Objective: $750,000 as a retirement plan. Rate 6% compounded annually.
Plan A: Deposit $15,000 at the end of each year until objective reached.
Plan B: Deposit $10,000 at the end of the first year, and an additional 5% over the previous year until objective reached. Question: which plan will get you there first and by how many years?
Any help will be appreciated. Thank you.
Plan A is a simple, straightforward FV of $1 per period, except here we need to solve for N in this formula: FV=P{[1 + R]^N - 1/ R}
750,000 = 15,000 x {[1.06]^N - 1 / 0.06} divide both sides by 15,000
50 = {[1.06]^N -1 / 0.06} cross multiply
3 = 1.06^N - 1 add 1 to both sides
4 = 1.06^N take the log of both sides
N = Log(4) / Log(1.06)
N = 23.79 years to save $750,000.
Plan B is a bit more involved, but there is a rarely-used formula for this type of investment, namely:
FV =P x {(1.05^N) - (1.06^N)} / (0.05 - 0.06)
750,000 = 10,000 x {(1.05^N) - (1.06^N)} / (0.05 -0.06) divide by 10,000
75 ={(1.05^N) - (1.06^N)} / (-0.01) cross multiply
-0.75 ={(1.05^N) - (1.06^N)}
As far as I know, there is no direct solution for N. Somebody should correct me on this. But, by simple iteration we get:
N = 23.0244 years to save $750,000
As you can see the two plans are quite similar. Plan B being slightly faster at getting to your objective of $750,000. The difference being:
23.79 - 23.0244 =0.7656 years - or about 9.2 months faster.
