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24 (1.09)5(1 + if = (1.08)° 1.09 Thus, if the investor expects the prevailing interest rate to be greater than 7.01% at the end of 5 years. Option A would be the better choice. Odierwise, Option would be preferred. 5.3 UNIQUENESS OF THE YIELD RATE Our intuition leads us to expect that the yield rate as defined in Section 5.2 will be unique; and, in fact, in most commonly encountered financial transactions the yield rate is unique. However, transactions are occasionally encountered in which a yield rate is not unique. As an example, consider a transaction in which a person makes payments of $100 immediately and $132 at the end of two years in exchange for a payment in return of $230 at the end of one year. An equation of value for this transaction is 100(1 It if + 132 = 230(1 + i) (1 f if2.3(1 + 0 + 1.32 = 0. Now factoring we obtain [(1 f 01.1][(1 + 01.2] = 0. Thus, the yield rate i is equal to either 10% or 20%! E The fact that transactions exist with multiple yield rates is difficult for most people to comprehend intuitively. However, it is not surprising when we recall formula (5.3) P(i) = E «f = 0 (5.3) pFormula (5.3) is an nth degree polynomial in v and could be written as an nth I"degree polynomial in i by simply multiplying both sides by (1 f 0". Of pcourse, it is wellknown that an nth degree polynomial has n roots (counting complex roots and roots of multiplicity m, where m > 1, as m roots). In the outflows in this project were directed to multiple parties and/or the cash inflows arose from multiple sources. If that is the case, the yield rate calculation for the investor (lender) is still valid. However, diere is not a single borrower on the other side of the transaction for which the same yield rate applies. Another important consideration in using yield rates is to consider the period of time involved. For example, consider investing a sum of money under Options A and B. Option A credits 9% effective for five years, while Option credits 8% effective for ten years. Which option should we choose as an investor? To say that Option A is better than Option because its yield rate is higher is naive. If we wish to invest for only five years, then the simple comparison of yield rates is valid. However, if we wish to invest for ten years, then we need to consider the rate at which we can reinvest the proceeds from Option A after the end of the first five years. Thus, it is valid to use yield rates to compare alternative investments only if the period of investment is the same for all the alternatives. Section 5.4 contains a more comprehensive treatment of reinvestment rates. The above definitions and formulas assume payments at integral periods of time. However, the results can easily be extended to include other regular or irregular intervals as well. Solving for yield rates is analogous to solving for unknown rates of interest on annuities. In fact, if the payments constitute a basic annuity, then the techniques discussed in Section 3.8 can be applied directly. If the payments do not constitute a basic annuity, the techniques would be similar. In general, an iteration method would typically be applied to the equation of value given by formula (5.3). The easiest way to solve such problems in practice would be to utilize a pocket calculator with builtin financial fiinctions or to use a personal computer with a financial software package. Both of these options contain subroutines which readily handle such calculations. Example 5.1 Find the yield rate for the investment project summarized in Table 5.1. The equation of value is 10O0(10  5v    v"*  +7v* + 8v + 9v* + lOv + 12v°) = 0. The yield rate computed on a pocket calculator with builtin financial functions is found to be .1296, or 12.96%. Example 5.2 Find the effective rate which the investment under Option A must earn for the second five years to be equivalent to the investment under Option for the entire ten years for the illustration given in this section. Let the rate in question be denoted by /. The equation of value is
Bo = Co > 0 Bi = Bq{1 +0 + Ci > 0 B2 = Bi(l + /) + C2 > 0 Bn = B„ i(l+i) + C„ =0. By successive substitution in the above equations we have b„ = Cod + 0" + Cid + 0" + • • • + C„.i(l + 0 + C„ = 0 . (5.6) This is the expected result, since the investment is exactly terminated at the end of n periods. Note that Cq > 0 and C„ < 0, but that C, for r = 1, 2, . . . , n  1 may be either positive, negative, or zero. To prove the uniqueness of /, let 7 > i be another yield rate. Let the outstanding investment at time / for interest rate j be denoted by . Then we have 0 = Co Co = Bo b\ = bq{\+j) + q > Bod+o + c, = Bj B2 = b\(i+j) + C2 > Bi(i+o + Cj = B2 Bnl = „2(1 +7) + C„ , > B„.2(l+0 + C„ , = B„ , B\ = B„ ,(i+;) + c„ > B„ ,(i + o + c„ = B„ = o. But this is a contradiction, since B„ must equal 0 if J is a yield rate. Thus j cannot be greater than /. The proof for 1 < j < 1 is analogous. This establishes the uniqueness of i. Thus, if the outstanding investment balance is positive at all points throughout the period of investm.ent, then the yield rate will be unique. example given immediately above, we have a quadratic with two distinct positive roots for i. Since the yield rate is widely used as a measure of the financial value of a transaction, it is quite important in practice to be able to ascertain whether or not a yield rate is unique. One very common situation in which the yield rate will be unique is when all cash flows in one direction are made before the cash flows in the other direction. Stated slightly more generally, this situation is one in which the net payments are all of one sign for the first