Retirement Plan Math

A few days ago, I posted the two basic steps of retirement planning.

Step 1: Figure out what quality of life you want during retirement (i.e., how much you want to live on each year). This will determine the size of the nest egg you need to have ready when you retire.

Step 2: Figure out how much you need to start saving now to get there.

Of course, those makes sense to me, but my inner math teacher feels compelled to elaborate for those who need it (even if they won’t ask for it). The calculation for step 1 is a present value formula for an annuity payment. In this case, the payment (PMT) is the amount you want to draw from your portfolio each year in retirement. The present value (PV) is the amount of the portfolio when you retire. (That could be confusing because it’s a present value, but we’re talking about future money; we’ll deal with that later.) You have to specify a real return for the portfolio (r) and the number of years you want to draw this benefit (n). So, if you start with PV in your portfolio at retirement, you can draw PMT from the portfolio each year for n years until you run out of money. You can solve this with the following formula:

Eqn09

If you’re scared of math, you can solve it in Excel: =PV(rate, nper, -pmt). Because Excel looks at time-value-of-money equations as cash flows, enter your PMT value as a negative number. If you don’t, you’ll get the same absolute value, but PV will be negative… no big deal.

For Step 2, you set that present value to a future value (FV), because that’s what you want your portfolio to be in the future. Now use the future value annuity formula and solve for the payment (PMT), which is the amount you need to invest each year (at a real return, r, for n years) to reach that FV. The formula is:

Eqn12

Again, Excel makes it easy: =PMT(rate, nper, pv, -fv). In this case, add a present value (pv) if you already have some savings started.

Note that the second r and n are not the same as the previous formula. This time n is the number of years you have to save for retirement. The previous n was the number of years you plan to draw on your savings during retirement. Similarly, the r in the PMT formula is the real return you expect to earn while you save for retirement. The r in the PV formula is the real return you expect to earn during retirement (which will probably be lower, assuming safer investments).

This (and lots more financial math) is covered in Basic Personal Finance, but now you have the two most important formulas. You can complicate this by simulating returns to add some realism, or you can simply pad your numbers by using a smaller r in either or both formulas. You can also use a larger n and/or larger PMT in the first formula to pad your nest egg.

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