I want to know how to calculate minimum fixed monthly payment needed in order to pay off a credit card balance within 12 months. I just want to understand the concept, and how to work around this problem. Any book suggestion on this kind of problem will also be helpful.
Example - The outstanding balance on the credit card = 4773 , Annual interest rate = 0.2
Thanks in advance!
I just want to add that, the annual interest rate is 20% (i.e. 0.2) My apologies for the confusion.
On
Let $B_1$ be your current balance that you will start paying this month, $B_i$ for month $i$, and $P$ the amount of the fixed payment. You'll pay off your bill in month 12, and want the 13th statement to be 0.
$$ B_1 = B_1$$ $$B_2 = \left(1+\dfrac{0.2}{12}\right)(B_1-P)$$ $$B_3 = \left(1+\dfrac{0.2}{12}\right)\left(\left(1+\dfrac{0.2}{12}\right)(B_1-P)-P\right) = \left(1+\dfrac{0.2}{12}\right)^2B_1 - \left(1+\dfrac{0.2}{12}\right)^2P -\left(1+\dfrac{0.2}{12}\right)P$$
$$B_4 = \left(1+\dfrac{0.2}{12}\right)^3B_1 - \left(1+\dfrac{0.2}{12}\right)^3P -\left(1+\dfrac{0.2}{12}\right)^2P - \left(1+\dfrac{0.2}{12}\right)P$$
$$\vdots$$
$$B_{13} = \left(1+\dfrac{0.2}{12}\right)^{12}B_1 - P\sum_{i=1}^{12}\left(1+\dfrac{0.2}{12}\right)^{i}$$
You want $B_{13}=0$ which gives:
$$P = \dfrac{\left(1+\dfrac{0.2}{12}\right)^{12}B_1}{\sum_{i=1}^{12}\left(1+\dfrac{0.2}{12}\right)^{i}} = \dfrac{\left(\dfrac{61}{60}\right)^{12}\left(\dfrac{1}{60}\right)}{\left(\dfrac{61}{60}\right)^{13}-\dfrac{61}{60}}B_1$$
With $B_1=4773$ your payments rounded to the nearest cent is $\$434.90$
On
So let me add how to get to the estimate formula: As usual, you start paying at the end of the first month, and as in Tylers answer you get $$ B = P⋅\frac{1 - (1+i^{month})^{-12}}{i^{month}} $$ however, as in David Peterson's answer, banks compute $i^{month}=i^{annual}/12$. Now cite Sir Isaac Newton for the binomial series $$ (1+i)^{-12}=1-12i+\frac{12⋅13}2i^2-\frac{12⋅13⋅14}6i^3\pm... $$ and insert to get $$ B=P⋅(12-6⋅13⋅i+2⋅13⋅14⋅i^2\mp...) $$ which when inverted gives \begin{align} P&=\frac B{12}(1 + \frac{13}2⋅i^{month} + \frac{143}{12}⋅(i^{month})^2+...)\\ &=\frac B{12}(1 + \frac{13}{12}\frac{i^{annual}}2 + \frac{143}{12^3}⋅(i^{annual})^2+...) \end{align} Which results in the rule: Divide balance by 12 and increase that number by one half of the annual interest rate. This will be too low, since $\frac{13}{12}$ is about $8\%$ bigger than $1$. For estimates, make that 10%.
Thus for the given balance, $B/12$ is $398$, 10% of that is 40, increase by 10% to 44, thus the first estimate is 438 and the corrected estimate is 442. The exact payment rate is 442,61.
A series of payments made at fixed points in time are called annuities.
Essentially, you are summing up a finite geometric series. If you want to calculate how much you'd have to pay per month to pay off a $ \$ 4773$ debt at an annual effective rate of $20 \%$, we'd first convert our annual effective interest rate to the effective interest rate per month:
$(1 + i) = (1 + i_{12})^{12} \implies i_{12} = (1.2)^{1/12}-1$.
Once we have that, we need to solve
$4773 = P\frac{1 - (1+i_{12})^{-12}}{i_{12}}$.
At this point you can just plug numbers in.
Annuities are really flexible and pretty interesting, so I'd encourage you to read the wikipedia page. Unfortunately, the calculations can become tedious, but many calculators will calculate the values for you.
EDIT: I want to make clear my assumptions: We start paying next month, and the annual effective rate of interest is $20\%$
To explain the $i_{12}$ part, here's a picture:
If we consider the total interest accrued in the year, it's $1 + i$. We want to change this into a monthly interest though, since we are making a payment every month. So, the payment made in the first month will accrue interest from time $1$ until time $12$, the second payment from time $2$ until time $12$, etc. Basically, we want to find $i_{12}$ so that $(1 + i) = (1 + i_{12})^{12}$ for that reason. We could just use $(1 + i)$, but that would introduce fractional exponents.
I should also apologize also. I was using the notation $i^{(12)}$ originally which is actually something called the nominal rate of interest, and the relationship is $i_{12} = \frac{1}{12}i^{(12)}$. The notation really isn't as horrible as I make it seem, I promise!