$$m$$ Total invested capital
$$m_0$$ Assets at $$t=0$$
$$I$$ Rate of income
$$E$$ Rate of expenses
$$r$$ Rate of return on invested capital
$$r_w$$ Rate of withdrawal in retirement
$$t_s$$ Saving time
$$S$$ Saving rate $$\frac{I-E}{I}$$

# Saving time until financial independence has a simple closed-form solution

The three most harmful addictions are heroin, carbohydrates, and a monthly salary. - Nassim Taleb in The Bed of Procrustes

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. Articles in this section will discuss how to think about saving and investing.

I first read about financial independence on the Mr. Money Mustache blog. I always had a vague concept of it in mind as a goal but reading about it in this article made me want to give his ideas an explicit mathematical basis. That project began a series of inquiries by friends and family which eventually led to this blog.

## Saving time to financial independence

At the end of the day, success in personal finance is purely behavior-based. A person's net worth (the sum of their bank accounts, assets, and credit liabilities) is a merciless, indifferent answer to one question: "Over a lifetime thus far, how much more was earned or received than was spent?" Playing with these equations is useful for setting goals but in the absence of actions to increase income and/or decrease spending, it does nothing to build wealth. The goal of presenting these equations is to inspire those actions among the kind of people who need more than handwaving arguments to be convinced.

As with loans, we begin with the familiar first order ODE.

[Rate of change of capital] $$=$$ [Rate of capital accumulation] $$-$$ [Rate of capital depletion]
$$\displaystyle \frac{\partial m}{\partial t}$$ $$\displaystyle =$$ $$\displaystyle rm+I$$ $$\displaystyle -$$ $$\displaystyle E$$

Separate and integrate the first order ordinary differential equation. Remember the limits, at $$m_{t=0} = m_0$$ and at $$m_{t_s} = E/r_w$$. The end limit states that for a withdrawal rate, $$r_w$$, we are looking for the interest on $$m$$ to cover our expenses, $$mr_w = E$$.

\begin{align} \int_{m_0}^{E/r_w} \frac{\partial m}{rm+I-E} &= \int_0^{t_s} \partial t \\ \frac{1}{r} \log\left(\frac{ \frac{rE}{r_w}+I-E}{rm_0+I-E}\right) &= t_s \end{align}

Now define the saving rate $$S = \frac{I-E}{I}$$ and recognize $$1-S = \frac{E}{I}$$.

$$t_s = \frac{1}{r} \log\left(\frac{ (1-S) \frac{r}{r_w}+S}{r\frac{m_0}{I}+S}\right)$$

And consider common simplifications.

\begin{align} \text{for } m_0=0 \quad t_s &=\frac{1}{r} \log\left[ \left( \frac{1-S}{S}\right) \frac{r}{r_w}+1\right] \\ \text{for } r_w=r \quad t_s &= -\frac{1}{r} \log\left(r\frac{m_0}{I}+S \right)\\ \label{eq:savingtime} \text{for } r_w=r \text{ and } m_0=0 \quad t_s &= -\frac{1}{r} \log\left(S \right) \end{align}

This is how long it takes a person to save for retirement starting from some multiple of their income ($$\frac{m_0}{I}$$). There is nothing that can change the indisputable fact that the number of years a person must work for retirement is a function of how aggressively they can save their income and how productively they can invest those savings. You can suppose other factors, windfalls of various kinds either as one time events or as unreasonable increases in income later in life, but assuming consistent income (which is by no means guaranteed to rise) and expenses (which are all but guaranteed to rise in healthcare costs alone), this is how long it takes to be free of the need to work for money.

[Caption] Plot of $$t_s$$ at various interest rates with $$m_0 =$$ 0 and $$r_w =$$ 0.04. Derivative as $$S \rightarrow$$ 1 shown as dashed line.

