The savings-goal formula, in three rearrangements.

1. The base equation

Start with a monthly compounding schedule. At month n, the balance is the starting balance grown forward at compound interest, plus the sum of contributions also grown forward:

FV = PV(1 + r)ⁿ + PMT · [(1 + r)ⁿ − 1] / r

Where FV is the future value (the goal), PV is the present value (starting balance), PMT is the monthly contribution, r is the periodic interest rate (APR / 12 for monthly compounding), and n is the number of periods (months). This is the future-value-of-an-annuity formula in its general form. The first term is the future value of the lump-sum starting balance; the second term is the future value of the contribution stream.

2. Rearrangement 1: solving for PMT

Given goal FV, starting balance PV, rate r, and time n, the required monthly contribution is:

PMT = [FV − PV(1 + r)ⁿ] / [((1 + r)ⁿ − 1) / r]

Numerator: the gap between the goal and what the starting balance grows to on its own. Denominator: the “annuity factor” that converts a series of monthly contributions into a single future value at rate r over n periods. The result is the monthly amount that, contributed for n months and compounded at r, exactly closes the gap.

Worked example. Goal S$50,000, starting balance S$5,000, time 36 months, APR 3.5 % (monthly r = 0.002917). PV grows to 5000 × 1.002917³⁶ = S$5,553. Gap = 50,000 − 5,553 = S$44,447. Annuity factor = (1.002917³⁶ − 1) / 0.002917 = 37.96. Required PMT = 44,447 / 37.96 = S$1,170.65 per month.

3. Rearrangement 2: solving for n

Given goal FV, starting balance PV, rate r, and contribution PMT, the time required is:

n = ln[(FV · r + PMT) / (PV · r + PMT)] / ln(1 + r)

Derivation. Set FV equal to the future-value formula and solve algebraically for n. The result requires that the bracketed argument of the logarithm be positive (otherwise the contribution stream cannot reach the goal at all) and that PMT or PV provide enough fuel for the goal to be reachable in finite time.

Worked example. Goal S$50,000, starting balance S$5,000, contribution S$1,000 per month, APR 3.5 %. Numerator: ln[(50,000 · 0.002917 + 1,000) / (5,000 · 0.002917 + 1,000)] = ln[1,145.83 / 1,014.58] = ln(1.1294) = 0.1217. Denominator: ln(1.002917) = 0.002912. n = 0.1217 / 0.002912 = 41.8 months ≈ 3 years 6 months.

Special case: when r = 0, the formula collapses to n = (FV − PV) / PMT, the simple division of remaining gap by monthly contribution.

4. Rearrangement 3: solving for r

Given FV, PV, n, and PMT, there is no closed-form solution for r. The equation is transcendental in r and must be solved numerically. The standard methods are bisection (slow but reliable) and Newton-Raphson (fast but sensitive to starting point); the calculator on this site uses bisection over the [−0.5 %, 60 %] APR range, which is fast enough at retail input scales and reliable across the parameter space.

The required-rate output is most useful as a sanity check. If the required APR comes back at 7 %, the plan is feasible only with equity-heavy investing — appropriate for a 15+ year goal but inappropriate for a 3-year wedding fund. If it comes back at 1 %, a high-yield savings account suffices. If it comes back at 12 %, the plan needs revising: extend the timeline, raise the contribution, or lower the goal.

5. Compounding frequency: monthly vs annual

The calculator uses monthly compounding (r = APR / 12, n in months). The most-quoted retail rate is APR. Monthly compounding gives a slightly higher effective annual yield than the APR alone:

APY = (1 + APR / 12)¹² − 1

For a 3.5 % APR, the APY is 3.557 %. The difference is small but real and matters for accurate long-horizon projections. For Singapore Savings Bonds, the rate paid is essentially the APY (annual coupon, semi-annual compounding); for high-yield savings accounts, the relationship between quoted rate and realised yield depends on the issuer’s posting convention.

6. The annuity-due variant

The standard formula assumes ordinary annuity timing: contributions made at the end of each period. The annuity-due variant assumes contributions at the beginning of each period — appropriate for payroll-deduction structures where the salary lands at the start of the month and the contribution is moved immediately. The annuity-due future value is simply the ordinary-annuity formula multiplied by (1 + r). For monthly contributions at retail rates, the difference is small (under 0.4 % of accumulated balance for typical r); the calculator uses the ordinary-annuity convention as the standard. For meaningful difference, contribute earlier in the period rather than later — payday is typically the right answer.

7. Multiple goals and the priority question

The single-goal formulation works for one goal at a time; real households have several concurrent goals competing for the same monthly capacity. The standard sequencing for households with limited monthly savings capacity is: emergency fund first (build to 3 months of expenses), then high-interest debt elimination, then employer-match retirement contributions to the match level, then the next-priority goal (housing deposit, children’s education), then taxable retirement contributions. Working through the calculator on each goal sequentially and comparing the required monthly contributions to total monthly savings capacity produces a feasible multi-goal plan; trying to fund every goal simultaneously without a sequence usually produces under-funding of all of them.