# DAY COUNT CONVENTIONS Global Money Markets

To those unfamiliar with the workings of financial markets, it may come as a shock that there is no wide spread agreement as to how many days there are in a year.The procedures used for calculating the number of days between two dates (e.g., the number of days between the settlement date and the maturity date) are called day count conventions.

## Day Count Convention by Country

Day count conventions vary across different types of securities and across countries. In this section, we will introduce the day count conventions relevant to the money markets.

Day Count Basis

The day count basis specifies the convention used to determine the number of days in a month and in a year. According to the Securities Industry Association Standard Securities Calculation Methods book, Volume2, the notation used to identify the day count basis is:

Although there are numerous day count conventions used in the fixed-income markets around the world, there are three basic types. All day count conventions used world wide are variations of these three types. The first type specifies that each month has the actual number of calendar days in that month and each year has the actual number of calendar days in that year or in a coupon period (e.g., Actual/Actual). The second type specifies that each month has the actual number of calendar days in that month but restricts the number of days in each year to a certain number of days regardless of the actual number of days in that year (e.g., Actual/360). Finally, the third types restricts both the number of days in a month and in a year to a certain number of days regardless of the actual number of days in that month/year (e.g., 30/360). Below we will define and illustrate the three types of day count conventions.

Actual/Actual

Treasury notes, bonds and STRIPS use an Actual/Actual (in period) day count convention. When calculating the number of days between two dates, the Actual/Actual day count convention uses the actual number of calendar days as the name implies. Let’s illustrate the Actual/Actual day count convention with a 3.625% coupon, 2-year U.S. Treasury note with a maturity date of August 31, 2003. The Bloomberg Security Display(DES) screen for this security is presented in Exhibit.In the “Security Information” box on the left-hand side of the screen, we see that the day count is specified as “ACT/ACT.” From the “Issuance Info” box on the right-hand side of the screen, we see that interest starts accruing on August 31, 2001 (the issuance date) and the first coupon date is February 28, 2002. Suppose this bond is traded with a settlement date of September11, 2001. How many days are there between August 31, 2001 and September 11, 2001 using the Actual/Actual day count convention?

To answer this question, we simply count the actual number of days between these two dates. To do this, we utilize Bloomberg’s DCX (Days Between Dates) function presented in Exhibit. The function tells us there are 11 actual days between August 31, 2001 and September 11,2001.4 In the same manner, we can also determine the actual number of calendar days in the full coupon period. A full 6-month coupon period can only have 181, 182, 183 or 184 calendar days. For example, the actual number of days between August 31, 2001 and February 28, 2002 is 184.

Bloomberg Security Description Screen for a2-Year U.S. Treasury Note Source: Bloomberg Financial Markets

Bloomberg DCX (Days Between Dates) Screen Source: Bloomberg Financial Markets

Bloomberg Security Description Screen of a 26-Week U.S. Treasury Bill Source: Bloomberg Financial Markets

Actual/360

Actual/360 is the second type of day count convention. Specifically, Actual/360 specifies that each month has the same number of days as indicated by the calendar. However, each year is assumed to have 360 days regardless of the actual number of days in a year. Actual/360 is the day count convention used in U.S. money markets. Let’s illustrate the Actual/360 day count with a 26-week U.S. Treasury bill which matures on March 7, 2002. The Bloomberg Security Display (DES) screen for this security is presented in Exhibit. From the “Security Information” box on the left-hand side of the screen, we see that the day count is specified as “ACT/360.” Suppose this Treasury bill is purchased with a settlement date on September 11, 2001 at a price of 98.466. How many days does this bill have until maturity using the Actual/360 day count convention?

Once again, the question is easily answered using Bloomberg’s DCX(Days Between Dates) function and specifying the two dates of interest.This screen is presented in Exhibit . We see that with a settlement date of September 11, 2001 there are 177 calendar days until maturity on March 7, 2002. This can be confirmed by examining the Bloomberg’s YA(Yield Analysis) screen in Exhibit . We see that with a settlement date of September 11, 2001 this Treasury bill has 177 days to maturity. This information is located just above the “Price” box in the center of the screen.

