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CFA Level I Exam · Yield-Based Bond Duration Measures and Properties

Macaulay, Modified and Money Duration Explained

Updated 7 October 2026 · Fact-checked

Macaulay duration is the weighted average time until a bond's cash flows arrive, with weights equal to each cash flow's share of price. Modified duration equals Macaulay duration divided by (1 + YTM per period) and estimates the percentage price change for a yield change. Money duration is modified duration times price; PVBP is money duration per basis point.

Understand Macaulay, Modified and Money Duration

A bond pays cash flows at different dates. Macaulay duration (MacDur) asks: on average, how long do you wait to get your money back, when each cash flow is weighted by its present value as a share of the bond's full price? It is measured in years. A zero-coupon bond has MacDur equal to its maturity. A coupon bond has MacDur below maturity, because coupons arrive early.

MacDur is a time measure. It does not directly tell you how much the price moves. To get price sensitivity, divide by (1 + YTM per period). The result is modified duration (ModDur). ModDur gives the approximate percentage price change for a 1-unit change in yield: %ΔPrice ≈ −ModDur × ΔYield. The minus sign matters. Yields up, price down.

You can also estimate ModDur without cash-flow weights. Approximate modified duration (ApproxModDur) reprices the bond at a slightly lower yield and a slightly higher yield, then uses the price difference. This is the same idea as a slope. It works for any bond you can price, including bonds with options (where the result is closer to effective duration).

Money duration converts the percentage measure into currency: ModDur × full price (or position value). It tells you the currency loss per 100% yield change. The price value of a basis point (PVBP) is the currency change in value for a 1 bp (0.01%) yield change. PVBP ≈ money duration × 0.0001.

These measures are linear estimates. They are good for small yield changes. For larger moves, the price-yield curve bends (convexity), and duration alone is less accurate.

Key formulas to remember

Macaulay duration
MacDur = Σ [t × PV(CF_t)] ÷ Full price, where PV(CF_t) = CF_t ÷ (1 + r)^t
t is measured in periods. If the bond pays m times a year, divide the result by m to express it in years.
Closed form for a bond on a coupon date (per-period rate r, coupon rate c per period, N periods)
MacDur = (1 + r) ÷ r − [(1 + r) + N × (c − r)] ÷ [c × ((1 + r)^N − 1) + r]
Result is in periods. Use it for bonds priced on a coupon date. Check: for a par bond (c = r) it reduces to (1 + r)/r × [1 − 1/(1 + r)^N].
Modified duration
ModDur = MacDur ÷ (1 + YTM ÷ m)
m = coupon periods per year. Use the annual MacDur with the periodic yield denominator as stated: YTM ÷ m.
Approximate modified duration
ApproxModDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔYTM)
PV₋ is price when yield falls by ΔYTM; PV₊ is price when yield rises by ΔYTM; PV₀ is the starting price. ΔYTM is in decimals (0.005 for 50 bps).
Price change estimate
%ΔPV ≈ −ModDur × ΔYTM
First-order estimate. Valid for small, parallel yield changes. Convexity improves it for larger moves.
Money duration
MoneyDur = ModDur × PV (full price or position market value)
Expressed in currency units.
Price value of a basis point
PVBP = MoneyDur × 0.0001, or (PV₋ − PV₊) ÷ 2 using a 1 bp yield change
Change in value for a 1 bp yield change. Quote in the same currency and size as the position.

How to solve Macaulay, Modified and Money Duration questions

Use this order for almost any question on Macaulay, modified or money duration.

  1. 1Identify what the question asks: time (MacDur), percentage price sensitivity (ModDur), or currency sensitivity (money duration or PVBP).
  2. 2Write down the yield, coupon frequency and whether the price needed is full price or position value.
  3. 3If MacDur is given, convert to ModDur: divide by (1 + YTM ÷ m). Do not skip this step when the question asks about price change.
  4. 4If prices at shifted yields are given, compute ApproxModDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔYTM). Convert basis points to decimals first.
  5. 5For a price change, apply %ΔPV ≈ −ModDur × ΔYTM. Check the sign: yield up means price down.
  6. 6For currency questions, multiply ModDur by the position value to get money duration, then by 0.0001 for PVBP.
  7. 7Check reasonableness: ModDur should be a bit below MacDur, and MacDur should not exceed maturity for a bond with positive coupons.

Quickest way: Convert, multiply, check the sign

When to use it: Use this when the stem gives a duration and a yield change and you need a price change under time pressure (about 90 seconds per question).

  1. Ask: is the given duration Macaulay or modified? If Macaulay, divide by (1 + periodic yield) first.
  2. Multiply by the yield change in decimals. 25 bps = 0.0025.
  3. Put a minus sign in front for a yield rise (plus for a yield fall).
  4. Eliminate options with the wrong sign first. Then eliminate the option that used MacDur directly instead of ModDur.
  5. For PVBP, estimate money duration × 0.0001 by moving the decimal four places.

Common mistakes in Macaulay, Modified and Money Duration

  • Using Macaulay duration directly to estimate price change.

    Both are called duration, and both are in years, so they look interchangeable.

    Fix: Price change needs modified duration. Divide MacDur by (1 + YTM ÷ m) first. Examiners often include the MacDur-based answer as a wrong option.

