Derivation of the Loan Repayment Formula

I once took a loan of #1,000,000 at an interest rate of 24% per annum (or 2% monthly) from my bank. I assumed at the time that I would end up paying the bank a total of #1,240,000, with the extra #240,000 being the total interest. I expected to pay the sum of #103,333.33 monthly, with my last payment being at the end of the 12th month. My calculation was as shown below;

Assuming Loan taken = L, total interest = ti, annual interest rate = i,  Total amount to be paid = T, and monthly payments = m,

T = L + ti

T = L + L x i  (where ti = L x i)

Recall that L = 1000000 and i = 24%

So T = 1000000 + 1000000 x 0.24 = #1,240,000

Therefore, my monthly repayments will be, m = #1,240,000/12 = #103,333.33.

However, at the end of the first month, my bank deducted approximately #94,559.60 from my account. At first, I thought they made a mistake. But then I imagined that these money-hungry entities might make mistakes crediting an account below the expected amount, but it is almost impossible for them to debit an account below the expected amount. I concluded that my thoughts on how to compute the monthly repayments were off. I studied my calculation and realized it was accurate up to T (the Total amount to be paid) only if I made a one-time payment of #1,240,000 at the expiration of the loan. The subsequent calculation for the monthly payment was totally wrong!

Since the bank was going to make monthly deductions from my account in order to get back their capital and make a profit (interest) at the expiration of the loan, it made no sense to use the initial loan (#1,000,000) to calculate the interest over the twelve months. The reason is that at the end of each month, the total amount owed to the bank reduces by a fixed amount such that after the first deduction or debit, the initial loan of #1,000,000 reduces to #1,000,000 minus the first deduction. The interest for the following month will be charged on what is left of the loan [(#1,000,000 - the deduction) x monthly interest rate] and so on. In other words, interest is owed on the remaining loan balance at the end of each month. I couldn't intuit what the fixed monthly amount to be deducted from my account should be, but as an Engineer with a strong mathematical background, I tried to see if I could derive a formula starting from a basic equation that accurately represents the situation as stated in the following word problem.

Assuming someone took a loan of amount L0 at a monthly interest rate of R, and the loan plus the interest is to be repaid over a period of n months (n can be 1, 3, 6, or 12 months), calculate the fixed amount (m) to be paid at the end of each month in order to completely repay the loan (amount borrowed plus interest) by the end of the nth month. 

The goal here is to find the only unknown which is m, the fixed monthly repayment.


At the end of the first month

With the initial loan of  L0, the interest to be paid is L0 x R or L0R.

The monthly repayment, m is a summation of the interest (L0R) and the amount that reduces the loan (let's call it rL) and this means that m = L0R + rL (or rL = m - L0R).

The amount of loan left (Let's call it L1) is related to  L0  by the equation L0 - rL = L1

And substituting rL = m - L0R in the preceding equation results in  L0 - (m - L0R) = L1 

we can manipulate the equation to get -m +  L0 (R + 1) = L1 --------- first month equation

L1 is the loan balance after the first end of month deduction.


At the end of the second month

Similarly, the loan balance, L2, after the second end of month deduction is related to L1 by the equation:

L1 - (m - L1R) = L2

The left hand side of the above equation can be manipulated to contain only the initial variables (L0, R)  and the constant (m) as follows:

L1 - m + L1R = L2

-m +  L0 (R + 1) -m + [-m +  L0 (R + 1)]R = L2   (replaced  L1  with -m +  L0 (R + 1))

-m(R + 2) + L0 (R2 + 2R + 1) = L2  (Arrived at this by expanding the brackets and taking like terms)

-m(R + 2) + L0 (R + 1)2 = L2  --------- second month equation


There are some similarities between the coefficients of the L0 term in the first and second months equations, but the coefficients of the m term are significantly different, so it's difficult to fully see the emerging pattern. We will continue with the third month to see if we can spot a clear relationship.


At the end of the third month

Similarly, the loan balance, L3, after the second end of month deduction is related to L2 by the equation:

L2 - (m - L2R) = L3

The left hand side of the above equation can be manipulated to contain only the initial variables (L0, R)  and the constant (m) as follows:

L2 - m + L2R = L3

-m(R + 2) + L0 (R + 1)2 - m + [-m(R + 2) + L0 (R + 1)2]R = L3  (replaced  L2 with -m(R + 2) + L0 (R + 1)2)

-m(R2 + 3R + 3) + L0 (R3 + 3R2 +  3R + 1) = L3  (Arrived at this by expanding the brackets and taking like terms)

-m(R2 + 3R + 3) + L0 (R + 1)3 = L3  --------- third month equation


The coefficients of the L0 term in all three equations show an obvious pattern which is that at the end of the nth month, the coefficient of the L0 term of the loan balance is always (R + 1)n. 

The pattern formed by the coefficients of the m term in the equations is not obvious, but it is present. Starting from the third month equation, we can manipulate the coefficient of the m term as follows:

R2 + 3R + 3 = R3 + 3R2 +  3R  =  R3 + 3R2 +  3R + 1 - 1 =  (R + 1)3 - 1
                                   R                                    R                             R

We can therefore rewrite the third month equation as -m[(R + 1)3 - 1] + L0 (R + 1)3 = L3
                                                                                                            R

Similarly, we can rewrite the second month equation as -m[(R + 1)2 - 1] + L0 (R + 1)2 = L2
                                                                                                            R

Similarly, we can rewrite the first month equation as -m[(R + 1) - 1] + L0 (R + 1) = L1
                                                                                                            R

The coefficients of the m term in all three equations now show an obvious pattern which is that at the end of the nth month, the coefficient of the m term of the loan balance is always (R + 1)n - 1
                                                                                                                                     R

This means that at the end of the last or nth month, the loan balance or amount left will be 

 -m[(R + 1)n - 1] + L0 (R + 1)n = Ln
                                                                                                R

Nothing will be left at the end of the last or nth month, so the loan balance, Ln, will be equal to zero. We can rewrite the previous equation as

                                                        -m[(R + 1)n - 1] + L0 (R + 1)n = 0
                                                                        R   

                                                         m[(R + 1)n - 1] = L0 (R + 1)n 
                                                                        R   
It can be seen that the fixed amount (m) to be paid at the end of each month in other to completely repay the loan (amount borrowed plus interest) by the end of the nth month is 

                                                          m =  L0R(R + 1)n 
                                                                    (R + 1)n - 1

So for a loan of #1,000,000 at an interest rate of 24% per annum ( or 2% monthly), the fixed monthly repayments required to completely repay the loan (amount borrowed plus interest) by the end of the 12th month is

                                                         m =  1000000 x 0.02(0.02 + 1)12  =  #94,559.60
                                                                            (0.02 + 1)12 - 1

 

The total amount to be paid to the bank = m x 12 = 94559.6 x 12 = #1,134,715.20

And the total interest to be paid is #134,715.20 (lower than my initial calculation by more than #100,000. Of course, I felt much better about the loan).




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