Buy Borrow Die

Calculate your maximum debt-to-equity ratio and the number of years it takes to reach it using the Buy Borrow Die strategy.

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Expected Returns %:
Margin Interest Rate %:
Inflation Rate %:
Starting Yearly Expenses:
Starting Equity Balance:
Maximum Debt-to-Equity Ratio %:
Years to Maximum Debt-to-Equity Ratio:

How Does This Work?

If you want a closer look at the math functions used in this tool, go to my Desmos Graph.

Say you have a starting equity balance of $1,000,000, then you have an expense of $50,000. Instead of selling $50,000 worth of your equity to pay for it, you elect to borrow $50,000.

You now have two balances: debt

d_{0} = 50000

and equity

q_{0} = 1000000

Equity

Your entire equity stays invested because you chose to borrow money to pay for your expenses. It grows according to your investments' rate of return, say 10%.

r = 1.1

After one year, your equity is $1,100,000. After two years it's $1,210,000. The function for your equity at year $x$ is simply

q (x) = q_{0} r^{x}

Debt

Your debt starts out at $50,000. After one year, you are charged interest on your entire debt amount, say 6%.

z = 1.06

If your debt was $50,000, you are charged $3,000 of interest. Again, instead of selling your equity to pay for this expense, you elect to borrow more money! Your debt is now $53,000.

Except now you have another year of living expenses you need to pay for and the price of those expenses has gone up according to your inflation rate, say 3%.

w = 1.03

So instead of another $50,000 expense, your living expense after one year is $51,500. If you add it all up, your previous debt times your inflation rate plus your new expense, you get $104,500. The formula for debt at year n is

d_{n} = z d_{n - 1} + w^{n} d_{0}

Explicit Formula

One problem with our debt equation is that it requires the previous year's debt $d_{n - 1}$ . We can find an explicit formula for debt like we have for equity. Let's write out the first few years of debt and see if we find a pattern. $d_{1} = z d_{0} + w d_{0}$

$d_{2} = z (z d_{0} + w d_{0}) + w d_{0}$ $= z^{2} d_{0} + z w d_{0} + w^{2} d_{0}$

$d_{3} = z (z^{2} d_{0} + z w d_{0} + w^{2} d_{0}) + w^{3} d_{0}$ $d_{3} = z^{3} d_{0} + z^{2} w d_{0} + z w^{2} d_{0} + w^{3} d_{0}$

A pattern is starting to appear. The power of the $z$ in each term goes down starting from $n$ to 0. Also, the power of $w$ in each term goes from 0 to $n$ . Finally, each term has a single $d_{0}$ in it. Let's factor that out and rewrite the equation with these patterns in mind.

d_{n} = d_{0} (z^{n} + z^{n - 1} w + z^{n - 2} w^{2} + z^{n - 3} w^{3} + \dots + w^{n})

Geometric Series

It turns out this equation is a type of geometric series. Each term has the same base, but the exponents change. Let's isolate the geometric series and call it $S$ .

S = z^{n} + z^{n - 1} w + z^{n - 2} w^{2} + z^{n - 3} w^{3} + \dots + w^{n}

Watch what happens when we multiply $S$ by $z^{- 1} w$

z^{- 1} w S = z^{n - 1} w + z^{n - 2} w^{2} + z^{n - 3} w^{3} + z^{n - 4} w^{4} + \dots + z^{- 1} w^{n + 1}

All the exponents changed, so what? Notice how this new series, is same as the old one minus $z^{n}$ and plus $z^{- 1} w^{n + 1}$ .

z^{- 1} w S = S - z^{n} + z^{- 1} w^{n + 1}

Now solve for $S$ .

\begin{matrix} z^{- 1} w S - S & = & - z^{n} + z^{- 1} w^{n + 1} \\ (z^{- 1} w - 1) S & = & - z^{n} + z^{- 1} w^{n + 1} \\ S & = & \frac{- z^{n} + z^{- 1} w^{n + 1}}{z^{- 1} w - 1} \\ = & \frac{z^{- 1} (- z^{n + 1} + w^{n + 1})}{z^{- 1} (w - z)} \\ = & \frac{- z^{n + 1} + w^{n + 1}}{w - z} \\ = & \frac{z^{n + 1} - w^{n + 1}}{z - w} \end{matrix}

Finally, if we plug everything back in, we have a closed-form solution for our debt.

d (x) mrow> = d_{0} \frac{z^{x + 1} - w^{x + 1}}{z - w}

Debt-to-Equity Ratio

Now to find our debt-to-equity ratio at time $x$ our function is simply debt over equity.

\frac{d (x)}{q (x)}

To be continued...

To find the maximum debt-to-equity ratio, you need to use some calculus. I'll write that tutorial another time.

Created by Log