SIP Calculation Formula & Derivation
The formal mathematical proof and step-by-step derivation of mutual fund compounding formulas.
The Standard SIP Future Value Formulas
1. Beginning of Month Formula (Annuity Due — Indian Mutual Fund Standard)
This formula assumes each monthly installment is invested on day 1 of the month, compounding for that entire month before the next installment arrives.
2. End of Month Formula (Ordinary Annuity)
This formula assumes installments are contributed on the last day of each month, earning zero return during their first month.
Formula Variables
| Symbol | Parameter | Description | Example Value |
|---|---|---|---|
| FV | Future Value | Projected accumulated maturity corpus at tenure end | ₹22,40,359 |
| P | Monthly Investment | Fixed periodic installment amount contributed each month | ₹10,000 |
| i | Periodic Monthly Rate | Monthly compounding rate expressed as a decimal | 0.00948879 (0.9489%) |
| n | Total Installments | Total number of monthly contributions (Years × 12) | 120 (10 years) |
Step-by-Step Proof from Geometric Progression
Consider n monthly payments of amount P, compounded at periodic monthly rate i:
- The 1st payment is invested for n months, growing to:
P × (1 + i)n - The 2nd payment is invested for n - 1 months, growing to:
P × (1 + i)n - 1 - The n-th (final) payment is invested for 1 month, growing to:
P × (1 + i)1
The total future value FV is the sum of this finite geometric progression:
Factoring out P × (1 + i):
Applying the standard sum formula for a finite geometric series with common ratio r = 1 + i:
This proves the Annuity Due SIP formula Q.E.D.
Arithmetic Walkthrough: ₹10,000 / 10 Years / 12% Return
Calculating the maturity value under the AMC Effective Annual CAGR convention:
i = (1 + 0.12)^(1/12) - 1 = (1.12)^0.083333 - 1 ≈ 0.0094887929
n = 10 years × 12 months = 120 installments
(1 + 0.0094887929)^120 = (1.12)^10 ≈ 3.1058482083
(3.1058482083 - 1) / 0.0094887929 = 2.1058482083 / 0.0094887929 ≈ 221.929948
FV = 10,000 × 221.929948 × 1.00948879 = ₹22,40,359
Frequently Asked Questions: SIP Formula & Math
Common questions regarding SIP calculations, compounding formulas, and mutual fund conventions.
The standard SIP formula for payments at the beginning of each month (Annuity Due) is FV = P × [((1 + i)^n - 1) / i] × (1 + i), where P is monthly investment, i is periodic monthly rate, and n is total number of monthly payments.
In mutual funds, SIP installments are debited at the start of each month (e.g. 1st or 5th) and compound for the entire month. Multiplying by (1 + i) accounts for beginning-of-period compounding, distinguishing an Annuity Due from an Ordinary Annuity.
In India, mutual fund asset management companies (AMCs) use Effective Annual CAGR where i = (1 + R)^(1/12) - 1. Many consumer broker portals use nominal rate division where i = R / 12.