## Income, wealth, and being rich

An article in Bloomberg recently told of how some Wall Street bankers were struggling to make ends meet because of the drop in bonuses. There was the obvious backlash and schadenfreude.  Although I agree that if you’re making upwards of 350 thousand per year you shouldn’t get much sympathy, most people are still confused about the difference between wealth and income.  Your are rich if you have lots of wealth.  Having a high income will help you attain wealth but if you spend more than your income then you will be struggling.  To be truly rich, you should be able to sustain your current lifestyle on just your wealth (e.g. savings and investments) and not require any additional income.  The easiest way to become “rich” is to reduce your expenses.

This can be made concrete with the simple model of wealth change:

$\frac{dW}{dt}= I +r W - E$

where W is wealth, I is annual income, r is the annual return on investments, and E is annual expenditure.  In the absence of income (I = 0), you can maintain your lifestyle and wealth, if $rW\ge E$.   You are rich if the annual return on your wealth exceeds your annual expenditures.  If you’re annual expenditure is 100 thousand dollars, and your annual rate of return is 5%, then you’ll need 2 million dollars to be rich.  However, this is a conservative estimate because you’ll still have $2 million when you die. Suppose, you just want to maintain your lifestyle while you’re alive and have nothing left when you die. Then, we need to solve the differential equation, yielding $W(t) = \frac{(E-I)}{r}(1-e^{rt}) + W(0)e^{rt}$ or $W(0)=W(t)e^{-rt}+\frac{E-I}{r}(1-e^{-rt})$ for the current wealth you need if you want to have $W(t)$ dollars in t years. If we suppose no income and zero wealth at time t, then we get $W(0) = E(1-e^{-rt})/r$, for the necessary current wealth, which gives the intuitive result that you need t times your annual expenditure if the growth rate is very slow. The above formula also works if we include the rate of inflation (and taxes) if we let r be the real annual rate of return that includes inflation and taxes. So putting in some numbers, if we suppose that you want to have zero wealth in 50 years time and you have a real rate of return of 3%, then you’ll need about 26 times your annual expenditure to be rich. If you get 5% real return then you’ll need 18 times the expenditure and if you can get the historical 7% from the stock market then you’ll only need 14 times the expenditure. If you can get your expenditure down to$50K per year then you’ll only need $700,000 to be self-sufficient. However, if you wanted to be really rich such that you could live basically anywhere, travel wherever you want, own a summer home, send your kids to private school, then let’s say you’ll need to spend about a million dollars a year. If you can get 7% real returns, then you’ll need$14 million dollars to be rich.

Mar 8: Corrected some errors