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  1. [Present Value Annuity Due][1] = 33.61 x (1 + 0.010416667) x { 1 - 1/(1 + 0.010416667)^36 }/0.010416667
  2. = 33.61 x 1.010416667 x { 1 - 1/(1.010416667)^36 }/0.010416667
  3. = 33.61 x 1.010416667 x { 1 - 1/1.45217196873 }/0.010416667
  4. = 33.61 x 1.010416667 x { 1 - 0.688623676487 }/0.010416667
  5. = 33.61 x 1.010416667 x { 0.311376323513/0.010416667 }
  6. = 33.61 x 1.010416667 x 29.8921261007
  7. = 33.61 x 30.2035024242
  8. PVAD = 1015.14
  9.  
  10. A' = PVAD / (1+i)
  11. A' = 1015.14 / 1.010416667
  12. A' = $1,004.67
  13.  
  14. Present Value Annuity Due = 33.61 x (1 + 0.0105) x { 1 - 1/(1 + 0.0105)^36 }/0.0105
  15. = 33.61 x 1.0105 x { 1 - 1/(1.0105)^36 }/0.0105
  16. = 33.61 x 1.0105 x { 1 - 1/1.45648978356 }/0.0105
  17. = 33.61 x 1.0105 x { 1 - 0.68658222755 }/0.0105
  18. = 33.61 x 1.0105 x { 0.31341777245/0.0105 }
  19. = 33.61 x 1.0105 x 29.8493116619
  20. = 33.61 x 30.1627294344
  21. PVAD = 1013.77
  22.  
  23. A'' = PVAD / (1+i)
  24. A'' = 1013.77 / 1.0105
  25. A'' = $1,003.23
  26.    
  27. Newton Raphson Method IRR Calculation with TVM equation = 0
  28.  
  29. TVM Eq. 1: PV(1+i)^N + PMT(1+i*type)[(1+i)^N -1]/i + FV = 0
  30.  
  31. f(i) = 0 + 33.61 * (1 + i * 0) [(1+i)^36 - 1)]/i + -1000 * (1+i)^36
  32.  
  33. f'(i) = (33.61 * ( 36 * i * (1 + i)^(35+0) - (1 + i)^36) + 1) / (i * i)) + 36 * -1000 * (1+0.1)^35
  34.  
  35. i0 = 0.1
  36. f(i1) = -20859.0286
  37. f'(i1) = -772196.0009
  38. i1 = 0.1 - -20859.0286/-772196.0009 = 0.0729873910496
  39. Error Bound = 0.0729873910496 - 0.1 = 0.027013 > 0.000001
  40.  
  41. i1 = 0.0729873910496
  42. f(i2) = -7274.5413
  43. f'(i2) = -301995.7711
  44. i2 = 0.0729873910496 - -7274.5413/-301995.7711 = 0.0488991687999
  45. Error Bound = 0.0488991687999 - 0.0729873910496 = 0.024088 > 0.000001
  46.  
  47. i2 = 0.0488991687999
  48. f(i3) = -2431.1344
  49. f'(i3) = -124187.6435
  50. i3 = 0.0488991687999 - -2431.1344/-124187.6435 = 0.0293228701788
  51. Error Bound = 0.0293228701788 - 0.0488991687999 = 0.019576 > 0.000001
  52.  
  53. i3 = 0.0293228701788
  54. f(i4) = -732.3776
  55. f'(i4) = -57078.0048
  56. i4 = 0.0293228701788 - -732.3776/-57078.0048 = 0.0164917006907
  57. Error Bound = 0.0164917006907 - 0.0293228701788 = 0.012831 > 0.000001
  58.  
  59. i4 = 0.0164917006907
  60. f(i5) = -167.5999
  61. f'(i5) = -32858.4347
  62. i5 = 0.0164917006907 - -167.5999/-32858.4347 = 0.0113910349433
  63. Error Bound = 0.0113910349433 - 0.0164917006907 = 0.005101 > 0.000001
  64.  
  65. i5 = 0.0113910349433
  66. f(i6) = -17.997
  67. f'(i6) = -26021.5726
  68. i6 = 0.0113910349433 - -17.997/-26021.5726 = 0.010699415611
  69. Error Bound = 0.010699415611 - 0.0113910349433 = 0.000692 > 0.000001
  70.  
  71. i6 = 0.010699415611
  72. f(i7) = -0.288
  73. f'(i7) = -25192.367
  74. i7 = 0.010699415611 - -0.288/-25192.367 = 0.0106879831887
  75. Error Bound = 0.0106879831887 - 0.010699415611 = 1.1E-5 > 0.000001
  76.  
