Core Path · L1.3.06

Day counts and rate conventions

10 minInterest ratesConventions

Know these first

By the end you will be able to

  1. 1Apply ACT/360, ACT/365F and 30/360 to accrue interest.
  2. 2Count the days between two dates under each convention and turn them into a year fraction.
  3. 3Name the convention used in the main money and bond markets.
  4. 4Compare rates quoted on different day counts.

Why it matters

Lend $10 million at 5% for 92 days. Is the interest $127,777.78 or $126,027.40? Both answers are right, depending on the market. The $1,750.38 difference comes down to one question: how many days are there in a year?

Core explanation

How many days in a year?

Interest rates are quoted per year. Most deals do not last exactly a year, so you need to know what fraction of a year the money is out for. Over short periods, money markets use simple interest (L1.3.02): interest is the notional times the rate times that fraction.

The fraction sounds easy: days divided by days in a year. But which days, and which year? Markets settled this long ago with day count conventions. Each one has two parts, written as “how to count days / days in a year”.

ACT/360

Actual calendar days, over a 360-day year.

US dollar and euro money markets

ACT/365F

Actual calendar days, over a fixed 365-day year (even in leap years).

Sterling and yen money markets

30/360

Every month counts as 30 days, over a 360-day year.

US corporate bonds, many swaps' fixed legs

Because a 360-day year is shorter, ACT/360 gives a slightly bigger fraction for the same dates, and so a bit more interest. The 30/360 count smooths out the calendar: every whole month counts the same, so coupon payments come out as neat equal amounts. Government bonds such as US Treasuries and UK gilts use a fourth convention, ACT/ACT, which divides by the actual length of the coupon period. It returns in the bond module.

The interest earned since the last payment, but not yet paid, is called accrued interest. When a bond is sold between coupon dates, the buyer pays the seller the accrued interest, worked out with the bond's day count.

Which convention?

0 / 6
  • A US dollar loan paying SOFR (the US overnight rate) plus a margin.
  • A sterling deposit paying SONIA (the UK overnight rate).
  • A euro deposit paying €STR (the euro short-term rate).
  • The semi-annual coupon on a US corporate bond.
  • The semi-annual coupon on a US Treasury note.
  • A Japanese yen loan paying TONA (the Tokyo overnight average rate).

The formal version

The arithmetic is simple, but two new symbols appear: NN for the notional and τ\tau (tau) for the day count fraction.

Counting 30/360 days. Write the start date as D1/M1/Y1D_1/M_1/Y_1 (day, month, year) and the end date as D2/M2/Y2D_2/M_2/Y_2. First fix the 31sts, then count:

  • If D1=31D_1 = 31, set D1=30D_1 = 30.
  • If D2=31D_2 = 31 and D1D_1 is now 30, set D2=30D_2 = 30.
days30/360=360(Y2−Y1)+30(M2−M1)+(D2−D1)\text{days}_{30/360} = 360(Y_2 - Y_1) + 30(M_2 - M_1) + (D_2 - D_1)

This is the “bond basis” version in the ISDA definitions. Other versions (such as 30E/360) treat month ends slightly differently, so always check the documentation.

Comparing rates on different bases. Two rates are equivalent if they pay the same interest for the same dates. With actual days dd on both sides:

N r360 d360=N r365 d365N \, r_{360} \, \frac{d}{360} = N \, r_{365} \, \frac{d}{365}
r365=r360×365360r_{365} = r_{360} \times \frac{365}{360}

The formulas, for reference

Interest=N×r×ττ=daysbasis\text{Interest} = N \times r \times \tau \qquad \tau = \frac{\text{days}}{\text{basis}}

ACT/360: actual days ÷ 360. ACT/365F: actual days ÷ 365. 30/360: 30-day-month days ÷ 360.

Try it

The calculator starts on 31 January to 31 July 2026. Notice that 30/360 counts 180 days while the calendar has 181. Now try a period that includes February, or a whole year, and watch the gap between ACT/360 and ACT/365F grow with the number of days.

Try itDay count calculator

Money market yields in the yield toolkit →
ACT/360τ = 181 ÷ 360 = 0.50277820,111.11
ACT/365Fτ = 181 ÷ 365 = 0.49589019,835.62
30/360τ = 180 ÷ 360 = 0.50000020,000.00

Actual days: 181. Days on a 30/360 count: 180. Most against least: 275.49 on this notional.

