Duration (finance)

Overview

In finance, duration is a measure of how the price of a fixed-income instrument responds to a change in interest rates. It is used to compare rate risk across bonds and to construct hedges, and is often paired with convexity and the price value of a basis point. Duration-based estimates work best for small, parallel shifts in the yield curve. Macaulay duration is the present-value-weighted average time to the cash flows and links payment timing to interest-rate risk. Modified duration expresses the first-order percentage price change for a stated compounding convention. When yields vary by maturity, Fisher–Weil duration discounts each payment at its own spot rate; Key rate duration isolates sensitivity at selected maturities; and effective or option-adjusted duration estimates sensitivity for instruments with cash flows that depend on rates.

Early development

The idea of duration was set out by Frederick Macaulay in a National Bureau of Economic Research study in 1938. He defined a time-weighted average of the present values of cash flows and used it to summarize a bond’s timing and rate sensitivity. In actuarial work, Frank Redington linked duration to immunization and added convexity to improve protection against larger moves in yields.

Extensions

With a term structure of rates, discounting each payment at its own spot rate preserves the present-value weighting and gives a first-order hedge for a small parallel shift of the zero curve. This is the Fisher–Weil formulation. To handle non-parallel moves, practitioners report localised sensitivities at selected maturities using key rate durations. Option features led to effective or option-adjusted duration, estimated by small curve shifts in a pricing model while the option-adjusted spread is held constant. These uses are standard in index and reporting methodologies.

Terminology and market usage

In modern texts “duration” can mean different but related measures. Macaulay duration is the present-value-weighted average time to payment. Modified duration is the first-order percentage change in price for a small change in the stated yield and compounding. Money or dollar duration is

P ×

D

mod

{\displaystyle P\times D_{\text{mod}}}

. DV01, PV01 and PVBP express the price change per basis point. In the UK gilts market, modified duration is often called “volatility” in index guides and factsheets.

Definition and intuition

This section uses the following conventions. A fixed-income instrument has

n

{\displaystyle n}

cash flows

C

i

{\displaystyle C_{i}}

at times

t

i

{\displaystyle t_{i}}

(in years). The last cash flow includes the bond redemption. The nominal yield to maturity is

y

{\displaystyle y}

with

m

{\displaystyle m}

compounding periods per year. The price as a function of yield is

P ( y )

=

Analogy

Imagine a long plank set along a timeline that begins today. Each future cash flow is a small weight placed on the plank at the cash flow time. Heavier weights correspond to cash flows with larger present values. If you slide a single support under the plank to the point where the system balances, that balance point is the time-centre of all the weights. If most of the weight lies far along the plank the balance point sits further from today and the bond is more sensitive to a change in yields. If weight is concentrated near the start through high coupons or short maturity the balance point moves inward and sensitivity falls. This time-centre corresponds to Macaulay duration. Now tilt the ground by a very small amount. The plank drops a little and, for such a small tilt, the vertical drop at the balance point is almost exactly proportional to the tilt. That proportional response mirrors modified duration, which gives the first-order change in price for a small change in yield. With a larger tilt the motion does not remain proportional because the plank follows a curve. The extra curvature in the response explains convexity and shows why the second-order term matters for larger yield moves or for cash-flow patterns that make the curve more pronounced. If the ground does not tilt uniformly but is raised or lowered under specific years, different parts of the plank move by different amounts. That picture matches shifts in the term structure and motivates measures such as Fisher–Weil duration and key-rate durations, where sensitivity depends on which maturities move.

Worked examples

Zero-coupon bond Assume maturity

T = 3

{\displaystyle T=3}

years and yield

y = 5 %

{\displaystyle y=5\%}

with annual compounding (

m = 1

{\displaystyle m=1}

). Then

D

Mac

= T = 3 ,

D

mod

=

T

1 + y

/

m

Term-structure intuition

When the term structure is not flat, discounting each payment at its own zero-coupon rate preserves the weighting idea in Macaulay’s statistic and leads to the Fisher–Weil refinement for parallel shifts of the zero-rate curve. Non-parallel movements are analysed with key-rate durations in later sections.

Formal derivation

Let a fixed-income instrument pay cash flows

C

i

{\displaystyle C_{i}}

at times

t

i

{\displaystyle t_{i}}

(years),

i = 1 , … , n

{\displaystyle i=1,\ldots ,n}

. The last cash flow

C

n

{\displaystyle C_{n}}

at time

t

n

{\displaystyle t_{n}}

includes the redemption. With a yield to maturity

Continuous compounding

If pricing uses a continuously compounded rate

r

{\displaystyle r}

, then

P ( r )

=

∑

i = 1

n

C

i

e

− r

t

i

,

d

P

d

r

Term-structure version (Fisher–Weil)

When the term structure is not flat, discount each cash flow at its own zero-coupon rate

z ( t )

{\displaystyle z(t)}

. For a parallel shift

a

{\displaystyle a}

to the zero curve,

P ( a )

=

∑

i = 1

n

C

i

exp

(

−

∫

0

t

Money duration and DV01

Money duration

=

P ×

D

mod

,

DV01

=

PVBP

=

P ×

D

mod

×

10

− 4

.

{\displaystyle {\text{Money duration}}\;=\;P\times D_{\text{mod}},\qquad {\text{DV01}}\;=\;{\text{PVBP}}\;=\;P\times D_{\text{mod}}\times 10^{-4}.}

These identities are widely used in portfolio reporting and regulation.