[Present Value Annuity Due][1] = 33.61 x (1 + 0.010416667) x { 1 - 1/(1 + 0.010416667)^36 }/0.010416667
= 33.61 x 1.010416667 x { 1 - 1/(1.010416667)^36 }/0.010416667
= 33.61 x 1.010416667 x { 1 - 1/1.45217196873 }/0.010416667
= 33.61 x 1.010416667 x { 1 - 0.688623676487 }/0.010416667
= 33.61 x 1.010416667 x { 0.311376323513/0.010416667 }
= 33.61 x 1.010416667 x 29.8921261007
= 33.61 x 30.2035024242
PVAD = 1015.14

A' = PVAD / (1+i)
A' = 1015.14 / 1.010416667
A' = $1,004.67

Present Value Annuity Due = 33.61 x (1 + 0.0105) x { 1 - 1/(1 + 0.0105)^36 }/0.0105
= 33.61 x 1.0105 x { 1 - 1/(1.0105)^36 }/0.0105
= 33.61 x 1.0105 x { 1 - 1/1.45648978356 }/0.0105
= 33.61 x 1.0105 x { 1 - 0.68658222755 }/0.0105
= 33.61 x 1.0105 x { 0.31341777245/0.0105 }
= 33.61 x 1.0105 x 29.8493116619
= 33.61 x 30.1627294344
PVAD = 1013.77

A'' = PVAD / (1+i)
A'' = 1013.77 / 1.0105
A'' = $1,003.23

Newton Raphson Method IRR Calculation with TVM equation = 0

TVM Eq. 1: PV(1+i)^N + PMT(1+i*type)[(1+i)^N -1]/i + FV = 0

f(i) = 0 + 33.61 * (1 + i * 0) [(1+i)^36 - 1)]/i + -1000 * (1+i)^36

f'(i) = (33.61 * ( 36 * i * (1 + i)^(35+0) - (1 + i)^36) + 1) / (i * i)) + 36 * -1000 * (1+0.1)^35

i0 = 0.1
f(i1) = -20859.0286
f'(i1) = -772196.0009
i1 = 0.1 - -20859.0286/-772196.0009 = 0.0729873910496
Error Bound = 0.0729873910496 - 0.1 = 0.027013 > 0.000001

i1 = 0.0729873910496
f(i2) = -7274.5413
f'(i2) = -301995.7711
i2 = 0.0729873910496 - -7274.5413/-301995.7711 = 0.0488991687999
Error Bound = 0.0488991687999 - 0.0729873910496 = 0.024088 > 0.000001

i2 = 0.0488991687999
f(i3) = -2431.1344
f'(i3) = -124187.6435
i3 = 0.0488991687999 - -2431.1344/-124187.6435 = 0.0293228701788
Error Bound = 0.0293228701788 - 0.0488991687999 = 0.019576 > 0.000001

i3 = 0.0293228701788
f(i4) = -732.3776
f'(i4) = -57078.0048
i4 = 0.0293228701788 - -732.3776/-57078.0048 = 0.0164917006907
Error Bound = 0.0164917006907 - 0.0293228701788 = 0.012831 > 0.000001

i4 = 0.0164917006907
f(i5) = -167.5999
f'(i5) = -32858.4347
i5 = 0.0164917006907 - -167.5999/-32858.4347 = 0.0113910349433
Error Bound = 0.0113910349433 - 0.0164917006907 = 0.005101 > 0.000001

i5 = 0.0113910349433
f(i6) = -17.997
f'(i6) = -26021.5726
i6 = 0.0113910349433 - -17.997/-26021.5726 = 0.010699415611
Error Bound = 0.010699415611 - 0.0113910349433 = 0.000692 > 0.000001

i6 = 0.010699415611
f(i7) = -0.288
f'(i7) = -25192.367
i7 = 0.010699415611 - -0.288/-25192.367 = 0.0106879831887
Error Bound = 0.0106879831887 - 0.010699415611 = 1.1E-5 > 0.000001

i7 = 0.0106879831887
f(i8) = -0.0001
f'(i8) = -25178.8435
i8 = 0.0106879831887 - -0.0001/-25178.8435 = 0.0106879801183
Error Bound = 0.0106879801183 - 0.0106879831887 = 0 < 0.000001

