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32 Let P be the purchase price and let D be the sinking fimd deposit. Now the total annual payment is spUt into interest on the purchase price .09 and the sinking fiind deposit D. Thus, we have OJOOO .09P. However, we know that the accumulated value of the sinking fiind after n years is P. Thus, we have P = 051.07 " 1.08  .09P 10005 .07 « 1.08(1 + 091.07) " 6.5 DIFFERING PAYMENT PERIODS AND INTEREST CONVERSION PERIODS In Sections 6.3 and 6.4 it has been assumed that the payment period and interest conversion period are equal and coincide. Section 6.5 examines the implications of removing this assumption. We shall first analyze amortization schedules in which payments are made at a different frequency than interest is convertible. Consider first a loan of / which is being repaid with payments of 1 at the end of each interest conversion periods for a total of n interest conversion periods. The number of payments is n/k, which is integral. Table 6.4 is a generalization of Table 6.1 for this situation. The reader should verify the entries in Table 6.4. Consider next a loan of aij which is being repaid with payments of 1/m at the end of each mth of an interest conversion period for a total of n interest conversion periods. The number of payments is mn, which is integral. Table 6.5 is a generalization of Table 6.1 for this situation. The reader should verify tiie entries in Table 6.5. The same observations made about the amortization schedules in Section 6.3 also apply to the amortization schedules in this secion. The reader is cautioned not to rely upon the memorization of formulas in amortization and sinking fimd Schedules, since the reasoning used in constructingthese schedules is of primary importance. Any amortization or sinking fund schedule can be constructed from basic principles. In the case of sinking fimds the approach is similar, but the situation may be a bit more complex, since the frequency of tie following may differ: (1) interest payments on the loan, (2) sinking funi deposits, and (3) interest conversion period on the sinking fund. Cases involving sinking fimds with differing frequencies can be handled fiom basic principles. Example 6.8 is illustrativeof" such a case. 1000 Thus, could charge an effective rate of 10.94%. In essence we have shown that "51 .10&.08 41 .1094 This example illustrates that to a borrower the total periodic payment is of primary concern and that the difference between the amortization method and the sinking fund method is somewhat artificial. The reader may find the answer of .1094 surprising, since the answer might be expected to lie between .08 and . 10. In general, if the equivalent rate of interest in the amortization method is denoted by J, we have the following approximate equaUty: This formula would produce an answer for this example equal to .10 + i(.10  .08) = .11, which is close to die true answer of .1094. The equivalent rate i is greater dian i because the borrower not only is paying j per unit borrowed but also is investing in a sinking fund on which interest is being sacrificed at rate i  j. Since the average balance in the sinking fiind per unit borrowed is 1/2, the extra interest cost is approximately i (/ f). Thus, the total interest cost per unit borrowed is approximately + \{i I). It is also instructive to consider Bs yield rate on this transaction. If A invests the sinking fiind with B, then Bs yield rate over the fouryear term of the loan is the same as the cost to A, i.e. 10.94%. However, if A invests the sinking fund elsewhere, then Bs yield rate is just 10% (ignoring reinvesbnent rates). Thus, Bs overall yield rate is affected by whether or not has the advantage of holding the sinking fund on which only 8% is credited. Example 6.7 Ati investor buys an nyear annuity with a present value of $1000 at 8% at a price which will permit the replacement of the original investment in asinkingfund earning 7% cmd will produce an overall yield rate of 9%. Find the purchase price of the annuity. The annual payment of the annuity is 1000
a I >o (Sirs a I 5 2 .o to :2 5 a. a, a If s p. 15 I a II iS I 15 15 15 15 II II 15 15 15 <N5 15 I 15 iRi Ri 15 15 <Ng 15 15 II II IRi iRi a a 15 15 15 I « a: sI5 I Table 6.4 Amortization Schedule for a Loan in Which Payments Are Made Less Frequently Than Interest Is Convertible Table 6.5 Amortization Schedule for a Loan in Which Payments Are Made More Frequently Than Interest Is Convertible
D =2000!lL22 nn 2.02 51.02 8.5830 = $470.70. The sinking fiind schedule is given in Table 6.6. Table 6.6 Sinking Fund Schedule for Example 6.8 Year Interest paid Sinking fund deposit Interest eamed on sinking fund Amount in sinking fiind Net amount of loan 1/4 1/2 3/4 1 1 1/4 1 1/2 1 3/4 2 0 0 0 200.00 0 0 0 200.00    470.70   470.70   9.41  480.11  470.70  9.60  960.41   19.21  979.62  470.70  19.59  1469.91   29.40  1499.31  470.70  29.99  2000.00 
2000.00 1529.30 1519.89 1039.59 1020.38 530.09 500.69 0 Example 6.9 A debt is being amortized by means of monthly payments at an annual effective rate of interest of 11%. If the amount of principal in the third payment is $1000, find the amount of principal in the 33rd payment. The principal repaid column in Table 6.5 is a geometric progression with common ratio (1 +/)"". The interval from the third payment to the 33rd payment is (33  3)/12 = 2.5years. Thus, the principal in the 33rd payment is 1000(1.11)25 = $1298.10. L = {RiiL)a +;)« + {Rjma +;)"2 + = E Rri+J)"iIn\j t=[ Example 6.S A borrower,takes out a loan of $2000 for two years. Construct a sinking fund schedule if the lender receives 10% effective on the loan and if the borrower replaces the amount of the loan with semiannual deposits in a sinking fund earning 8% convertible quarterly. In this case all three frequencies differ: (1) interest payments on the loan are made annually, (2) sinking fund deposits are made semiannually, and (3) interest on the sinking fiind is convertible quarterly. The intprest payments on die loan are $200 at die end of each year. Let die sinking fiind deposit be D. Then DILH = 2000 2l.02 6.6 VARYING SERIES OF PAYMENTS If a loan is being repaid by the amortizaUon method, it is possible that the borrower repays the loan with installments which are not level. The case in which all the payments are level except for an irregular final payment was considered in Section 6.3. In this section we consider more general patterns of variation. We will assume that the interest conversion period and the payment period are equal and coincide. Consider a loan L to be repaid with n periodic installments Ry, Rj, . . ., Rf. Then we have " L = JvR,. (6.8) 1 = 1 Often the series of payments follows some regular pattern so that the results of Section 4.6 can be used. If it is desired to construct an amortization schedule, it can be constructed from basic principles as in Section 6.3. Alternatively, the outstanding loan balance column can be found prospectively or retrospectively as in Section 6.2, and then the interest paid and principal repaid columns can be determined. One pattern of variation that is fairly common involves the borrower making level payments of principal. Since successive outstanding loan balances decrease, successive interest payments will also decrease. Thus, successive total payments consisting of principal and interest will decrease. Example 6.12 is an illustration of this type. It is possible that when a loan is amortized with varying payments, the interest due in a payment is larger than the total payment. In this case, the principal repaid would be negative and the outstanding loan balance would increase instead of decrease. The increase in the outstanding loan balance arises from interest deficiencies being capitalized and added to the amount of the loan. This situation is often called negative amortization. Example 6.13 is an illustration of this type. It is also possible to have a varying series of payments with the sinking fund method. We will assume that the interest paid to the lender is constant each period so that only the sinking fund deposits vary. Assume that the varying payments by the borrower are Ry, 7?2> • • >Rn that i j . Let the amount of the loan be denoted by L. Then the sinking fund deposit for the rth period is R,  iL. Smce the accumulated vilue of the sinking fund at the end of n periods must be L, we have • + {Rn iL)
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