Use reasonable estimates of the rate of return on investment. There is a considerable difference between the returns in a good year (which can exceed 40%) and the expected returns over the long term (about 7-9% depending on optimism). It is also important to be skeptical of those who promise unreasonable returns. Bernie Madoff promised 10%-12% so let that be an upper bound, a guarantee of 10% before inflation is where people start going to jail for life. Inflation averages 1-3% per year depending on who is asked and the stock market gives 6-8% per year. We can then conservatively say 4-5% is our safe withdrawal rate but you can do the calculation with any rate you like.

Bloggers in the early retirement world frequently own rental properties which produce returns in the 10-20% range. This happens in favorable circumstances but like all high reward strategies, it is risky (no shortage of testimonials to that fact, the famed Mr. Money Mustache began early retirement with a \$200,000 loss on an investment in real estate).

## How does adjusting the saving rate change the length of the saving phase?

Let us reconsider

\begin{align} t =& \frac{1}{r} \log\left(\frac{ (1-S) \frac{r}{r_w}+S}{r\frac{m_0}{I}+S}\right) \\ \frac{\partial t}{\partial S} =& \frac{1}{r} \left(\frac{1}{ (1-S) \frac{r}{r_w}+S}\right) \left( \frac{\frac{rm_0}{I} - \frac{r^2m_0}{r_wI} -\frac{r}{r_w} }{\left( r\frac{m_0}{I}+S\right)}\right)\\ \text{for } m_0=0 \quad \frac{\partial t}{\partial S} =& \frac{1}{r} \left(\frac{1}{ (1-S) \frac{r}{r_w}+S}\right) \left( -\frac{\frac{r}{r_w}}{S}\right) \\ \text{for } r_w=r \quad \frac{\partial t}{\partial S} =& \frac{1}{r} \left( \frac{-1 }{r\frac{m_0}{I}+S}\right)\\ \text{for } m_0=0 \text{ and } r_w=r \quad \frac{\partial t}{\partial S} =& -\frac{1}{rS} \end{align}

For all variations shown, in the limit as $$S \rightarrow$$ 0

$$\left. \frac{dt}{dS} \right\rvert_{\lim S \to 0} = -\infty$$

In the limit as $$S \rightarrow$$ 1 things change depending on what simplifications are available.

\begin{align} \left. \frac{dt}{dS} \right\rvert_{\lim S \to 1} &= \frac{1}{r} \left( \frac{\frac{rm_0}{I} - \frac{r^2m_0}{r_wI} -\frac{r}{r_w} }{\left( r\frac{m_0}{I}+1\right)}\right)\\ \label{eq:FI-derivative} \text{for } m_0=0 \quad \left. \frac{dt}{dS} \right\rvert_{\lim S \to 1} &= -\frac{1}{r_w}\\ \text{for } r_w=r \quad \left. \frac{dt}{dS} \right\rvert_{\lim S \to 1} &= -\frac{1}{r^2\frac{m_0}{I}+r}\\ \text{for } r_w=r \text{ and } m_0=0 \quad \left. \frac{dt}{dS} \right\rvert_{\lim S \to 1} &= -\frac{1}{r}\\ \end{align}

The derivative $$-\frac{1}{r_w}$$ is plotted on the graph above.

Notice that when the saving rate is small (near 0), small adjustments result in radical changes in the length of saving time. At high savings rates, incremental changes to $$S$$ result in linear changes in the saving time. If that fails to persuade low savers to change their habits, no logical-based argument will succeed.

## What savings rate do I need to retire in ___ years?

Alteratively, if you want to solve for your savings rate given a certain retirement goal (x years away), we can solve for $$S$$.

\begin{align} t_s &=\frac{1}{r} \log\left[ \left( \frac{1-S}{S}\right) \frac{r}{r_w}+1\right]\\ S &= \left[ \left(e^{rt_s}-1\right) \frac{r_w}{r} +1 \right]^{-1} \end{align}

This equation may be interpreted graphically by swapping the x and y axes on the figure above. If your goal is to retire in $$t_s$$ years and a target expense level $$E$$, use the equation above to solve for savings rate and then solve for $$I$$.

$$I = \frac{E}{1-S}$$