$$m$$ 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 $$\frac{I-E}{I}$$

# How long until I can earn my current income with my savings?

This is a question often asked by beginners who correlate their income and their spending. Such people intuitively know that if their income doubled tomorrow, their expenses would double in short order. While such a correlation between income and expenses is the result of insufficient financial discipline (at least in first world countries), the solution to this question answers a more relevant problem - "If I would like an income of X in my retirement, how long will I have to work if I save some fraction of X and invest at an interest rate $$r$$?" (X is in dollars per time in all parts of that question.) This is a good way to set goals for those people who fantasize about a luxurious jet setting retirement. Regardless of the sensibility of the question and the motivation, it can be a powerful goal generating tool. Start again with a balance on our savings.

$$\frac{\partial m}{\partial t} = rm +I-E$$

Solve the first order ODE by separation of variables.

$$\int_{m_0}^{I/r_w} \frac{\partial m}{rm +I-E} = \int_0^{t_s} \partial t$$ $$t_s = \frac{1}{r} \left[ \log\left(r \frac{I}{r_w}+I-E\right)- \log\left(r m_0+I-E\right)\right]$$ $$t_s = \frac{1}{r} \log\left( \frac{r \frac{I}{r_w}+I-E}{ r m_0+I-E} \right)$$

The basic equation with some initial capital $$m_0$$, withdrawal rate $$r_w$$, and saving rate $$S = \frac{I-E}{I}$$.

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

Assume $$r=r_w$$ and $$m_0=$$ 0 for the basic case.

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