# Calculating interest rate from Loan Amount, Loan Term and Monthly Payment

I am trying to find the equation that the below website uses to calculate interest rate.

https://www.calculator.net/interest-rate-calculator.html?cloanamount=15000&cloanterm=3&cmonthlypay=475.5312917&x=12&y=28

I am able to calculate the repayment amount by following the directions here What's the math formula that is used to calculate the monthly payment in this mortgage calculator?, but have been unable to rearrange the equation to get the interest rate from the repayment amount, loan term and loan amount.

• There is generally no closed form solution for the interest rate; numerical methods or a financial calculator must be used Mar 31, 2021 at 2:10
• I ended up using the Excel RATE function to get the result I was looking for. Mar 31, 2021 at 5:34

Starting from @python_enthusiast's answer, the equation to be solved for $$i$$ is $$\text{PV}=\frac{\text{FV}}{(1+i)^n}+\frac{\text{PMT}}{i}\Bigg[1-\frac 1{(1+i)^n} \Bigg]$$ which does not show any explicit solution but which does not make any problem using a numerical method.

However, we can make good approximations. For conveniency, I shall use $$a=\text{PV}$$, $$b=\text{FV}$$ and $$c=\text{PMT}$$ and I shall take advantage of the fact that $$i \ll 1$$.

Using Taylor expansion for the rhs and series reversion, we have $$i=t+\frac {(3 b+2 c)+3 n (b+c)+c n^2 } { 3(2 b+ c)+3 c n} t^2+O(t^3)$$ where $$t=2\frac{(b-a)+c n}{ (2 b+c)n+c n^2 }$$

Let me try for $$a=2000$$, $$b=1000$$, $$c=100$$ and $$n=15$$. This would give $$i=\frac{823}{39366}=0.0209064$$ while the "exact" solution is $$i=0.0213729$$.

If this is not sufficiently good, add the next term which is $$\frac {(n+1)\left(\left(24 b^2+36 b c+14 c^2\right)+n \left(48 b^2+78 b c+31 c^2\right)+n^2 \left(30 b c+22 c^2\right)+5 c^2 n^3 \right)}{36[(2 b+c)n+c n^2]^2}\,t^3$$ This would give $$i=\frac{1221967}{57395628}=0.0212902$$

Edit

Another formulation

$$i=\frac{-6 ((a-b) (2 b+c))+6 c (-a+3 b+c)n+6 c^2 n^2 } {2 (a-b) (3 b+2 c)+ \left(6 b (a+b)+6 a c-c^2\right)n+2 c (a+2 b)n^2+c^2 n^3 }$$ gives, for the worked example $$i=\frac{27}{1270}=0.0212598$$