## WWWFinance -- Day Counting for Bonds

```Latest Revision: December 1996.
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The ability to measure time is an important part of determining the value of an investment. In
BA 350, most present value calculations are computed using whole periods. But real life bond transactions are only executed on coupon dates occasionally. The same rules developed earlier for finding the present value or future value of a cash flow can be used to find the value of flows when the number of periods to a flow is not a whole number. For example, if the periodic rate is i and the cash flow is to be received in three and a half periods, then the money multiplier is:

M3.5 = (1+i)3.5

When a bond is traded between coupon dates, then the purchaser must pay accrued interest in addition to the price of a bond. Suppose, for example, that you buy a 12% bond at par on the day it is issued, but decide to sell it ten days later. Let us suppose that interest rates stay the same, so this bond is still priced at par. When you sell it, you will want to receive the interest due for the two days that you held the bond -- so you will want more than just the price of the bond. In general the accrued interest will be equal to the number of days interest has accrued (in this case, ten) divided by the number of days in the coupon period times the future coupon payment (in this case, 6% of the face value, if the bond pays semiannually).

It turns out that the amount of interest paid will depend on the kind of bond that you bought. Different types of bonds use different conventions for counting the time between two dates. The three most common methods are:

• Actual/Actual.
• 30/360.
• Actual/360.

### Actual/Actual Day Counting

Actual/Actual day counting is used for Treasury bonds and notes.

The Actual/Actual day counting method is the most intuitive of the day counting schemes. To determine the number of days between any two dates, we just count the actual number of days. For example, the number of days between February 25 and March 5 will be five in most years, and six in leap years.

In our case above, where we sold a 12% bond ten days after the coupon date, we would need to know the number of days between coupons. This can vary. Many Treasury bonds pay coupons on the 15th of the month. If the bond pays coupons in January and July, then the coupon periods can be 181, 182 or 184 days. This means that you will either be paid 0.05525, 0.05495, or 0.05435 times the amount of the coupon payment.

### 30/360 Day Counting

In the U.S., 30/360 day counting is used for corporate bonds, U.S. Agency bonds, and all mortgage backed securities.

The 30/360 day counting scheme was invented in the days before computers to make the computations easier. The premise is that for the purposes of computation, all months have 30 days, and all years have 360 days. This way, tables could be used to look up the value of accrued interest.

Using the 30/360 day count, we could again compute the number of days between February 25 and March 5. This time the answer is ten days, and it doesn't matter if it is a leap year or not. Since we assume all months are 30 days long, there will always be ten days between the 25th of any month and the 5th of the next month.

Similarly, there will always be 180 days between any two coupon dates, for bonds which pay semiannually.

The Public Securities Association (PSA) publishes the following rules for calculating the number of days between any two dates:

The number of days from M1/D1/Y1 to M2/D2/Y2 is computed according to the following procedure:

1. If D1 is 31, change D1 to 30.

2. if D2 is 31 and D1 is 30 or 31, then change D2 to 30.

3. If M1 is 2, and D1 is 28 (in a non-leap year) or 29, then change D1 to 30.

Then the number of days, N is:

N = 360(Y2 - Y1) + 30(M2 - M1) + (D2 - D1)

### Actual/360 Day Counting

Actual/360 Day Counting is used for bank deposits and in calculating rates pegged to some indices, such as LIBOR.

Bank deposits compound interest daily (using an actual day count), but assume 360 days per year to calculate the daily rate. So, if the stated rate is 6%, then the daily rate is 6%/360 = 0.0166667%.

Acknowledgement

This note was written by Mark Taranto, August 1996.