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39 One final issue to consider is the amount of premium or discount between coupon payment dates. It is evident that these values should be based on the marlcet price or book value rather than on the flat price. For example, a $1000 par value bond with a market value of $980 and accrued coupon of $30 is still a discount bond despite the fact it is selling for $1010. Thus we have Premium = B,"  if g > / (7.2i) Discount = CBrif f > g (7.22) for which B," can be calculated by any of the three methods described above. Example 7.5 Compute the flat price, accrued interest, and market price (book value) five months after purchase fi)r the bond in Table 7.2. Use all three methods. For convenience in mustrating the three methods, we set jfc = 5/6. For die theoretical method, we have: Bsi = 1037.17(1.03)5* = $1063.04 Frs/6 = 40 (1.03)5* 1 .03 = $33.25 Bs% = 1063.04  33.25 = $1029.79. For die practical mediod, we have: b = 1037.17 1 + i(.03) 5/6 = (40) = $33.33 6 = $1063.10 B% = 1063.10  33.33 = $1029.77. For die semidieoretical mediod, we have: Bjfe = $1063.04 Frs,6 = $33.33 BsZ = $1029.71. In practice, an actual day count would be used based on die calendar date of purchase. For sake of iUustration, assume diat die bond has a January 1 issue and redemption date and is being purchased on June 1. A pocket calculator widi budtin financial fiinctions gives die following answers to diis question: b{ = $1063.06 Fr = $33.37 Bf = $1029.69. The number of days in the first five months of die year is 151, while the number of days in the first six months of the year is 181. The reader should verify that using A = 151/181 in the formulas for the semitheoretical method produces the above answers. Thus, the day count basis is actual/actual. 7,6 DETERMINATION OF YIELD RATES As mentioned in Section 7.1, one of the three primary questions under consideration in this chapter is the determination of the yield rate to an investor, given the purchase price of a security. This section discusses this question for bonds. The determination of the yield to maturity on a bond is similar to the determination of an unknown rate of interest for an annuity, discussed in Section 3.8. We first consider die determination of the yield rate for a bond purchased on a coupon payment date immediately after the coupon is paid. One approach is linear inte oIation in bond tables. These are tables of bond prices for a wide variety of terms, coupon rates, and yield rates. Extensive tables of this type have been published. This method is not as widely iised as it once was due to the widespread availability of computers with financial software and pocket calculators with builtin financial fiinctions. Nevertheless, bond tables are still used in practice. We will not use this method in this book. Another approach is to develop approximate formulas for the yield rate by an algebraic technique. We start with formula (7.2) P = + (  ) = + ( 1) (7.2) and letting =  we have \ or I g (7.23) To solve formula (7.23) for /, we can use formula (3.25) ««1 2 12 (3.25)
P(l +/)" = Frsj +C (7.21) since both sides represent the value of the investment at the end of n periods. An illustration of yield rates taking into account reinvestment rates is given in Example 7.8. One of the reasons for the popularity of zero coupon bonds in recent years is that such bonds do not present a reinvestment risk to the investor. Since there are no coupons to reinvest, the yield rate is lockedin at the date of purchase. One final consideration in practical applications is the effect of expenses. If commissions and other fees are involved in buying and selling bends, then the actual yield rate realized will be reduced. However, the procedures described above can readily be applied to reflect such expenses. This would be achieved by increasing die purchase price by any expenses at purchase and reducing die sale price by any expenses at sale. It should be noted that generally no expenses are charged when a bond is redeemed or matures. Example 7.6 A $100par value 10year bond with 8% semiannual coupons is selling for $90. Find the yield rate convertible semiannually. We will illustrate die mediods discussed in diis section. We have P  90100 100 The bond salesmans mediod formula (7.25), gives die following approximate semiannual yield rate •04 + 4: 20 1 +(.5)(.l) = .0474 ie. 4.74%, or 9.48% convertible semiannually. The more refmed formula (7.24) gives .04 + 4: 20 = .0475 21 40 (1) i.e. 4.75%, or 9.50% convertible semiannually, which should be closer to die mie yield rate. We next illustrate an ad hoc iteration wddi formula (7.23) usmg a startng value « = 0475. We obtain die following series of values: / = .0475 j, = .04786 /2 = .04788 ,•3 = .04788 ithus, die yield rate convertible semiannually is 2(.04788) = .09576, or 9.576%. Iformula (7.24) is indeed more accurate dian formula (7.25), as expected. The rate of I Convergence using diis ad hoc mediod is quite acceptable. Finally, we use die NewtonRaphson mediod widi formula (7.26). Since dus mediod is so powerful, we will carry die calculation to two more decimal places of accuracy. We obtain die foUowing series of values: io = .0475 i§ /, = .04788 /2 = .0478807 ,3 = .0478807 IJhus, die yield rate convertible semiannuaUy is 2(.0478807) = .0957614, or 9.57614%. answer was