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31 = (1+ «  . However, prospectively B2 must equal the present value of the future payments. Thus, we have (1 +J)ar = Ji 1 + i)a. = (1 +J)a. which gives 1 + i 1 + ( The development is identical to case 1 above, except that die prospective value for die future payments, which equals Bj is computed at ratey instead of (. Thus, we have (1 +J)aiX = Xa 1+ ; ,. = (1+ 1, which gives 6.4 SINKING FUNDS <" Rather than repay a loan in installments by the amortization method, a borrower may choose to repay it by means of one lumpsum payment at the end Sof a specified period of time. In many such cases the borrower will accumulate fiind which will be sufficient to exactly repay the loan at the end of the cified period of time. In fact, in some cases the lender will insist that the [borrower accumulate such a fund, which is often called a sinking fimd. In some cases payments into a sinking fiind may vary irregularly at the discretion of the borrower. However, we shall be primarily interested in those leases in which the payments follow a regular pattern, i.e. where they are some I form of an aimuity. It is usually required that the borrower pay interest on the loan periodically ver the term of the loans. Such interest is sometimes called service on the poan. Thus, the amount of the loan remains constant. Since the balance in the sinking fund at any point could presumably be applied against the loan, the net amount of the loan is equal to the original amount of the loan minus the accumulated value of the sinking ftmd. This 2. The outstanding loan balance on Bs original amortization schedule at the end of two years is 528.71 02 = (528.71)(13.5777) = $7178.67. The total payments made by A over the first two years are (8)(528.71) = $4229.68. The total principal repaid by A over this period is 10,000  7178.67 = $2821.33 . Thus, the total interest received by apparendy is 4229.68 2821.33 = $1408.35. However, die total interest paid by A over die entire loan is equal to (24X528.71) 10,000 = $2689.04 which does not equal die sum of die interest received by and B, i.e. 1557.05 + 1408.35 = $2965.40 . What has happened is diat incurred a loss at die end of two years equal to die difference between die outstanding loan balance and die price to C, i.e. 7178.67 6902.31 = $276.36. If diis loss is offset against the interest received by B, dien Bs net investment income from this transaction is 1408.35 276.36 = $1131.99. The system now balances, since adding this amount to die amount of interest received by is equal to the total interest paid by A, i.e. 1557.05 + 1131.99 = $2689.04 . Example 6.5 An amount is invested at an annual elective rate of interest i which is just sufficient to pay 1 at the end of each year for n years. In the first year the fund actually earns rate i and 1 is paid at the end of the year. However, in the second year the fund earns rate j where j > i. Find the revised payment which could be made at the ends of years 2 through n: (1) assuming ffie rate earned reverts back to i again after this one year, and (2) assuming the rate earned remains at j for the rest of the nyear period. 1. The initial investment is Bq = a, and the account balance at die end of die fust year is J3, = a,. Let X be die revised payment. We now construct die next line of die amortization schedule on die revised basis, obtaining h = J<ir \
> 1 Consider a loan of amount 1 repaid over n periods. The expression 1/a is the amount of each payment necessary to repay the loan by the amortization method. However, the expression 1/s is the periodic sinking fund deposit necessary to accumulate the amount of the loan at the end of n periods, while i is the amount of interest paid on the loan each period. Thus, the two methods are equivalent. It is instructive to consider this equivalence from an alternate viewpoint. Consider a loan of amount being repaid with installments of 1 at the end of each period for n periods. The amount of interest each period is i . Thus, 1  ia is left to go into the sinking fund each period. However, the sinking fund will accumulate to which is the original amount of the loan. At first glance it might seem that the two methods cannot be equivalent, since from Table 6.1, the interest paid in the amortization method decreases 1  v", 1  v"\ . . . , 1  v; while the interest paid in the sinking fijnd metiiod is a constant = 1  v " each period. However, each period the sinking fimd earns interest which exactly offsets the seeming discrepancy, so that the net amount of interest is the same for the sinking fund method as for die amortization method. For example, during the rth period, 1 < r < , tiie amount of interest in the amortization schedule is = 1  V nr+1 The net amount of interest in the sinking fund method is the amount of interest paid 1 less the amount of interest earned on the sinking fund. The amount in the sinking fund is the accumulated value of the sinking fund deposits of 1  ia at die end of f  1 periods, i.e. {\1 )sj=. Amortization schedules and sinking funds Ml Thus, the net amount of interest in the sinking fund method in the rth period is = (1  v")  v[(l +0" 1 ia )j = 1  v"  v" + v" = 1  V nt+\ Therefore, the net amount of interest in the sinking fund method is equal to the amount of interest in the amortization method if the rate of interest on the loan equals the rate of interest earned on the sinking fund. The equivalence in the methods can be seen from a consideration of a sinking fund schedule. Table 6.3 is a sinking fund schedule for the same example considered in Table 6.2. Let the sinking fund deposit be denoted by D. Then we have 10( 4.5061 = $221.92. 1 The reader should verify the entries in Table 6.3. Table 6.3 Sinking Fund Schedule tor a Loan of $1000 Repaid Over Four Years at 8% Interest  Interest  Sinking  eamed on  Amount in  Net amount  Year  paid  fiind deposit  sinking fimd  sinking fimd  of loan       1000.00   80.00  221.92   221.92  778.08   80.00  221.92  17.75  461.59  538.41   80.00  221.92  36.93  720.44  279.56   80.00  221.92  57.64  1000.00  
