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27 Table 5.2 (29p6 x 20p9) Calendar year Investment year rates Portfolio Calendar year of original investment       rates  of portfolio rate + 5   8.00%  8.10%  8.10%  8.25%  8.30%  8.10%  Z + 5  z+ 1  8.25  8.25  8.40  8.50  8.50  8.35  z + 6   8.50   8.70    8.75   8.90   9.00  8.60  Z+ 7   9.00  9.00  9.10  9.10  ,9.20  8.85  z + 8  z + 4  9.00  9.10  9.20  ,9.30  9.40  9.10  z + 9   9.25  9.35  ,9.50  9.55  9.60  9.35  z+ 10   9.50  9.50  9.60  9.70  9.70    Z+ 7  io!oo  10.00  9.90  9.80      1000  9.80  9.70      z + 9  9.50  9.50       z+ 10  9.00       
The pattern of interest rates for a particular year of investment follows a horizontal line to the righthand column and then downward. For example, in Table 5.2 the solid lines are the successive interest rates credited for deposits made at the beginning of calendar year z + 2. The interest rates credited in any particular calendar year appear on an upward diagonal to the right. For example, in Table 5.2 the dashed lines are the various rates of interest credited in calendar year z + 7. Competition among investment funds generally is focused on new money rates to be credited in the first year in order to entice depositors to invest in that particular fiind. The new money rates appear in the column headed i\ and are connected by dotted lines in Table 5.2 for calendar years z through z + 10 inclusive. In practice, implementing an investment year method is generally more complex than may be implied by Table 5.2. One source of complexity is that investment funds often change their credited rates more frequentiy than annually, e.g. monthly or quarterly. Another complication is the need to handle deposits or witiidrawals at any date. Typically, the credited rates are for calendar periods and the interest credited would be based on the periods funds are invested at the various rates. investment year method to attract ncv/ deposits and discourage vithdravals during periods of rising interest rates. Of course, vhen interest rates decline the situation will reverse and the portfolio method will be more attractive than the investment year method. When interest rates fluctuate a lot up and down, as they did during the 1980s, it becomes an interesting guessing game as to which method will produce the more favorable results. In applying the investment year method an immediate problem arises in connection with reinvestment rates. Two general approaches to this problem have been developed in practice. Under the declining index system, the funds associated with a particular investment year decline as the need to reinvest the money occurs. The interest rate credited under the investment year method reflects the investment rate on the remaining assets which are dwindling. By contrast, under the fixed index system the funds associated with a particular investment year remain fixed in amount. The interest rate credited under the investment year method reflects the investment rate on the original investment modified by subsequent reinvestment rates. Another consideration in implementing the investment year method is the need to truncate the process at some point. To illustrate in the extreme, it makes littie sense to attempt to maintain an investment year method for 100 years! Normally, an arbitrary period is chosen after which time the process stops and reverts to the portfolio method. For example, if the period for which the investment year method is applicable is chosen to be 10 years, then any funds on deposit more than 10 years will be credited on a portfolio basis. In practice, what is normally done to implement the investment year method is to specify a twodimensional table of interest rates by date of original investment and time elapsed since that date. In order to simplify the presentation, we will assume that these periods are measured in calendar years and that all deposits and withdrawals are made on January 1. Let be the calendar year of deposit and let m be the number of years for which the investment year method is applicable. The rate of interest credited for the rth year of investment is denoted by if for / = 1,2,..., m. For r > m the portfolio method is applicable and interest rates vary by calendar year only. The portfolio rate of interest credited for calendar year is denoted by i. This notation is a generalization of the notation developed in formula (1.4*) and used subsequendy in Chapter 1. Table 5.2 is an illustrative array of rates credited under the investment year method with m = 5. The first year in the table is calendar year z and the most recent year is calendar year z + 10.
