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38 Table 7.1 Bond Amortization Schedule for a $1 nperiod Bond with Coupons at Rate g Bought to Yield Rate i. II II I 1 + I JZ. oc As an example of a bond bought at a premium, consider a $1000 par value twoyear 8% bond with semiannual coupons bought to yield 6% convertible semiannually. The price of the bond is computed to be $1037.17. The semiannual coupon is $40. Table 7.2 is a bond amortization schedule for this example. Table 7.2 Bond Amortization Schedule for a $1000 TwoYear Bond with 8% Coupons Paid Semiannually Bought to Yield 6% Convertible Semiannually Amount for Half   Interest  amortization of  Book  year  Coupon  eamed  premium  value      1037.17   40.00  31.12  8.88  1028.29   40.00  30.85  9.15  1019.14   40.00  30.57  9.43  1009.71   40.00  30.29  9.71  1000.00  Total  160.00  122.83  37.17  
[vt \i In the first line of Table 7.2 we have the following calculations. The lifaterest earned portion of the first coupon is /, = IBq = .03(1037.17) = $31.12. .The principal adjustment portion of the first coupon is >J = FrIi = 40.00  31.12 = $8.88. he book value at the end of the first period is B = BjjPj = 1037.17  8.88 = $1028.29. ach successive line of the schedule is calculated in similar fashion. As an example of a bond bought at a discount, consider the same $1000 par ilue twoyear 8% bond with semiannual coupons bought to yield 10% convertible semiannually. The price of the bond is computed to be $964.54. : semiannual coupon is $40. Table 7.3 is a bond amortization schedule for 1 example. The reader should verify the entries in Table 7.3. It should be noted that the amounts for accumulation of discount are really the negatives of the numbers shown in Table 7.3, i.e. they are increments to the [book value rather than decrements. However, they are usually written as ositive numbers to avoid minus signs. Thus, the reader should be careful to [ascertain in any bond authorization schedule whether the bond is selling at a
   Amount for   Half   Interest  accumulation  Book  year  Coupon  earned  of discount  value      964.54   40.00  48.23  8.23  972.77   40.00  48.64  8.64  981.41   40.00  49.07  9.07  990.48   40.00  49.52  9.52  1000.00  Total  160.00  195.46  35.46  
It should also be noted that if it is desired to find the interest earned or principal adjustment portion of any one coupon, it is not necessary to construct the entire bond amortization schedule. The book value at the beginning of the period in question is equal to the price at that point computed at the original yield rate (which will almost certainly differ from the current market price) and can be determined by the methods of Section 7.3. Then that one line of the schedule can be calculated. The bond amortization schedule discussed in this section is closely related to the amortization schedule described in Chapter 6. Additional insight into the bond amortization schedule can be obtained from the sinking fimd method. For example, in Table 7.2 the investor can be considered to be investing $1037.17 on which there is a semiaimual return of $31.12. This leaves $8.88 each period to place into a sinking fiind to replace the premium paid for the bond, since there will be a loss of the premium upon redemption. If the sinking fimd can be invested at the yield rate, then the balance in the sinking fiind at the end of two years is 8.8855, 03 = (8.88)(4.1836) = $37.15 which is the amount of premium (with .02 roundoff error). Example 7.4 illustrates a situation in which the sinking fimd earns a rate of interest different from the yield rate. Similarly, in Table 7.3 we have 8.2305 = (8.23)(4.3101) = $35.47 cm P = BqXo = Pt = for r = 1, 2, . . . , n. (7.7) (Jotethat P, > 0 for premium bonds and P, < 0 for discount bonds. The erest earned column also is constant (7.8) P, for r = 1, 2, . . . , n. ime additional properties of the straight line method are explored in the iercises. Clearly, the larger the amount of premium or discount and the longer term of the bond, the greater the error in using this method. The straight line method will also be used in Section 7.6 to derive a formula ich is useful in connection with the determination of an unknown yield rate a bond. Example 7.4 Find the price of a $1000 par value twoyear 8% bond with fmiannual coupons bought to yield 6% convertible semiannually if the estor can replace the premium by means of a sinking fund earning 5% mvertible semiannually. The bond will sell at a premium, i.e. P > 1000. The semiannual coupon is $40, the