Sep 11

Accrued Bond Interest. 2007 FRM.

by David Harper, CFA, FRM, CIPM

FRM |

Learning Outcome

  • LO 21.5: Calculate the accrued interest and invoice price on a coupon bond.

Interest accrues between coupon payments

In the chart below, a bond is settled in September, between semiannual coupons paid on July 1st and January 1st. The bond seller earns the accrued interest; i.e., the interest earned between the last coupon and the settlement date. The bond buyer earns the going-forward interest:

accruedInterest_2

 

Accrued interest = (1-w) x coupon

In order to calculate the accrued interest, we need to multiply the coupon payment by the fraction of time between the last coupon and the settlement (1-w since w is the going-forward fraction of the between-coupons period). In the spreadsheet below, we assume a 360 day year and the following:

  • The coupons are paid on July 1st and January 1st
  • The settlement date is 9/11/2007

Under a 360-day year, the number of days between coupons is 180:

  • The Excel function =DAYS360("7/1/07","1/1/08") gives 180
  • The number of days until the next coupon is given by (w) and equals 110. That is, =DAYS360("9/11/07","1/1/08") gives 110
  • The number of w periods is therefore 0.61 = 110/180. w periods represents the fractional going forward accrual period
  • (1-w) is the number of fractional periods that accrues to the seller. In this case, 1- w = 0.39
  • If our coupon, for example, pays $4 semiannually ($8 per year), then the accrued interest is $4 x 0.39 = $1.56

The price the buyer pays to the seller is called the full or dirty price; this full price includes the accrued interest owed to the seller.

Clean (flat) price = Full (invoice) price - accrued interest

The EditGrid spreadsheet below illustrates the relationship between clean price, full price and accrued interest. The input assumptions are shown in yellow:

  • Bond par value of $100
  • Yield (YTM) of 6%
  • Coupon pays 8% semiannually on January and July 1st. Each coupon pays $4.
  • Settlement is 9/11/2007

The first thing I did is calculate the full price, which is the present value (PV) of the bond on the settlement date, two ways:

  1. I calculated the price of the bond on July 1st, 2007 (when the last coupon is paid) because you can use the calculator or the =PV() function. Then I compounded that forward by (1-w) or 0.39 periods to bring it forward to the settlement date. That value is labeled "Grow P0 to settlement."
  2. I also discounted each of the cash flows (i.e., five coupons and one par) to the present value. That is labeled "Present value street method"

Both methods produce the same result: $105.79 is the full price (aka, invoice or dirty price).

Then we subtract the accrued interest to calculate the clean (flat) price:

  • $105.79 (full) - $1.56 (accrued interest) = $104.23 (clean)

Here is the EditGrid spreadsheet:

EditGrid Spreadsheet by bt/admin.

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