Last week I helped my dad rearrange a maturing deposit. The family version of the plan was "4 percent a year, left for 20 years, it more than doubles." I promised him a set of numbers he could take and check against someone else's, and got stuck on the first step: the same "4%, 20 years" gives four different results depending on whether you compound annually, compound monthly, quote only the nominal rate, or use simple interest. Every number below comes from one set of assumed inputs. It is not a quote and it does not forecast any return.
1. Pin down the inputs and the convention first
The inputs are: principal 100,000, 3,000 added each month, nominal annual rate 4%, interest calculated monthly, 20 years. In formula terms that is a periodic rate of i = 0.003333 and N = 240 periods. The principal follows FV = P × (1 + r/m)^(m×n) and reaches 222,258.21 after 20 years; the contributions follow FV = A × ((1 + i)^N - 1) / i and come to 1,100,323.88. Total 1,322,582.09, contributions paid in 820,000.00, gain 502,582.09, a return of 61.29%.
61.29% is the figure most often misread: it is the 20-year cumulative gain divided by total contributions, not an annual rate. The nominal rate I entered is 4% throughout. To compare across different terms you have to switch to the effective annual rate EAR = (1 + r/m)^m - 1; a nominal 4% compounded monthly is 4.0742%, and the two percentages cannot be used interchangeably. Frequency itself matters little: a nominal 6% is 6.1678% effective when compounded monthly and 6.1831% when compounded daily, a gap of 0.0153 percentage points. Percentage points and percent are two different things, which I checked with Percentage Calculator: up 10% then down 10% leaves 99.0000%, a net -1.0000%.
2. The same inputs along three routes
| Convention | Value after 20 years | Difference |
|---|---|---|
| Compound, contribution at each month end | 1,322,582.09 | Baseline |
| Compound, contribution at each month start | 1,326,249.83 | 3,667.75 more |
| Simple interest | 1,186,800.00 | 135,782.09 less |
In the simple-interest row only the principal earns interest, and each contribution earns for the months it is present. Stretch the term and the gap becomes impossible to ignore: under the same conditions over 40 years, compound interest gives 4,039,871.13 and simple interest 2,849,600.00, a gap of 1,190,271.13, which is 8.77 times the 135,782.09 at 20 years. The comparison without contributions: principal 100000, 3.5%, 30 years, compound 280679.37 against simple 205000.00, a gap of 75679.37, confirmed by 1.035^30 = 2.806794.
To decide whether to trust the period-by-period sum or the formula, I reconciled once: 1000 at each month end for 20 years at a nominal 6% compounded monthly gives 462040.90, with 240000 paid in and 222040.90 of gain; 1000*((1.005^240-1)/0.005) also gives 462040.90, a difference of 0.0000. Moving contributions to the start of each month gives 464351.10, 2310.20 more. What I trust is the difference in displayed values, not the last binary digit: 0.1 + 0.2 equals 0.30000000000000004 in a script, and 1.005 * 100 is 100.49999999999999, which only returns to 100.50 at two decimals.
3. I use the rule of 72 only for mental arithmetic
- At 1%: 72/1 = 72.00 years, exact 69.661 years, error 3.36%
- At 4%: 72/4 = 18.00 years, exact 17.673 years, error 1.85%
- At 8%: 72/8 = 9.00 years, exact 9.006 years, error -0.07%, the most accurate of the whole-number rates, while 69.3/8 = 8.6625 years is actually further off
- At 15%: 72/15 = 4.80 years, exact 4.959 years, error -3.22%
- At 20%: 72/20 = 3.60 years, exact 3.802 years, error -5.31%
My mother's fund went from 50 to 120 in six years, which by (end / start)^(1 / n) - 1 is 15.709%; exact doubling takes 4.75 years, the rule of 72 says 4.58 years, 61.1 days short. So I only use it to screen how far a sales pitch is off, never as a planning scale.
4. The three reverse questions are what he actually wanted
Once the forward calculation was done, what he asked was the reverse of three things. In Compound Interest Calculator I solved for term, principal and rate one at a time. How many years: reached in month 247, about 20.58 years, so planning in whole years takes 21, with a balance of 1,031,254.78 by then. How much principal: against a target of one million, the contribution line still leaves 579,699.67 to cover, which equals 282,676.80 set aside today as a lump sum. How high a rate: reaching one million needs a nominal annual rate of 9.513%, an effective 9.939%; at a nominal 3% the same period only reaches 532,316.97.
I read these three numbers as requirements, not forecasts: the reverse calculation assumes a constant rate and full, uninterrupted contributions, and the tool does not support a changing rate. None of the amounts deduct tax, fees or inflation; that has to be done outside the result. At 3% inflation, 100 yuan in 20 years is worth 55.3676 yuan today.
Q: Is an annual rate of 4% the same as 0.33% a month?
The periodic rate uses the displayed value of 4% divided by 12, i = 0.003333, but each month's interest starts earning within the same year, so the effective annual rate becomes 4.0742%, no longer the 4% you entered.
Q: How much difference does contributing at the start rather than the end of each month make?
It affects only the contribution stream, equivalent to moving it one period earlier: 3,667.75 in the main example and 2310.20 in the 1000-a-month case. Card 3 in the tool defaults to end-of-period, and every number in this article uses that convention.
Q: Can the rule of 72 serve as a budget basis?
No. The most accurate whole-number rate is 8%, and even that is -0.07%; the 1% case is off by 3.36% and the 20% case by -5.31%, and the direction of the error flips from positive to negative. Using it as a scale means planning with a number that keeps changing.
Q: Is 502,582.09 an expected return?
No. It is just the arithmetic result of that set of assumed inputs: with simple interest it is 1,186,800.00, stretched to 40 years of compound interest it is 4,039,871.13. Change an input and the number changes; it does not vouch for any product.
Q: Should I deduct inflation before reading it?
Yes, but the tool does not do that step: every amount above is nominal, and tax, fees and inflation all have to be deducted outside the result. The totals I gave do not include them.
Back to the opening line, "4 percent for 20 years more than doubles." That treats 4% as annual compounding, where 72/4 = 18.00 years against an exact 17.673 years, an error of 1.85%. My dad's case is a nominal 4% compounded monthly, 4.0742% effective, which is a different convention, so I did not copy that conclusion straight to him. On paper I set out three numbers side by side: 1,186,800.00, 1,322,582.09 and 1,326,249.83, so he could see how far the number moves when the convention shifts one notch. He only needs to say yes to two things: whether 3,000 a month can keep going for twenty years, and whether this figure, with no tax, fees or inflation deducted, holds its purchasing power after the 55.3676 yuan discount.