interest earned is .03P. The difference is placed into the sinking fimd which accumulate to the amount of premium. Thus, the fiindamental equation of value is (40  .03P)j 025 = P  1000 1000 + 40541.025 1000 b(40)(4.1525) .jj 3 1+ .0355,025 1 +(.03)(4.1525) price is smaller dian die bond in Table 7.2. The reader should justify die relative itude of the two prices from general reasoning. In general, the price of a bond these conditions is given by p = (1 Ssj) i is die yield rate of interest and j the sinking fund rate of interest. nich is the amount of discount (with .01 roundoff error). It should be noted that the sinking fund in this case is a negative sinking fund. * Another method of writing up or writing down the book values of bonds is straight line method. This method does not produce results which are sistent with compound interest. However, the method is very simple to In the straight line method the book values are linear, grading Thus, the principal adjustment column is constant premium or at a discount, so that the entries in the principal adjustment colutnn can be appropriately inte reted. Table 7.3 Bond Amortization Schedule for a $1000 TwoYear Bond with 8% Coupons Paid Semiannually Bought to Yield 10% Convertible Semiannually
7.5 VALUATION BETWEEN COUPON PAYMENT DATES The preceding sections have assumed that the price or book value of a bond is being calculated just after a coupon has been paid. It remains to consider die determination of prices and book values between coupon payment dates. Let B, and j be the prices or book values of a bond on two consecutive coupon payment dates. Let Fr be the amount of the coupon. As an exercise, the reader will be asked to prove the recursion formula = fi,(l+ 0Fr (7.10) assuming a constant yield rate i over this interval. We need to analyze the behavior of B,. for 0 < it < 1. When a bond is bought between coupon payment dates, it is necessary to allocate the coupon for the current period between the prior owner and the new owner. Since the new owner will receive the entire coupon at the end of the period, the purchase price should include a payment to the prior owner for Uie portion of the coupon attributable to the period from the prior coupon payment date to the date of purchase. This value is called the accrued coupon and is denoted by Frj. Clearly, at the end points of the interval we would have Fro ~ and.Frj = Fr, where Frj is computed just before the coupon is paid. We define the flat price of a bond as the money which actually changes hands at the date of sale (ignoring expenses) and denote it by {. We define the market price of a bond as the price excluding accrued coupon and denote it m by B, t*k It is clear that Bt{k = <k + Fr,forO <k<\. (7.11) In practice, bond prices are quoted as market price plus accrued coupon. As we shall see, this leads to a smooth progression of market prices (assuming a constant yield rate 0 The relationship between die flat price and the market price is illustrated in Figure 7.1. This figure uses the book values for the bond in Table 7.3 as prices. The flat price is denoted by the solid fine, while the market price is denoted by the dotted line. The accrued coupon at any date is equal to the vertical distance between the solid line and the dotted line. The book value of a bond is the asset value assigned to a bond after its purchase. It is common practice for book values to be equal to market values computed at the original yield rate at date of purchase. Of course, these values are quite likely to differ from the current market price if the bond were to be purchased anew. If this practice is followed, then any accrued coupon must be handled as a separate item in the financial statements of the owner of the bond. 1040 1020 h 1000 Figure 7.1 Comparison of flat price and market price There are three methods used to compute die values in formula (7.11). I Note that the values on the coupon payment dates are known, so that differences yunong the three methods arise only for the interim values between coupon fpayment dates. The first mediod is an exact method based on compound interest and is ..often called die theoretical method. The flat price on the interim date is die value on the preceding coupon date accumulated with compound interest at the iyield rate for the fractional period = ,(1+0*. (7.12) accrued coupon is computed as in formula (3.22) Fr = Fr (1 +o*i iThe market price or book value then is the difference 5 = B,(l +0* (1 + 0*  1 (7.13) (7.14)
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