FRM Exam Part I · Pricing Conventions, Discounting, and Arbitrage

Bond Pricing Conventions and Day Counts for FRM Part I

Updated 11 October 2026 · Fact-checked

Bond pricing conventions fix how prices are quoted and interest is counted. The quoted clean price excludes accrued interest. You pay the dirty price, which is clean plus accrued. Accrued interest = coupon × (days since last coupon ÷ days in coupon period), with the day count set by the market: 30/360, Actual/Actual or Actual/360.

Understand Bond Pricing Conventions and Day Counts

A bond pays coupons on set dates, but you can buy it on any day. If you buy halfway through a coupon period, the seller has earned half of that coupon. The buyer pays the seller for it. That amount is accrued interest.

Markets quote the clean price, which leaves accrued interest out. The clean price moves only when yields move. If it included accrued interest, the quote would jump down on every coupon date even though nothing changed in value. The cash you actually pay at settlement is the dirty price (also called the full or invoice price): dirty price = clean price + accrued interest.

To measure accrued interest you need a day count convention. It says how to count the days between two dates and how long the coupon period or year is. The common ones are:

30/360: every month has 30 days and the year has 360. Used for US corporate, agency and municipal bonds.
Actual/Actual: real days elapsed over the real days in the coupon period. Used for US Treasury notes and bonds.
Actual/360: real days over a 360-day year. Used for money market instruments such as T-bills and many floating-rate deals.

Treasury bills pay no coupon. They are quoted on a discount rate basis: the discount is taken off face value and annualised on 360 days. This is not a true yield. To compare a bill with a bond, convert it to a bond equivalent yield, which divides the gain by the price paid and annualises on 365 days.

Key formulas to remember

Dirty price: Dirty price = Clean price + Accrued interest; Dirty price is the cash paid at settlement. Clean price is the quote.
Accrued interest: AI = Coupon per period × (Days from last coupon to settlement ÷ Days in coupon period); Coupon per period = annual coupon ÷ payments per year. Use the day count of the bond for both day numbers.
30/360 day count: Days = 360 × (Y2 − Y1) + 30 × (M2 − M1) + (D2 − D1); Before computing: if D1 = 31, set D1 = 30. If D2 = 31 and D1 is 30 or 31, set D2 = 30. This is the standard US rule.
Accrued interest, 30/360 shortcut: AI = Annual coupon × Days (30/360) ÷ 360; Equivalent to the period formula when the period is 180 days (semiannual) or 360 days (annual).
T-bill price from discount rate: P = 100 − d × (n ÷ 360) × 100, with d as a decimal; n is actual days to maturity. P is per 100 face value.
T-bill bond equivalent yield: BEY = ((100 − P) ÷ P) × (365 ÷ n); Uses price as the base and 365 days. For bills of up to six months this is the usual simple-interest form. BEY is higher than the discount rate.

How to solve Bond Pricing Conventions and Day Counts questions

Use this order for any question on quotes, accrued interest or bill yields. Most lost marks come from using the wrong day count or the wrong base.

1Identify the instrument and its convention: Treasury bond (Actual/Actual), corporate or municipal (30/360), or money market or T-bill (Actual/360).
2Write down the dates: last coupon date, settlement date and next coupon date. Do not count the last coupon date itself as a day you earned interest beyond the stated difference.
3Count the accrual days with the right method: real calendar days, or the 30/360 formula with the 31st-day adjustments.
4Count the denominator: days in the coupon period for Actual/Actual, 180 or 360 for 30/360, or 360 for Actual/360.
5Compute accrued interest = coupon per period × (accrual days ÷ period days). Remember semiannual coupons are half the annual coupon.
6Convert between prices: dirty = clean + accrued, or clean = dirty − accrued. Scale to the face amount at the end.
7For T-bills, price from the discount rate with 360 days, then compute BEY with price as the base and 365 days.
8Check reasonableness: accrued interest must be between zero and one full coupon, and the BEY must exceed the discount rate.

Quickest way: Fraction-of-period shortcut

When to use it: Use for accrued interest and clean/dirty conversions when the question gives the dates or the number of days directly.

Compute the fraction f = accrual days ÷ period days. Sanity check it is between 0 and 1.
Accrued interest = f × coupon per period. On a calculator, enter days ÷ period days first, then multiply.
Add to the clean price for dirty, subtract from the dirty price for clean.
For T-bills, compute discount = d × n ÷ 360 first, subtract from 100 for price, then divide the discount by the price and multiply by 365 ÷ n.
Eliminate options: if an answer has accrued interest above one coupon, or a BEY below the discount rate, it is wrong.

