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Perpetuities
A function that can give a good idea of how the value accumulates over time is accumulate from the itertools library. This function takes a list and gets a rolling sum. Let’s see what happens when we use it.
In [4]:
from itertools import accumulate
#We will get back an object that is not helpful at first look.
total_PV = accumulate(PV)
print(total_PV)
<itertools.accumulate object at 0x10d8760c0>
You can convert this to a list and then it will be more helpful.
In [5]:
#Convert it to a list to use it
total_PV = list(total_PV)
print(total_PV)
[98.0392156862745, 194.15609381007306, 288.3883272647775, 380.77286986742894, 471.34595085042054, 560.1430890690398, 647.1991069304312, 732.5481440494423, 816.2236706367081, 898.2585006242236, 978.6848045335526, 1057.5341220917182, 1134.8373745997237, 1210.6248770585526, 1284.9263500574045, 1357.770931428828, 1429.1871876753214, 1499.2031251718838, 1567.8462011489057, 1635.1433344597115, 1701.120916136972, 1765.8048197421294, 1829.2204115118916, 1891.392560305776, 1952.3456473586039, 2012.1035758417684, 2070.6897802370277, 2128.1272355264978, 2184.4384662024486, 2239.6455551004397, 2293.7701520592545, 2346.833482411034, 2398.8563553049353, 2449.8591718675834, 2499.861933203513, 2548.8842482387386, 2596.945341410528, 2644.0640602064, 2690.258882555294, 2735.5479240738173, 2779.948945170409, 2823.4793580102046, 2866.1562333433376, 2907.9963071993507, 2949.015987450344, 2989.231360245435, 3028.658196319054, 3067.311957175543, 3105.2078011524927, 3142.360589365189, 3178.784891534499, 3214.494991700489, 3249.504893824009, 3283.82832727844, 3317.4787522337647, 3350.4693649350634, 3382.813102877513, 3414.5226498799143, 3445.6104410587395, 3476.0886677046465, 3505.9692820633786, 3535.26400202292, 3563.9843157087453, 3592.141485988966, 3619.746554891143, 3646.810347932493, 3673.343478365189, 3699.3563513384206, 3724.8591679788437, 3749.8619293910233, 3774.3744405794346, 3798.4063142935634, 3821.966974797611, 3845.065661566285, 3867.711432908123, 3889.9131695177675, 3911.6795779585955, 3933.019194077054, 3953.940386350053, 3974.4513591667182, 3994.560156045802, 4014.274662790002, 4033.602610578433, 4052.551578998464, 4071.1289990181017, 4089.3421559000994, 4107.1981920589205, 4124.704109861686, 4141.866774374203, 4158.69291605314, 4175.189133385431, 4191.361895475912, 4207.217544584228, 4222.762298611988, 4238.0022535411645, 4252.943385824671, 4267.591554730069, 4281.952504637323, 4296.031867291493, 4309.835164011267, 4323.367807854183, 4336.635105739396, 4349.642260528819, 4362.394373067469, 4374.896444183793, 4387.153376650777, 4399.169977108605, 4410.9509579496125, 4422.500939166287, 4433.824450163026, 4444.925931532378, 4455.809736796449, 4466.480134114166, 4476.941307955064, 4487.197360740259, 4497.252314451234, 4507.110112207092, 4516.774619810874, 4526.249627265563, 4535.5388502603555, 4544.6459316278, 4553.574442772352, 4562.327885070933, 4570.909691246013, 4579.323226711777, 4587.5717908939, 4595.658618523431, 4603.5868809053245, 4611.359687162083, 4618.980085453022, 4626.45106416963, 4633.77555310748, 4640.956424615177, 4647.996494720762, 4654.89852423604, 4661.665219839255, 4668.299235136525, 4674.803171702475, 4681.1795801004655, 4687.430960882809, 4693.559765571381, 4699.568397619001, 4705.459213351962, 4711.23452289408, 4716.896591072627, 4722.4476383064975, 4727.889841476958, 4733.225334781331, 4738.456210569932, 4743.584520166601, 4748.612274673138, 4753.541445757978, 4758.37396642939, 4763.1117317935195, 4767.756599797568, 4772.3103919584, 4776.774894076862, 4781.1518569381005, 4785.4429969981375, 4789.649997056998, 4793.774506918625, 4797.818144037868, 4801.782494154772, 