  77. i7 = 0.0106879831887
  78. f(i8) = -0.0001
  79. f'(i8) = -25178.8435
  80. i8 = 0.0106879831887 - -0.0001/-25178.8435 = 0.0106879801183
  81. Error Bound = 0.0106879801183 - 0.0106879831887 = 0 < 0.000001
  82.  
  83. IRR = 1.07%
  84.  
  85.  
  86. Newton Raphson Method IRR Calculation with TVM equation = 0
  87.  
  88. TVM Eq. 2: PV + PMT(1+i*type)[1-{(1+i)^-N}]/i + FV(1+i)^-N = 0
  89.  
  90. f(i) = -1000 + 33.61 * (1 + i * 0) [1 - (1+i)^-36)]/i + 0 * (1+i)^-36
  91.  
  92. f'(i) = (-33.61 * (1+i)^-36 * ((1+i)^36 - 36 * i - 1) /(i*i)) + (0 * -36 * (1+i)^(-36-1))
  93.  
  94. i0 = 0.1
  95. f(i1) = -674.7726
  96. f'(i1) = -2860.8622
  97. i1 = 0.1 - -674.7726/-2860.8622 = -0.135863356364
  98. Error Bound = -0.135863356364 - 0.1 = 0.235863 > 0.000001
  99.  
  100. i1 = -0.135863356364
  101. f(i2) = 46220.4067
  102. f'(i2) = -1361282.2783
  103. i2 = -0.135863356364 - 46220.4067/-1361282.2783 = -0.101909776386
  104. Error Bound = -0.101909776386 - -0.135863356364 = 0.033954 > 0.000001
  105.  
  106. i2 = -0.101909776386
  107. f(i3) = 14472.9891
  108. f'(i3) = -417070.1913
  109. i3 = -0.101909776386 - 14472.9891/-417070.1913 = -0.0672082095036
  110. Error Bound = -0.0672082095036 - -0.101909776386 = 0.034702 > 0.000001
  111.  
  112. i3 = -0.0672082095036
  113. f(i4) = 4620.5467
  114. f'(i4) = -136713.9676
  115. i4 = -0.0672082095036 - 4620.5467/-136713.9676 = -0.0334110286059
  116. Error Bound = -0.0334110286059 - -0.0672082095036 = 0.033797 > 0.000001
  117.  
  118. i4 = -0.0334110286059
  119. f(i5) = 1412.836
  120. f'(i5) = -50859.7324
  121. i5 = -0.0334110286059 - 1412.836/-50859.7324 = -0.00563196002357
  122. Error Bound = -0.00563196002357 - -0.0334110286059 = 0.027779 > 0.000001
  123.  
  124. i5 = -0.00563196002357
  125. f(i6) = 345.5376
  126. f'(i6) = -24366.4494
  127. i6 = -0.00563196002357 - 345.5376/-24366.4494 = 0.00854891782087
  128. Error Bound = 0.00854891782087 - -0.00563196002357 = 0.014181 > 0.000001
  129.  
  130. i6 = 0.00854891782087
  131. f(i7) = 37.705
  132. f'(i7) = -17208.0395
  133. i7 = 0.00854891782087 - 37.705/-17208.0395 = 0.010740042325
  134. Error Bound = 0.010740042325 - 0.00854891782087 = 0.002191 > 0.000001
  135.  
  136. i7 = 0.010740042325
  137. f(i8) = -0.8934
  138. f'(i8) = -16335.3764
  139. i8 = 0.010740042325 - -0.8934/-16335.3764 = 0.0106853483863
  140. Error Bound = 0.0106853483863 - 0.010740042325 = 5.5E-5 > 0.000001
  141.  
  142. i8 = 0.0106853483863
  143. f(i9) = 0.0452
  144. f'(i9) = -16356.5205
  145. i9 = 0.0106853483863 - 0.0452/-16356.5205 = 0.0106881114105
  146. Error Bound = 0.0106881114105 - 0.0106853483863 = 3.0E-6 > 0.000001
  147.  
  148. i9 = 0.0106881114105
  149. f(i10) = -0.0023
  150. f'(i10) = -16355.4516
  151. i10 = 0.0106881114105 - -0.0023/-16355.4516 = 0.010687973564
  152. Error Bound = 0.010687973564 - 0.0106881114105 = 0 < 0.000001
  153.  
  154. IRR = 1.07%
  155.    
  156. sum(1/(1+.0104)**i for i in xrange(36)) == 30.2116668761916
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