Calculating pay off for credit card with multiple APRs
"@Joe's original answer and the  example with proportionate application of the payment to the two balances is not quite what will happen with US credit cards. By US law (CARD Act of 2009), if you make only the minimum  required payment (or less), the credit-card company can choose which part of the balance that sum is applied to. I am not aware of any company that chooses to apply such payments to anything other than that part of the balance which carries the least interest rate  (including the 0% rate that ""results"" from acceptance of balance transfer  offers). If you make more than the minimum required payment, then the excess must, by law, be applied to paying off the highest rate balance. If the highest rate balance gets paid off completely, any remaining amount must be applied to second-highest rate balance, and so on.  Thus, it is not the case that that $600 payment (in Joe's example) is applied proportionately to the $5000 and $1000 balances owed. It depends on what the required minimum payment is. So, what would be the minimum required payment? The minimum payment is the total of (i) all finance charges incurred during that month, (ii) all service fees and penalties (e.g. fee for exceeding credit limit, fee for taking a cash advance, late payment penalty) and other charges (e.g. annual card fee) and (iii) a fraction of the outstanding balance that (by law) must be large enough to allow the customer to pay off the entire balance in a reasonable length of time. The law is silent on what is reasonable, but most companies use 1% (which would pay off the balance over 8.33 years).  Consider the numbers in Joe's example together with the following assumptions: $5000 and $1000 are the balances owed at the beginning of the month, no new charges or service fees during that month, and the previous month's minimum monthly payment was made on the day that the statement paid so that the finance charge for the current month is on the balances stated). The finance charge on the $5000 balance is $56.25, while the finance charge on the $1000 balance is $18.33, giving a minimum required payment of $56.25+18.33+60 = $134.58. Of the $600 payment, $134.58 would be applied to the lower-rate balance ($5000 + $56.25 = $5056.25) and reduce it to $4921.67. The excess $465.42 would be applied to the high-rate balance of $1000+18.33 = $1018.33 and reduce it to $552.91. In general, it is a bad idea to take a cash advance from a credit card. Don't do it unless you absolutely must have cash then and there to buy something from a merchant who does not accept credit cards, only cash, and don't be tempted to use the ""convenience checks"" that credit-card companies send you from time to time. All such cash advances not only carry larger rates of interest (there may also be upfront fees for taking an advance) but any purchases made during the rest of the month also become subject to finance charge. In other words, there is no ""grace period"" for new charges, and this state of affairs will last for one month beyond the first credit-card statement whose statement is paid off in full in timely fashion. Finally, turning to the question asked, viz. "" I am trying to determine how much I need to pay monthly to zero the balance, ...."", as per the above calculations, if the OP makes the minimum required payment of $134.58 plus $1018.33, that $134.58 will be applied to the low-rate balance and the rest $1018.33 will pay off the high-rate balance in full if the payment is made on the day the statement is issued. If payment is made later, but before the due date, that $1018.33 will be accruing finance charges until the date the payment is made, and these will appear as 22% rate balance on next month's statement. Similarly for the low-rate balance. What if several monthly payments will be required? The best calculator known to me is at https://powerpay.org (free but it is necessary to set up a username and password).  Enter in all the credit card balances and the different interest rates, and the total amount of  money that can be used to pay off the balances, and the site will lay out a payment plan.  (Basically, pay off the highest-interest rate balance as much as possible while making minimum required payments on the rest). Most people are surprised at how much can be saved (and how much shorter the time to be debt-free is) if one is willing to pay just a little bit more each month." The first thing you need to do is look at your terms and conditions of your credit card, or ask your bank, how they will apply the payments.  As Dilip notes in his answer, in the US, they will likely apply the minimum payment to the lower rate balance, and then must apply the rest above the minimum to the higher rate balance.  In other countries, this will vary by law and custom.  Do not assume it will pay off the higher balance, or proportionally, without asking. Let's take the following example. You owe $6000.  $5000 is at 13.5% (normal purchase rate) and $1000 is at 22% (cash advance rate).  If your bank applies payments to both balances proportionally, then a payment of $600 will reduce your purchase balance by $500 and your cash advance balance by $100. The average APR, then, is simply sum of the product of the APR times balance.  So here, (.135*5000 + .225*1000)/6000 = 15%.  