  • Forgetting to divide by m (or use the periodic yield) for semiannual bonds.

    Students divide by (1 + annual YTM) when the bond pays twice a year.

    Fix: Use 1 + YTM ÷ m. Also, if you computed MacDur in periods, divide by m to get years.

  • Entering the yield change as a whole number, such as 50 instead of 0.005.

    Basis points and percentages get mixed up under time pressure.

    Fix: Convert first: 50 bps = 0.50% = 0.0050. Write the decimal before multiplying.

  • Dropping the negative sign or applying it the wrong way.

    Duration is quoted as a positive number, so the inverse relationship is easy to lose.

    Fix: Write %ΔPV ≈ −ModDur × ΔYTM every time. A yield rise gives a negative price change.

  • Using only the price change on one side in approximate duration, or forgetting the 2 in the denominator.

    The formula is confused with a simple one-sided percentage change.

    Fix: Use both PV₋ and PV₊, and divide by 2 × PV₀ × ΔYTM. Sanity check against the likely size of duration.

  • Treating money duration or PVBP as a percentage.

    Students carry over the percentage idea from ModDur.

    Fix: Money duration and PVBP are currency amounts tied to the position size. Multiply ModDur by the market value, then by 0.0001 for PVBP.

Worked examples

Example 1

A 3-year bond pays a 5% annual coupon, has a par value of 100 and a YTM of 5%, so it is priced at 100. Its Macaulay duration is 2.859 years. Which is the best estimate of the percentage price change if the YTM rises by 50 bps? A) −1.43% B) −1.36% C) +1.36%

Show the solution
  1. Check what is given: MacDur = 2.859, not ModDur. Annual coupons, so m = 1.
  2. ModDur = 2.859 ÷ (1 + 0.05) = 2.859 ÷ 1.05 = 2.723.
  3. Yield change = +50 bps = +0.005.
  4. %ΔPV ≈ −2.723 × 0.005 = −0.0136, or −1.36%.
  5. Eliminate C: the sign is wrong for a yield rise. Eliminate A: it is 2.859 × 0.005 = 1.43%, which wrongly uses MacDur.
  6. Optional check of MacDur: PVs are 4.762, 4.535 and 90.703. Weighted sum = 4.762 + 9.070 + 272.109 = 285.941. Divide by price 100 to get 2.859.

Answer: B) −1.36%

Example 2

A bond is priced at 100.00. If its yield falls by 0.5%, the price rises to 102.80. If its yield rises by 0.5%, the price falls to 97.30. An investor holds a position with a market value of 2,000,000 USD. What is the approximate PVBP of the position? A) 550 USD B) 1,100 USD C) 11,000 USD

Show the solution
  1. ΔYTM = 0.5% = 0.005.
  2. ApproxModDur = (102.80 − 97.30) ÷ (2 × 100 × 0.005) = 5.50 ÷ 1.00 = 5.50.
  3. Money duration = 5.50 × 2,000,000 = 11,000,000 USD.
  4. PVBP = 11,000,000 × 0.0001 = 1,100 USD.
  5. Cross-check: for 1 bp, value changes by about 0.055% of 2,000,000, which is 1,100 USD.
  6. Eliminate C: that is money duration ÷ 1,000, not × 0.0001 (a decimal slip). Eliminate A: it is half the correct value, as if the denominator 2 was applied twice.

Answer: B) 1,100 USD

Exam tips

  • Read the word carefully: Macaulay, modified, effective and money duration each answer different questions. Underline it in the stem.
  • Basis point conversions cost marks. Write 0.0001 per bp and scale up.
  • Use the sign test first. With three options, wrong-sign answers can usually be eliminated at once, and guessing carries no penalty.
  • If a question gives prices at yield ± Δ, use the approximate duration formula. Do not try to build the cash-flow table.
  • Know the direction of properties: MacDur of a zero-coupon bond equals maturity, and a coupon bond's MacDur is below its maturity. Quick conceptual items often test only this.

Practice questions from Yield-Based Bond Duration Measures and Properties

Macaulay, Modified and Money Duration in other exams

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

Macaulay, Modified and Money Duration: frequently asked questions

What is the difference between Macaulay duration and modified duration?

Macaulay duration is the weighted average time to receive a bond's cash flows, measured in years. Modified duration equals Macaulay duration divided by (1 + YTM per period) and measures price sensitivity to yield changes. Use modified duration to estimate price change.

How do I calculate modified duration for the CFA Level I exam?

If Macaulay duration is given, divide it by (1 + YTM ÷ m). If prices at higher and lower yields are given, use the approximate formula (PV₋ − PV₊) ÷ (2 × PV₀ × ΔYTM). Then multiply by the yield change and add a negative sign for a price estimate.

What is money duration and how does it relate to PVBP?

Money duration is modified duration times the bond's full price or position value, so it is stated in currency. PVBP is the change in value for a 1 basis point yield change, approximately money duration × 0.0001.

Can I compute bond prices for approximate duration on the calculator?

Yes. On the TI BA II Plus, enter N, I/Y, PMT and FV, then press CPT PV for each yield. Remember the PV sign is negative, so use its absolute value. Compute the price at the lower yield and at the higher yield, then apply the formula.