Bloomberg DCX (Days Between Dates) Screen Source: Bloomberg Financial Markets

Bloomberg Yield Analysis for a 26-Week U.S. Treasury Bill Source: Bloomberg Financial Markets

When computing the number of days between two dates, Actual/360 and Actual/Actual will give the same answer. What then is the importance of the 360-day year in the Actual/360 day count? The difference is apparent when we want to compare, say, the yield on 26-week Treasury bill with a coupon Treasury which has six months remaining to maturity.U.S.Treasury bills, like many money market instruments, are discount instruments. As such, their yields are quoted on a bank discount basis which determine the bill’s. The quoted yield on a bank discount basis for a Treasury bill is not directly comparable to the yield on a coupon Treasury using an Actual/Actual day count for two reasons. First, the Treasury bill’s yield is based on a face-value investment rather than on the price. Second, the Treasury bill yield is annualized according to a 360-day year while a coupon Treasury’s yield is annualized using the actual number of days in a calendar year (365 or 366). These factors make it difficult to compare Treasury bill yields with yields on Treasury notes and bonds. We demonstrate how these yields can be adjusted to make them comparable shortly.

Another variant of this second day count type is the Actual/365. Actual/365 specifies that each month has the same number of days as indicated by the calendar and each year is assumed to have 365 days regardless of the actual number of days in a year. Actual/365 does not consider the extra day in a leap year. This day count convention is used in the UK money markets.

30/360

The 30/360 day count is the most prominent example of the third type of day count convention which restricts both the number of days in a month and in a year to a certain number of days regardless of the actual number of days in that month/year. With the 30/360 day count all months are assumed to have 30 days and all years are assumed to have360 days. The number of days between two dates using a 30/360 day will usually differ from the actual number of days between the two dates.

To determine the number of days between two dates, we will adopt the following notation:

Y1 = year of the earlier date

M1 = month of the earlier date

D1 = day of the earlier date

Y2 = year of the later date

M2 = month of the later date

D2 = day of the later date

Since the 30/360 day count assumes that all months have 30 days,some adjustments must be made for months having 31 days and February which has 28 days (29 days in a leap year). The following adjustments accomplish this task:

1. 1. If the bond follows the End-of-Month rule and D2 is the last day of February (the 28th in a non-leap year and the 29th in a leap year) andD1 is the last day of February, change D2 to 30.
2. 2. If the bond follows the End-of-Month rule and D1 is the last day of February, change D1 to 30.
3. 3. If D2 is 31 and D1 is 30 or 31, change D2 to 30.
4. 4. If D1 is 31, change D1 to 30.

Once these adjustments are made, the formula for calculating the number of days between two dates is as follows:

Number of days = [(Y2 − Y1) × 360] + [(M2 − M1) × 30] + (D2 − D1)

To illustrate the 30/360 day count convention, let’s use a 4% coupon bond which matures on August 15, 2003, issued by Fannie Mae. The Bloomberg Security Description (DES) screen for this bond is presented in Exhibit .We see that in the “Security Information” box that the bond has a 30/360 day count. Suppose the bond is purchased with a settlement date of September 11, 2001. We see from the lower left-hand corner of the screen that the first coupon date is February 15, 2002 and the first interest accrual date is August 27, 2001. How many days have elapsed in the first coupon period from August 27, 2001 until the settlement date of September 11, 2001 using the 30/360 day count convention?

Referring back to the 30/360 day count rule, we see that adjustments1 through 4 do not apply in this example so no adjustments to D1 and D2 are required. Accordingly, in this example,

Y1 = 2001

M1 = 8

D1 = 27

Y2 = 2001

M2 = 9

D2 = 11

Inserting these numbers into the formula, we find that the number of days between these two dates is 14, which is calculated as follows:

Number of days = [(2000 – 2000) × 360] + [(9 – 8) × 30] + (11 – 27)

= 0 + 30 + (–16)

= 14

To check this, let’s employ Bloomberg’s DCX (Days Between Dates)function presented in Exhibit .The function tells us there are 14 days between August 27, 2001 and September 11, 2001 using a 30/360 day count. Note that the actual number of days between these two dates is 15.

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