confirmed on a pocket calculator widi buUtin financial fiinctions. Example 7.7 Assume that the bond in Example 7.6 was issued on March i. Slightly over two years later on May 15 the market price of the bond is 88, compute the yield rate if the bond is bought on that date. 4, A pocket calculator widi buUtin financial fiinctions produces die answer 10.2694%. wiU attempt to confirm tius answer. The price of die bond on die immediately Ipreceding March 1 at a yield rate of 10.2694% convertible semiannuaUy is computed to [be 87.8194. The number of days from March 1 to May 15 is 75, while die number of fdays from March 1 to September 1 is 184. Using formula (7.20), die semiflieoretical I method produces 87.8194(1.051347) ,75/184 75 184 • 4 = 87.9998 functions are able to handle such iterations directly. This approach will be illustrated in Example 7.7. Thus far in Chapter 7 we have computed the yield rate i ignoring reinvestment rates. However, investors in bonds need to consider the rates at which coupons from the bonds can be reinvested. If die investor can only reinvest die coupons at a lower rate than /, then / overstates the yield rate actually realized on the transaction taking into account reinvestment rates. However if the investor is fortunate enough to be able to reinvest the coupons at a higher rate than /, then the overall yield rate on the transaction will actually exceed /. Consider the situation in which a bond is purchased for P, coupons of Fr are paid at the end of each period for n periods, the bond is redeemed for Cat the end of n periods, and the coupons are reinvested at rate j. If we denote die yield rate considering reinvestment by / (to distinguish it from /), then / would satisfy the following equation of value
232 The theory of imerest It is evident that the lowest price in each category will result when the lowest of n is used, i.e. n = 10, 20, and 30. Thus, die prices to compare are: 109.00 + ( 50) = 109.00 + (.3650)(9.2222) = $112.37 104.50+ (.4325)a5g, = 104.50 + (.4325)(17.1686) = $111.93 100.00 + (.5000) 015 = 100.00 + (.5000)(24.0158) = $112.01. The lowest price $111.93 occurs for n = 20, i.e. for redemption 10 years issue. This example illustrates diat die least favorable redemption date can between die earliest and latest possible redemption dates when die redemption are not equal at all die possible redemption dates. It is fairly common in for a callable bond to have redemption values which decrease as the term of die 1 increases, as in this example. The excess of die redemption value over the value, i.e. $9 and $4.50 in this example, is often termed the call premium. 7.8 SERIAL BONDS As mentioned in Section 7.2, a borrower in need of a large amount of fun may issue a series of bonds with staggered redemption dates instead of usingl common maturity date. These types of bonds are called serial bonds and > considered further in this section. tg If the redemption date of each mdividual bond is known, then the valu of any one bond can be performed by methods already described. The value I the entire issue of bonds is merely the sum of the values of the individual 1 It will be shown that the value of the entire issue of bonds can often be fo more efficiently than by summing the individual bond values, however. In some cases, the bonds to be redeemed on each successive redemptio date are not known in advance but are chosen randomly. The value of any oa bond is impossible to determine in advance with certainty, since its reden date depends upon chance. However, the value of the entire issue of bonds i be determined with certainty. Of course, values for any one bond can be determined on the same basis,] that used for callable bonds in Section 7.7. Also, the yield rate actually realii after redemption can be determined. The valuation of an issue of serial bonds can most efficiently be perfor using Makehams formula P = K+l(CK). i Assume that the serial bonds are redeemable at m different redemption dab Let the purchase price, redemption value, and present value of the redemptio value for the first redemption date be denoted by Pj, Cj, and K; for second redemption date by P2, C, and K2, and so forth until the redemption date when we have P, C, and K. Thus we have Pi = + (CiAi) P2 = K+{C2K2) I summing, we obtain P K + 1{ K) i (7.28) = E /=1 r = E /=1 ; reader should note that tiie symbol P, was also used in Section 7.4 to present the amount of principal adjustment in tiie fth bond coupon. pThus, tiie price of an entire issue of serial bonds is denoted by P and is by formula (7.28). This formula is usually more efficient than a ation of the prices of each individual bond, since and K are usually simple to obtain. The sum is merely the sum of Uie redemption I for tiie entire issue of bonds, a figure readily obtainable. The sum K is pie form if die bonds are redeemed according to a systematic pattern at : intervals. In tiiis case, K is some form of annuity for which a simple session is possible. Example 7.10 illustrates tiiis technique. pie 7.10 Find the price of a $1000 issue of 5 1/4% bonds with \ coupons which will be redeemed in 10 annual installments at the end ? 11th through the 20th years from the issue date at 105. The bonds are ght to yield 7% effective. I this example we have: 100 105 .0525 i(.0525) = 105
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