Bond Duration and Convexity: How Bond Prices Respond to Interest Rate Changes

Duration is a measure of how sensitive a bond's price is to changes in interest rates. Specifically, MODIFIED DURATION tells you the approximate PERCENTAGE change in the bond's price for a 1 percentage point (100 basis point) change in interest rates. A bond with modified duration of 7 will lose approximately 7% of its value if interest rates rise 1%, and gain approximately 7% if rates fall 1%. This is the most important single concept in fixed income investing: bond prices and interest rates move in OPPOSITE directions. When rates rise, bond prices fall. When rates fall, bond prices rise. Duration quantifies HOW MUCH they fall or rise. The formula for price change using modified duration: ΔP/P ≈ -D × Δy Where ΔP/P is the percentage change in price, D is modified duration, and Δy is the change in yield (interest rate) expressed as a decimal. **Worked example**: a 10-year Treasury bond with a 4% coupon has a modified duration of 8.2. If interest rates rise from 4% to 5% (a 100 basis point increase), the bond price will fall by approximately: ΔP/P ≈ -8.2 × 0.01 = -0.082 = -8.2% A bond selling at par ($1,000) would fall to approximately $918. If rates fell from 4% to 3%, the bond would rise by approximately 8.2% to $1,082. **The word 'approximately' matters** because duration is a LINEAR approximation of a nonlinear relationship. For small rate changes (less than 50 basis points), duration gives a very accurate estimate. For large rate changes (100+ basis points), duration alone understates the bond's price increase when rates fall and overstates the price decline when rates rise. The correction for this nonlinearity is called CONVEXITY, which we cover in a later section. **Intuition for duration**: duration is essentially the weighted average time until you receive the bond's cash flows. Longer-term bonds have higher duration because more of the cash flows are in the distant future and are more affected by discount rate changes. Zero-coupon bonds have the highest duration for their maturity because all the cash flow comes at the end. High-coupon bonds have lower duration than low-coupon bonds of the same maturity because high-coupon bonds return more cash earlier. Key relationships: - Longer maturity = higher duration - Lower coupon = higher duration - Lower yield = higher duration - Zero-coupon bond duration = its maturity (exactly) Snap a photo of any bond duration problem and FinanceIQ calculates Macaulay duration, modified duration, the predicted price change from a given rate movement, and the convexity-adjusted estimate for larger changes. This content is for educational purposes only and does not constitute financial advice.

Two types of duration appear in finance courses and CFA exams: Macaulay Duration and Modified Duration. They are closely related but represent different things. **Macaulay Duration** (developed by Frederick Macaulay in 1938): the weighted average time (in years) until the bond's cash flows are received. It is measured in years and represents the 'effective maturity' of the bond considering both the coupon payments and the final principal repayment. The formula: Macaulay Duration = Σ (t × PV(CFt) / Bond Price) Where t is the time of each cash flow, PV(CFt) is the present value of the cash flow at time t, and the sum is taken over all cash flows. **Worked example**: a 3-year bond with 5% annual coupon at a yield of 5%. Face value $1,000. Cash flows: $50, $50, $1,050 (coupon + principal in year 3). Present values at 5%: Year 1: $50 / 1.05 = $47.62 Year 2: $50 / 1.05² = $45.35 Year 3: $1,050 / 1.05³ = $907.03 Bond price = $47.62 + $45.35 + $907.03 = $1,000.00 (selling at par) Macaulay Duration = (1 × $47.62 + 2 × $45.35 + 3 × $907.03) / $1,000 = ($47.62 + $90.70 + $2,721.09) / $1,000 = $2,859.41 / $1,000 = 2.859 years Interpretation: the weighted average time until cash flows are received is 2.859 years. The bond behaves as if it were a 'zero-coupon bond' with maturity of 2.859 years. **Modified Duration**: an adjustment of Macaulay Duration that gives the percentage price change for a change in yield. Formula: Modified Duration = Macaulay Duration / (1 + y/n) Where y is the yield and n is the number of compounding periods per year (2 for semi-annual coupons, 1 for annual). For our example (annual compounding, y = 5%): Modified Duration = 2.859 / (1 + 0.05/1) = 2.859 / 1.05 = 2.723 This means the bond's price will change by approximately 2.723% for each 1% change in yield. **Why two versions exist**: Macaulay Duration has a clean economic meaning (weighted average time to cash flows) but is not directly useful for estimating price changes. Modified Duration is the practical tool for estimating price sensitivity but has a less intuitive interpretation. On exams and in practice, when someone says 'duration' without qualification, they usually mean MODIFIED DURATION because that is the number used for price sensitivity calculations. **For small yield changes**, Macaulay and Modified Duration are very close. The difference only becomes significant for high-coupon, high-yield bonds. For most practical purposes, knowing modified duration is sufficient. FinanceIQ calculates both Macaulay and Modified Duration from bond specifications and explains which one to use for different types of problems.

