Personal budgeting is an important part of achieving financial independence (FI). Assume financial independence is defined as having achieved 25X your annual budget (or 300X your monthly budget) in your investment account.

Show a brief derivation of the expression you derived earlier this summer to calculate time to achieve financial independence (FI) with monthly compounding when you deposit R fraction of your monthly income M. .

Note: Am = R(M) or R = Am/M, and your monthly budget Bm = (1-R) M.

Regrettably, the median retirement account equates to less than about one year of income for the person who is 10-years out of school. By contrast, we have previously shown that a person can achieve FI after about a decade of working if that individual saves about 50% of their annual income in to their FI account. Use equation you derived above to create a plot of time-to-FI based on saving {5% to 50%, steps of 5%} of your monthly income deposited to FI account. Create two curves with 8.5% and 12.5% annual returns and assume monthly compounding.