This is called a weighted average. If the bank applies the payment differently - such as to the lower rate first, or some specified part to the lower rate and the rest to the higher rate - then this will be misleading if you enter it into a calculator, because your average APR will rise over time as you pay off the purchase balance but don't pay off the cash advance balance, or may decrease if the opposite happens. The weighted average is probably reasonably close in the circumstance that you describe, even if you have rules applying the balance differently, so long as they don't 100% pay down the lower rate - so it may be the simplest option for you in terms of rough calculations (where it's not critical to be correct, just close). One approach using the online calculators that might be better, is to treat these like two separate loans/cards.  Many calculators exist for multiple balances.  Then you can allocate funds differently to the two 'cards'.  This would allow you to see how long you will need until you've paid off the higher balance, for example, although it probably won't perfectly match things - unless you find a site that has this specific option available you probably will have to either live with a small error in your calculations or do the math by hand.
"There is no formula that can be applied to most variations of the problem you pose.  The reason is that there is no simple, fixed relationship between the two time periods involved: the time interval for successive payments, and the time period for successive interest compounding. Suppose you have daily compounding and you want to make weekly payments (A case that can be handled).  Say the quoted rate is  4.2% per year, compounded daily Then the rate per day is 4.2/365, or 0.0115068 % So, in one week, a debt would grow through seven compoundings.  A debt of $1 would grow to 1 * (1+.000225068)^7, or 1.000805754 So, the equivalent interest rate for weekly compounding is 0.0805754% Now you have weekly compounding, and weekly payments, so the standard annuity formulas apply. The problem lies in that number ""7"", the number of days in a week.  But if you were trying to handle daily / monthly, or weekly / quarterly, what value would you use? In such cases, the most practical method is to convert any compounding rate to a daily compounding rate, and use a spreadsheet to handle the irregularly spaced payments." Depending on who you have the loan through and how they figure the interest charges (whether daily, monthly, bi-monthly, etc. normally monthly I would assume), your interest is probably figured either daily or once a month.   Let's assume that it is figured daily, otherwise it wouldn't make sense to make bi-weekly payments.   At 4% Annual Interest on a $150,000 home loan the interested added each day is about $16.44, but it doesn't stop there because it is compounding interest daily so the next day it becomes 4% of 150,0016.44 (which is negligibly larger amount) and they will tack on another $16.44. So what will happen is that the amount of interest you owe grows rather quickly, especially if you miss a monthly payment.  Everyone knows that the faster you pay something off the less interest you pay, but not everyone knows the formula for compounding interest.  a quick Google search rendered this site with a simple explanation Compound Interest Formula  unfortunately this formula doesn't take into account the payments being made. The big thing with making your payments bi-weekly rather than a bigger payment once a month is that you pay off some of that principle right away and it won't collect interest for 14 more days. if the interest is only calculated once a month, make your full payment before the interest is calculated, the same goes for your credit cards. "Given the exact formula that goes into the 'Fair Issac"" calculation is a closely guarded trade secret, AND that each agency has their own formula, I'm not sure there's really any 100% for sure authoritative answer to the question in terms of which option would be best.  There are a lot of balancing factors, like how long you've had accounts, payment history over time, etc.  Stuff that is known to HURT a score can include things like closing a longstanding account.   If you have a very low interest loan (like some car companies offer now and then) I'd just make the normal payments.  If you have something at a higher interest rate, especially above 6-7%, then I'd worry less about credit score and more about how much I'm going to pay in interest and pay it off as rapidly as I could. The big key is 'never pay late' more than anything else,  Followed by how much of your debt capacity is used (which paying down any account, loan or credit card, will help), and long standing relationships (length of history)  See this (approximate) chart and notice that any early payoff is basically going lower your 'capacity used', possibly reduce the types of credit used (if it's the last loan of that type), all of which should help your score." Thanks for this, great in depth answer. I had previously calculated a WACC and have used it for my discount rate.   As part of your last point on revenue vs. cash, I've set a accounts receivable period of 30 days, and then applied a factor of 30/365 * revenue to understand what portion of my revenue is not cash in hand. Does that make sense? In addition to the answer from CQM, let me answer your 'am I missing anything?' question. Then I'll talk about how your approach of simplifying this is making it both harder and easier for you. Last I'll show what my model for this would look like, but if you aren't capable of stacking this up yourself, then you REALLY shouldn't be borrowing 10,000 to try to make money on the margin. Am I missing anything? YES. You're forgetting (1) taxes, specifically income tax, and (2) sales commissions//transaction fees.  