Math question

Question

Suppose you want to have enough money in the bank 35 years from now (when you retire) so that the interest on the account would be enough to live on each year from then on. If you'll need $75,000 yearly to live on, how much will need to be sitting in your bank account when you start retirement? Assume an average APR of 5.0%. How much would you need to save each month to have this much saved in 35 years?

1)Savings when you start retirement:

???$ (Round to the nearest dollar and do not use commas.)

2)Monthly Payment:

???$ (Round to the nearest dollar and do not use commas.)

Answer 1

1. Assuming that you would like your retirement savings to last indefinitely, then you should only draw interest from your account each year and not touch the principal. in order for 5% interest to amount to $75,000, your account should hold $75,000 / 0.05 = $1,500,000.00. Of course, you need to be careful when talking about an average rate of return. Slightly higher rates of return earlier will offset similar lower returns later, but the converse is not true.
2. There is a formula for future value: FV = P*((1+r)^t - 1) / r, which you can easily solve for P. In this case the future value of your account (at retirement) needs to be $1,500,00.00, the annual interest rate, r, is 0.05, and the time, t, is 35 years. In this case, your annual payment into your account, P, is given by: P = r* FV / ((1+r)^t - 1) or 0.05 * $1,500,000.00 / ((1+0.05)^35 - 1) = $16,607.56. Note that this formula assumes that you will be depositing your $16,607.56 payment at the end of each year. That means that your final payment of $16,607.56 would be deposited at the end of year 35, so you would have to wait throughout your entire first year of retirement before you are able to withdraw your first retirement payment. That's not much to live on that first year! Instead, I would think that you would make each payment into your account at the start of each year so that you can withdraw your first retirement payment at the start of your first year of retirement. This simply means that each payment will benefit from one additional year of interest. If you divide that previously computed amount by (1 + r), you get the adjusted payment amount of $16,607.56 / (1 + 0.05) = $15,816.72. Of course, my previous caveat about sequencing of returns still applies. We assumed that you would earn a 5% return every year. If the returns are a little higher early on and a little lower later on, your overall return will be improved. But, if slightly lower return occurs earlier in your savings period, you will need relatively larger returns later to offset the shortfall.