portion of the transaction and then have the opposite sign for the remainder of the transaction. Stated in mathematical terms, this situation can be characterized as one in which some value of exists, Q <k <n, such that J?, < 0 for / = 0, 1, 2,. . . , A: and Rj > 0 for / = k+l, k+2, . . . ,n. The financial transaction given in Table 5.1 is of this type with n = 10 and = 5. It can easily be shown that a yield rate in this situation will be unique. Looking at formula (5.3) as an nth degree polynomial, we see that there is only one sign change. From Descartes rule of signs we know that there will be at most one positive real root. Since v > 0, then i > 1. Thus, the uniqueness will hold not only for positive values of i, but also for negative values of i >  I. This covers all values of concern, since values of / < 1 have no practical significance. Descartes rule of signs will also give us an upper bound on the number of multiple yield rates which may exist. The maximum number of yield rates is equal to the number of sign changes in the cash flows. Of course, the actual number of yield rates may well be less than the maximum. Actually, yield rates are unique under a broader set of conditions than given above. It is possible to show that if the outstanding investment balance is positive at all points throughout the period of investment, then the yield rate will be unique. Let be the outstanding investment balance at time t where / = 0, 1, 2, . . . , n. Then we have Bq = Co (5.4) Bj = fi, i(l + 0 + C, for r = 1, 2, . . . , n. (5.5) It is possible to show that if 1. >Oforr=0,1.....nl, and 2. I > 1 exists such that formula (5.3) is satisfied, then i is unique. The proof of this result is as follows. The condition / > 1 is necessary to ensure that 1 + / is positive. Now rewrite formula (5.3) as Cod + 0" + Ci(l + 0"" + • • • + C„.,(l + /) + C„ = 0. We know that
Figure 5.1 Time diagram for reinvestment rates involving an investment of 1 for periods The accumulated value at the end of n periods is equal to the principal plus the accumulated value of the interest, i.e. 1 + is (5.7) Formula (5.7) simplifies to the familiar (1 + 0" if = J Second, consider the investment of 1 at the end of each period for n periods at rate i such that the interest is reinvested at rate j. It is desired to find the accumulated value of this aimuity at die end of n periods. This situation is illustrated in Figure 5.2. However, if the outstanding investment balance ever becomes negative at any one point, then a yield rate is not necessarily unique. The situations in which multiple yield rates can occur may strike the reader as somewhat artificial and not very realistic of typical financial transactions. Although such situations are not common, they do occur in practice. A realistic example would be an investment in a physical plant which requires major renovation expenses midway through the period of investment. The resulting sign change in net cash flows may lead to multiple yield rates. The discussion in this section has focused on the possibility of multiple yield rates. However, it is also possible that no yield rate exists or that all yield rates are imaginary. These possibilities are illustrated in Examples 5.3 and 5.4, respectively. Readers who are interested in a more extensive discussion of the uniqueness of the yield rate are referred to papers by W. H. Jean (1968) and D. S. Promislow (1980) listed in the bibliography. Example 5.3 A is able to borrow $1000 from for one year at 8% effective and lend it to for one year at 10% effective. What is As yield rate on this transaction? In this example, A is able to make a $20 profit at the end of one year in exchange for no net invesUnent at all. Thus, no finite yield rate exists. We could say die yield rate is infinite. However, such a statement would not distinguish this transaction from an even more favorable one in which A is able to lend the $1000 to a fourth party D at 12% effective. Example 5.4 What is the yield rate on a transaction in which a person makes payments of $100 immediately and $101 at the end of two years, in exchange for a payment of $200 at the end of one year? An equation of value is 100(1 + if + 101 = 200(1 + 0 100j2 =  1. Thus, die yield rates are all imaginary numbers! 5.4 REINVESTMENT RATES In Section 5.3 and the earlier chapters we have not directly considered the reinvestment by die lender of payments received from the borrower. This is equivalent to die implicit assumption that the lender can reinvest payments received from the borrower at a reinvestment rate equal to the original investment rate. This may or may not be a valid assumption in practice depending upon the particular circumstances involved. If the lender is not able to reinvest the payments from the borrower at rates as high as the original investment, then the overall yield rate considering reinvestment will be lower than die stated yield rate. On the other hand, if the lender is able to reinvest such payments at even higher rates, then the overall yield rate will be higher than that stated. Actually, we have already seen an example of a problem considering reinvestment rates. Example 5.2 considers the reinvestment of proceeds from Option A at the end of five years in order to make a valid comparison with Option over the tenyear period in question. We now analyze two other situations in which reinvestment rates are directiy taken into account. First, consider the investment of 1 for n periods at rate / such that the interest is reinvested at rate J. It is desired to find the accumulated value at the end of n periods. This situation is illustrated in Figure 5.1.
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