Duration · RiskAtoms Academy

By the end you will be able to

1Calculate Macaulay duration as the PV-weighted average time to a bond's cash flows.
2Convert Macaulay duration to modified duration.
3Use modified duration to estimate a price change, and calculate DV01.
4Explain what makes duration higher or lower: maturity, coupon and yield.

Why it matters

A pension fund holds £1bn of bonds. The board asks one question: “How much do we lose if interest rates rise by 1%?” One number answers it. If the portfolio behaved like the bond in this lesson's worked example, the answer would be about £27.2m. That number comes from duration.

Core explanation

The average wait, and why it measures risk

A bond's price depends on its yield (L1.4.03), and when yields rise, prices fall (L1.4.06). The question is: by how much? Duration is the standard answer.

First idea: the average wait. Think of a bond as a stream of payments. Some arrive soon, some late. How long do you wait, on average, to get your money back? A simple average of the dates is not good enough: a payment of 105 in year 3 matters far more than a payment of 5 in year 1. So we weight each date by how much of the bond's value arrives on it, measured by the payment's present value.

This weighted average wait is the Macaulay duration, in years. Picture the present values as weights on a seesaw laid along a timeline: the duration is where the seesaw balances.

Second idea: from time to sensitivity. The longer you wait for the money, the more a change in the discount rate hurts it. A small adjustment turns Macaulay duration into modified duration, which gives the percentage fall in price for a 1 percentage point rise in yield. If modified duration is 2.7, a 1% rise in yield cuts the price by about 2.7%.

DV01 asks the same question in money: how much does the price change when the yield moves by one basis point? Traders use it to size and hedge positions.

What drives duration. Predict each answer for a 10-year 4% bond at a 4% yield, then reveal it:

Think firstChange one thing

20-year instead of 10-year: does the duration of 8.44 years go up or down?
8% coupon instead of 4%: does the duration of 8.44 years go up or down?
8% yield instead of 4%: does the duration of 8.44 years go up or down?
Zero coupon: does the duration of 8.44 years go up or down?

A zero-coupon bond has all its value on one date, so its duration equals its maturity. Any bond that pays coupons has a duration shorter than its maturity.

Duration draws a straight line through a curve. It works well for small moves in yield; for large ones it is less accurate, and L1.4.09 shows how to fix that.

The formal version

Duration brings new notation, so let's build it one piece at a time. We keep the letters from L1.4.03 and add $CF_t$ , the cash flow at time $t$ , and $PV_t = CF_t / (1+y)^t$ , its present value.

The formulas, for reference

D_{\text{Mac}} = \frac{1}{P}\sum_{t=1}^{T} t \times PV_t \qquad D_{\text{mod}} = \frac{D_{\text{Mac}}}{1+y}

\Delta P \approx -D_{\text{mod}} \times P \times \Delta y \qquad \text{DV01} = D_{\text{mod}} \times P \times 0.0001

With m coupons a year, measure t in years and divide by 1 + y/m instead of 1 + y.

Optional: why it is the slope (Math Atom: Derivatives)

Differentiate the price with respect to the yield, one term at a time:

\frac{dP}{dy} = \sum_{t=1}^{T} \frac{-t \times CF_t}{(1+y)^{t+1}} = -\frac{1}{1+y}\sum_{t=1}^{T} t \times PV_t = -\frac{D_{\text{Mac}}}{1+y}\,P = -D_{\text{mod}}\,P

So $D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}$ : modified duration is the slope of the price-yield curve, as a fraction of the price. The approximation follows the tangent line.

Try it

It opens on the worked example below: a 3-year 5% bond at 5%. Test a +100 bp move and compare the estimate with the actual price. Then stretch the maturity to 30 years and try +300 bp.

Try itDuration explorer

Open the full bond pricer →

Coupon rate5.00%

Years to maturity3

Yield5.0%

Coupons a year

Yield change to test+100 bp

Macaulay duration

2.859

years

Price100.0000

Modified duration2.723

DV01 per 1000.0272

Duration estimate97.2768

Actual price97.3270

Error0.0502

Present value of each payment, balanced on a seesawThe triangle sits at the balance point: the Macaulay duration. Raise the coupon and watch it move left.