On the first: You have not considered anything in your financial model for taxes. You should include at least 25% of your expected returns going to taxes, because anything that you buy... and then sell within 12 months... is taxed as income. Not capital gains. On the second: you will incur sales commissions and/or transaction fees depending on the brokerage you are using for your plan. These tend to vary widely, but I would expect to spend at least $25 per sale. So if I were building out this model I would think that your break-even would have to at least cover: monthly interest + monthly principal payment income tax when sold commissions and broker's fees every time you sell holdings On over-simplifying: You have the right idea with thinking about both interest and principal in trying to sketch this out. But as I mentioned above, you're making this both harder and easier for yourself. You are making it harder because you are doing the math wrong. The actual payment for this loan (assuming it is a normal loan) can be found most easily with the PMT function in Excel: =PMT(rate,NPER,PV,FV)... =PMT(.003, 24, -10000, 0). That returns a monthly payment  (of principal + interest) of 432.47.  So you actually are over-calculating the payment by $14/month with your ballpark approach. However, you didn't actually have all the factors in the model to begin with, so that doesn't matter much.  You are making it artificially easier because you have not thought about the impact of repaying principal. What I mean is this--in your question you indicate:  I'm guessing the necessary profit is just the total interest on this loan = 0.30%($10000)(24) = $720 USD ? So I'll break even on this loan - if and only if - I make $720 from stocks over 24 months (so the rate of return is 720/(10000 + 720) = 6.716%). This sounds great-- all you need is a 6.716% total return across two years. But, assuming this is a normal loan and not an 'interest-only' loan, you have to get rid of your capital a little bit at a time to pay back the loan. In essence, you will pay back 1/3 of your principal the first year... and then you have to keep making the same Fixed interest + principal payments out of a smaller base of capital.  So for the first few months you can cover the interest easily, but by the end you have to be making phenomenal returns to cover it.  Here is how I would build a model for it (I actually did... and your breakeven is about 1.019% per month. At that outstanding 12.228% annual return you would be earning a whopping $4.) At least as far as the variables are concerned, you need to be considering: Your current capital balance (because month 1 you may have $10,000 but month 2 you have just 9,619 after paying back some principal).  Your rate of return (if you do this in Excel you can play with it some, but you should save the time and just invest somewhere else.) Your actual return that month (rate of return * existing capital balance).  Loan payment = 432 for the parameters you gave earlier.  Income tax = (Actual Return) * (.25). With this kind of loan, you're not actually making enough to preserve the 10,000 capital and you're selling everything you've gained each month.  Commission = ($25 per month) ... assuming that covers your trade fees and broker commissions. I guarantee you that this is not the deal breaker in the model, so don't get excited if you think I'm over-estimating this and you realize that Scottrade or somewhere will let you have trades at $7.95 each.   Monthly ending balance == next month's starting capital balance.  Stack it all up in Excel for 24 months and see for yourself if you like. The key thing you left out is that you're repaying each month out of capital that you'd like to use to invest with. This makes you need much higher returns.  Even if your initial description wasn't clear and this is an interest-only loan, you're still looking at a rate of about 7.6% annually that you need to hit in order to just break even on the costs of holding the loan and transferring your gains into cash. If you know the amounts that were combined ($5,000 and $7,500 in your example) -- NOT the original loan amounts necessarily -- then you can calculate a payment schedule (in Excel, Google Sheets, online, etc.) using that amount and the interest rate.  You can then apply your payments ($100) to that payment schedule, making sure to either accrue interest if your $100 didn't cover the monthly payment, or pay down extra principal if your $100 more than covered the payment.   The outstanding principal is the amount left or remaining balance. A program like GnuCash or Quicken makes doing the payment schedule, and applying payments relatively easy to handle.  Spreadsheets will require you to have 36 lines (3 years x 12 months) of payment and recalculation detail, but that shouldn't be too much work. To be fair to your mother, make sure you include any partially accrued interest on the full balance when paying it off.  Or even better, include a full month's interest in the pay-off amount. Closing your oldest revolving account will lower your average age of accounts and hurt your score.  No ifs, ands, or buts.  The amount it drops is hard to tell, and it may only be a few points if your other cards are fairly old as well. While the FICO scoring algorithm is proprietary and hard to predict, you can use the official FICO Simulator to estimate the impact.  