FRM Exam Part I · Interest Rate Futures

Day Count Conventions and Treasury Quoting Explained

Updated 11 October 2026 · Fact-checked

A day count convention sets how interest accrues between two dates. US Treasury bonds use actual/actual, corporate and municipal bonds use 30/360, and money market instruments such as T-bills use actual/360. Accrued interest = coupon × days since last coupon ÷ days in coupon period. Treasury prices are quoted in 32nds.

Understand Day Count Conventions and Quoting Conventions

Interest is earned over time, so you need a rule for counting the time. That rule is the day count convention. It is written as (days in the period) / (days in the reference year). Two markets can have the same coupon rate and still pay different cash amounts because they count days differently.

The three conventions you must know are:
- Actual/actual: count the real days. Used for US Treasury bonds. Accrued interest uses actual days since the last coupon over actual days in the coupon period.
- 30/360: every month is treated as 30 days and the year as 360. Used for US corporate and municipal bonds.
- Actual/360: count real days, divide by 360. Used for money market instruments such as T-bills.

When you buy a bond between coupon dates, you pay the seller for the interest earned so far. That is accrued interest. The price quoted in the market is the clean price (no accrued interest). What you actually pay is the cash (dirty) price = clean price + accrued interest. Quotes are clean so that the price does not jump on each coupon date.

Treasury bond and note prices are quoted in dollars and 32nds of a dollar per $100 face value. A quote of 95-16 means 95 + 16/32 = 95.5. Some quotes add fractions of a 32nd, but 32nds are the base unit. Treasury bills are quoted on a discount rate basis: cash price = 100 − discount rate × (n ÷ 360) × 100, where the rate is a decimal, or equivalently 100 − Y × n ÷ 360 with Y in percent. This rate is not a true yield, so it is often converted to a bond equivalent yield for comparison.

Key formulas to remember

Accrued interest (actual/actual): Accrued = (Coupon per period) × (Days since last coupon ÷ Days in coupon period); Both day counts are actual calendar days. Coupon per period is the annual coupon ÷ 2 for semiannual bonds.
30/360 day count: Days = 360 × (Y2 − Y1) + 30 × (M2 − M1) + (D2 − D1); Adjust first: if D1 = 31, set D1 = 30. If D2 = 31 and D1 is 30 or 31, set D2 = 30. Accrual fraction = days ÷ 360 for a year, or days ÷ 180 for a semiannual period.
Actual/360 accrual: Interest = Rate × (Actual days ÷ 360) × Principal; Gives slightly more interest than actual/365 for the same stated rate.
Cash (dirty) price: Cash price = Quoted (clean) price + Accrued interest; Quoted prices exclude accrued interest.
32nds quote: Price = whole dollars + (32nds ÷ 32); Example: 102-08 = 102 + 8/32 = 102.25. One 32nd = 0.03125 per $100.
T-bill price from discount rate: P = 100 − Y × (n ÷ 360); P per $100 face, Y is the quoted discount rate in percent, n is days to maturity.
Bond equivalent yield of a T-bill: BEY = [(100 − P) ÷ P] × (365 ÷ n); Uses the actual price paid as the base and a 365-day year, so it exceeds the discount rate.

How to solve Day Count Conventions and Quoting Conventions questions

Use the same sequence for any day count or quoting question. Identify the instrument first, because that tells you the convention.

1Identify the instrument: Treasury bond or note (actual/actual), corporate or municipal bond (30/360), or T-bill or money market (actual/360).
2Write down the dates: last coupon date, settlement date and next coupon date.
3Count the days using the right convention. For 30/360, apply the 31st-day adjustments before using the formula.
4Find the coupon per period and compute accrued interest as coupon × (days elapsed ÷ days in period).
5Convert the quote. For a 32nds quote, change it to a decimal. For a T-bill, convert the discount rate to a price.
6Add accrued interest to the clean price to get the cash price, and scale by face value (for example, × 1,000 for $100,000).
7Check reasonableness: accrued interest must be less than one coupon, and a T-bill price must be below 100.

Quickest way: Fraction-of-a-coupon shortcut

When to use it: Use it for any accrued interest question with a semiannual bond when the answer options are spaced widely.

Estimate the fraction of the coupon period elapsed (for example, 45 of 181 days is about one quarter).
Multiply by the semiannual coupon to get a rough accrued interest.
Eliminate options that are larger than the full coupon or far from your estimate.
Only then compute exactly with the day counts given.
For 32nds, convert by multiplying the 32nds figure by 0.03125 and adding it to the whole number.

