Day-count convention

The same day count convention, in this case actual/actual ... 

The accrued interest calculation for a bond is dependent on the day-count basis specified for the bond in question. We have already seen that when bonds are traded in the market the actual consideration that changes hands is made up of the clean price of the bond together with the accrued that has accumulated on the bond since the last coupon payment these two components make up the dirty price of the bond. When calculating the accrued interest, the market will use the appropriate day-count convention for that bond. A particular market will apply one of five different methods to calculate accrued interest these are ... 

In order to value a bond with the settlement date between coupon payments, we must answer three questions. First, how many days are there until the next coupon payment date From Chapter 1, we know the answer depends on the day count convention for the bond being valued. Second, how should we compute the present value of the cash flows received over the fractional period Third, how much must the buyer compensate the seller for the coupon earned over the fractional period This is accrued interest that we computed in Chapter 1. In the next two sections, we will answer these three questions in order to determine the full price and the clean price of a coupon bond. 

The last step in this process is to find the bond s value without accrued interest (called the clean price or simply price). To do this, the accrued interest must be computed. The first step is to determine the number of days in the accrued interest period (i.e., the number of days between the last coupon payment date and the settlement date) using the appropriate day count convention. For ease of exposition, we will assume in the example that follows that the actual/actual calendar is used. We will also assume there are only two bondholders in a given coupon period— the buyer and the seller. 

See the discussion on day count conventions and accrued interest in Chapter 1. 

We will illustrate this process using a hypothetical 4-year floater that deliver cash flows quarterly with a coupon formula equal to 3-month LIBOR plus 15 basis points and does not possess a cap or a floor. The coupon reset and payment dates are assumed to be the same. For ease of exposition, we will invoke some simplifying assumptions. First, the issue will be priced on a coupon reset date. Second, although floaters typically use an ACT/360 day-count convention, for simplicity we will assume that each quarter has 91 days. Third, we will assume initially that the LIBOR yield curve is flat such that all implied 3-month LIBOR forward rates are the same. (We will relax this assumption shortly.) Note the same principles apply with equal force when these assumptions are relaxed. 

While our illustrations assume that the timing of the cash flows for both the fixed-rate payer and floating-rate payer will be the same, this is rarely the case in a swap. An agreement may call for the fixed-rate payer to make payments annually but the floating-rate payer to make payments more frequently (semi-annually or quarterly). Also, the way in which interest accrues on each leg of the transaction differs, because there are several day count conventions in the fixed-income markets as discussed in Chapter 3. 

The quarterly floating-rate payments are based on an actual/360 day count convention. Recall that this convention means that 360 days are assumed in a year and that in computing the interest for the quarter the... 

In our illustration we will assume that the frequency of settlement is quarterly for the fixed-rate payments, the same as with the floating-rate payments. The day count convention is the same as for the floating-rate payment, actual/360. The equation for determining the euro amount of the fixed-rate payment for the period is... 

Currency/Rate Payment Freq. Compounding Freq. Day Count Convention... 

To derive the swap term structure, observed market interest rates combined with interpolation techniques are used also, dates are constructed using the applicable business-day convention. Swaps are frequently con-strncted nsing the modified following bnsiness-day convention, where the cash flow occurs on the next business day unless that day falls in a different month. In that case, the cash flow occurs on the immediately preceding business day to keep payment dates in the same month. The swap curve yield calculation convention frequently differs by currency. Exhibit 20.2 lists the different payment frequencies, compounding frequencies, and day count conventions, as applicable to each currency-specific interest rate type. 

The short end of the swap curve, out to three months, is based on the overnight, 1-month, 2-month, and 3-month deposit rates. The short-end deposit rates are inherently zero-coupon rates and need only be converted to the base currency swap rate compounding frequency and day count convention. The following equation is solved to compute the continuously compounded zero-swap rate (r ) ... 

The denominator of this ratio is the number of calendar days between the last coupon date and the next one. This figure depends on the day-count convention (see below) used for that particular bond. Using /, the price formula is modified as (1.17) (for annual-coupon-paying bonds for bonds with semiannual coupons, r/2 replaces r). 

In calculating the accrued interest on a bond, the market uses the day-count convention appropriate to that bond. These conventions govern both the number of days assumed to be in a calendar year and how the days between two dates are figured. FIGURE 1.7 shows how the different... 

The expression also assumes an actual/365 day count. If any other day-count convention is used, the 1/A factor must he replaced by a fraction whose numerator is the actual number of days and whose denominator is the appropriate year base. 

FIGURE 1.8 Accrued interest day-count convention rules... 

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