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28 Figure 5.8 Net present values for Example 5.12 The possible nonuniqueness of the yield rate is often cited as a reason to favor die net present value metiiod of capital budgeting over die yield rate mediod. However, we will now show that problems also exist in using the net present value method. In Figure 5.8 P(i) is a decreasing fiinction in die vicinity of die 20% yield rate, which is the normal pattern from the lenders vantage point. However, P(i) is an increasing function in die vicinity of die 10% yield rate. Thus, if die lender in diis example requires only a 5% rate of return, the invesUnent is a poor one since P(i) is negative. However, if die lender niples die required rate of return to 15%, die investment somehow becomes a good one since P{i) is now positve! This result is illogical, showing tiiat using the net present value mediod of capital budgeting does not really solve the problems inherent when multiple yield rates exist. 5.9 MORE GENERAL BORROWING/LENDING MODELS Sections 5.3 and 5.8 have illustrated some of the difficuUies in finding rational interpretations of certain financial calculations and in making valid comparisons among different financial transactions when multiple yield rates exist. Various approaches have been suggested in practice to circumvent the problems inherent with multiple yield rates. One such approach is to discount the future cash outflows at a prescribed rate of interest and Uien perform the rest of the calculations based only on die future cash inflows. The prescribed rate of interest is generally a conservative rate which the investor can earn with safety. In effect, the investor is "prefunding" die future cash outflows, i.e. a fund equal to the present value of the future cash outflows computed at die prescribed rate could be established out of which such cash outflows could be paid. The yield rate now computed using ("AS the fuUire cash inflows will be unique. Whether or not such a fund is achially established by the investor is irrelevant to the validity of the calculation. The remainder of this section discusses another approach which has been developed. In Section 5.3 we showed that if the outstanding investment balance is positive throughout the period of investment, then the yield rate will be unique. We can generalize this result and define a pure investment project as one in which 0 for = 0, 1, 2..... . A pure investment project is one in which the investor has money invested in the project throughout the period of investment. , We now switch to the perspective of a borrower and define a pure financing project as one in which < 0 for f = 0, 1, 2, . . . , n. A pure financing project is one in which the investor owes money to the project throughout die period of investment. Thus, the investor has actually become a borrower in this case. Multiple yield rates can arise when some outstanding balances are positive and others are negative during the period of investment. We will call such a project a mixed project, since the investor is a lender during some portions of the period of investment and a borrower during other portions. I This more general model is based on the premise that a different rate of interest should be used during those portions of the period of investment during which the investor is in lender status than the rate used during those portions in borrower status. The project return rate is denoted by r and is the interest preference rate during those portions of the period of investment during which Uk investor is in lender status, i.e. the investment balance B, > 0. The project financing rate is denoted by / and is the interest preference rate during those portions of the period of investment during which the investor is in borrower status, i.e. the investment balance fi, < 0, Typically, r will be greater dian /, since an astute investor will have a higher interest preference rate as a lender than as a borrower. However, die madiematical development does not require that r > /. We generalize the approach used in developing forinulas (5.4) through (5.6) to fit this situation. The initial fund balance is Bq  Cq. Successive fund balances are developed as a recursion formula B, = B, i(l +r) + C,, if B, i > () B, = S,.i(l +/)+ C,, if < 0 (5.23) (5.24a) (5.24*)
for f = 1,2, . . . , n. The ending ftind balance is a polynomial in r and /of the form B = C,(I +"(1 + )" + c,(I +r)"4l +ff"" ++ (5.25) where the rrijs are integers such that n > rtiQ > > . . . nt > 0. In formula (5.25) is the total number of periods from time j to time n for which interest rate r is used, with rate / being used for the remainder of the periods. If = /, formulas (5.23) through (5.25) simplify to formulas (5.4) through (5.6). Recall that fi = 0 at the yield rate. If ? /, the concept of yield rate can still be utilized. However, in this more general case the yield rate is not a single number, but rather a functional relationship between r and /. In other words, for a given value of /, if a value of r can be found such that B = 0, then r and / are a yield rate pair for the transaction. A transaction for which yield rates exist will typically have an infinite number of r,/pairs and a functional relationship between r and /can be found. The above constihites a generalization of the yield rate method of capital budgeting. It is also possible to generalize the net present value method. Recall that formula (5.2) defining net present value is based on which are the negatives of the Cs used above. Thus, positive values of net present value correspond to negative values of B, and conversely. The fact that negative values of B are favorable to the investor can be interpreted as reflecting the fact that a negative balance in the investment balance at the end of the investment period is, in effect, a positive balance to the investor. Readers who are interested in a more extensive analysis of this borrowing/lending model are referred to two papers both by D. Teichroew, A. A. Robichek, and M. Montalbano (both 1965) listed in the bibliography. Example 5.13 An investor is required to make a coriribution of $1600 immediately and $10,000 at the end of two years in exchange for receiving $10,000 at the end of one year. (1) Find the yield rates, if r = f. (2) Express r as a function offifr and f are a yield rate pair. (3) Would the investor accept or reject the transaction if r = 70% andf = 30%? (4) Rework (3) if f = 50%. 1. Let i = r = f. The equation of value is 1600(1 + if + 10,000 = 10,000(1 + 0This is a quadratic whose two roots are I = .25, or 25% i = 4, or 400%. Thus, we have a Uansaction with multiple yield rates. The investment balance is positive for the first year and negative for the second year. Thus, interest rate r is applicable for the first year and /is applicable for second year. We have: Bq = 1600 i5] = 1600(1 + r)  10,000 2 = [1600(1 + r)  10,000](1 +/) + 10,000 = 0. This defines a functional relationship between r and /. Solving for r as a function of/ J 10,000 1600 1 +/ r = 6.25 = 5.25  I  1+/ 6.25 !+/ Some sample values of rand/are given in Table 5.5. Table S.S Project Rates for Example 5.13 Financing rate f  Return rate r     212.5         
Note that r = /at the two yield rates which would be expected. Also, note that r > / between the two yield rates, which is the normal relationship. However, t outside djis range r < f. Finally, note diat r increases as/increases. Using formulas (5.23) dirough (5.25) we have: Bn = C = 1600 Bi = Boil + r) + Ci = 1600(1.7) 10,000= 7280 Bz = S,(l +/) + C2 = (7280)(1.3)+ 10.000 = 536. Since B2 > 0, the investor rejects this transaction.
