Loan Calculation Formulas
Mathematical formulas for loan calculations, EMI, amortization, and payment schedules
EMI (Equal Monthly Installment) Formula
Standard EMI Formula
Where:
- EMI = Equal Monthly Installment (monthly payment)
- P = Principal loan amount
- r = Monthly interest rate (Annual rate ÷ 12 ÷ 100)
- n = Total number of monthly payments (Years × 12)
Alternative EMI Formula
Same formula, different arrangement for clarity.
EMI Factor Method
Useful when calculating EMI for different principal amounts with same rate and term.
Principal and Present Value Formulas
Principal from EMI
Calculate loan amount when EMI, rate, and term are known.
Present Value of Annuity
Present value of a series of equal payments (annuity formula).
Loan Balance at Any Time
Outstanding balance after t payments.
Interest Rate Formulas
Converting Annual to Monthly Rate
For annual rate as percentage (e.g., 6% becomes 0.005 per month).
Effective Annual Rate (APR)
True annual cost including compounding effect.
Finding Interest Rate (Iterative)
To find r when P, EMI, and n are known, solve:
This requires numerical methods (Newton-Raphson) as there's no direct algebraic solution.
Amortization Formulas
Monthly Interest Payment
Interest portion of payment t, where Bt-1 is balance before payment.
Monthly Principal Payment
Principal portion = Total payment - Interest portion.
Remaining Balance After Payment t
Total Interest Paid
Cumulative Interest Up to Payment t
Different Loan Type Formulas
Simple Interest Loan
Where t is loan term in years, n is number of payments.
Interest-Only Loan
Pay only interest each month, principal due at end.
Balloon Payment Loan
Where B is the balloon payment (remaining balance at end).
Adjustable Rate Mortgage (ARM)
EMI recalculated each adjustment period:
Refinancing Analysis Formulas
Break-Even Point
Net Present Value of Refinancing
Where d is the discount rate (opportunity cost of money).
Total Cost Comparison
Prepayment Analysis Formulas
Time Savings from Extra Payments
For additional payment A each month, solve for new term n:
Interest Savings from Prepayment
Lump Sum Prepayment Impact
New balance after lump sum payment, then recalculate term or EMI.
Loan Comparison Formulas
Annual Percentage Rate (APR)
Approximation where F is total finance charges, P is loan amount, n is number of payments.
Effective Interest Rate with Fees
Cost per $1000 Borrowed
Useful for comparing loans of different amounts.
Formula Applications
Example 1: Home Mortgage EMI
Given: Loan = $300,000, Rate = 4.5% annually, Term = 30 years
Example 2: Car Loan with Different Terms
Compare: $25,000 car loan at 6% APR
3-year term:
EMI = $760.42
Total cost = $27,375
5-year term:
EMI = $483.32
Total cost = $29,000
Longer term = lower EMI but $1,625 more in total interest.
Example 3: Outstanding Balance Calculation
After 5 years of $300k, 30-year mortgage at 4.5%:
After 5 years, still owe $273,044 of the original $300,000.