I’m going to start this lesson by introducing a set of cash flows. These cash flows are equally likely events in one year, and I will import numpy so I can their mean (expected payoff)

import numpy as np
returns = [1000,1100,1200]
expected = np.mean(returns)
print(expected)

Now let’s say this is a company’s projected payoffs, and you have a rate of return of 10%, how much is it worth to you?

print(expected/1.1)

Now let’s just check to make sure the rate of equity does not need to be unlevered if there is 0 debt.

def unlever(r_a,r_d,E,D):
    return r_a + (r_a-r_d)*(D/E)
print(unlever(.1,.04,1000,0))

The rate of equity checks out, now what if this project were half debt financed and half equity financed. How much would the equity financers expect?

print(unlever(.1,.04,500,500))

According to our equation, the equity investors are going to want 16%. Why is this though? Let’s take a look at what happens in the next year. For this example we will pretend the debt matures in one year.

The payout in period 1 will be 500*1.04 = 520. Bond holders have first dibs so all this money gets taken out before equity investors have a shot at a return. The returns after debt payments is:

returns_after = [x-520 for x in returns]
print(returns_after)

Now, to find the equity return, we would need to find the expectation and divide it by what the equity investors put in.

expected = np.mean(returns_after)
print(expected)
print(580.0/500.0)

We get the same number as our equation, 16%.

Something you might think when you look at these equations is what is the optimal amount of debt and equity. I won’t go into detail, but if you are interested there is a theory about it. Modigliana and Miller said that the amount of debt vs. amount of equity does not matter overall for the valuation of a company. The last example we did shows how with or without debt the overall value summed to 1000. The thing is, their theorem assumes there is no taxes, no costs if the company goes bankrupt and no transaction costs. If we just considered taxes, we would find as much debt as possible is the best because of the tax shield. If we added in the cost of bankruptcy (lawyers and dissolving assets), we would find there is a point that too much debt has a negative effect on company value.

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