Future value

For a single amount, the future value is the amount times (1 + r) raised to the number of periods, n. $10,000 at 6% a year is $10,600 after one year and $21,329.28 after thirteen. That is compound interest run forward to a date: compound interest is how a balance earns interest on its interest, and future value is the balance it reaches.

For a series of equal deposits, an annuity, the future value adds up the deposits, each grown for the periods it has left. The first deposit grows the longest and the last barely at all. The sum has a closed form: the deposit times ((1 + r)^n − 1) ÷ r, the annuity factor. A savings projection with a starting balance and regular contributions is the two parts added together.

The result depends on conventions as well as inputs: how often the rate is applied, and whether a deposit lands at the start or the end of a period. A projection that credits a year's contributions at the year's end comes out a little lower than one that credits a monthly deposit as it arrives at a monthly rate. The rate is an assumption, so the future value is what the balance would be if that rate held in every period.

A 529 holds $10,000 for a child who is five and enrolls at 18, thirteen years out. $250 goes in a month, $3,000 a year, credited at the end of the year, and the balance is assumed to earn 6% a year. These are the college savings calculator's defaults.

The $10,000 on its own grows by 1.06 to the 13th power, a factor of 2.1329, to $21,329.28. The contributions are 13 deposits of $3,000; the annuity factor at 6% over 13 years is 18.8821, so they grow to $56,646.41. Together the balance at enrollment is $77,975.69, of which $49,000 was paid in and $28,975.69 is growth.

At 4% a year the same deposits reach $66,531.25, and at 8% they reach $91,682.13. Credited monthly at 0.5% a month, with the $250 added at the end of the month, the 6% case reaches $80,634.20. The figures here are this example's arithmetic.

Balance today

$10,000

Grown 13 years at 6% (× 2.1329)

$21,329.28

Contributions, $3,000 a year × 18.8821

+ $56,646.41

Future value at enrollment

$77,975.69

A goal with a date on it, such as a college bill or a down payment, is met or missed by the balance on that date, and the balance today understates it. In the example, $49,000 paid in becomes $77,975.69 at 6%; the $28,975.69 of growth is more than a third of the total.

It also shows what each input is worth. The starting $10,000 more than doubles because it has all 13 years to grow, while the last $3,000 deposit has none. Two points of return either way move the result by $11,444 down or $13,706 up, which is why a projection needs to say what rate it assumes and how it applies it.

Taiga Finance's college planner, under Scenarios, runs this sum for each beneficiary. It takes a “Savings today”, a “Monthly contribution” and an “Enrolls at age”, and under Assumptions a “Nominal annual return” in “%/yr”. The “Balance vs cost” chart draws the “Projected balance” year by year with the college years marked, and “Year by year” lists the “Contributions”, “Growth”, “College bill” and “End balance” for the years to enrollment and through college. The end balance in the year before enrollment is the future value in the example: the planner grows the balance at the flat return and credits twelve months of contributions at the end of the year.

The math behind “Extra monthly to hit target” starts from the same sum: “529 today”, the line “+ $250/mo until enrollment”, and the row “growing at 6% nominal · at enrollment”, which reads the balance on the day college starts. Contributions stop at enrollment; from then the planner draws the year's bill from the balance at the start of the year. The public college savings calculator runs the same projection on figures typed in, with nothing saved.