Common mistakes in Day Count Conventions and Quoting Conventions

Using 365 days in the denominator for Treasury accrued interest.
Students think actual/actual means actual days over 365.
Fix: Divide by the actual number of days in the coupon period (about 181 to 184 for a semiannual bond), not the year.
Reading 95-16 as 95.16.
The hyphen looks like a decimal point.
Fix: The digits after the hyphen are 32nds. 95-16 = 95 + 16/32 = 95.50.
Forgetting to add accrued interest to the quoted price.
The quote looks like the price you pay.
Fix: Quoted price is clean. Cash price = quoted price + accrued interest. Check the question for which one it asks.
Skipping the 31st-day adjustment in 30/360.
Students plug dates directly into the formula.
Fix: Set D1 to 30 if it is 31. Set D2 to 30 if it is 31 and D1 is already 30 or 31.
Treating the T-bill discount rate as the yield.
Both are quoted as percentages.
Fix: The discount rate is based on face value and a 360-day year. Compute the price, then use BEY if a true yield is needed.
Applying the wrong convention to the instrument.
Students memorize conventions without linking them to markets.
Fix: Link them: Treasury bonds actual/actual, corporates 30/360, T-bills actual/360.

Worked examples

Example 1

A US Treasury bond has a 6% coupon paid semiannually on January 15 and July 15. Settlement is March 1 of a non-leap year. The quoted price is 98-08. Find the accrued interest and the cash price per $100 face value.

Show the solution

Treasury bond, so actual/actual.
Semiannual coupon = 6 ÷ 2 = 3 per $100.
Days from January 15 to March 1: 16 (rest of January) + 28 (February) + 1 = 45.
Days from January 15 to July 15: 16 + 28 + 31 + 30 + 31 + 30 + 15 = 181.
Accrued interest = 3 × 45 ÷ 181 = 0.7459.
Quote 98-08 = 98 + 8/32 = 98.25.
Cash price = 98.25 + 0.7459 = 98.9959.

Answer: Accrued interest is about 0.7459 per $100 and the cash price is about 98.9959 per $100 (about $98,995.86 for $100,000 face).

Example 2

A 91-day Treasury bill is quoted at a discount rate of 4.8%. Find the cash price per $100 face value and the bond equivalent yield.

Show the solution

T-bills use actual/360.
Price = 100 − 4.8 × 91 ÷ 360 = 100 − 1.2133 = 98.7867.
Discount amount = 100 − 98.7867 = 1.2133.
BEY = (1.2133 ÷ 98.7867) × (365 ÷ 91).
1.2133 ÷ 98.7867 = 0.012282.
365 ÷ 91 = 4.0110.
BEY = 0.012282 × 4.0110 = 0.04926, or about 4.93%.

Answer: The price is about 98.7867 per $100 face and the bond equivalent yield is about 4.93%, higher than the 4.8% discount rate.

Exam tips

Match the convention to the instrument first. Many questions are solved by this alone.
Read whether the question asks for the clean (quoted) or cash (dirty) price. Distractor options often include the wrong one.
Do the 32nds conversion before anything else. Mistakes here carry through the whole answer.
A financial calculator handles the arithmetic, but it will not count days for you. Count the days by hand and check both day totals.
Expect the T-bill discount rate versus yield contrast. The bond equivalent yield is always above the discount rate.

Practice questions from Interest Rate Futures

Day Count Conventions and Quoting Conventions: frequently asked questions

What is the difference between actual/actual and 30/360?

Actual/actual uses the real number of days in both the accrual period and the reference period. 30/360 treats every month as 30 days and the year as 360. Treasury bonds use the first, and corporate and municipal bonds use the second.

How do I calculate accrued interest on a Treasury bond?

Take the semiannual coupon and multiply by the days since the last coupon divided by the days in the current coupon period. Both counts are actual days. Add the result to the clean price to get the cash price.

How do I read a Treasury price quote in 32nds?

The number after the hyphen is the count of 32nds. So 101-12 means 101 + 12/32 = 101.375 per $100 of face value. Each 32nd equals 0.03125.

Why is actual/360 used for money market instruments?

It is the market convention for T-bills and similar short-term instruments. It gives slightly more interest than a 365-day year for the same stated rate, so check which convention a quoted rate uses before comparing.