It is also worth noting that the investment year m.ethod illustrated in Table 5.2 is probably based on die fixed index system. If the declining index system were being utilized, the rates going horizontally dirough the fiveyear period for each calendar year of original investment would be more nearly constant than in Table 5.2. Example 5.10 Ati investment of $1000 is made at the beginning of calendar year z + 4 in an investment fund crediting interest according to the rates contained in Table 5.2. How much interest is credited in calendar years z + 7 through z + 9 inclusive? We can readily adapt die approach taken in formula (1.38) to compute accumulated values at varying rates of interest to diis situation. The accumulated value of die investment at die beginning of calendar year + 7 is 1000(1.09)(1.091)(1.092) = $1298.60. The accumulated value of die investment at die beginning of calendar year + 10 is 1000(1.09)(1.091)(1.092)(1.093)(1.094)(1.091) = $1694.09. Thus, die total amount of interest credited in calendar years z + 7 dirough + 9 is 1694.09  1298.60 = $395.49. 5.8 CAPITAL BUDGETING A problem facing bodi individual and corporate investors is the need to determine the amount of capital to invest and die allocation of that capital among various alternative investments. The process of making such financial decisions is often called capital budgeting. In practice, two major approaches to capital budgeting are most commonly encountered. The first of these is die yield rate method. In this method die investor computes die yield rate for each alternative investment using formula (5.3). The investor establishes an interest preference rate, which is die minimum acceptable rate of return. Setting die interest preference rate is a matter of business judgment involving considerations such as die cost of raising capital and the investors profit objectives. Invesdnents widi yield rates higher than die interest preference rate are considered further, while investments widi yield rates that are lower are rejected. The various alternative invesdnents with rates higher dian die interest preference rate are ranked and those widi die highest yield rates are selected in descending order until die amount of capital available for investment is exhausted. The second approach is the net present value method. In this method the investor computes Pii) for each alternative investment using formula (5.2). P{i) is calculated at the interest preference rate as described above. Investments with a positive {1) are considered further, while investments with a negative P(i) are rejected. Capital is then allocated among those investments with a positive P(i) in such a manner that the total present value of retiirns from the investment minus contributions to die investment is maximized. The present values are computed at the interest preference rate. If a unique yield rate exists, then these two approaches will produce consistent results. In other words, investments with yield rates higher than the interest preference rate will have a positive P(i) and conversely. However, the feet that the yield rate may not always exist and be unique has led many writers in finance to favor the net present value mediod over the yield rate method. Another argument that has been advanced to favor the net present value method is diat it automatically maximizes dollar rehirns to the investor as part of the decision process. On the other hand, die yield rate method has die appeal of using numbers that are very easy to grasp and compare. However, the use of the yield rate method does not lead directiy to financial results measured in terms of dollars without making additional calculations. The above description of capital budgeting has been viewed from the perspective of an investor (lender), which is typically the manner in which capital budgeting is applied. However, it is possible to adapt the procedure for use by a borrower. In this case, the rules for the yield rate method work in the opposite direction, i.e. a "favorable" transaction has a low yield rate while an "unfavorable" transaction has a high yield rate. On the other hand, the net present value method can be applied in the same fashion for borrowers as for lenders as long as the R,s in formula (5.2) are from the borrowers perspective. The discussion of capital budgeting in this section has not considered the comparative risk involved in the various alternative investments. In essence, we are assuming that risk is nonexistent in the alternative investments being compared. Consideration of risk may well modify the decision process. For example, it is doubtful that many investors would (or should) prefer a highrisk investment with a projected yield rate of 15% to lowrisk investment with a projected yield rate of 14%. The subject of financial calculations involving risk is considered further in Section 9.5. The description of capital budgeting given in this section is quite brief. Readers interested in a more extensive treatment are referred to any of several standard textbooks in finance.
Example 5.11 Analyze the investment project given in Table 5.1 as a capital budgeting exercise. We know from Example 5.1 that the yield rate on this project is 12.96%. Table 5.3 tabulates the net present value, ?(/), at a wide range of illustrative interest rates using formula (5.2). Table 5.3 Net Present Values for Example 5.11 Rate of interest i Net presem value P(i)  $27,000   12,675   3,695   2,046    5,778    8,236 
Let us assume that an investor has an interest preference rate equal to 10%. Using the yield rate method the investor would accept this project for further consideration, since 12.96% > 10%. Using die net present value mediod die investor also would accept it, since P(.l) = 3695 > 0. Now, consider an investor with an interest preference rate equal to 15%. Using the yield rate mediod die investor would reject diis project since 12.96% < 15 %. Using die net present value mediod die investor also would reject, since P(.15) = 2046 < 0. It is instructive to analyze Table 5.3 graphically. These results are displayed as the solid line in Figure 5.6. Borrower Figure 5.6 Net present values for Example 5.11 It is clear that P{i) is a decreasing function of the rate of interest. Moreover, it is positive to the left of the yield rate 12.96%, and is negative to the right. The solid Une in Figure 5.6 is a typical graph for the value of a financial ttansaction with a unique yield rate to the investor (lender). The reader should note that the graph for the value of a financial transaction with a unique yield rate from the borrowers side of the ttansaction will be an increasing function rather than decreasing. In order to illustrate this, assume that there is only one party on the borrowers side of the transaction for the project summarized in Table 5.3. The corresponding graph for the borrower is given by the dashed line in Figure 5.6. Note that positive values of P(/) lie to the right of the yield rate rather than to the left. Example 5.12 Analyze the illustration given at the beginning of Section 5.3 as a capital budgeting exercise. Figure 5.7 is a time diagram for this transaction. The two cash flows into the investment are shown at die top of the diagram and the one cash flow out of the investment is shown at the bottom. Figure 5.7 Time diagram for Example 5.12 From formula (5.2) Pit) = 100 + 230v132v Which is shovra in Section 5.3 to have two yield rates, 10% and 20%. Table 5.4 labulates P(i) at a wide range of illustrative interest rates. Table 5.4 Net Present Values for Example 5.12 Rate of imerest i Net presem value P(i) These results are displayed graphically in Figure 5.8. The maximum value of P(i) <>ccurs at 14.78%, but is equal to P(i) at 15% to die two decimal places used in Table 5.4. The verification of this result is left as an exercise.
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