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42 Note that the total payments over 30 years are more than three times the amount of the loan! The effect of compound interest over long periods of time is truly dramatic. Finally, we need to compute the APR. From formula (8.8) we have 1044.23 = 117,800. Using iteration as described in Section 3.8, we find that J = .( 8433. Thus, the APR is APR = i = 12(.008433) = .1012, or 10.12%. The APR is greater dian 9.9%, as expected. Example 8.5 A borrower takes out a 30year adjustable rate mortgage for $65,000. The interest rate for the first year is 8%. If the interest rate increases to 10% for the second year, find the increase in the monthly payment. The monthly payment for the first year is = 65,000 5, 5 03501.08/12 1 36.2835 The outstanding loan balance after one year is 476.9508/12 = 476.95(135.1450) = $64,457.42. The revised monthly payment for the second year is 6M542 64,457.42 g g. 10/12 113.3174 Thus, the increase in the monthly payment is 568.82  476.95 = $91.87 which is an increase of 19.3%. 8.4 APPROXIMATE METHODS The calculation of an exact annual percentage rate is somewhat complicated since it involves an iteration. Several approximate methods for calculating unknown rate of interest on installment loans have been developed. Tl methods do not utilize any interest functions and can be calculated directly. Although these mediods are not as important as they were in the days befon the development of pocket calculators with builtin financial functions, they stil have considerable educational and conceptual value. Moreover, they are usefiij in providing the background necessary for computing the unearned finance charge defined in Section 8.2 and for the various depreciation methods to considered in Section 8.5. Finally, one of these approximate methods has sucl? a high level of accuracy that it has value in other situations. We will examine four such approximate methods. All of these methods are similar in that they replace the true division of die installment payments betwem] die] principal and interest with an arbitrary division between principal and interest. It is assumed that the amount of principal invested during each mth of a year is invested at rate i/m for that mth of a year. Thus, if we denote the outstanding loan balance at time t/m by B,/, we have . nl m ,=0 X) t/m t=Q (8.10) The derivation of formula (8.10) reflects the fact that the finance charge list equal the sum of the amounts of interest earned during each mth of a year 1 the outstanding loan balances for those periods. The various methods differ ily the arbitrary division of installment payments between principal and crest, which will be reflected in the denominator of formula (8.10). The first method is often called the maximum yield method, since it duces larger answers than any of the other methods. The value of / duced by this method is labelled i". This method assumes that all ailment payments are applied entirely to principal until it is fully paid and ereafter are applied entirely to interest. We also assume that the finance ige is less than one installment payment, i.e. his latter assumption leads to the result that all installment payments except ; of the last one are used for principal repayment. With these assumptions Table 8.1 is a modified amortization schedule for method. The reader should verify the entries in Table 8,1 and note the ious relationships in the table. We now need to sum the outstanding loan balance column to evaluate ala (8.10) t=0 « J max Ln{L + ) 2m L(n + 1)  K(n  1) (8.11)
264 The theory of Interest SO that n + 1 2mK L(n + iy (8.13) Table 8.3 Amortization Schedule Using the Constant Ratio Method Period  Payment amount  Interest paid  Principal repaid  Outstanding loan balance  \ m 2 nl  L+K n L +K n  n n    n 1 m n m  L+K n L+K n     Toial  L + K    
It is interesting to note that formula (8.13) can be derived by an alternative argument. The amount of interest per year is mKln. The average outstanding loan balance can be found by averaging the outstanding loan balances during die first period and the last period, i.e. , . n + 1 n n 2 n The rate of interest is computed as the annual amount of interest divided by the average outstanding loan balance, i.e. 2mK + !) (8.13) 0 = 1 2 Widi this assumption the rate of interest is 2mK Ln (8.14) Formula (8.14) replaces the term (n + l)/2n with 1/2 . The reader should note that the relationship between formulas (8.13) and (8.14) is quite analogous to the relationship between formulas (7.24) and (7.25), the two versions of the bond salesmans formula. In general, formula (8.14) produces less accurate results than formula (8.13). It is interesting to note that i  is equal to the harmonic mean of / and /. The proof of this result is left as an exercise. The fourth method is the direct ratio method. The value of i produced by this method is labelled / . This method uses an approximate division into principal and interest which is closest to the exact division by the actuarial method. In the true amortization schedule the interest paid column decreases over time while the principal repaid column increases. The direct ratio method reflects this pattern while none of the other approximate methods do. In general, the direct ratio method will produce more accurate results than the other approximate methods. The direct ratio method can best be illustrated by example. Consider a oneyear loan repaid with 12 monthly installments. The sum of the positive integers from 1 through 12 is 78. The direct ratio method assumes that the interest paid is 12/78 of the finance charge in the first month, 11/78 in the second month, . . . , 1/78 in the last month. This decreasing pattern of interest payments will produce a corresponding increasing pattern of principal repayments. The direct ratio method is often called the rule of 78, despite the fact that the number 78 is valid only for a term of 12 months. Table 8.4 is a modified amortization schedule for this method. We define to be the sum of the first r positive integers, i.e. S= 1+2+ (8.15) The reader should verify the entries in Table 8.4 and note the various relationships in the table. The two methods are equivalent, since the outstanding loan balance is linear under these assumptions. On occasion an even simpler version of the constant ratio method is encountered in which the average outstanding loan balance is taken to be the average of the beginning balance and the ending balance, i.e.
