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\( t_s \) Saving time
\( r \) Rate of return on invested capital
\( r_w\) Rate of withdrawal in retirement
\( S \) Saving rate
\( m \) Total invested capital
\( V_0 \) Value of windfall at \(t=\) 0
\( F \) Savings per time
\( t_\text{crossing} \) Time when the windfall and consistent savings are equal
\( \text{NPV} \) Net present value
\( V_t \) Value of cash received at time \(t\)
\( r_r \) Hypothetical rate of return

All the equations about saving money in one place

Photo credit to Pixabay

This article summarizes the relationships relevant to financial independence, investing, and net present value calculations. It also contains the results of a comparison study between steady saving and windfalls.

How long does it take to reach financial independence?

The acronym FIRE (financial independence, retire early) is the name of a small movement trying to save large fractions of income so the investment returns will cover their expenses. Philosophical reasons vary but generally, these people are not hoping to stop working completely, they are hoping to remove compensation from the decision-making process about what kind of work to do and whom to do it with. From the financial independence article, this equation and graph show how long one must save before their investment income covers their expenses (for constant income and expenses and starting with a net worth of $0).

$$ t_s =\frac{1}{r} \log\left[ \left( \frac{1-S}{S}\right) \frac{r}{r_w}+1\right] $$
Plot of saving time to FI vs savings rate. All plots represent a 4% rate of return.

[Caption] Plot of \(t_s\) at various interest rates with \(r_w =\) 0.04. Derivative as \(S \rightarrow \) 1 shown as dashed line.

How long until I save $1 million?

From the saving $1 million article, these equations and graphs show how a steady rate of investment grows over time [left] and how long it takes to save $1 million [right].

$$ m(t) = \frac{P}{r}\left[ e^{rt}-1\right] \quad\quad t_s = \frac{1}{r} \log\left(\frac{rm_\text{target}+P}{P} \right) $$
[Left] Plot of the growth of a continuous savings rate.  [Right] Plot of the length of time required to save $1,000,000.

Windfall or slow and steady?

From the windfalls article, this equation and graphs show how long it takes a steady saver to overtake someone with a windfall. \(V_0\) and \(F\) in dollar and dollar per year, respectively.

$$ t_\text{crossing} = -\frac{1}{r} \log \left( 1-r\frac{V_0}{F} \right) $$
[Left] Growth of continual saving compared to a windfall.  [Right] The crossing-point where saving overtakes the windfall.

Net present values

From the net present values article, for a pile of cash received at a future time, the NPV is the size of the pile of cash required today to generate that future pile of cash at a given rate of return.

For \(V_t\) dollars received at time \(t\):

$$ \text{NPV} = V_t e^{-r_rt} $$

For \(F\) dollars per time, received between now and time \(t\):

If \(F\) is a function of \(t\) If \(F\) is constant
\(\displaystyle \text{NPV} = \int^{t_\text{total}}_{0} e^{-r_r\tau} F(\tau) d\tau \) \(\displaystyle \quad\quad \text{NPV} = \frac{F}{r_r} \left( 1- e^{-r_rt} \right) \)

And rules of thumb for back-of-the-envelope calculations on investments

From the rule of 72 article:

Doubling time Tripling time Quadrupling time
\(\displaystyle \frac{72}{r_\text{as percent}}\) \(\displaystyle\frac{108}{r_\text{as percent}}\) \(\displaystyle\frac{144}{r_\text{as percent}} \)
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© MC Byington