Accrued interest calculation

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 ... 

The calculation embodied in Table 3 shows that the total amount of capital required for the project is somewhere around six times the cost of the purchased and delivered equipment (a ratio of 5-7 can be expected, depending upon the complexity of the plant). However, it is equally important that the way in which this capital is to be spent be determined - in order that the demands on the company s funds and borrowing capacity can be foreseen well ahead of their requirement, and also so that the way in which any loan has to be drawn upon can be seen, and hence any accrued interest be calculated. 

It is the full price the bond s buyer pays the seller at delivery. However, the very next cash flow received and included in the present value calculation was not earned by the bond s buyer. A portion of the next coupon payment is the accrued interest. From Chapter 1, we know that accrued interest is the portion of a bond s next coupon payment that the bond s seller is entitled to depending on the amount of time the bond was held by the seller. Recall, the buyer recovers the accrued interest when the next coupon payment is delivered. 

It should be noted that the convention in the European bond markets is to quote the clean price and then calculate the accrued interest to obtain the dirty price. However, we want to emphasize that the answer is the same regardless of the order in which clean and dirty prices are calculated. ... 

The most obvious source of dollar return is the annual coupon interest payments. For the 1 million par value of this 5-year bond, the annual coupon payments consist of five payments of 30,000 with the first occurring on April 11, 2004. Since this bond has a settlement date that does not fall on a coupon payment date, the buyer pays the seller accrued interest. There are 89 days the first interest accrual date (11 April 2003) and the bond s settlement date of 9 July 2003. In addition, there are 366 days in the annual coupon period. At settlement, the buyer will pay the seller 7,295.08 (per 1 million in par value) in accrued interest which is calculated as follows ... 

The procedure for calculating the yield to call is the same as that for the yield to maturity determine the interest rate that will make the present value of the expected cash flows equal to the market price plus accrued interest. The expected cash flows are the coupon payments to a particular call date in the future and the call price. 

To illustrate this calculation, we use our hypothetical floater with a coupon rate of 0.0545 (in decimal) assuming a market price is 99.99. Assume the accrued interest is 0.3179 (per 100 of par value) and the... 

For settlement amounts, real accrued interest is calculated as for ordinary OATs. Clean price and accrued are each multiplied by the Index Ratio to arrive at a cash settlement amount. For actual coupons paid, the (real) annual coupon rate is multiplied by the Index Ratio for the payment date, and likewise for the par redemption amount (with the cash value subject to a par floor). 

Real interest is accrued on a European 30/360 basis. To calculate settlement amounts, real accrued interest and clean price are multiplied by the indexation ratio for the settlement date, as for France s issues. Also, as in France, coupon and redemption amounts are calculated by multiplying the real value of the payment by the indexation ratio for the payment date. All five coupon-paying bonds pay on 1 December each year. 

The settlement price and the accrued interest for Swedish inflation-linked bonds are calculated in the following way A reference CPI value is calculated for every day of the year based upon the CPI values CPI 3 and CPIf 2 for three months and two months prior to the month of settlement, respectively. These reference CPI values apply to the first day of the month containing the settlement date and the first day of the following month, respectively. The reference CPI for any day between these two dates is calculated by linear interpolation. 

For settlement purposes there are separate rounding conventions applied to the zero-coupon and coupon-paying issues. For zero-coupon bonds there is no rounding in the calculation, but the settlement price is rounded to the nearest krona. Coupon bonds are rounded once before you get to the settlement price, the clean price is rounded to three decimal places before adding on accrued interest. The settlement price is then rounded to the nearest krona. 

The coupon arrived at, now in money terms, is truncated to two decimal places for the two oldest existing linkers (2006 and 2011), and truncated to four decimal places for all others bar one. The exception is the new 2035 issue, which uses natural rounding to six decimal places. Accrued interest is calculated on the money value, not real value, of the coupon to be paid on an actual/actual basis. 

The two terms are not necessarily synonymous. The value date in a trade is the date on which the transaction acquires value for example, the date from which accrued interest is calculated. As such it may fall on a nonbusiness day such as a weekend or public holiday. The settlement date is the day on which the transaction settles or clears and so can only fall on a business day. 

In this example, one counterparty sells 10 million nominal of the UKT 5.75% 2009 at the spot price of 104.60, this being the market price of the bond at the time. The consideration for this trade is the market value of the stock, which is 10,505,560 as before. Repo interest is calculated on this amount at the rate of 5.75% for one week, and from this the termination proceeds are calculated. The termination proceeds are divided by the nominal amount of stock to obtain the forward dirty price of the bond on the termination date. For various reasons, the main one being that settlement systems deal in clean prices, we require the forward clean price, which is obtained by subtracting from the forward dirty price the accrued interest on the bond on the termination date. At the start of the trade the 5.75% 2009 had 29 days accrued interest, therefore on termination this figure will be 29 + 7 or 36 days. 

The cash proceeds in a repo are typically no more than the market value of the collateral. This minimises credit exposure by equating the value of the cash to that of the collateral. The market value of the collateral is calculated at its dirty price, not clean price—that is, including accrued interest. This is referred to as accrual pricing. To calculate the accrued... 

Calculation of the coupon income is the difference between the accrued interest bought in at the time of purchasing the cash market bond subtracted from the accrued interest received when the bond is sold. Market conventions play an important role here, for example How many days between trade and settlement How many days in a month How many days in a year In the United Kingdom the market convention is actual number of days in a month and a 365 days in a year. In Germany the convention is actual number of days in month and 360 days in a year. 