4805.669111916443, 4809.479521486709, 4813.215217143832, 4816.877663866502, 4820.468297908335, 4823.9885273611135, 4827.439732706974, 4830.823267359778, 4834.14045819586, 4837.392606074372, 4840.580986347424, 4843.70684936022, 4846.7714209413925, 4849.775902883718, 4852.7214734154095, 4855.609287662166, 4858.440478100163, 4861.21615500016, 4863.937406862902, 4866.605300845982, 4869.2208831823355, 4871.785179590524, 4874.299195676985, 4876.763917330377, 4879.180311108213, 4881.549324615895, 4883.871886878328, 4886.148908704243, 4888.381283043375, 4890.569885336642, 4892.715573859453, 4894.819190058287, 4896.881558880674, 4898.9034890986995, 4900.885773626176, 4902.829189829585, 4904.734499832926, 4906.602450816595, 4908.433775310387, 4910.229191480772, 4911.989403412522, 4913.715101384825, 4915.406962141986, 4917.06564915881, 4918.691812900794, 4920.28609107921, 4921.849108901186, 4923.381479314888, 4924.88380324989, 4926.356669852833, 4927.800656718464, 4929.216330116141, 4930.604245211903, 4931.964946286179, 4933.2989669472345, 4934.606830340426, 4935.889049353359, 4937.146126817019, 4938.37855570296, 4939.586819316627, 4940.771391486889, 4941.932736751852, 4943.071310541031, 4944.187559353953, 4945.281920935247, 4946.354824446321, 4947.4066906336475, 4948.437931993772, 4949.44895293507, 4950.440149936343, 4951.411911702297, 4952.3646193159775, 4953.2986463882135, 4954.214359204131, 4955.112116866795, 4955.992271438035, 4956.855168076504, 4957.701145173043, 4958.530534483376, 4959.343661258212, 4960.140844370796, 4960.922396441957, 4961.688623962703, 4962.439827414414, 4963.17630138668, 4963.8983346928235, 4964.60621048316, 4965.300206356039, 4965.980594466705, 4966.647641634025, 4967.301609445122, 4967.942754357963, 4968.571327801925, 4969.187576276397, 4969.791741447448, 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4989.424118247529, 4989.6314884779695, 4989.83479262546, 4990.0341104171175, 4990.229520016782, 4990.421098055668, 4990.60891966242, 4990.793058492568, 4990.973586757419, 4991.150575252372, 4991.324093384678, 4991.494209200665, 4991.660989412417, 4991.824499423938, 4991.984803356801, 4992.141964075296, 4992.296043211074, 4992.4471011873275, 4992.595197242478, 4992.74038945341, 4992.882734758245, 4993.022288978671, 4993.159106841834, 4993.293242001798, 4993.424747060586, 4993.55367358881, 4993.680072145892, 4993.803992299894, 4993.925482646955, 4994.044590830348, 4994.161363559165, 4994.275846626632, 4994.38808492807, 4994.498122478501, 4994.606002429902, 4994.711767088139, 4994.815457929548, 4994.917115617203, 4995.016780016866, 4995.114490212613, 4995.21028452217, 4995.304200511931, 4995.396275011697, 4995.486544129115, 4995.575043263838, 4995.66180712141, 4995.746869726872, 4995.83026443811, 4995.912023958931, 4995.992180351893, 4996.070765050875, 4996.147808873407, 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In [6]:
#Now we can also graph this
plt.plot(t,total_PV)
plt.xlabel("t")
plt.ylabel("Cummulative Present Value")
plt.title("Cummulative PV of $100 Perpetuity with r=2%")
plt.show()
As you can see, as we get to a larger and larger number of years the values approach $5000. There is an actual equation which we can use to find this value that a perpetuity should approach. This equation is:
$$ PV = \frac{P}{r} $$
where
$$ PV = \text{Perpetuity Present Value} $$$$ P = \text{Annual Payment} $$$$ r = \text{Discount Rate} $$
In [7]:
print(100/.02)
5000.0
If we instead have a perpetuity due, meaning we also get a payment right away, we can convert it to be:
$$ PV = \frac{P}{r} + P $$
where
$$ PV = \text{Perpetuity Present Value} $$$$ P = \text{Annual Payment} $$$$ r = \text{Discount Rate} $$
In [8]:
#Present value for a perpetuity due
print(100+100/.02)
5100.0