Based on the information you provided (5+ cards, oldest card 5 years), your estimate is 750-800.  Performing the same estimate and only changing the number of cards and age (2-4 cards, oldest card 2-4 years), the score estimate drops to 735-785.  Both of these estimates assume you have 9% or less utilization.  You can probably estimate that your score will drop at least 15 points. However, it may not matter to you whether your score is maximized.  Once you get above a certain FICO score, it doesn't matter.  For example, I recently refinanced a vehicle and asked the loan officer about their lowest APR, and found out that they required a 780 FICO for it.  Kind of like the difference between getting a 91 or a 99 in a class, an A is an A. Some other factors you may want to consider before you make your choice: Should I pay off the credit cards now or do the monthly payments thing? It need not be a binary choice; you could take the middle ground. Assuming that an emergency fund of $10-15k is sufficient, you could pay off one credit card and do the monthly payment for the other cards. With one card cleared, you may feel better. I would choose to clear the card with the highest interest, followed by the smallest debt first as it's the mathematically optimum way. I should also let you know that there's this debt-snowball method which is not as optimum but has some valid reasons for existing. Choose the optimum way if you're financially disciplined. p.s. I'm not trained financially, nor am I in the industry. Just my two cents. Part 2 = 12%  -5000/0.12 (1 - 1/(1.12^2 ) = -8450.26 Then subtract year 0s 5000 payment if you havent = -13450.26 and then compare to option 2 -2000/0.12 (1 - 1/(1.12^10) = -11300.46 and subtract 20000 -31300.46  As you can see both values are substantially smaller compared to a 4% discount rate. Basically you need to use a time-value-of-money equation to discount the cashflows back to today. The Wikipedia formula will likely work fine for you, then you just need to pick an effective interest rate to use in the calculation. Run each of your amounts and dates though the formula (there are various on-line calculators to pick from, and sum up the values. You did not mention your location or jurisdiction, but a useful proxy for the interest rate would be the average between the same duration mortgage rate and fixed-deposit rate at your bank; it should be close enough for your purposes - although if an actual lawsuit is involved and the sums high enough to have lawyers, it might be worth engaging an accountant as well to defend the veracity of both the calculation and the interest rates chosen. Ditto to Victor. The simple rule is: Pay the minimums on all so you don't get any late fees, etc, then pay off the highest interest rate loan first. A couple of special cases do come to mind: If one or more of these are credit cards, then, here in the U.S. at least, credit cards charge you interest on the average daily balance, unless you pay off the balance entirely, in which case you pay zero interest. So for example say you had two credit cards, both with 1% per month interest, with debt of $2000 and $1000. You have $1500 available. Ignoring minimum payments for the moment, if you put that $1500 against the larger balance, you would still pay interest on the full amount for the current month, or $30. But if you paid off the smaller and put the difference against the larger, then your interest for the current month would be only $15. (Either way, your interest for NEXT month would be the same -- 1% of the $1500 remaining balance or $15 -- assuming you couldn't pay off the other card.)  If one or more of the loans are mortgage loans on which you are paying mortgage insurance, then when you get the balance below a certain point -- usually 80% of the original loan amount -- you no longer have to pay mortgage insurance premiums. Thus the amount you are paying on such premiums needs to be factored into the calculation. There may be other special cases. Those are the ones that I've run into. I think is an excellent idea. Use free money or almost free to do a lump sump payment. My recommendation is to have a reminder to pay credit card before, almost finishing, the 0% APR period. "By the sounds of things, you're not asking for a single formula but how to do the analysis... And for the record you're focusing on the wrong thing. You should be focusing on how much it costs to own your car during that time period, not your total equity.  Formulas:  I'm not sure how well you understand the nuts and bolts of the finance behind your question, (you may just be a pro and really want a consolidated equation to do this in one go.) So at the risk of over-specifying, I'll err on the side of starting at the very beginning. Any financial loan analysis is built on 5 items: (1) # of periods, (2) Present Value, (3) Future Value, (4) Payments, and (5) interest rate.  These are usually referred to in spreadsheet software as NPER, PV, FV, PMT, and Rate. Each one has its own Excel/google docs function where you can calculate one as a function of the other 4. I'll use those going forward and spare you the 'real math' equations.  Layout: If I were trying to solve your problem I would start by setting up the spreadsheet  up with column A as ""Period"". I would put this label in cell A2 and then starting from cell A3 as ""0"" and going to ""N"". 