5.4 Reinvestment rates It is desired to accumulate a fund of $1000 at die end of 10 years by equal deposits at die beginning of each year. If the deposits earn interest at 8% effective but die mterest can be reinvested at only 4% effective, show diat die deposit necessary is 1000 .04  12 A loan of $10,000 is being repaid with payments of $1000 at the end of each year for 20 years. If each payment is immediately reinvested at 5% effective, find the effective annual rate of interest earned over die 20year period. An investor purchases a fiveyear financial instrument having the following features: (0 The investor receives payments of $1( at the end of each year for five years. ((/) These payments earn interest at an effective rate of 4% per annum. At the end of the year, this interest is reinvested at the effective rate of 3% per annum. Find die purchase price to the investor to produce a yield rate of 4%. An investor deposits $1( at the beginning of each year for five years in a fund earning 5% effective. The interest from this fund can be reinvested at only 4% effective. Show that die total accumulated value at die end of ten years is 1250(5n, 0,55, 041). A invests $2(XX) at an effective interest rate of 17% for 10 years. Interest is payable annually and is reinvested at an effective rate of 11 %. At die end of 10 years, the accumulated interest is $5685.48. invests $150 at die end of each year for 20 years at an effective interest rate of 14%. Interest is payable annually and is reinvested at an effective rate of 11%. Find Bs accumulated interest at die end of 20 years. 5.5 Interest measurement of a fund """" ° «500,000 and at d« end of die year $680,000. Gross interest eamed was $60,000, against which diere were 7. Payments of $100 now and $108.15 two years from now are equivalent to a payment of $208 one year from now at eidier rate ( or j. Find the absolute difference of the two rates. 8. Show that we cannot guarantee uniqueness of the yield rate if Cq and C have the same sign. 9. An investor pays $100 immediately and %X at the end of two years in exchange for $200 at the end of one year. Find X such that two yield rates exist which are equal in absolute value but opposite in sign. 4. Following the same approach used immediately above, Bq and B, are unchanged. The new ending balance is Bj = (7280X1.5) + 10,000 = 920. Since B2 < 0, the investor accepts this transaction. It is instructive to consider why (4) is accepted and (3) is rejected. The only difference between the two examples is in the borrowing interest preference rate /. If die maximum rate at which the investor is willing to borrow is 30%, die transaction should be rejected. However, if the investor is wilting to borrow at rates up to 50%, then the transaction should be accepted. EXERCISES 5.2 Discounted cash flow analysis 1. a) Express the present value of the contributions in Table 5.1 in terms of annuity immediate symbols. b) Express die present value of the retiims in Table 5.1 in terms of annuityimmediate symbols. 2. The internal rate of reUim for an investinent in which Cq = $3000, Cj = $1000, /Jj = $2000, and R2 = $4000 can be expressed as 1/ . Find n. 3. A tenyear investinent project requires an initial investinent of $1( ,( at inception and maintenance expenses at die beginning of each year. The maintenance expense for the first year is $3000, and is anticipated to increase 6% each year diereafter. Projected annual reUims fiom the project are $30,000 at the end of die first year decreasing 4% per year diereafter. Find R. 4. An investor enters into an agreement to contribute $7000 immediately and $1000 at die end of two years in exchange for the receipt of $4000 at die end of one year and $5500 at die end of fliree years. Find: a) P(.09). b) P(.10). 5. Find die NewtonRaphson iteration formula based on formula (5.2) which can be used in general to calculate an unknown yield rate. 5.3 Uniqueness of the yield rate 6. a) In Exercise 4 what is the maximum number of possible yield rates using Descartes rule of signs? b) From the two answers it is apparent diat a yield rate exists between 9% and 10%. Is diis rate unique? c) Explain your answer to ).
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