CFA Level I Exam · Yield-Based Bond Convexity and Portfolio Properties

Portfolio Duration and Convexity for Bond Portfolios

Updated 7 October 2026 · Fact-checked

Portfolio duration is the market-value-weighted average of the durations of the bonds in the portfolio, and portfolio convexity is found the same way. Multiply each bond's weight by its measure and add the results. The estimate holds only for a parallel yield curve shift; twists and steepening call for key rate duration.

Understand Portfolio Duration and Convexity

A bond portfolio reacts to interest rates as a whole. You want one number that tells you roughly how much its value falls when yields rise. That number is portfolio duration.

The usual shortcut is the weighted-average approach. Each bond's weight is its market value (full price, including accrued interest) divided by the total market value of the portfolio. Portfolio duration is the sum of weight × duration. Portfolio convexity is the sum of weight × convexity. Use the same type of duration for every bond, for example modified or effective.

This works because a percentage price change is a value-weighted blend of the percentage changes of the bonds. But it carries a hidden assumption. It treats every bond as if its yield moves by the same amount, a parallel shift. If short-term yields rise more than long-term yields, or the curve twists, the single portfolio number can misstate the price change.

There is a second method: the cash flow yield approach. You pool all the portfolio's cash flows, find the single yield (IRR) that equates them to the portfolio market value, and compute duration from that yield. It is theoretically better than the weighted average, but it still assumes one yield change applies to all cash flows. Neither method handles non-parallel shifts. For that you need measures such as key rate duration, which are covered elsewhere.

Convexity adds a correction. Duration alone is a straight-line estimate, and convexity captures the curve. For a given yield change, convexity raises the estimated price gain when yields fall and cuts the estimated loss when yields rise.

Key formulas to remember

Weight of bond i: wᵢ = Market value of bond i ÷ Total portfolio market value; Use full price including accrued interest, not par value.
Portfolio duration (weighted average): D_p = Σ wᵢ × Dᵢ = w₁D₁ + w₂D₂ + … + wₙDₙ; Use one type of duration throughout. Valid as an estimate for parallel yield shifts.
Portfolio convexity (weighted average): C_p = Σ wᵢ × Cᵢ; Same market-value weights as for duration.
Approximate percentage price change: %ΔPV ≈ −D_p × Δy + ½ × C_p × (Δy)²; Enter Δy as a decimal. 50 bps = 0.005. The convexity term is always positive for an option-free bond portfolio.
Portfolio money duration or BPV: Portfolio BPV = Σ BPVᵢ; Money measures are additive across bonds because they are in currency units.
Cash flow yield approach: Find the IRR of the pooled portfolio cash flows, then compute duration at that yield; Better theoretically than the weighted average, but still assumes a single yield change.

How to solve Portfolio Duration and Convexity questions

Use this order for any question on portfolio duration and convexity.

1Read what is given: market values or weights, and the type of duration (modified, effective or Macaulay).
2Compute total market value, then each weight as a decimal. Check that the weights add up to 1.
3Multiply each weight by its duration and add. This is the portfolio duration.
4If convexity is asked or given, multiply each weight by its convexity and add.
5Convert the yield change to a decimal. Then compute −D_p × Δy for the duration effect.
6Add ½ × C_p × (Δy)² if convexity is part of the question. Combine the two effects with their signs.
7Check the assumption. If the question says yields at different maturities move by different amounts, a single portfolio duration is not enough. Apply each bond's own yield change instead, or state the limitation.
8Pick the option that matches your answer. Eliminate options with the wrong sign or those that ignore convexity.

Quickest way: Weights, multiply, add, then sign check

When to use it: Use it for any weighted-average question when options are far enough apart to separate by estimation.

Get weights straight from market values. If values are in equal-sized units, such as 40, 35 and 25 out of 100, use them as decimals directly.
On the TI BA II Plus, the chain calculation 0.4 × 3 + 0.35 × 6 + 0.25 × 9 = 5.55 follows the order of operations only if the calculator is set to AOS mode (2nd, Format, change to AOS, then 2nd, Quit). In the default chain (Chn) mode it works left to right and gives a wrong result. If you are unsure of the mode, compute each product separately and add them. The HP 12C also needs each product computed and added step by step.
Do the sign check first. Rising yields mean a price fall, so the duration effect is negative and the convexity effect is positive.
The convexity term is usually tiny: ½ × C × (Δy)². Estimate it and see which option it separates. Many wrong options leave it out or subtract it.

