# formula to calculate the monthly repayments of this contract

I know that the interest rate is constant through the whole period and the interest method is declining balance. By declining balance mean that the interest at period t is calculated on the balance of period t. In the table below, interest of the 1st repayment is base on the 2748 and the interest of the second repayment is based on 2299.555.

Details :
Loan amount borrowed = 2748 The column principal = the part of the principal repaid in each period.

So far I've come up with this

 |   Date   |Principal|Balance |PayInt|#DaysPM|IntRate0|
0|26/01/2011|         |        |      |       |        |        |
1|20/02/2011|448.445  |2748    |25    |31     |1.33%   |
2|20/03/2011|446.759  |2299.555|28    |28     |1.667%  |
3|20/04/2011|454.205  |1852.796|31    |31     |1.667%  |
4|20/05/2011|461.775  |1398.591|30    |30     |1.667%  |
5|20/06/2011|469.471  |936.816 |31    |31     |1.667%  |
6|20/07/2011|467.345  |467.345 |30    |30     |1.667%  |


Definition:

PayInt = # Interval between payment in days
IntRate0 = $\frac{Interest_t}{Balanct_t}$

The interest is supposed to be constant and 1.33% was obtained by 1.66%*$\frac{24}{30}$

I just would like to know if there is a formula to calculate the monthly repayments of this contract.

Thanks

• It can be anything. The data you provide do not contain information about either the interest amount charged per period, or the total (principal + interest) payment made per period, so the interest can be 1% or 1000%, or whatever. So the fact that you know that interest is constant does not help. What exactly do you mean by "declining balance method"? Sometimes these terms have different meaning in different corners of the world. Sep 2, 2013 at 9:12
• The problem is that i don't have information about the interest or the total(principal + interest) payment per period. I'll give more details about the explanation.
– DJJ
Sep 2, 2013 at 9:44
• After the clarification, my comment still stands: the interest rate can be anything. Consider it yourself. Say interest rate is 10% per month(...). So for first month it will be 274.8. But it could be 5% per month - so for the first month it will be 137.4. There is no evidence in the data you provide to select one or the other case, or any other interest rate. Knowing how many months will it take for the principal to be repaid, does not tell us anything about the interest rate. Sep 2, 2013 at 10:00
• There seems to be a typo in the data, for payment #2, the principal column shows 448.759 but the balance only decreases by 446.759
– Joe
Sep 8, 2013 at 2:57
• You are right. Thanks
– DJJ
Sep 9, 2013 at 9:30

If the interest compounds once per month, not once per day, the formula would be

$L_t = L_{t-1} + RL_{t-1} - P$
Where $L_t$ is the loan balance at month $t$ ($L_0$ being the original amount), $R$ is the monthly interest rate and $P$ is the monthly installment(which you don't seem to have divulged), and the payment to interest is $RL_{t-1}$ (which you seem to have also not divulged)

so the payment to principal for month $t$ will be

$P_{pt} = P - RL_{t-1}$

And the monthly installment from the given information is

$P = P_{pt} + RL_{t-1}$

Assuming the monthly rate is constant, we can select any 2 adjacent months with the same installment amount:

$P_{p1} + RL_0 = P_{p2} + RL_1$

and solve for $R$

$R = \frac{P_{p2} - P_{p1}}{L_0 - L_1}$

Substitute $L_0-P_{p1}$ for $L_1$ (thank you @DJJ) to get

$R = \frac{P_{p2} - P_{p1}}{P_{p1}}$

Which should give the monthly rate, multiply by 12 to get the APR.

If the interest compounds daily, the monthly rate would be

$R = (1 + \frac{r}{n})^n - 1$

where $r$ is the daily rate and $n$ is the number of days in the month.

Edit:

With the new data you have posted (the interest payments), your APR is 20% ($1\frac{2}{3} * 12$), with the first month being $\frac{4}{5}$ (24 days out of 30) of the regular monthly rate

• I apoligise for not having been clear enough. I don't have information about interest. (neither the interest rate nor the interst payment). I only assume that the interest rate is constant through out the repayment period and the interest method is declining balance.
– DJJ
Sep 7, 2013 at 14:27
• I totally agree with your formula though. You can simplify it further by replacting L1 by $L_0-P_{p1}$ to get $(P_{t+1}}/{P_t})-1$. In fact that's what i have written if you take $a_t$ = 1 for all t. But you can see that the fomula you gave does work for all the observations of the table above.
– DJJ
Sep 7, 2013 at 14:32
• You are right, the formula does simplify further, I'll add that to the answer. But that formula is conditional on the interest begin compounded monthly at a constant rate. Since your monthly rate is not constant (probably compounded daily) the formula will be somewhat different.
– Joe
Sep 8, 2013 at 1:40
• The interest rate is constant. And as far as i understand it, the interest rate is compounded monthly. I manage to get the interest paid for this month for this one
– DJJ
Sep 9, 2013 at 8:36
• I'm posting it in the question.
– DJJ
Sep 9, 2013 at 9:13