The following relationships between Table 6.2 and Table 6.3 should be noted: 1. The total payment in the sinking fund method, i.e. interest paid on the loan plus the sinking fund deposit, equals the payment amount in the amortization method. 2. The net interest paid in the sinking fund method, i.e. interest paid on the loan minus interest earned on the sinking fund, equals the interest paid in tiie amortization method. 3. The annual increment in the sinking fund, i.e. the sinking fund deposit plus the interest earned on the sinking fund, equals the principal repaid in the amortization method. concept plays the same role for the sinking fund method that the outstanding loan balance, discussed in Section 6.2, does for the amortization method. It is possible to show that if the rate of interest paid on the loan equals the rate of interest earned on the sinking fund, then the sinking fund method is equivalent to the amortization method. Recall formula (3.6) (3.6,
"ii\i&.j + i. (6.5) We can now fmd an expression for gj as follows: Thus, n\i&j 7i\i&J + aJ) (6.6) (6.7) It should be noted tiiat if i =j , then al(&; = °n\i would be expected. The construction of a sinking fund schedule at two rates of interest is very similar to the construction of a sinking fiind schedule at a single rate of interest. As an example, consider a $10( loan for four years on which an annual effective rate of interest of 10% is charged if the borrower accumulates the amount necessary to repay the loan by means of four annual sinking fund 1000 = $321.92 4l.l0&.08 \ shown earlier in this section. Thus, on the amortization offer, could charge / where 321.9251 I = 1000 «41 = 3.1064. We use the techniques of Section 3.8 to solve for the unknown rate of interest on the annuity by iteration. Formula (3.29) is used to obtain the starting value for the iteration .. = 2(nk) 2(4  3.1064) ° k(n + 1) (3.1064)(5) Successive applications of the NewtonRaphson iteration formula (3.28) produce i, = .1093 /2 = .1094 /3 = .1094. 4. The net amount of the loan in the sinking fiind method, i.e. the original amount of the loan minus the amount in the sinking fimd, equals the outstanding loan balance in the amortization method. It now remains to consider the operation of the sinking fund method when the rate of interest earned on the sinking fund differs from the rate of interest paid on the loan. We will denote the rate of interest paid on the loan by and the rate of interest earned on the sinking fund by j. In practice j is usually less than or equal to i. It would be unusual for a borrower to be able to accumulate money in a sinking fund at a higher rate of interest than is being paid on the loan. However, this is not necessarily the case mathematically, and the following analysis is valid even if j is greater than i. The same basic approach will be used for the case in which j as was previously used in the case in which the two rates were equal. The total payment will be split into two parts. First, interest at rate / will be paid on the amount of the loan. Second, the remainder of the total payment not needed for interest will be placed into a sinking fund accumulating at rate j. Let agj represent die present value of an annuity of 1 at the end of each period for n periods under the conditions just described. Then if a loan of 1 is made, the periodic installment under the amortization method will be 1/ 1("&;However, from the sinking fiind method this payment must pay interest at rate i on the loan and provide for a sinking ftind deposit which will accumulate at rate j to the amount of the loan at the end of n periods. Thus deposits in a ftind earning an annual effective rate of 8%. The total semiannual payment made by the borrower is = + looo(.io) = 221.92 + 100.00 = $321.92. «41.10&.08 •41.08 This example is a generalization of the example considered in Table 6.3. Note .that the sinking fund schedule is identical to Table 6.3, except that each entry in the interest paid column is $1( instead of $80. In general, the sinking ftmd schedule at two rates of interest is identical to the sinking ftind schedule at one rate of interest equal to the rate of interest earned on the sinking fiind, except that a constant addition of ( ]) times the amount of the loan is added to the interest paid column. 11 Example 6.6 A wishes to borrow $1000. Lender offers a loan in which principal is to be repaid at the end of four years. In the meantime 10% keffe<ve is to be paid on the loan and A is to accumulate the amount %ecessary to repay the loan by means of annual deposits in a sinking fund earning 8% effective. Lender offers a loan for four years in which A "repays the loan by the amortization method. What is the largest effective rate of interest that can charge so that A is indifferent between the two offers? ; Under either method A will make four equal payments at the end of each year to lrepay the loan. Thus, A will be indifferent between the two offers if the annual payment is equal on both of them. On the sinking ftind offer from the annual payment is
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