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CFA Level I · CFA Level I Exam

Yield-Based Bond Convexity and Portfolio Properties for CFA Level 1

This chapter measures how bond prices react to yield changes. Duration gives the first-order estimate of the percentage price change, and convexity corrects it for curvature. You solve questions by computing duration, adding the convexity adjustment, then extending the measures to portfolios and bonds with embedded options.

What this chapter covers

This chapter is about interest rate risk. A bond's price moves opposite to its yield, but not in a straight line. Duration gives you the slope of that relationship. Convexity gives you the curvature. Together they let you estimate a price change without repricing the bond.

You start with Macaulay duration, modified duration and effective duration. Then you move to money duration and the price value of a basis point (PVBP), which turn percentage risk into currency risk. Next comes convexity and the convexity adjustment. Finally you apply the ideas to portfolios and to callable, putable and option-free bonds.

This chapter connects to the rest of the Fixed Income topic, including bond pricing, yield measures, the yield curve and credit analysis. It also links to Portfolio Construction and to Derivatives, where hedging a rate exposure uses the same sensitivity ideas. Questions are often short calculations, and you can finish them quickly if the formulas are automatic.

Fixed Income carries a topic weight of 11-14% in the 2027 curriculum, and this chapter holds some of its most testable calculations. Duration and convexity questions are compact, numerical and easy to practise, so they are reliable marks if you are accurate. Because every question is a standalone three-option item with no penalty for wrong answers, knowing the direction of an effect lets you eliminate two options even when the arithmetic runs out of time. The concepts also help in portfolio and derivatives questions, so the effort pays back across topics.

Yield-Based Bond Convexity and Portfolio Properties: topics in the order to study them

  1. 1Macaulay, Modified and Effective DurationDuration is the base measure. Everything else in the chapter builds on it, so learn what each version measures and when to use it.
  2. 2Money Duration and Price Value of a Basis PointThese convert modified duration into currency terms, which is a short step once you are comfortable with duration.
  3. 3Bond Convexity and Convexity AdjustmentConvexity refines the duration estimate. You need duration first, because the adjustment is added to the duration-based result.
  4. 4Portfolio Duration and ConvexityPortfolio measures are weighted averages of the single-bond measures, so learn the single-bond versions first.
  5. 5Bond Price Behavior: Callable, Putable and Option-FreeThis closes the chapter by showing how embedded options change duration and convexity, using all the earlier ideas.

How to prepare Yield-Based Bond Convexity and Portfolio Properties

Plan about a week of short sessions. Mix concept reading with calculator practice, because most marks here come from applying a few formulas correctly.

  1. Read the definitions of Macaulay, modified and effective duration. Write one line for each saying what it measures and when it applies, for example effective duration for bonds with embedded options.
  2. Memorise the core estimate: %ΔPrice ≈ −ModDur × ΔYield + ½ × Convexity × (ΔYield)². Practise it with yield changes in decimals, such as 0.0050 for 50 basis points.
  3. Practise money duration and PVBP. Money duration = ModDur × full price, which gives the currency price change per 1.00 (100%) change in yield. For a 1 percentage point change, use about money duration × 0.01. PVBP is the price change for a one basis point yield change, about money duration × 0.0001. Check that your answer is in currency units, not a percentage.
  4. Do portfolio problems by weighting each bond's measure by its market value share. Remember that a portfolio duration is a good guide for small parallel yield shifts only.
  5. Draw the price-yield curves for an option-free, a callable and a putable bond. Mark where convexity turns negative for the callable bond (at low yields, when the call is likely) and note that it is positive at high yields. The putable bond has positive convexity. At high yields the put moves into the money, so its price falls less than an option-free bond's. At low yields the put is far out of the money, so the bond behaves like an option-free bond.
  6. Finish with timed sets of standalone questions. Aim for about 90 seconds each. For numerical items, estimate the direction and size first, then use that to eliminate wrong options.

Common mistakes in Yield-Based Bond Convexity and Portfolio Properties

  • Using yield changes in percentage points inside the formula.

    Fix: Convert to decimals first, so 50 bps is 0.0050. Square the decimal in the convexity term.

  • Forgetting the sign on the duration term.

    Fix: Write the minus sign every time: price change ≈ −ModDur × ΔYield. A yield rise must give a price fall.

  • Using Macaulay duration where modified duration is needed.

    Fix: Macaulay is a weighted average time to cash flows. For price sensitivity use modified or effective duration.

  • Applying modified duration to bonds with embedded options.

    Fix: Use effective duration for callable and putable bonds, since it allows for changing cash flows.

  • Treating portfolio duration as exact for any yield move.

    Fix: Remember it assumes a parallel yield shift and is an approximation. Add convexity for larger moves.

  • Mixing up money duration and PVBP units.

    Fix: Money duration = ModDur × price is the currency price change per 1.00 (100%) change in yield, so a 1 percentage point change is about money duration × 0.01. PVBP is for 1 basis point, so it is about money duration × 0.0001 and much smaller.

Last-day revision: Yield-Based Bond Convexity and Portfolio Properties

  • Bond prices and yields move in opposite directions, and the relationship is convex for option-free bonds.
  • Modified duration = Macaulay duration ÷ (1 + YTM per period), with YTM per period = YTM ÷ periods per year.
  • Effective duration uses price changes from a curve shift and suits bonds with embedded options.
  • %ΔPrice ≈ −ModDur × ΔYield for the first-order estimate.
  • Convexity adjustment = ½ × Convexity × (ΔYield)², and it is positive for positive convexity.
  • Money duration = ModDur × full price of the position. It is the currency price change per 1.00 (100%) change in yield, so for a 1 percentage point (100 basis point) change use about money duration × 0.01.
  • PVBP is the price change for a one basis point change in yield, about money duration × 0.0001.
  • Portfolio duration is the market-value-weighted average of bond durations.
  • A callable bond has negative convexity at low yields, when the call is likely, and its price rises less than an option-free bond's. At high yields it has positive convexity.
  • A putable bond has positive convexity. At high yields the put moves into the money, so its price falls less than an option-free bond's. At low yields the put is far out of the money and the bond behaves like an option-free bond.
  • Higher convexity is favourable: it gains more when yields fall and loses less when yields rise.

Yield-Based Bond Convexity and Portfolio Properties practice questions

Yield-Based Bond Convexity and Portfolio Properties in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Yield-Based Bond Convexity and Portfolio Properties: frequently asked questions

What is the difference between modified duration and effective duration?

Modified duration is calculated from a bond's fixed cash flows and its yield to maturity. Effective duration comes from price changes when the benchmark curve shifts, so it handles bonds with embedded options. Use modified for option-free bonds and effective when cash flows can change.

Why do we add a convexity adjustment?

Duration gives a straight-line estimate, but the price-yield curve bends. The convexity adjustment corrects for that bend. It makes the estimate more accurate, especially for large yield changes.

Can a bond have negative convexity?

Yes. A callable bond has negative convexity when yields are low enough that a call becomes likely, because its price rises less than an option-free bond's. At high yields it has positive convexity. An option-free bond and a putable bond have positive convexity. For the putable bond, the put moves into the money at high yields, so its price falls less than an option-free bond's. At low yields the put is far out of the money and the bond behaves like an option-free bond.

How do I calculate portfolio duration?

Weight each bond's duration by its share of the portfolio's market value and add the results. This works as a guide for small parallel yield shifts. Convexity can be combined the same way.