Common mistakes in Bond Pricing Conventions and Day Counts

Using the wrong day count, for example real days on a corporate bond that uses 30/360.
Students default to calendar days because they are easier to count.
Fix: Read the instrument first. Treasury notes and bonds use Actual/Actual, corporates use 30/360, bills use Actual/360. Use the convention the question names.
Dividing by 365 or 360 for a Treasury bond's accrued interest instead of the days in the coupon period.
Mixing up the bond rule with the money market rule.
Fix: For Actual/Actual, the denominator is the actual days between the two coupon dates, such as 181 to 184 for a semiannual bond.
Using the annual coupon instead of the coupon per period.
Students skip the step of dividing by the payment frequency.
Fix: Write coupon per period first: a 6% semiannual bond pays 3 per 100 each period.
Treating the T-bill discount rate as a yield or dividing the discount by face value in the BEY.
The discount rate looks like an interest rate, but it is based on face value and a 360-day year.
Fix: Convert to price first. BEY = (100 − P) ÷ P × 365 ÷ n. The result is always higher than the discount rate.
Paying the clean price at settlement, or quoting dirty when asked for clean.
Students forget that the quote leaves accrued interest out.
Fix: Cash paid = clean + accrued. If the question asks for the quote, subtract accrued from the full price.
Forgetting the 30/360 adjustments when a date falls on the 31st.
The formula looks mechanical, and the 31st-day rule is easy to skip.
Fix: Change D1 = 31 to 30 first. Change D2 = 31 to 30 only if D1 is then 30 or 31.

Worked examples

Example 1

A US Treasury bond with a 6% annual coupon pays semiannually, on March 5 and September 5. Settlement is July 3. The clean price is 98.50 per 100 face. Find the accrued interest, the dirty price, and the amount paid for $1,000,000 face value.

Show the solution

Convention: Treasury bond, so Actual/Actual.
Coupon per period = 6% ÷ 2 = 3 per 100.
Days from March 5 to July 3: March 5 to July 5 is 31 + 30 + 31 + 30 = 122 days, so July 3 is 120 days.
Days in the coupon period, March 5 to September 5: 31 + 30 + 31 + 30 + 31 + 31 = 184 days.
Accrued interest = 3 × 120 ÷ 184 = 1.9565 per 100.
Dirty price = 98.50 + 1.9565 = 100.4565 per 100.
For $1,000,000 face: 100.4565% × 1,000,000 = $1,004,565 (rounded to the nearest dollar).

Answer: Accrued interest is 1.9565 per 100, the dirty price is 100.4565, and the buyer pays about $1,004,565.

Example 2

A 91-day Treasury bill has a quoted discount rate of 4.80%. Find its price per 100 face value and its bond equivalent yield.

Show the solution

Convention: discount rate uses Actual/360.
Discount = 4.80% × 91 ÷ 360 × 100 = 1.2133 per 100.
Price = 100 − 1.2133 = 98.7867.
BEY = (100 − 98.7867) ÷ 98.7867 × 365 ÷ 91.
(1.2133 ÷ 98.7867) = 0.012282.
365 ÷ 91 = 4.0110, so BEY = 0.012282 × 4.0110 = 0.04926, or 4.93%.
Check: BEY 4.93% is above the discount rate of 4.80%, as expected.

Answer: Price = 98.7867 per 100. Bond equivalent yield ≈ 4.93%.

Exam tips

Read the convention in the question before touching numbers. A wrong day count gives a plausible but wrong answer, and wrong options are built from it.
Write the fraction (days ÷ period days) on its own line. Check it is between 0 and 1 before multiplying by the coupon.
For T-bills, always go through price. Discount rate leads to price, and price leads to BEY. Do not shortcut between rates.
Check whether the question asks for the quote (clean), the cash paid (dirty) or the accrued interest. Underline it so you answer the right one.
Use the calculator memory for the intermediate price. Rounding the price early shifts the BEY in the third decimal and can change your answer choice.

Practice questions from Pricing Conventions, Discounting, and Arbitrage

Bond Pricing Conventions and Day Counts: frequently asked questions

What is the difference between a clean price and a dirty price?

The clean price is the quoted price and excludes accrued interest. The dirty price is the clean price plus accrued interest, and it is the cash the buyer pays at settlement. The clean price changes only when yields change, while the dirty price rises steadily between coupons and drops on the coupon date.

How do I calculate accrued interest on a bond?

Multiply the coupon per period by the days since the last coupon divided by the days in the coupon period. For a 30/360 bond, you can also use annual coupon × 30/360 days ÷ 360. Count days using the bond's own convention.

Which day count convention applies to which instrument?

As the exam presents them, US Treasury notes and bonds use Actual/Actual, US corporate and municipal bonds use 30/360, and money market instruments such as T-bills use Actual/360. If the question names a convention, always use that one.

Why is the T-bill bond equivalent yield higher than the discount rate?

The discount rate divides the gain by face value, which is larger than the price paid, and it uses a 360-day year. The bond equivalent yield divides by the lower price and uses 365 days. Both changes raise the number.