Savings goal calculator: monthly deposit

Give it a target, a deadline, the rate the account pays and anything you have saved already. It returns the deposit you need at the end of every period, then splits the target into the money you paid in and the money the interest supplied.

The deposit frequency follows the compounding frequency, so monthly compounding means a monthly deposit and 60 of them over 5 years. That pairing is deliberate. A monthly deposit into an account that adds interest once a year is a different calculation, and treating one as the other is how a plan quietly misses by a few hundred dollars.

This is the future value of an annuity, rearranged. Instead of asking what a stream of deposits grows into, it asks what stream reaches a number you have already fixed:

$$PMT = \frac{\left(FV - P\left(1 + \frac{r}{n}\right)^{nt}\right)\frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1}$$

Read it in two halves. The top is what the deposits still have to supply: the target, less whatever your existing balance grows into on its own. The bottom is the annuity factor, which is what one unit of deposit is worth by the deadline once its own interest is counted. At 4 percent compounded monthly over 5 years that factor is 66.298978, so every dollar a month is worth 66.30 at the finish line.

The rate $r$ goes in as a decimal, and $r/n$ is the rate for one period, never the annual rate. At 4 percent compounded monthly the period rate is 0.00333333 and there are 60 periods, not 0.04 and 5.

Run the same factor forwards rather than backwards and you get the future value of an annuity calculator, which answers the other half of the question: not what to save, but what a given deposit reaches.

Three things change the answer: the deadline, the head start, and the rate. They are nowhere near equal.

Time is the strongest, and most of what it does has nothing to do with interest. Going from 5 years to 8 on a $30,000 target cuts the deposit from $452.50 to $265.68, about 41 percent. Spreading the same target over 96 deposits instead of 60 would cut it by 37.5 percent in an account paying nothing at all, so the extra interest earned by holding those deposits longer supplies under 4 points of the 41. Time works on the count of deposits first and on the interest second.

A head start comes next. $5,000 sitting in the account at the outset takes the same 5-year deposit from $452.50 to $360.41, roughly a fifth off, because that money is both already there and growing.

The rate is the weakest lever over a short horizon, and the reason is in the numbers above. Over 5 years at 4 percent, interest supplies $2,850.26 of the $30,000, under a tenth of the goal, so a better rate only acts on that tenth. Sweeping the rate across the entire 0 to 15 percent range the slider allows moves the 5-year deposit by about 32 percent, less than three extra years do on their own. Stretch the same goal to 30 years and interest supplies about 48 percent of it, so the rate finally rivals the deadline, though at 4 percent it does not pass half the target until around year 32. The compound interest calculator shows where the crossover sits.

The figure you type in should be the money you will need on the day, not the money that sounds right now. Three adjustments are worth making first.

Prices. A target costed at today's prices buys less by the deadline, so either raise it by the inflation you expect over the term, or enter a real rate and read the whole answer in today's money. The real rate is $(1 + r)/(1 + f) - 1$, where $r$ is the nominal rate and $f$ the inflation rate, not $r - f$: the subtraction always flatters the answer. And the deposit a real rate gives back is itself in today's money, so the standing order behind it has to rise with prices each year for the plan to hold.

Tax. In an ordinary savings account, interest is usually taxed in the year it is earned rather than when you withdraw, so the rate that matters is the one you keep after tax, not the one advertised. Inside a tax-sheltered account the advertised rate is the one you keep, which is why the wrapper can be worth more than a better headline rate. Which wrappers exist, and how they are taxed, depends on where you live.

The account itself. A rate quoted for 12 months is not a rate you hold for 5 years. If the plan only works at a rate you are not sure of, drag the rate slider down to one you are sure of and look at the deposit that comes back. A plan that survives the lower rate is a plan you can start this month.