Bond pricing annual coupons

Considering the example in which a 5-year bond pays an annual coupon of 1%, with a discount rate of 2%. The bond price is given by the following steps ... 

Exhibit 16.5 shows very clearly the price responses for three 20-year bonds offering 10%, 6% annual coupon, and zero-coupon payments. One feature worth noting is that the curvilinear, price/yield relationship is not constant. It varies at different points on the curve for each type of... 

When the rates have been established they must then be calibrated. The calibration procedure is achieved using the observed market price of a bullet government bond and pricing the bond using the tree calculated rates to obtain the appropriate discount factors. As an example, consider a government bond trading at par and offering a coupon of 4.625% paid semi-annually. On maturity the bond will be redeemed for 102.3125, which is made up of the bond s face value, say 100, and one half of the annual coupon, 2.3125. 

We now revisit the earlier Vasicek example for short interest rates to consider the case where the underlying bond pays an annual coupon at a 5% rate (p = 0.05), all the other characteristics remain as before. In order to calculate the call price of the coupon-bond European option first we need to calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This is done by trial and error using equation (18.48) and the value we get here is = 22.30%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon payments considered as zero-coupon bonds and calculate the value of the European call options contingent on those zero-coupon bonds as in the above example. The calculations are described in Exhibit 18.7. 

The fair price of a bond is the sum of the present values of all its cash flows, including both the coupon payments and the redemption payment. The price of a conventional bond that pays annual coupons can therefore be represented by formula (1.12). 

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). 

A bond paying a semiannual coupon has a dirty price of 98.50, an annual coupon of 3 percent, and exactly one year before maturity. The bond therefore has three remaining cash flows two coupon payments of 1.50 each and a redemption payment of 100. Plugging these values into equation (1.20) gives... 

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. 

C, = the bond cash flow at time t P = the bonds fair price C = the annual coupon payment rm = the redemption yield n = the number of years to maturity... 

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. 

C = annual coupon rate F - face, par, or maturity value of the bond P = quoted price for the bond P = price per dollar of face value N = number of coupon periods remaining... 

The current yield of a bond is calculated by dividing the security s annual dollar coupon payment by the market price. The formula for the current yield is... 

To illustrate, consider once again the 5.25% coupon BMW Finance described in Exhibit 3.10. From the yield analysis screen in Exhibit 3.11, we can locate the bond s full price under the heading Payment Invoice on the right-hand side of the screen. The full price is 1,117,726.69 (labeled Total ) for a 1 million par value position. The cash flows of the bond are (1) annual payments of 5,250 for the next four years and (2) a payment of 1,000,000 at maturity. The interest rate that makes these cash flows equal to the full price is 2.793%. 

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. 

Without getting into too much detail, these bonds pay a smaller fixed coupon (between 2% and 4%) to which the accumulated inflation in that period is added. By mid-2003 the French Tresor had already issued five different inflation-linked OATs, three of them referenced to French inflation and two of them linked to Eurozone retail prices. The success of these bonds has recently permitted their launch via normal auctions, rather than syndicate issues. As of July 2003, they represent more than 10% of the French bond annual supply. 

B. Calculate the price of a French government zero-coupon bond with precisely five years to maturity, with the same required yield of 5.40 percent. Note that French government bonds pay coupon annually. 

As noted in chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bond s cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,rg,.rj f—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (16.7). 

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, is the price of the convertible bond is the price of the underlying equity C is the bond coupon r is the risk-free interest rate N is the time to maturity a is the annualized share price volatility c is the call option feature rd is the dividend yield on the underlying share... 

In order to sell the bond, it must be competitively priced. After all who would buy our investor s bond with a 10 percent coupon rate, when there are other bonds out there paying 12 percent To adjust, our investor will sell the bond at less than its face value of 1,000—and by doing so achieve a coupon rate and yield that is competitive with the then-prevailing market conditions. In this case, if the investor sells the bond for 830, the 100 that the new buyer of the bond would receive each year would result in a coupon rate with an annual yield of 12 percent ( 100/ 830). Conversely, if interest rates had generally gone down, the investor s bond could command more than the face amount when... 

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