Annuity on Principal paid monthly

I have to calculate the annuity amount to be paid monthly on a loan of 5000 with 5% interest rate in 2 years. The annuity should be calculated monthly. The interest will change on the basis of remaining amount. My problem is that every calculator and formulae I am using is giving me the amount of 362, while the actual annuity amount should be 219.36. I do not know how I am getting this amount. Only this Calculator has given me the correct amount. But they have not shown the process they have used to calculate the annuity payout.

This is the loan details:

Loan Amount 5000 €
Nominal Interest Rate 5.00 %
Duration 2 years
Start-Date 01.01.2018

I will have to use only above input parameters to calculate below result.

This is the repayment plan of loan that I will get:

+------------+---------+-----------+----------+-------------------------------+---------------------------------+
| Date       | Annuity | Principal | Interest | Initial Outstanding Principal | Remaining Outstanding Principal |
+------------+---------+-----------+----------+-------------------------------+---------------------------------+
| 01.01.2018 | 219.36  | 198.53    | 20.83    | 5000                          | 4801.47                         |
+------------+---------+-----------+----------+-------------------------------+---------------------------------+
| 01.02.2018 | 219.36  | 199.35    | 20.01    | 4801.47                       | 4602.12                         |
+------------+---------+-----------+----------+-------------------------------+---------------------------------+
| ...        | ...     | ...       | ...      | ...                           | ...                             |
+------------+---------+-----------+----------+-------------------------------+---------------------------------+
| 01.12.2019 | 219.28  | 218.37    | 0.91     | 218.37                        | 0                               |
+------------+---------+-----------+----------+-------------------------------+---------------------------------+
• Is the interest rate on a yearly basis? Is it compounded? In the link you provide you also take an inflation rate into account.
– Jan
Commented Sep 18, 2018 at 14:20
• @Jan Interest will be yearly basis. And I do not know why they have used the inflation. But main result should be the annuity payout. Commented Sep 18, 2018 at 14:22
• The repayment plan you show is wrong. Try calculating the interest for the first month. It should be $20.37$ since we pay $((1+0.05)^{(1/12)}-1)*5000$ as interest for the first month. My basic excel calculation gives the correct value to be $219.11$ (well in between 219.10 and 219.11 but rounding).
– Jan
Commented Sep 18, 2018 at 14:35
• @Jan yes you are almost there how did you get this 219.11 what is that amount. Do not worry about the repayment plan. Commented Sep 18, 2018 at 14:37
• I quickly build it in excel using the 24 payments and used the solver. The analytic answer I do not know. Usually the problem is the other way around, with the amount increasing by the interest instead of ignoring the interest paid.
– Jan
Commented Sep 18, 2018 at 14:57

I don't know what formulas you have, but if the effective rate for one month is $i$, the annuity will be $$A=M_0\frac{i(1+i)^N}{(1+i)^N-1},$$ whit $M_0$ being the initial debt (that is the capital to pay at the beginning) and $N$ is the number of periods to pay. You have $N=2\times 12=24$ and $i=\frac{.05}{12}$ if, as I suppose, the $5\%$ is a yearly nominal rate. $M_0=5000$. Try those numbers in the formula and you'll get the answer you are looking for.

To get the formula you can think as follows:

• The annuity $A_n$ is constant in each period (let's say $A$) and it pays the period's interests $I_n$ and devolves part of the capital $C_n$. That is $$A=I_n+C_n.$$
• The interests to pay at $n$ are proportional to the ammount $M$ of the debt at $n-1$, that is $$I_n=iM_{n-1}.$$
• The debt at $n$ equals the debt at $n-1$ minus the capital paid at period $n$: $$M_n=M_{n-1}-C_n=M_{n-1}-(A-I_n)=$$ $$=M_{n-1}-(A-iM_{n-1}),$$ which gives $$M_n=(1+i)M_{n-1}-A.$$

Now, we can use the formula to see that $$M_1=(1+i)M_0-A;$$

$$M_2=(1+i)M_1-A=$$ $$=(1+i)\big((1+i)M_0-A\big)-A=$$ $$=(1+i)^2M_0-A\big(1+(1+i)\big);$$

$$M_3=(1+i)M_2-A=$$ $$=(1+i)\Big((1+i)^2M_0-A\big(1+(1+i)\big)\Big)-A=$$ $$=(1+i)^3M_0-A\big(1+(1+i)+(1+i)^2\big);$$

$$\vdots$$

$$M_n=(1+i)^nM_0-A\big(1+(1+i)+(1+i)^2+\cdots+(1+i)^{n-1}\big)$$ $$=(1+i)^nM_0-A\frac{(1+i)^n-1}i.$$ (in the last step I used the formula $$1+q+q^2+\cdots+q^{n-1}=\frac{q^n-1}{q-1},\quad q\neq1$$ for the sum with the dots.)

Now, you want that at the last period $N$ your remaining debt be zero, that is $M_N=0$, so $A$ has to be such that $$M_N=(1+i)^NM_0-A\frac{(1+i)^N-1}i=0,$$ and solving the equation for $A$ gives the formula $$A=M_0\frac{i(1+i)^N}{(1+i)^N-1}.$$

• I get this formulae. Please can you help me with the explanation of this formulae on how it is derived. Commented Sep 18, 2018 at 14:54
• Sure. I added a derivation of the formula to the answer. Commented Sep 18, 2018 at 15:12
• Great answer. One thing though that if the effective yearly interest rate is $5\%$ we should not use $0.05/12$ but rather compounded interest rate such that $(1+r)^{12}=1.05$. So we get $r=1.05^{(1/12)}-1$.
– Jan
Commented Sep 18, 2018 at 15:13
• That depends on whether it is a nominal or an effective interest rate. The answer was not very precise, but as I put at the beggining I assumed ('correctly', as it seems, since I got the 'right' answer) that it was an yearly nominal rate. It is common use in my country to understand that this is the case when nothing else is said, and it seem this was the case, too. Commented Sep 18, 2018 at 15:23
• I meant the question... not the answer. =S Commented Sep 18, 2018 at 23:05

The 5% interest is an annual stated figure, but the repayment period is monthly. The standard convention is that this refers to a 24-period loan, where each period is a month and the monthly interest rate is $r=.05/12$. So the question is to compute the monthly payment $C$ such that a 24-period annuity with discount rate $r$ has present value 5000.

Recall that the present value of an n-period annuity with first payment paid one period from now is

$$PV=\frac{C}{r}(1-\frac{1}{(1+r)^n}).$$