• max (1( )(13)(80)(11) 2. Using formula (8.12) imin (2)(12)(80) (1( )(13) + (80)(11) 3. Using formula (8.13) 4. Using formula (8.16) = .1584, or 15.84%. = .1383, or 13.83%. 1 = (2)(12)(80) = .1444, or 14.445 (1000)03)+ I(80)(11) The exact answer produced by die achiarial mediod was found to be 14.45% in Example 8.1. The direct ratio method produces an answer closer to that obtainedj by die actuarial mediod dian do any of die otfier approximate mediods; and, in 1 it achieves a remarkable level of accuracy. Example 8.7 Compare the accuracy of formulas (3.29), (3.32), and (8.17 in obtaining starting values to solve for i, if  8.5136. = .0962, or 9.62%. Formula (8.17) gives k(n+ 1)+ l(nfc)(nl) 2(20 8.5136) 8.5136(21) + 1(20 8.5136)(19) = .0913, or 9.13%. Thus Our highlytouted direct ratio (rule of 78) formula (8.17) beats die constant ratio formula (3.29) as expected, but loses out to the obscure formula (3.32). However, if we apply formula (3.32) to the loan considered in Examples 8.1 and fe.6, we obtain = .1541, or 15.41 This result is considerably inferior to that produced by the direct ratio method. ; Thus, no one approximate formula will always produce the best starting values to use in S solving for unknown rates of interest on annuities. j Experience has shown, however, diat generally die direct ratio formula (8.17) will de die best results for "shorter" terms, while formula (3.32) works best for "longer" s. This difference is attributable to die fact diat formula (3.32) more accurately cts the compounding of interest over longer time periods. Example 8.8 Compute the unearned finance charge recovered by the brrower for the consumer loan in Example 8.1 if the loan is repaid in full er six regular installments are made: (1) using the actuarial method, and 1) using the rule of 78. The outstanding loan balance at the end of sue months, prospectively, is 90«g:o,2O43 = $517.95. The total that would have been paid over the final six months on the original payment schedule is 6 • 90 = $540.00. Thus, the unearned finance charge is It can be shown in advance that i = 10%. Using a contrived number will make it easier to compare the effectiveness of the three formulas. Formula (3.29) gives , = 2(nkl 2(20  8.5136) j2.85%. " k(n + 1) 8.5136(21) Formula (3.32) gives Formula (8.16) produces the following analogous formula for starting values using Chapter 3 notation 2{nk) ,. k{n + \) + l(n  k)(n  I) The derivation of formula (8.17) is left as an exercise. In general, formula (8.17) will produce results superior to those produced by formula (3.29). In Section 8.2 calculation of the unearned finance charge on loans which are repaid early was discussed. There are two methods in widespread use. The first is based on an exact application of the actuarial method. The second is based on the rule of 78. It will be shown in Example 8.8 that the use of the rule of 78 favors the lender. The reader should be careful not to confuse diis applicafion of die rule of 78 with the derivation of formula (8.16). Two entirely different situations are involved. Example 8.6 Find the rate of interest on the installment loan in Example i 8.1: (1) by the maximum yield method, (2) by the minimum yield method, (3) by the constant ratio method, and (4) by the direct ratio method (rule of 78).] 1. Using formula (8.11) (2)(12)(80)
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