Why Duration and Convexity Matter
Fixed income is one of the heaviest-weighted topics on the CFA exam, and within fixed income, duration and convexity are the two metrics that appear most frequently. They measure how sensitive a bond's price is to changes in interest rates, and mastering them is non-negotiable for any serious candidate.
Duration tells you the first-order effect: roughly how much a bond's price will move for a small parallel shift in yields. Convexity captures the second-order effect: how the price-yield relationship curves, which becomes important for larger rate moves. Together they give you a powerful toolkit for estimating bond price changes, managing portfolio risk, and comparing instruments.
Macaulay Duration
Macaulay duration is the weighted-average time until a bond's cash flows are received, where the weights are the present values of each cash flow divided by the bond's total price. A five-year coupon bond might have a Macaulay duration of 4.3 years because the interim coupons pull the weighted average forward from the maturity date.
Zero-coupon bonds are the simplest case: their Macaulay duration equals their maturity, since all cash flow arrives at the end. For coupon-paying bonds, higher coupons mean shorter duration because a larger share of value is received earlier.
Key relationships to remember: duration rises with maturity (but at a decreasing rate for coupon bonds), falls with coupon rate, and falls with yield to maturity. These relationships hold for standard option-free bonds and are tested frequently.
Modified Duration
Modified duration converts Macaulay duration into a direct measure of price sensitivity. The formula divides Macaulay duration by one plus the yield per period. If a bond has a Macaulay duration of 7.0 years and a yield of 6% paid semiannually, its modified duration is 7.0 divided by 1.03, which equals 6.80.
The interpretation is straightforward: for a 1% (100 basis point) increase in yield, the bond's price will decline by approximately 6.80%. This is a linear approximation, which is why convexity is needed for accuracy when rate changes are large.
Dollar Duration and DV01
Dollar duration (also called PVBP or DV01) translates the percentage sensitivity into dollar terms. It equals modified duration multiplied by the bond's market value, divided by 10,000. If a portfolio has a market value of $50 million and a modified duration of 5.5, the dollar duration is $27,500 per basis point.
This metric is essential for hedging. To immunize a portfolio against a parallel yield shift, you need offsetting positions whose dollar durations cancel out. Portfolio managers use DV01 to size hedges with futures, swaps, or other fixed-income instruments.
The Convexity Adjustment
Duration provides a linear estimate of price change, but the actual price-yield relationship for an option-free bond is a convex curve. For small yield changes (under 25 basis points), the linear approximation is adequate. For larger moves, you need convexity.
The full price change formula combines both effects: the percentage price change equals negative modified duration times the yield change, plus one-half times convexity times the yield change squared. The convexity term is always positive for option-free bonds, meaning duration alone understates the price increase when yields fall and overstates the price decrease when yields rise.
Consider a bond with modified duration of 8.0 and convexity of 90. For a 200 basis point rate decline, duration alone predicts a 16% price increase. Adding the convexity adjustment (0.5 times 90 times 0.02 squared equals 1.8%) gives a total estimated increase of 17.8%. That 1.8% difference matters when managing large portfolios.
Callable and Putable Bonds: Negative and Enhanced Convexity
Callable bonds introduce negative convexity at low yields. When rates fall below the coupon rate, the issuer is likely to call the bond, capping the price near the call price. The price-yield curve flattens and eventually bends downward, creating a region where duration actually increases as yields fall — the opposite of what happens with option-free bonds.
Effective duration (rather than modified duration) must be used for bonds with embedded options, because the cash flows change when rates change. Effective duration is calculated numerically: shock the yield curve up and down by a small amount, reprice the bond at each scenario using an option model, and compute the slope of the resulting price-yield curve.
Putable bonds exhibit enhanced positive convexity. When yields rise sharply, the put option gains value and establishes a price floor, reducing downside price sensitivity. Investors effectively benefit from more favorable curvature in both directions.
Portfolio Duration Management
At the portfolio level, duration is the primary tool for interest rate risk management. The portfolio's effective duration is the market-value-weighted average of the individual bond durations.
A pension fund with liabilities that have a duration of 12 years might target a portfolio duration of 12 to immunize against parallel rate shifts. If the current portfolio duration is only 8, the manager can extend duration by buying longer-maturity bonds, entering receive-fixed interest rate swaps, or buying Treasury futures.
Key-rate duration refines this analysis by measuring sensitivity to shifts at specific maturities (2-year, 5-year, 10-year, 30-year) rather than assuming parallel shifts. This is critical for managing exposure to yield curve reshaping — steepening, flattening, or butterfly shifts that affect different maturities differently.
Common Exam Pitfalls
Several duration-related mistakes appear repeatedly in candidate forums. First, confusing Macaulay and modified duration: remember that Macaulay is measured in years while modified is a sensitivity measure. Second, forgetting to annualize when working with semiannual yields: divide by (1 + y/2), not (1 + y). Third, applying modified duration to bonds with embedded options: always use effective duration for callable or putable bonds.
Finally, remember that duration assumes a parallel shift in the yield curve. In reality, curves twist and reshape. For the exam, use key-rate durations when the question specifies non-parallel shifts.
Putting It Into Practice
Duration and convexity are tested across CFA Levels I, II, and III, with increasing sophistication at each level. Level I focuses on definitions and basic calculations. Level II adds effective duration for complex securities and the convexity adjustment formula. Level III applies these tools to portfolio construction and liability-driven investing.
For FRM candidates, duration and convexity appear in the market risk readings, particularly in the context of interest rate risk management and hedging with derivatives.
Ready to test your understanding? Try our CFA fixed income practice questions, or explore the community Q&A for discussions on tricky duration scenarios.