EQUATED MONTHLY INSTALLMENT(EMI)
DEFINITION

Equated Monthly Installment (EMI) refers to a fixed amount of monthly repayment made by a borrower to a lender, such that every (equal) monthly installment consists of two portions – repayment of principal and interest on remaining principal.

CALCULATOR

#### Enter the following details:

Loan Amount:

Duration of loan: (years)

Annual effective interest rate: (%)

Monthly installment amount:

Total amount paid in years:

Total interest paid:

Flat rate of interest:

FORMULA AND DERIVATION
Most loans are generally amortisation loans which include repayments consisting of both principal and interest payments. Equated Monthly Instalments too consist of interest and principal amounts in every instalment. When a loan is given, the lender loses out on the time value of the money that has been lent. So, the amount lent will be equivalent to the amount that will be repaid by the borrower in the future.

Let us first assume a simple case, where a bank provides a loan of amount $$\100$$ to a borrower. Let us assume (for simplicity) that the loan will be repaid in one instalment. The bank is losing $$10\ \%$$ interest that it could have earned on that amount if it hadn’t lent it to the borrower. So, by the end of the year, $$\\ 100$$ would have accumulated to $$\\ 110$$ at $$10\ \%$$ interest. The borrower, therefore, has to make a payment of $$\\ 110$$ to the bank at the end of the year. In simple terms, the amount lent by the bank was the present value of what it would receive from the borrower at the end of the year.

The same logic is applied in calculating the amount of each instalment that the borrower has to pay. If $$L$$ is the amount of loan granted for n years at effective interest rate $$i$$, then the annual instalment amount $$X$$ can be calculated from the following equation:
$L=X{\left(1+i\right)}^{-1}+X{\left(1+i\right)}^{-2}\dots \dots \dots X{\left(1+i\right)}^{-n}$ $\ \ \ =Xv+Xv^2+Xv^3\dots \dots \dots Xv^n$ $\ \ \ =Xa_{\overline{n|}}$
where,
$$a_{\overline{n|}}=\frac{1-v^n}{i}$$ (This is the annuity function used to calculate the present value of a series of level cash flows)

The above equation can be rearranged to give:
$X=\ \frac{L}{a_{\overline{n|}}}$
or
$X=\ \frac{L\times i}{1-v^n}$
The above equation considers that $$X$$ is the annual instalment amount. But in case of EMI’s, there are $$12$$ instalments every year. Now let us consider a loan of amount L for n years at effective interest rate $$i$$ p.a. and $$12$$ instalments per year. If $$X$$ is the amount of each monthly instalment, we can write the loan equation as follows:
$L=X{\left(1+i\right)}^{{-1}/{12}}+X{\left(1+i\right)}^{{-2}/{12}}\dots \dots \dots X{\left(1+i\right)}^{-n}\$ $\ \ \ =12Xa^{\left(12\right)}_{\overline{n|}}$
where,
$$a^{\left(12\right)}_{\overline{n|}}=\ \frac{1-v^n}{i^{\left(12\right)}}$$ (This is the annuity function used to calculate the present value of a series of level cash flows, made 12 times a year)
$i^{\left(12\right)}=12\left[{\left(1+i\right)}^{{1}/{12}}-1\right]$
The above equation can be rearranged to give:
$X=\ \frac{L}{12a^{\left(12\right)}_{\overline{n|}}}$
or
$X=\ \frac{{L\times i}^{\left(12\right)}}{12\left(1-v^n\right)}$
EXAMPLES
Example 1

A bank lends $$\textrm{₹}\ 100000\$$ in loan for $$10$$ years, to be repaid through monthly instalments, with interest charged at $$10\ \%$$ p.a. effective. The monthly instalment amount needs to be determined.

Here,
$$L=100000$$
$$n=10$$
$$i=10\ \%$$
$X=\ \frac{L}{12a^{\left(12\right)}_{\overline{n|}}}=\ \frac{{L\times i}^{\left(12\right)}}{12\left(1-v^n\right)}$ $\ \ \ =\ \frac{100000\times 0.095689}{12\left(1-1.1^{-10}\right)}$ $\ \ \ =1297.75$