Bond duration and convexity for the CFA® Program Level I

Duration estimates a bond's sensitivity to interest rates; convexity corrects for the curvature that a duration-only estimate misses. Macaulay duration measures payment timing, modified duration measures sensitivity to yield to maturity, and effective duration uses repricing when cash flows may change. Money duration and PVBP express sensitivity in currency. (RBI, bond sensitivity (secondary source)) (OSFI, duration and value changes)

This guide supports study for the CFA® Program Level I, run by CFA Institute. (CFA Institute, Level I exam) Our questions complement CFA Institute's own curriculum; they do not replace it. See the course page and the time value of money guide for the discounting foundations.

What does Macaulay duration measure?

Macaulay duration is the average time to receive a bond's cash flows, weighted by their present values. For positive payments, each weight is the payment's present value divided by the bond's full price. The weights sum to one. It is not a promised holding period. (IMF, duration definitions (secondary source))

Here, $t$ measures years from settlement, $CF_t$ is a payment, and $P_0$ is full price, including accrued interest. A zero-coupon bond's Macaulay duration equals its remaining maturity; earlier positive coupons pull the weighted average below maturity. (RBI, bond sensitivity (secondary source))

Our original example assumes £100 face value, four years remaining, a 4.5% annual coupon and a 5.25% annual yield. Settlement is immediately after a coupon, so accrued interest is zero. Payments are £4.50 after each of the first three years and £104.50 after year four. All inputs are illustrative, with no tax, fees, default or embedded option.

The present values are £4.275534, £4.062266, £3.859635 and £85.158474. Multiplying them by years one to four and adding gives 364.612867 pound-years. Divide by the unrounded price to obtain Macaulay duration of 3.745154 years. Keep full precision until the final answer.

The final payment contributes most of this bond's present value, so the weighted timing lies close to year four. Duration averages payment dates; it does not identify the date when cumulative cash receipts recover the purchase price. (IMF, duration definitions (secondary source))

How does modified duration estimate a price change?

Modified duration converts payment timing into sensitivity to the bond's own yield, holding contractual cash flows fixed. If $y$ is nominal annual yield compounded $m$ times annually, use Macaulay duration in years: (RBI, bond sensitivity (secondary source))

Enter the yield change as a decimal. One basis point is 0.0001; a rise of 50 basis points is +0.005. The minus sign reflects the inverse price–yield relationship for this fixed-payment bond. (RBI, bond sensitivity (secondary source)) (CME Group, basis-point value (secondary source))

Our annual bond has modified duration 3.745154 ÷ 1.0525 = 3.558341. A rise from 5.25% to 5.75% therefore gives:

The estimated price is £95.623781 per £100 face value. For semi-annual coupons, match the yield convention: an assumed Macaulay duration of 3.8 years and nominal yield of 6% give 3.8 ÷ (1 + 0.06/2) = 3.689320. Using 1.06 in that denominator would mix annual and semi-annual compounding.

What are money duration and PVBP?

Money duration scales modified duration by full market value. Here it means currency sensitivity per unit change in decimal yield. Multiplying by a one-basis-point move gives the approximate price value of a basis point (PVBP). VBP and BPV are alternative labels; DV01 is commonly used for dollar exposures. (OSFI, duration and value changes) (CME Group, basis-point value (secondary source))

For £250,000 face value of our bond, multiply the price per £100 by 2,500. Full market value is £243,389.77; money duration is £866,063.82 per unit decimal-yield change; PVBP is about £86.61. These results use unrounded inputs. A one-basis-point yield rise implies an approximate £86.61 loss; a fall implies a similar gain.

PVBP is quoted here as a positive sensitivity magnitude. The direction belongs in the price-change calculation. Using £250,000 face value instead of £243,389.77 market value would give the wrong sensitivity. Always state whether a price or sensitivity refers to £100 face value or the whole position.

For a direct repricing check, we take half the difference between prices at yields one basis point below and above the starting yield. Our bond gives £0.034642554 per £100 face value, closely matching the duration-based £0.034642553. This symmetric estimate averages the upward and downward price effects; individual moves differ slightly because of curvature. This follows from the symmetric duration formula below, using fixed cash flows and a 0.0001 yield change. (OSFI, duration and value changes)

How does convexity improve the estimate?

Duration follows a tangent to the price–yield curve. Convexity adds a second-order term. For an option-free bond with positive fixed payments, positive convexity raises the estimated price relative to the duration-only result for either direction of yield change. (IMF, convexity and curve assumptions)

With annual payments and annual compounding, differentiating our present-value formula twice gives the following normalised convexity. Use this convention with the one-half factor above; do not halve it again.

For the 50-basis-point rise, the correction is ½ × 16.527266 × 0.005² = 0.000206591, or 0.020659 percentage points. Estimated price change becomes −1.758511%, giving £95.643894. Directly discounting the unchanged payments at 5.75% gives £95.643707. The correction improves the approximation without making it exact.

For the 50-basis-point fall, the duration term becomes +1.779171%; the convexity correction stays positive. Together they predict +1.799830%, versus +1.800025% from exact repricing. Squaring the yield change is why the correction retains its sign. These are immediate price changes, excluding coupon income over a holding period.

When should you use effective duration?

Use effective duration when interest rates can alter expected cash flows, as with mortgage prepayments or callable bonds. (OSFI, duration and value changes) For the example below, reprice under equal upward and downward parallel shifts of a benchmark curve, allowing the model's cash flows to respond. Hold other valuation assumptions constant. Models of prepayment behaviour matter to the result. (San Francisco Fed, cash flows and options (secondary source))

Here, $P_-$ follows the downward shift, $P_+$ the upward shift, and $\Delta r$ is the positive size of either shift. This symmetric repricing calculation measures sensitivity around the starting value. (OSFI, duration and value changes)

Separately, suppose model prices per £100 face value are 100.40 initially, 102.15 after a 50-basis-point fall and 98.05 after a 50-basis-point rise. Effective duration is (102.15 − 98.05) ÷ (2 × 100.40 × 0.005) = 4.083665. A 25-basis-point upward shift from the original curve gives a first-order estimate of −1.020916%.

Bumping only a bond's yield to maturity while keeping its cash flows fixed approximates modified duration. Effective duration allows for rate-dependent cash flows; it can also be calculated for option-free bonds. (OSFI, duration and value changes) Mortgage prepayment options can also create negative convexity; a positive correction is not universal. (San Francisco Fed, cash flows and options (secondary source))

What should you check before using the answer?

Check the rate convention, decimal shock, full-price basis and position size. Match modified duration to a yield change and effective duration to its modelled curve shift. A single duration cannot describe every change in the curve's shape, and large moves can leave substantial approximation errors. Repricing tests the estimate under the chosen assumptions. (IMF, convexity and curve assumptions)

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