For small moves (10–25bp), it's negligible. For large moves (100bp+), it can represent several percentage points of the price estimate. Always include convexity for stress testing and scenario analysis.

Immunization is a strategy that protects a fixed-income portfolio from interest rate risk by matching the duration of assets to the duration of liabilities. The goal is to ensure that no matter which way rates move, the portfolio will still have enough value to meet its obligations.

How it works:

• Step 1: Determine the duration of your liability (e.g., a pension payment due in 8 years has a duration of approximately 8)
• Step 2: Build a bond portfolio whose dollar-weighted duration equals the liability duration
• Step 3: Ensure the present value of the portfolio at least equals the present value of the liability
• Step 4: Rebalance periodically as time passes and yields change (duration drifts)

Why it works: When rates rise, the portfolio loses value (bad), but the liability also loses value (good) because its present value is also discounted at a higher rate. If durations match, these two effects offset each other. The same logic applies in reverse when rates fall.

Limitations of basic duration matching:

• It assumes parallel yield curve shifts — all maturities move by the same amount. In reality, short and long rates often move differently.
• Convexity mismatch can cause problems. Even with matched durations, if the asset portfolio has lower convexity than the liability, large rate moves can create a shortfall. For better protection, the asset convexity should exceed the liability convexity.
• Rebalancing is required because duration changes over time as coupons are received and maturities shorten. The portfolio needs periodic adjustment to maintain the match.

Pension funds, insurance companies, and endowments use duration-based immunization as a core risk management tool. Understanding how to calculate and interpret duration is essential for anyone managing fixed-income assets against future liabilities.

Duration is affected by several bond characteristics, and understanding these relationships helps you predict which bonds will be most sensitive to rate changes.

Maturity and duration:

• Longer maturity = higher duration (almost always). Cash flows that are further in the future contribute more to duration because they are weighted by time.
• For zero-coupon bonds, Macaulay duration exactly equals maturity. For coupon bonds, duration is always less than maturity because some cash flows arrive before maturity.
• Exception: For deep-discount bonds with very long maturities, duration can actually decrease slightly past a certain point, though this is rare in practice.

Coupon rate and duration:

• Higher coupon = lower duration. Higher coupons mean more cash flow arrives sooner, pulling the weighted average time closer to the present.
• A zero-coupon bond has the maximum duration for any given maturity. A 10% coupon bond maturing in 20 years will have a significantly shorter duration than a 2% coupon bond with the same maturity.

Yield and duration:

• Higher yield = lower duration. Higher discount rates make distant cash flows worth less in present value terms, shifting the weight toward earlier cash flows.
• This means the same bond becomes slightly less rate-sensitive in a high-yield environment and more sensitive in a low-yield environment.

Payment frequency and duration: More frequent coupon payments (quarterly vs. annual) slightly reduce duration because you receive cash earlier within each year. The effect is modest compared to maturity and coupon rate, but it's measurable.

Rule of thumb: If you want to quickly rank bonds by rate sensitivity, look at maturity first, then coupon rate, then yield. Bonds that are long-maturity, low-coupon, and low-yield will have the highest duration and the most interest rate risk.

Duration and convexity are among the most heavily tested topics in the CFA Fixed Income curriculum. They appear at both Level 1 and Level 2, with increasing complexity.

CFA Level 1 — What you need to know:

• Macaulay duration — Be able to calculate it from individual cash flows and understand it as the weighted average time to receive cash flows.
• Modified duration — Know the formula (Macaulay / (1 + y)) and be able to use it to estimate price changes for a given yield shift.
• Duration properties — Understand how coupon rate, maturity, and yield level affect duration. Expect qualitative questions ("Which bond has higher duration?").
• Price-yield relationship — Know that the relationship is convex (not linear) and why duration underestimates price increases and overestimates price decreases.
• Money duration and PVBP — Dollar duration (modified duration × price × 0.01) and price value of a basis point may be tested.

CFA Level 2 — What gets harder:

• Effective duration and effective convexity — Required for bonds with embedded options (callable, putable, MBS). You must understand why analytical formulas break down and numerical methods are needed.
• Key rate duration — Measures sensitivity to changes at specific points on the yield curve, not just parallel shifts. Important for structured products.
• Duration contribution and portfolio duration — Calculate the duration of a multi-bond portfolio as the weighted average of individual durations.
• Convexity adjustment — Be comfortable adding the 0.5 × C × (Δy)² term to duration estimates. Exam questions will test whether you include it.

Exam tips: Always double-check whether the question gives you annualized or per-period values. Many errors come from not dividing by the payment frequency when converting Macaulay duration. Use this calculator to practice by verifying your hand calculations — if your number doesn't match, you likely have a frequency conversion error.

In practice, portfolio managers use convexity alongside duration to make better investment decisions and manage risk more precisely. Convexity is not just a theoretical refinement — it has real dollar implications.

Why positive convexity is valuable:

• A bond with positive convexity benefits from rate volatility. If rates move significantly in either direction, the bond will outperform what duration alone predicts. Bondholders gain more when rates fall than they lose when rates rise.
• Because this asymmetry is valuable, investors are willing to accept a slightly lower yield for bonds with higher convexity. This is sometimes called the convexity premium.

Barbell vs. bullet strategies:

• A bullet portfolio concentrates holdings around a single maturity (e.g., all 7-year bonds). It has lower convexity.
• A barbell portfolio splits holdings between short and long maturities (e.g., 2-year and 15-year bonds) to achieve the same average duration. It has higher convexity.
• The barbell will outperform the bullet during large parallel rate shifts, but may underperform if the yield curve flattens or steepens in unfavorable ways. The tradeoff between these structures is a classic portfolio management decision.

Negative convexity and callable bonds: Callable bonds and mortgage-backed securities exhibit negative convexity when rates fall below the coupon rate, because the issuer (or borrower) is likely to refinance. The bond's price gets "capped" near the call price, eliminating the upside that positive convexity provides. Managing negative convexity is a major challenge for MBS portfolio managers.

Bottom line: Duration gets you 90% of the way to understanding rate risk. Convexity handles the other 10% — and that 10% becomes increasingly important as rates become more volatile or as you manage larger portfolios where small pricing errors translate to significant dollar amounts.