The final component of the default swap is the accrued premium that may be payable by the buyer to the seller. If a default occurs somewhere in between two premium payment dates, which is likely considering there are only four payment dates a year on a quarterly default swap, then it is standard market practice for the buyer of protection to pay the accrued premium from the most recent premium payment date to and including the date of default. The value of this accrued on default is calculated in a similar manner to the value of the default protection above. However, instead of receiving 1 - R upon a default, the buyer will be paying a certain amount of accrued interest. 

This formula calculates the fair price on a coupon payment date, so there is no accrued incorporated into the price. Accrued interest is an... 

Different markets have different settlement conventions. U.K. gilts, for example, normally setde on T + 1 one business day after the trade date, T. Eurobonds, on the other hand, settle on T + 3- The term value date is sometimes used in place of settlement date, however, the two terms are not strictly synonymous. A setdement date can fall only on a business day a bond traded on a Friday, therefore, will settle on a Monday. A value date, in contrast, can sometimes fall on a non-business day—when accrued interest is being calculated, for example. 

All bonds except zero-coupon bonds accrue interest on a daily basis that is then paid out on the coupon date. As mentioned earlier, the formulas discussed so far calculate bonds prices as of a coupon payment date, so that no accrued interest is incorporated in the price. In all major bond markets, the convention is to quote this so-called clean price. 

When a bond is traded, accrued interest is calculated from and including the last coupon date up to and excluding the value date, usually the settlement date. Interest does not accrue on bonds whose issuer has defaulted. 

As noted earlier, for bonds that are trading ex-dividend, the accrued coupon is negative and is subtracted from the clean price. The negative accrued interest is calculated using formula (1.26). 

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... 

Duration is a measure of price sensitivity to interest rates—that is, how much a bond s price changes in response to a change in interest rates. In mathematics, change like this is often expressed in terms of differential equations. The price-yield formula for a plain vanilla bond, introduced in chapter 1, is repeated as (2.1) below. It assumes complete years to maturity, annual coupon payments, and no accrued interest at the calculation date. 

The price of a TIPS comprises its real price plus any accrued interest, both of which are adjusted for inflation by multiplying them times the index ratio for the settlement date. The bond s unadjusted accrued interest, as explained in chapter 1, is calculated using expression (12.5). 

Once the CPIs have heen forecast, the bond s yield can be calculated. Assuming that the analysis is carried out on a coupon date so that accrued interest is zero, the money yield of a bond paying semiannual coupons is calculated by solving equation (12.9) for ri. 

On the value date, the bond has precisely four interest periods to maturity and no accrued interest. Its cash flows are 2.50, 2.50, 2.50, and 102.50. Assuming that the yield cunre on December 7 is flat at 5 percent, its price is calculated as follows ... 

Most corporate bonds, as well as municipals and Treasury notes and bonds, pay interest on a semiannual basis. To find the interest paid during the year, multiply the par value (face or maturity value) of the bond by the annual coupon rate. This amount is then divided by 2 to determine the amount of interest paid every 6 months. It is important to know that corporate securities use a 180-day coupon period—a commercial year of 360 days or 30 days per month, whereas government securities use an exact year of 365 days (366 days for a leap year). In addition, corporate securities are delivered 5 business days after the sale, whereas government securities are delivered the same day or the day after the sale. Prices for these securities are calculated as of the delivery date. Finally, if a bond is sold between coupon dates, it will have accrued interest since the last coupon date. This accrued interest must be added to the quoted price to determine the actual amount that the investor is required to pay. 

Y = yield to maturity or the discount rate used in present value calculations AI = amount of accured interest Ar = accrued interest per dollar of face value AY = approximation of the actual yield CY = approximation of the yield to call... 

When a bond is sold between coupon dates, the price actually paid by an investor is equal to the present value of all future cash flows to be received. This value is greater than the quoted price by the amount of accrued interest, where the accrued interest is equal to the portion of the next coupon to be received that is owed to the original owner of the bond. The actual calculation of accrued interest depends on whether the security is a corporate security or a government security. 

There are 181 days between November 15 and May 15, and 184 days between May 15 and November 15. If the date of sale is included, there are 51 days between November 15 and January 5. The amount of accrued interest may be calculated as follows ... 

Step 2. The amount of accrued interest (given In this example) can be calculated as... 

The delivery date of June 15, 1992 falls halfway between the last coupon payment and the next coupon payment (90 days). Accrued interest on this bond may be calculated as... 

This formula calculates the fair price on a coupon payment date, so there is no accrued interest incorporated into the price. Accrued interest is an accounting convention that treats coupon interest as accruing every day a bond is held this accrued amount is added to the discounted present value of the bond (the clean price) to obtain the market value of the bond, known as the dirty price. The price calculation is made as of the bond s settlement date, the date on which it actually changes hands after being traded. For a new bond issue, the settlement date is the day when the investors take delivery of the bond and the issuer receives payment. The settlement date for a bond traded in the secondary market—the market where bonds are bought and sold after they are first issued—is the day the buyer transfers payment to the seller of the bond and the seller transfers the bond to the buyer. 

Consider a bond with a dirty price—including the accrued interest the seller is entitled to receive—of 97-89, a coupon of 6 percent, and five years to maturity. FIGURE 1.6 shows the gross redemption yields this bond would have under the different yield-calculation conventions. 

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