Clean 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. 

For bonds that are trading ex-dividend, the accrued coupon is negative and would be subtracted from the clean price. The calculation is given below ... 

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. 

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. ... 

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. 

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. 

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. 

Index-linked gilts, like all other linkers covered in this chapter, are known as capital indexed bonds, where the income and principal are adjusted for changes in a consumer price index, subject to a lag. In the United Kingdom, the index is the RPI and the lag is eight months. The market trades on a clean price basis, with the quoted price a cash price (not a real price), including inflation accretion. 

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

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. 

Matnrity date Clean price Accrued interest Dirty price... 

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... 

Let s illustrate variation margin as it is applied to during the term of a trade. Exhibit 10.18 shows a 60-day repo in the 5% Treasury 2004, a UK gilt, where a margin of 2% is taken. The repo rate is 5V2%. The start of the trade is 5 January 2000. The clean price of the gilt is 95.25. 

The accrued interest and the clean price premium or discount compared to par value. Thus when pricing the asset swap it is necessary to compare the par value and to the underlying bond price. 

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 investors buy a bond in the market, what they pay is the bond s all-in price, also known as the dirty, or gross price, which is the clean price of a bond plus accrued interest. 

Accrued interest compensates sellers for giving up all the next coupon payment even though they will have held their bonds for part of the period since the last coupon payment. A bond s clean price moves with market interest rates. If the market rates are constant during a coupon period, the clean price will be constant as well. In contrast, the dirty price for the same bond will increase steadily as the coupon interest accrues from one coupon payment date until the next ex-dividend date, when it falls by the present value of the amount of the coupon payment. The dirty price at this point is below the clean price, reflecting the fact that accrued interest is now negative. This is because if the bond is traded during the ex-dividend period, the seller, not the buyer, receives the next coupon, and the lower price is the buyer s compensation for this loss. On the coupon date, the accrued interest is zero, so the clean and dirty prices are the same. 

As explained in chapter 1, yield is conventionally defined as the rate at which a bond s expected cash flows must be discounted so that the sum of their present values will equal the bond s clean price— that is, the price excluding any accrued interest. This is known as the bond s redemption... 

From this it is easy to derive the definition of true yield, which is the discount rate that equates a bond s current market price to the present value of its cash flows. The bond s market price is its dirty price—that is, the price including accrued interest. This is represented in (16.4), where PV is replaced by P, representing the clean price plus AI, representing accrued interest. 

The left side of equation (16.6) is the bond s market price broken into clean price plus accrued interest, as it was in (16.4). In fact, it can be shown that rmu is identical to the initial yield in (16.4), mt. The value of % that results in a stable future bond value is the Macaulay duration. At this point, assuming the existence of only one, parallel yield shift, a change in yield will not impact the future value of the bond. The bond s cash flows are immunized, and the instrument can be used to match a liability existing on that date. 

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. 

An FRN with a par value of 100, a quoted margin of 10 basis points over six-month LIBOR is currently trading at a clean price of 98.50. The previous LIBOR fixing was 5.375%. There are 90 days of accrued interest, 92 days until the next coupon payment and five (10) years from the next coupon payment before maturity. Therefore we have ... 

As of the trade date, this bond has three more coupons to pay plus its final maturity payment. The time to payment of each cash flow is shown in column G of FIGURE 17.4, which is a spreadsheet calculation of its yield. At our trade date, from Bloomberg page YA we see that the bond has a clean price of par and a redemption yield of 7.738 percent. This is shown at FIGURE 17.5. Can we check this on Excel ... 

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