$$V_0$$ Value of windfall at $$t=$$ 0
$$V_w$$ Value of windfall over time
$$V_{ss}$$ Value of steady saving over time
$$F$$ Savings per time
$$r$$ Rate of return on invested capital
$$t_\text{crossing}$$ Time when the windfall and consistent savings are equal

# Is it better to have a windfall or save steadily?

Photo credit to Serpstat

Usually, we do not have a choice in the matter of windfalls, whether they come from wealthy relatives passing, insurance settlements, or winning the lottery, we tend to have little say in when and how they happen. Often I hear people justify their lack of saving habits by claiming their future windfalls will solve everything. This is lousy logic for two reasons: first and most obviously, the windfall is never guaranteed. Second, the power of habitual saving is far greater than most people believe. The equations below were derived here.

Windfall Steady saving Ratio Ratio ($$\lim t \rightarrow \infty$$)
$$\displaystyle V_{w} = V_0 e^{rt}$$ $$\displaystyle V_{ss} = \frac{F}{r} \left( e^{rt} - 1\right)$$ $$\displaystyle \frac{V_w}{V_{ss}} = \frac{V_0 e^{rt}}{\frac{F}{r} \left( e^{rt} - 1\right)}$$ $$\displaystyle \left.\frac{V_w}{V_{ss}}\right|_{\lim t \rightarrow \infty} = \frac{V_0}{\frac{F}{r}}$$

The windfall required to put you in a better situation than someone with a steady saving habit is $$F/r$$. Windfalls or not, long term if you want to be ahead with money you have to cultivate a habit of saving aggressively. Unless your windfall is many millions of dollars, you will always lose out against an aggressive middle-class saver who can put away 10,000/year. For the windfall recipient to come out ahead of the steady saver, the windfall ($$V_w$$) must be greater than the saving rate over the return ($$F/r$$). If not, there will always be a cross over point between the two curves. Where is that point? \begin{align} V_w & = V_{ss} \\ V_0 e^{rt_\text{crossing}} & = \frac{F}{r} \left( e^{rt_\text{crossing}} - 1\right)\\ \left( V_0 - \frac{F}{r} \right) e^{rt_\text{crossing}} & = - \frac{F}{r}\\ \log \left( V_0 - \frac{F}{r} \right)+rt_\text{crossing} & = \log \left(- \frac{F}{r}\right)\\ t_\text{crossing} & = \frac{1}{r} \log \left( \frac{ \frac{F}{r}}{ \frac{F}{r}-V_0 } \right)\\ t_\text{crossing} & = -\frac{1}{r} \log \left( 1-r\frac{V_0}{F} \right)\\ \end{align} This section was written in direct response to a friend who inherited30,000. He asked for my advice; I told him he should be saving aggressively, that he could afford to put $5-10k/yr away, that he should have started a decade ago when he first asked for my advice, and that even if he started today, the money from consistent saving would dwarf his windfall rather rapidly. I am not qualified to speculate on the origin of the psychological deficiencies that would lead someone to conclude their$30k windfall is large enough to render saving inconsequential. It would only take 5 or 6 years of saving for him to equal his windfall in the absence of investing. Even with rewards on a relatively short timescale, he could not be persuaded to modify his behavior to improve his long term well being.