Periodic coupon payment

In the United States, all bonds make periodic coupon payments except for one type the sero-coupon. Zero-coupon bonds do not pay any coupon. Instead investors buy them at a discount to face value and redeem them at... 

TIPS periodic coupon payments and their final redemption payments are both calculated using an inflation adjustment. Known as the inflation compensation, or IC, this is defined as in expression (12.2). 

After the first coupon payment, Barro attributes a double weight on the periodical inflation rate before the coupon payment. For instance, in the case of perfect indexation, the second coupon payment should be adjusted in the following way ... 

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. 

Note for the first coupon payment subsequent to the settlement date, t = 1 so the exponent is just w. This procedure for calculating the present value when a bond is purchased between coupon payments is called the Street method. In the Street method, as can be seen in the expression above, coupon interest is compounded over the fractional period w. 

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... 

The periodic interest payments made by the issuer (i.e., coupon payments). 

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 source of dollar return called reinvestment income represents the interest earned from reinvesting the bond s interim cash flows (interest and/or principal payments) until the bond is removed from the investor s portfolio. With the exception of zero-coupon bonds, fixed income securities deliver coupon payments that can be reinvested. Moreover, amortizing securities (e.g., mortgage-backed and asset-backed securities) make periodic principal repayments which can also be invested. 

Exhibit 3.16 presents the calculation of the discount margin for this security. Each period in the security s life is enumerated in Column (1), while the Column (2) shows the current value of the reference rate. Column (3) sets forth the security s cash flows. For the first 11 periods, the cash flow is equal to the reference rate (10%) plus the quoted margin of 80 basis points multiplied by 100 and then divided by 2. In last 6-month period, the cash flow is 105.40—the final coupon payment of 5.40 plus the maturity value of 100. Different assumed margins appear at the top of the last five columns. The rows below the assumed margin indicate the present value of each period s cash flow for that particular... 

To see the significance of the second drawback, it is useful to partition the value of an option-free floater into two parts (1) the present value of the security s cash flows (i.e., coupon payments and matnrity value) if the discount margin equals the quoted margin and (2) the present value of an annuity which pays the difference between the quoted margin and the discount margin mnltiplied by 100 and divided by the number of periods per year. 

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. 

If the futures contract is held until 10 December 2002, a total of 159 days will have elapsed since the last coupon payment. This period will generate interest to the value of 2.208. The coupon income will be 2.208 -0.986 = 1.222. 

Consider a bond paying a periodic cash payment p at times Ti,T2,...,T , and the principal at maturity T = T j. A coupon bond can be mapped into a portfolio of discount bonds with corresponding maturities (under one source of uncertainty, that is one factor model). The value of a coupon bearing bond at time t [Pg.594]

A bond s term to maturity is crucial because it indicates the period during which the bondholder can expect to receive coupon payments and the number of years before the principal is paid back. The principal of a bond—also referred to as its redemption value, maturity value, par value, or face value—is the amount that the issuer threes to repay the bondholder on the maturity, or redemption, date, when the debt ceases to exist and the issuer redeems the bond. The coupon rate, or nominal rate, is the interest rate that the issuer agrees to pay during the bond s term. The annual interest payment made to bondholders is the bond s coupon. The cash amount of the coupon is the coupon rate multiplied by the principal of the bond. For example, a bond with a coupon rate of 8 percent and a principal of 1,000 will pay an annual cash amount of 80. 

Another assumption embodied in the standard formula is that the bond is traded for settlement on a day that is precisely one interest period before the next coupon payment. If the trade takes place between coupon dates, the formula is modified. This is done by adjusting the exponent for the discount factor using ratio i, shown in (1.16). 

Bonds trade either ex-dividend or cum dividend. The period between when a coupon is announced and when it is paid is the ex-dividend period. If the bond trades during this time, it is the seller, not the buyer, who receives the next coupon payment. Between the coupon payment date and the next ex-dividend date the bond trades cum dividend, so the buyer gets the next coupon payment. 

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. 

Up to this point the discussion has involved plain vanilla bonds. But duration applies to all bonds, even those that have no conventional maturity date, the so-called perpetual, or irredeemable, bonds (also known as annuity bonds), which pay out interest for an indefinite period. Since these make no redemption payment, the second term on the right side of the duration equation disappears, and since coupon payments can stretch on indefinitely, n approaches infinity. The result is equation (2.12), for Macaulay duration. 

The unit in which convexity, as defined by (2.18), is measured is the number of interest periods. For annual-coupon bonds, this is equal to the number of years for bonds with different coupon-payment schedules, formula (2.19) can be used to convert the convexity measure from interest periods to years. 

A bond issued at time i and maturing at time Tmakes w payments (C,. .. CJ on w payment dates (tj,. .. t j, T). In the academic literature, these coupon payments are assumed to be continuous, rather than periodic, so the stream of coupon payments can be represented formally as a positive... 

To provide precise protection against inflation, interest payments for a given period would need to be corrected for actual inflation over the same period. Lags, however, exist between the movements in the price index and the adjustment to the bond cash flows. According to Deacon and Derry (1998), such lags are unavoidable for two reasons. First, inflation statistics for one month are usually not known until well into the following month and are published some time after that. This causes a lag of at least one month, as shown in FIGURE 12.3. Second, in some markets the size of a coupon payment must be known before the start of the coupon period in... 

Capital indexation. Capital-indexed bonds have been issued in the United States, Australia, Canada, New Zealand, and the United Kingdom. Their coupon rates are specified in real terms, meaning that the coupon paid guarantees the real amount. For example, if the coupon is stated as 2 percent, what the buyer really gets is 2 percent after adjustment for inflation. Each period, this rate is applied to the inflation-adjusted principal amount to produce the coupon payment amount. At maturity, the principal repayment is the product of the bond s nominal value times the cumulative change in the index since issuance. Compared with interest-indexed bonds of similar maturity, these bonds have longer durations and lower reinvestment risk. 

N= the number of coupon payments (interest periods) up to maturity M = the bond principal C = the unadjusted coupon payment... 

C , = the coupon payment plus the principal repayment mi = the number of days from the value date to maturity fii = the number of interest periods from the value date until C, = mj 182 or 183... 

A bond s yield to maturity will understate (or overstate) the realized compounded yield when the true reinvestment rate is greater than (or less than) the calculated yield to maturity. Figure A4-6 illustrates this relationship for a 10 percent coupon bond that pays 30 in interest every 6 months, has 10 years until it matures, and is originally priced to sell at par (that is, its yield to maturity is equal to the coupon rate). If the annual reinvestment rate is also 10 percent (5 percent per 6-month period), the terminal value of the cash flows received plus the interest earned from the reinvestment of those cash flows will be equal to 2,653.30 1,000 from the maturity value of the bond, 1,000 to be received in the form of coupon payments, and 653.30 from reinvesting the coupons every 6 months to earn a 5 percent, 6-month rate. Given the starting value of 1,000 and the terminal value of 2,653.30, the terminal value ratio is equal to... 

P = the bond s fair price C = the annual coupon payment r = the discount rate, or required yield N - the number of years to maturity, and so the number of interest periods for a bond paying an annual coupon... 

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