DRV-005

Why two SIP calculators give you different answers

A ₹23,004 discrepancy on identical inputs, caused by a convention that neither provider states. Here is the algebra and which one is right for an actual SIP.

The setup

A fixed amount P is invested every period at periodic rate i for n periods. The only question is whether the money arrives at the start of the period or the end, and that one question changes the answer by a factor of (1+i).

The derivation

Step 1: Follow each instalment separately

The first instalment compounds for the whole term, the second for one period less, and so on down to the last.

FV = P(1+i)^n + P(1+i)^(n-1) + ... + P(1+i)

Step 2: Sum the geometric series

Factoring out P(1+i) leaves a standard geometric sum, giving the annuity-due form.

FV = P(1+i) * ((1+i)^n - 1) / i

Step 3: Compare with the ordinary annuity

If instead each instalment arrives at the end of its period, every one of them compounds for exactly one period less, and the whole expression loses its factor of (1+i).

FV_ordinary = P * ((1+i)^n - 1) / i

FV = P * (((1+i)^n - 1) / i) * (1+i) [annuity due]

Checks that prove it is right

A formula you cannot test is a formula you have to trust. These take seconds, and they are what separate a derivation from a formula restatement.

n = 1 → FV = P(1+i) for the due form, P for the ordinary form

A single instalment invested at the start earns one period of growth; invested at the end it earns nothing. The whole difference between the two conventions, at its smallest.

i → 0 → FV → nP under both conventions

With no growth, timing cannot matter. Any implementation that does not converge here has the geometric sum wrong.

Worked example

₹10,000 a month at 12% for 10 years.

The annuity-due form gives ₹23,23,391. The ordinary-annuity form gives ₹23,00,387. That is a ₹23,004 gap on identical inputs. An actual SIP debits your account and then the money grows, which is the annuity-due case, so the larger figure is the right one for a real SIP. Providers split roughly evenly between the two and almost none of them say which they used, so a user comparing two calculators sees a discrepancy and reasonably concludes one is broken.

Computed by the compound-growth engine; both conventions asserted as golden cases.

What it assumes

The return is constant every period.
Instalments are equal and never missed.
No expense ratio, exit load or tax is deducted.

Where the formula stops being valid

Stating limits plainly is more useful than pretending there are none, and it is the item competitors most consistently omit.

Real equity returns are nothing like constant, and the sequence matters as much as the average. A poor first five years and a poor last five years produce very different outcomes from the same mean.
The NAV allocation date is not always the debit date, so a real SIP can sit a day or two either side of the clean annuity-due assumption.