Duration is a linear approximation of the bond price-yield relationship. The actual relationship is curved (CONVEX) — the curve bends upward, meaning prices rise MORE than duration predicts when yields fall and fall LESS than duration predicts when yields rise. This curvature is measured by CONVEXITY. **Why convexity matters**: for small yield changes, duration alone is accurate enough. For large changes (100+ basis points), duration UNDERSTATES the bond's response to falling yields and OVERSTATES its response to rising yields. The bigger the rate change, the more convexity matters. **The convexity-adjusted price change formula**: ΔP/P ≈ -D × Δy + (1/2) × C × (Δy)² Where D is modified duration, C is convexity, and Δy is the change in yield. The first term is the duration effect. The second term is the convexity adjustment. For large rate changes, the second term matters. **Worked example**: a bond has modified duration of 8 and convexity of 120. Yields fall by 100 basis points (1%). Using duration alone: ΔP/P ≈ -8 × (-0.01) = +0.08 = +8.0% Adding the convexity adjustment: ΔP/P ≈ -8 × (-0.01) + (1/2) × 120 × (-0.01)² = 0.08 + 0.006 = +8.6% Without convexity: predicts +8.0%. With convexity: predicts +8.6%. The actual price change will be closer to 8.6% — the duration-only estimate understates the gain by 0.6%. Now the same bond if yields RISE by 100 basis points: Using duration alone: ΔP/P ≈ -8 × 0.01 = -0.08 = -8.0% Adding convexity: ΔP/P ≈ -8 × 0.01 + (1/2) × 120 × (0.01)² = -0.08 + 0.006 = -7.4% Without convexity: predicts -8.0%. With convexity: predicts -7.4%. The convexity adjustment reduces the loss by 0.6%. **The asymmetric advantage**: notice that convexity HELPS the bondholder in both directions. Gains are larger than duration predicts when rates fall. Losses are smaller than duration predicts when rates rise. Bonds with higher convexity are more valuable than bonds with the same duration but lower convexity — they offer better price performance in both directions. **Calculating convexity**: the formula is mathematically complex. For a bond with cash flows CFt at times t: Convexity = (1/P) × Σ [CFt × t × (t+1) / (1+y)^(t+2)] This is rarely calculated by hand even on exams. Usually you are given the convexity and asked to apply it, or you use financial calculator or spreadsheet software. **Key relationships for convexity**: - Longer maturity = higher convexity - Lower coupon = higher convexity - Lower yield = higher convexity (same factors that increase duration) - Callable bonds can have NEGATIVE convexity at low yields (the call option caps their upside) **When convexity matters most**: for bond portfolio managers expecting LARGE yield changes (during Federal Reserve rate cycles, financial crises, or near major economic turning points), convexity becomes critical. For individual retail investors holding bonds to maturity, convexity is less important because they receive all their cash flows anyway. FinanceIQ handles the convexity adjustment automatically when calculating bond price changes for any given yield movement — and flags when the rate change is large enough that ignoring convexity would produce meaningful errors.

Duration is not just an academic concept — it is the main tool that bond portfolio managers use to manage interest rate risk. Understanding how duration is used practically is essential for CFA candidates, MBA students, and anyone pursuing a career in fixed income. **Portfolio duration**: the duration of a bond portfolio is the weighted average of the individual bond durations, weighted by the market value of each bond: Portfolio Duration = Σ (wi × Di) Where wi is the weight of bond i in the portfolio (market value of bond i / total portfolio value) and Di is the modified duration of bond i. **Worked example**: a portfolio has three bonds: - Bond A: $1M market value, modified duration 3 - Bond B: $2M market value, modified duration 7 - Bond C: $1M market value, modified duration 12 - Total portfolio: $4M Portfolio Duration = (1/4) × 3 + (2/4) × 7 + (1/4) × 12 = 0.75 + 3.5 + 3.0 = 7.25 years If interest rates rise 1%, the portfolio will lose approximately 7.25% of its value. If rates rise 50 basis points, the portfolio will lose 3.625%. **Duration as a hedging tool**: bond portfolio managers hedge interest rate risk by matching the duration of their assets and liabilities. A pension fund with liabilities having 10-year duration wants its asset portfolio to also have 10-year duration — this way, if rates change, both assets and liabilities change by the same percentage, leaving the funded status unchanged. This is called 'duration matching' or 'immunization.' **Active portfolio management**: managers who expect rates to fall may extend duration (buy longer-term bonds) to benefit from larger price gains. Managers who expect rates to rise shorten duration (buy shorter-term bonds) to reduce price losses. This is called 'rate anticipation' and is one of the main active strategies in fixed income. **Barbell vs bullet strategies**: - **Bullet portfolio**: concentrate bonds around a target maturity (e.g., 7-year bonds for a 7-year target duration). Simple, intuitive, exposure concentrated at one point on the yield curve. - **Barbell portfolio**: split bonds between short-term (e.g., 2-year) and long-term (e.g., 15-year) to achieve the same average duration (7). Has higher CONVEXITY than the bullet portfolio for the same duration, so it performs better in volatile rate environments but requires rebalancing as the short bonds mature. **Duration drift**: as time passes, a bond's duration changes. Time to maturity decreases (lowers duration), coupon payments are received (lowers duration slightly), and yields change (affects duration). Portfolio managers must rebalance periodically to maintain their target duration. A portfolio built to have 7-year duration today may have 6.5-year duration in 6 months without any trades. **Key-rate duration**: for complex portfolios, the single 'parallel shift' assumption of modified duration can be too simple. Real yield curves do not always move in parallel — short rates may rise while long rates fall (flattening) or vice versa (steepening). Key-rate duration measures sensitivity to changes at specific points on the curve (2-year, 5-year, 10-year, 30-year) rather than assuming parallel shifts. This is advanced fixed income material covered in CFA Level 2. **Practical application for finance students**: when you see a bond duration problem on an exam, the typical workflow is: 1. Calculate or look up the duration (usually given) 2. Identify the yield change 3. Apply the formula ΔP/P ≈ -D × Δy 4. If the problem mentions convexity or a large rate change, apply the convexity adjustment 5. Calculate the new bond price (old price × (1 + ΔP/P)) FinanceIQ handles portfolio duration calculations, duration matching for immunization problems, and the bullet-vs-barbell comparisons that show up in fixed income courses and CFA study material.