Worked example

A deposit and a bond trade

Two deals on the same desk. Try each step before revealing it.

Part 1: a euro deposit (ACT/360)

Step 1 of 4
  1. 1

    The deal

    A company in Spain deposits €5,000,000 at 3.25% from 15 January to 15 April 2026. Euro deposits use ACT/360.

Part 2: accrued interest on a US corporate bond (30/360)

Step 1 of 3
  1. 1

    The deal

    An investor buys $1,000,000 face value of a corporate bond with a 6% coupon, paid semi-annually. The last coupon was on 15 January 2026, and the trade settles on 10 June 2026.

Regulatory lens

After the LIBOR scandal, regulators pushed markets to replace LIBOR with risk-free rates built from real overnight transactions: SOFR for the dollar, SONIA for sterling, €STR for the euro and TONA for the yen. Most LIBOR settings stopped at the end of 2021 and the last US dollar settings after 30 June 2023. Each replacement rate keeps its market's day count, which is why the conventions in this lesson still matter on every new contract.

Looking ahead: the ISDA definitions that fix these conventions for derivatives return with swaps in later levels.

Sparks

SparkWhy a 360-day year?
The usual explanation is convenience before calculators: 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10 and 12, so a month is 30 days and a quarter 90. The habit stuck in money markets and never left.
SparkACT/360 quietly pays more
Hold a 5% ACT/360 deposit for a full 365-day year and you earn 5% × 365/360 = 5.0694%. Treasurers comparing a dollar deposit with a sterling one need to put both on the same basis first.

Key takeaways

  • Interest for part of a year: notional × rate × day count fraction τ.
  • ACT/360 and ACT/365F count actual days; 30/360 treats every month as 30 days.
  • Dollar and euro money markets use ACT/360; sterling and yen use ACT/365F; US corporate bonds use 30/360; government bonds usually ACT/ACT.
  • For the same quoted rate and dates, ACT/360 pays the most.
  • Convert between bases by scaling the rate: r₃₆₅ = r₃₆₀ × 365/360.

Reaction

Check what stuck

Question 1 · objective 1

A bank lends $2,000,000 for 91 days at 4% a year, ACT/360. How much interest is due, in $?

Question 2 · objective 1

Same deal in sterling: £2,000,000 for 91 days at 4%, ACT/365F. How much interest, in £?

Question 3 · objective 2

Under 30/360 (bond basis), how many days are there from 15 March 2026 to 31 August 2026?

days

Question 4 · objective 4

Same notional, same quoted rate, same dates. Which convention pays the most interest for a 6-month deposit?

Question 5 · objective 4

A dollar deposit pays 3% ACT/360. What ACT/365F rate pays the same interest? Answer in %, to four decimals.

%

Glossary

Day count convention
The rule that turns a period between two dates into a fraction of a year, for working out interest.
Day count fraction (τ)
The day count fraction: the part of a year the money is out for, under a given convention. Written τ.
Notional (N)
The amount on which interest is calculated. Written N.
Accrued interest
Interest earned since the last payment date but not yet paid.
ACT/360
Actual days divided by 360.
ACT/365F
Actual days divided by 365, also in leap years.
30/360
Days counted with 30-day months, divided by 360.
Risk-free rate (RFR)
An overnight interest rate benchmark based on actual transactions, such as SOFR, SONIA or €STR.

Deep Dives & Origins

Deep Dives

  • John C. Hull, Options, Futures, and Other Derivatives (Pearson), chapter 6, on day count conventions. Intermediate.
  • OpenGamma, Interest Rate Instruments and Market Conventions Guide (2nd edition, 2013). Advanced.

Origins

  • ISDA, 2006 ISDA Definitions, section 4.16 (Day Count Fraction).
  • Hull, Options, Futures, and Other Derivatives (Pearson).
  • Financial Conduct Authority, announcements on the future cessation of LIBOR (5 March 2021).
  • Federal Reserve Bank of New York, SOFR reference rate methodology; Bank of England, SONIA key features.

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Continuous compounding

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Real vs nominal rates