Worked example

A 3-year 5% bond at a 5% yield

Face value 100, annual coupon 5, three years to maturity, annual yield 5%. The coupon equals the yield, so the price is 100 (par). Try each step before revealing it.

Duration by the table method

Step 1 of 8

1
Cash flows and present values
5, 5 and 105. Discounted at 5%: 4.7619 + 4.5351 + 90.7029 = 100.0000, the price.

Regulatory lens

Duration-style sensitivities run through bank and insurer rules. Later levels build on this lesson:

Interest rate risk in the banking book (IRRBB): changes in economic value under rate shocks, L3.5.
Solvency II interest rate risk for insurers, L4.6.
FRTB, the market risk framework for trading books, which uses interest rate sensitivities, L4.3.

Each asks how much value changes when rates move. Duration and DV01 are the basic tools for answering.

Sparks

SparkMacaulay, 1938

The economist Frederick Macaulay introduced the measure in a 1938 study of US interest rates and bond yields for the National Bureau of Economic Research. He wanted a better way to describe the “length” of a bond than its maturity, because two bonds with the same maturity but different coupons behave differently.

SparkThe 2022 UK LDI crisis

Many UK pension funds use liability-driven investment (LDI) to match the long duration of their pension promises, often with borrowed money. In September 2022, after the government's fiscal statement, long-dated gilt yields jumped by more than a percentage point within days. With durations of 20 years or more, that meant huge price falls and urgent collateral calls. The Bank of England stepped in to buy long-dated gilts. L1.4.10 returns to this episode.

Key takeaways

Macaulay duration is the PV-weighted average time to a bond's cash flows, in years.
Modified duration = Macaulay duration ÷ (1 + y). It is the approximate percentage price change for a 1% change in yield.
ΔP ≈ −D_mod × P × Δy is a straight-line estimate. It works best for small moves in yield.
DV01 = D_mod × P × 0.0001 is the money change in price for one basis point.
Duration rises with maturity and falls with a higher coupon or a higher yield. A zero's duration equals its maturity.

Reaction

Check what stuck

Question 1 · objective 1

A 2-year bond has face value 100, a 5% annual coupon and a yield of 5%, so its price is 100. Using the table method, what is its Macaulay duration in years, to three decimal places?

Question 2 · objective 2

A bond has a Macaulay duration of 4.50 years and a yield of 5%, with annual coupons. Its modified duration is closest to:

A4.29B4.50C4.73D0.23

Question 3 · objective 3

A bond priced at 100 has a modified duration of 6.0. Its yield rises by 0.5% (0.005). Using the duration approximation, what is the estimated new price?

Question 4 · objective 4

All four bonds yield 4%. Which has the highest duration?

A5-year bond, 6% couponB10-year bond, 6% couponC10-year bond, 2% couponD5-year bond, 2% coupon

Question 5 · objective 3

A bond is priced at 98 per 100 and has a modified duration of 7.5. What is its DV01 per 100 of face value, to four decimal places?

Glossary

Macaulay duration: The average time to a bond's cash flows, with each date weighted by the present value of the payment on it. Measured in years.
Modified duration: Macaulay duration divided by one plus the yield. The approximate percentage price change for a one percentage point change in yield.
DV01 (PV01): The change in a bond's price, in money, when its yield moves by one basis point. Also called PV01 or basis point value.
Basis point: One hundredth of a percentage point: 0.01%, or 0.0001 as a decimal.
Zero-coupon bond: A bond that pays no coupons, only its face value at maturity.

Deep Dives & Origins

Deep Dives

Bruce Tuckman and Angel Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd edition, Wiley, 2011, chapter 4. Intermediate.
Frederick R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856, NBER, 1938. Advanced, historical.

Origins

Macaulay (1938), as above.
Frank J. Fabozzi and Francesco A. Fabozzi, Bond Markets, Analysis, and Strategies, 10th edition, Pearson, 2021.
Tuckman and Serrat (2011), as above.
Bank of England, Financial Stability Report, December 2022 (LDI episode).

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