Introduction
Perpetuities¶
Perpetuities are in general a more theoretical idea but are very important for many equations. The basic idea is that a perpetuity is an infinite payment stream, so one example would be $100 every single year. The value, however, is not infinite because eventually the time value of money would make the payments virtually worthless. To show this, look how much this payment is worth after 1, 10, 100, 500 and 1000 years with a 2% interest rate used.
In [1]:
#What $100 worth after 1 year, 10 years, 100 years, 500 years, and 1000 years when r=2%
print(100/(1.02)**1)
print(100/(1.02)**10)
print(100/(1.02)**100)
print(100/(1.02)**500)
print(100/(1.02)**1000)
We can visualize these payments to more easily understand how the value behaves.
In [2]:
import matplotlib.pyplot as plt
t = list(range(1,1001))
PV = [100/((1.02)**x) for x in t]
plt.plot(t,PV)
plt.xlabel("t")
plt.ylabel("Payment PV")
plt.title("PV of $100 with r=2%")
plt.show()
The value after 1000 years is a good estimate because by this time the rest of the payments are worth such a small amount, so what would the present be?
In [3]:
print(sum(PV))