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Compound Interest Part 1
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Compound Interest Part 2
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Present Value
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Annuities
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Perpetuities
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Bonds
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Dividend Discount Model
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Risk
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Capital Asset Pricing Model
Dividend Discount Model
In the case of a $5 dividend paid every year when the discount rate is 5%, the value can easily be found to be:
#What would we pay for a $5 dividend that we think we could get forever?
#This is easy, it's a perpetuity
r = .05
print(5/r)
In the case that we have a growing dividend we will see that it a bit more complicated but still easy to model in the end with a modification to the formula. To begin with, let's show how the dividend will grow if we have a 2% growth rate.
import matplotlib.pyplot as plt
#Track the first 25 years
t = list(range(1, 26))
#After the first year, the dividend grows by 2% each year
dividends1 = [5*(1.02)**(x-1) for x in t]
plt.plot(t,dividends1)
plt.xlabel("t")
plt.ylabel("Nominal Dividend Value")
plt.title("Nominal Dividend Value with 2% Growth")
plt.show()
Now what about the present value for each dividend?
#Calculate the present value
dividends2 = [d/(1.05**x) for x, d in zip(t, dividends1)]
#Plot the two together
plt.plot(t,dividends1,label="Nominal")
plt.plot(t,dividends2,label="Present Value")
plt.legend()
plt.xlabel("t")
plt.ylabel("Nominal Dividend Value")
plt.title("Dividend Value with g=2% r=5%")
plt.show()
Since the present value of the dividend will eventually reach the point where it is worthless, the perpetuity formula can still work. It is important to understand that in any case where g > r, the formual no longer works because it would essentially be trying to represent the case where a company keeps having dividends that are worth more and more every single year even in terms of present value. The stock price would be infinity if that were true.