Common mistakes in Portfolio Duration and Convexity

Weighting by par value or number of bonds instead of market value.
Par amounts are the easiest numbers in a question, and the word 'weight' sounds like a simple count.
Fix: Use market value including accrued interest. A discount or premium bond has a different weight than its par suggests.
Leaving out the ½ in the convexity adjustment, or forgetting to square Δy.
Students memorise 'convexity × Δy' without the structure of the second-order term.
Fix: Write the full formula first: ½ × C × (Δy)². Convert Δy to decimals before squaring.
Applying the portfolio duration to a non-parallel shift.
The weighted-average number looks precise, so students treat it as valid for any yield move.
Fix: Remember that it assumes the same yield change for all bonds. With a twist or steepening, work out each bond's price change separately or use key rate duration.
Mixing Macaulay duration for some bonds with modified for others, or mixing effective with modified.
The question gives different measures and students average whatever number is listed.
Fix: Convert to one consistent measure before averaging. Use effective duration for bonds with embedded options.
Getting the sign of the convexity adjustment wrong.
Students link a yield rise with a negative result and apply the negative sign to both terms.
Fix: The duration term is −D × Δy. The convexity term is positive for both rises and falls in yield, for option-free bonds.
Using basis points directly in the formula, such as 50 instead of 0.005.
Rushing under time pressure.
Fix: Always write Δy as a decimal. 50 bps = 0.50% = 0.005.

Worked examples

Example 1

A portfolio holds three option-free bonds with market values of USD 40 million, USD 35 million and USD 25 million. Their modified durations are 3.0, 6.0 and 9.0, and their convexities are 12, 45 and 100. Assuming a parallel upward shift of 50 bps, the approximate percentage change in portfolio value is closest to: A) −2.83%, B) −2.78%, C) −2.72%.

Show the solution

Total market value = 40 + 35 + 25 = USD 100 million. Weights are 0.40, 0.35 and 0.25.
Portfolio duration = 0.40 × 3.0 + 0.35 × 6.0 + 0.25 × 9.0 = 1.20 + 2.10 + 2.25 = 5.55.
Portfolio convexity = 0.40 × 12 + 0.35 × 45 + 0.25 × 100 = 4.80 + 15.75 + 25.00 = 45.55.
Duration effect = −5.55 × 0.005 = −0.02775, or −2.775%.
Convexity effect = ½ × 45.55 × (0.005)² = 0.5 × 45.55 × 0.000025 = 0.000569, or +0.057%.
Total = −2.775% + 0.057% = −2.718%, about −2.72%.
Option A subtracts the convexity term: −2.775% − 0.057% = −2.832%, about −2.83%. Option B is the duration-only result, −2.775%, rounded to −2.78%, because it ignores convexity.

Answer: C) −2.72%

Example 2

A portfolio has equal market values in a 2-year bond with modified duration 1.9 and a 10-year bond with modified duration 8.1. The weighted-average duration is 5.0. Over a period, the 2-year yield rises by 0.40% and the 10-year yield rises by 0.10%. Ignoring convexity, the approximate percentage change in portfolio value is closest to: A) −1.25%, B) −0.79%, C) −0.39%.

Show the solution

The shift is not parallel, so apply each bond's own yield change.
2-year bond: −1.9 × 0.0040 = −0.76%. Weighted by 0.5 this gives −0.38%.
10-year bond: −8.1 × 0.0010 = −0.81%. Weighted by 0.5 this gives −0.405%.
Portfolio change = −0.38% − 0.405% = −0.785%, about −0.79%.
A parallel-shift estimate using the average yield change of 0.25% gives −5.0 × 0.0025 = −1.25%. Option A is that trap.
The two estimates differ because the portfolio duration assumes the same yield change for both bonds. Here the 10-year bond, which carries most of the rate sensitivity, moved much less than the 2-year bond, so the portfolio duration with the average change overstates the loss.

Answer: B) −0.79%. The weighted-average duration is reliable only for parallel shifts.

Exam tips

Check whether the question states a parallel shift. If it does, the weighted average is valid. If yields move differently by maturity, use each bond's own change.
Read which duration is supplied. Effective duration is the right measure for bonds with embedded options, and the weighted average needs one consistent measure.
Watch the units. Yield changes in bps must be converted to decimals before you apply the formula, and the final answer is a percentage.
If the options are close, calculate the convexity term. It is small but often decides between two answers.
For theory questions, remember that the cash flow yield method is more accurate than the weighted average but still does not solve non-parallel shifts.

Practice questions from Yield-Based Bond Convexity and Portfolio Properties

Portfolio Duration and Convexity in other exams

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

Portfolio Duration and Convexity: frequently asked questions

How do you calculate the duration of a bond portfolio?

Compute each bond's weight as its market value divided by the portfolio market value. Multiply each weight by that bond's duration and add the products. Use the same type of duration for all bonds.

What is the main limitation of weighted-average portfolio duration?

It assumes a parallel shift in the yield curve, so every bond's yield changes by the same amount. When short-term and long-term yields move differently, the portfolio duration can misestimate the price change. Key rate duration handles those cases better.

Is portfolio convexity also a weighted average?

Yes. Portfolio convexity is commonly estimated as the market-value-weighted average of the individual bond convexities. You then use it in the second-order term ½ × C × (Δy)².

What is the cash flow yield approach to portfolio duration?

You pool all the portfolio's cash flows, find the IRR that equates them to the portfolio's market value, and compute duration from that yield. It is considered theoretically better than the weighted average, but it also assumes a single yield change for all cash flows.