IRR = 1.07%


Newton Raphson Method IRR Calculation with TVM equation = 0

TVM Eq. 2: PV + PMT(1+i*type)[1-{(1+i)^-N}]/i + FV(1+i)^-N = 0

f(i) = -1000 + 33.61 * (1 + i * 0) [1 - (1+i)^-36)]/i + 0 * (1+i)^-36

f'(i) = (-33.61 * (1+i)^-36 * ((1+i)^36 - 36 * i - 1) /(i*i)) + (0 * -36 * (1+i)^(-36-1))

i0 = 0.1
f(i1) = -674.7726
f'(i1) = -2860.8622
i1 = 0.1 - -674.7726/-2860.8622 = -0.135863356364
Error Bound = -0.135863356364 - 0.1 = 0.235863 > 0.000001

i1 = -0.135863356364
f(i2) = 46220.4067
f'(i2) = -1361282.2783
i2 = -0.135863356364 - 46220.4067/-1361282.2783 = -0.101909776386
Error Bound = -0.101909776386 - -0.135863356364 = 0.033954 > 0.000001

i2 = -0.101909776386
f(i3) = 14472.9891
f'(i3) = -417070.1913
i3 = -0.101909776386 - 14472.9891/-417070.1913 = -0.0672082095036
Error Bound = -0.0672082095036 - -0.101909776386 = 0.034702 > 0.000001

i3 = -0.0672082095036
f(i4) = 4620.5467
f'(i4) = -136713.9676
i4 = -0.0672082095036 - 4620.5467/-136713.9676 = -0.0334110286059
Error Bound = -0.0334110286059 - -0.0672082095036 = 0.033797 > 0.000001

i4 = -0.0334110286059
f(i5) = 1412.836
f'(i5) = -50859.7324
i5 = -0.0334110286059 - 1412.836/-50859.7324 = -0.00563196002357
Error Bound = -0.00563196002357 - -0.0334110286059 = 0.027779 > 0.000001

i5 = -0.00563196002357
f(i6) = 345.5376
f'(i6) = -24366.4494
i6 = -0.00563196002357 - 345.5376/-24366.4494 = 0.00854891782087
Error Bound = 0.00854891782087 - -0.00563196002357 = 0.014181 > 0.000001

i6 = 0.00854891782087
f(i7) = 37.705
f'(i7) = -17208.0395
i7 = 0.00854891782087 - 37.705/-17208.0395 = 0.010740042325
Error Bound = 0.010740042325 - 0.00854891782087 = 0.002191 > 0.000001

i7 = 0.010740042325
f(i8) = -0.8934
f'(i8) = -16335.3764
i8 = 0.010740042325 - -0.8934/-16335.3764 = 0.0106853483863
Error Bound = 0.0106853483863 - 0.010740042325 = 5.5E-5 > 0.000001

i8 = 0.0106853483863
f(i9) = 0.0452
f'(i9) = -16356.5205
i9 = 0.0106853483863 - 0.0452/-16356.5205 = 0.0106881114105
Error Bound = 0.0106881114105 - 0.0106853483863 = 3.0E-6 > 0.000001

i9 = 0.0106881114105
f(i10) = -0.0023
f'(i10) = -16355.4516
i10 = 0.0106881114105 - -0.0023/-16355.4516 = 0.010687973564
Error Bound = 0.010687973564 - 0.0106881114105 = 0 < 0.000001

IRR = 1.07%

sum(1/(1+.0104)**i for i in xrange(36)) == 30.2116668761916