# Calculating Interest Rate From EMI

Suppose there is a loan with equated monthly installments. If I know the loan period, loan amount and equated monthly installment amount, is there a way to calculate the interest rate?

EMI= a * (1 + r)^p / [(1 + r)^p - 1]

a: loan amount

r: interest rate

p: loan period

You need to solve for the interest rate. Also your formula should be

``````EMI = a*r*(1 + r)^p/((1 + r)^p - 1)
``````

Derived from the sum of the discounted payments being equal to the principal

$a=\sum_{k=1}^{p}\frac{EMI}{(1+r)^k}$

$\therefore a=\frac{EMI(1-(1+r)^{-p})}{r}$

$\therefore EMI=\frac{a\cdot r(1+r)^p}{(1+r)^p-1}$

Example solution

``````a = 1000
p = 3
r = 0.05

∴ EMI = 367.209
``````

Suppose the interest rate is unknown

``````EMI = 367.209
a = 1000
p = 3
``````

solve for varying `r` until this equation is satisfied:

$\frac{EMI(1-(1+r)^{-p}}{r}-a=0$

Plot of `(EMI (1 - (1 + r)^-p))/r - a` over a range of `r`

or use an online solver, e.g.

• Thank you so much. Sorry for late reply. – Tolga Karahan Feb 12 at 6:57