Accrued interest

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. 

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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. 

We assume we have constructed a market curve of Libor discount factors where Df(t) is the price today of 1 to be paid at time t. From the perspective of the asset swap seller, it sells the bond for par plus accrued interest. The net up-front payment has a value 100 F where P is the market price of the bond. If we assume both parties to the swap are interbank credit quality, we can price the cash flows off the Libor curve. 

In order to obtain the full or invoice price, we need to take into account the accrued interest. The accrued interest should be adjusted with the inflation for each period. The adjusted accrued interest is given by Equation (6.23) ... 

Convertible instruments are usually issued with attached call or put options. Such features can be implemented into the valuation model. If a soft call feature has been implemented, it enables the issuer to force the conversion when the share price overcomes a percentage or trigger level above the conversion price. However, this option cannot be called in the first years hard call . Differently, after the protection period, the issuer can exercise the option. This second time is referred to soft call . Using the same example shown in Section 9.3.1, we assume that the bond may be redeemed in whole but not in part at their principal amount plus accrued interest on the last 2 years, in which the maturity date is at 20 February 2019. On and after this call date , if the share price exceeds 130% of the conversion price the issuer can force the conversion. Figure 9.23 shows the stock price tree in which at years 4 and 5 the stock price is above the threshold. 

The reverse convertible bonds have increased popularity in Europe and United States. This type of instrument gives to the issuer (not the bondholder) at maturity the right to exchange the bond into shares or to redeem it at par value plus accrued interests. In the first case, the bond is exchanged if the share price is less than conversion price, or if the conversion value is less than par value. Conversely, the issuer can redeem the bond. They typically have a domestic stock as underlying security, but they can also include foreign shares and indexes. 

All bonds coupon-paying bonds accrue interest on a daily basis, and this is then paid out on the coupon date. In determination of the fair price... 

In all major bond markets the convention is to quote price as a clean price. This is the price of the bond as given by the present value of its cash flows, but excluding coupon interest that has accrued on the bond since the last dividend payment. As all bonds accrne interest on a daily basis, even if a bond is held for only one day, interest will have been earned by the bondholder. However, we have referred already to a bond s all-in price, which is the price that is actually paid for the bond in the market. This is also known as the dirty price (or gross price), which is the clean price of a bond plus accrued interest. In other words, the accrued interest must be added to the quoted price to get the total consideration for the bond. 

Interest accrues on a bond from and including the last coupon date up to and excluding what is called the value date. The value date is almost always the settlement date for the bond, or the date when a bond is passed to the buyer and the seller receives payment. Interest does not accrue on bonds whose issuer has subsequently gone into default. Bonds that trade without accrued interest are said to be trading flat or clean. By definition therefore,... 

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. 

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. 

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. 

As an illustration, we return to the previous example with the 2% German government bond. Since there are 366 days in the coupon period and 345 days from the settlement date to the next coupon period, there are 21 days (366 - 345) in the accrued interest period. Therefore, the percentage of the next coupon payment that is accrued interest is... 

Given the value of w, the amount of accrued interest (A7) is equal to... 

Accordingly, using a 2% German government bond with a settlement date of 8 July 2003, the portion of the next coupon payment that is accrued interest is... 

Once we know the full price and the accrued interest, we can determine the clean price. The clean price is the price that quoted in the market and represents the bond s value to the new bondholder. The clean price is computed as follows... 

Clean price = Full price - Accrued interest In our illustration, the clean price is... 

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. ... 

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

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 accrued interest is located on the right-hand side of the screen under Payment Invoice and is labeled 89 days accrued int. ... 

The investor s tenure as a bond s owner ends as a result of one of the following circumstances. First, the investor may simply sell the bond and will receive the bond s prevailing market price plus accrued interest. Next, the issuer may call the bond in which case the investor receives the call price plus accrued interest or the investor may put the bond and receive the put price plus accrued interest. Lastly, if the bond matures, the investor will receive the maturity value plus the final coupon payment. Regardless of the reason, if the proceeds received are greater than the investor s initial purchase price, a capital gain is generated, which is an additional source of dollar return. Similarly, if the proceeds received are less than the investor s initial purchase price, a capital loss is gener-... 

The most common measure of yield in the bond market is the yield to maturity. The yield to maturity is simply a bond s internal rate of return. Specifically, the yield to maturity is the interest rate that will make the present value of the bond s cash flows equal to its market price plus accrued interest (i.e., the full price). To find the yield to maturity, we must first determine the bond s expected future cash flows. Then we search by trial and error for the interest rate that will make the present value of the bond s cash flows equal to the market price plus accrued interest. 

Yield to next call is the yield to call for the next call date after the current settlement date. For the DZ Bank bond, the next call date is 10 April 2004. The yield to next call is 2.154%. Specifically, an annual interest of 2.154% makes the present value of the next coupon payment and the call price of 100 (i.e., the bond s cash flows assuming it will be called on 10 April 2004) equal to the current market price of 102.5198 plus the accrued interest. 

A yield can be calculated given the projected cash flows based on an assumed prepayment rate. The yield is the interest rate that will make the present value of the assumed cash flows equal to the clean price plus accrued interest. A yield calculated in this manner is called a cash flow yield. 

The zero-volatility spread, also referred to as the Z-spread or static spread, is a measure of the spread that the investor would realize over the entire benchmark spot rate curve if the bond were held to maturity. Unlike the nominal spread, it is not a spread at one point on the yield curve. The Z-spread is the spread that will make the present value of the cash flows from the nongovernment bond, when discounted at the benchmark rate plus the spread, equal to the nongovernment bond s market price plus accrued interest. A trial-and-error procedure is used to compute the Z-spread. 

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... 

The zero-volatility or static spread is the spread that when added to the government spot rate curve will make the present value of the cash flows equal to the bond s price plus accrued interest. When spread is defined in this way, spread dnration is the approximate percentage change in price for a 100 basis point change in the zero-volatility spread holding the government spot rate curve constant. 

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. 

Swedish bonds, like Australian and UK bonds, have a short period before the coupon is paid where the buyer will not be entitled to receive that coupon. In Sweden this ex-dividend period is five working days, during which the accrued interest will be negative. 

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. 

Gilts are registered securities. All gilts pay coupon to the registered holder as at a specified record date the record date is seven business days before the coupon payment date. The period between the record date and the coupon date is known as the ex-dividend or ex-div ( xd ) period during the ex-dividend period the bond trades without accrued interest. This is illustrated in Exhibit 9.1. 

Future price x Price factor + Accrued interest at delivery + Coupon income) (- Long gilt clean price + Accrued interest at purchase + Financing cost)... 

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