5 year loans will give you the highest purchase value w lowest payments, so n=60 months... but you also said 48 months so do whatever you want.  Then I would set up two tables side-by-side with 7 columns each. (Yes, seven.) Starting in C2, label the cells/columns as: ""Rate"", ""Car Value"", ""Loan Balance"", ""Payment"", ""Paid to Interest"", ""Principal"", and ""Accumulated Equity"". Then select and copy cells C2:I2 as the next set of column headers beginning in K2. (I usually skip a column to leave space because I'm OCD like that :) ) Numbers: Now you need to set up your initial set of numbers for each table. We'll do the older car in the left hand table and the newer one on the right. Let's say your rate is 5% APR. Put that in cell C1 (not C3). Then in cell C3 type =C$1/12. Car Value $12,000 in Cell D3. Then type ""Down Payment"" in cell E1 and put 10% in cell D1. And last, in cell E3 put the formula =D3*(1-D$1). This should leave you with a value for the first month in the Rate, Car Value, and Loan Balance columns. Now select C1:E3 and paste those to the right hand table. The only thing you will need to change is the ""Car Value"" to $20,000. As a check, you should have .0042 / 12,000 / 10,800 on the left and then .0042 / 20,000 / 18,000 on the right.  Formulas again: This is where spreadsheets become amazing. If we set up the right formulas, you can copy and paste them and do this very complicated analysis very quickly.  Payment The excel formula for Payment is =PMT(Rate, NPER, PV, FV). FV is usually zero. So in cell F3, type the formula =PMT(C3, 60, E3, 0). Obviously if you're really doing a 48 month (4 year) loan then you'll need to change the 60 to 48. You should be able to copy the result from cell F3 to N3 and the formula will update itself.  For the 60 months, I'm showing the 12K car/10.8K loan has a pmt of $203.81. The 20K/18K loan has a pmt of 339.68.  Interest The easiest way to calculate the interest is as =E3*C3. That's (Outstanding Loan Balance) x (Periodic Interest Rate). Put this in cell G4, since you don't actually owe any interest at Period 0.  Principal If you pay PMT each month and X goes to interest, then the amount to principal is ""PMT - X"". So in H4 type =-F3 - G3. The 'minus' in front of F3 is because excel's PMT function returns a negative amount. If you want to, feel free to type ""=-PMT(...)"" for the formula that's actually in F3. It's your call.  I get 159 for the amount to principal in period 1.  Accumulated Equity As I mentioned in the comment, your ""Equity"" comes from your initial Loan-to-Value and the accumulated principal payments. So the formula in this cell should reflect that. There are a variety of ways to do this... the easiest is just to compare your car's expected value to your loan balance every time. In cell I3, type =(D3-E3). That's your initial equity in the car before making any payments. Copy that cell and paste it to I4. You'll see it updates to =(D4-E3) automatically. (Right now that is zero... those cells are empty, but we're getting there) The important thing is that as JB King pointed out, your equity is a function of accumulated principal AND equity, which depreciates. This approach handles those both.  Finishing up the copy-and-paste formulas I know this is long, but we're almost done.  Rate // Period 1 In cell C4 type =C3.  Payment // Period 1 In cell F4 type =F3.  Loan Balance // Period 1 In cell E4 type =E3-H4. Your loan balance at the end of period is reduced by the principal you paid. I get 10,641.  Car Value // Period 1 This will vary depending on how you want to handle depreciation. If you ignore it, you're making a major error and it's not worth doing this entire analysis...  just buy the prettiest car and move on with life. But you also don't have to get it scientifically accurate. Go to someplace like edmunds.com and look up a ballpark. I'm using 4% depreciation per year for the old (12K) car and 7% for the newer car. However, I pulled those out of my ass so figure out what's a better ballpark.  In G1 type ""Depreciation"" and then put 4% in H1. In O1 type ""Depreciation"" and then 7% in P1. Now, in cell D4, put the formula =D3 * (1-(H$1/12)).  Paste formulas to flesh out table As a check, your row 4 should read 1 / .0042 / 11,960 / 10,641 / 203.81 / 45 / 159 / 1,319. If so, you're great. Copy cells C4:I4 and paste them into K4:Q4.  These will update to be .0042 / 19,883 / 17,735 / 339.68 / 75 / 265 / 2,148.  If you've got that, then copy C4:Q4 and paste it to C5:C63. You've built a full amortization table for your two hypothetical loans. Congratulations.  Making your decision I'm not going to tell you what to decide, but I'll give you a better idea of what to look at. I would personally make the decision based on total cost to own during that time period, plus a bit of ""x-factor"" for which car I really liked.  Look at Period 24, in columns I and Q. These are your 'equities' in each car. If you built the sheet using my made-up numbers, then you get ""Old Car Equity"" as 4,276. ""New Car Equity"" is 6,046. If you're only looking at most equity, you might make a poor financial decision.  The real value you should consider is the cost to own the car (not necessarily operate it) during that time... Total Cost = (Ending Equity) - (Payment x 24) - (Upfront Cash).  For your 'old' car, that's (4,276) - (203.81 * 24) - (1,200) = -1,815.75 For the 'new' car, that's (6,046) - (339.68 * 24) - (2,000) = -4,106.07.  Is one good or bad? Up to you to decide. There are excel formulas like ""CUMPRINC"" that can consolidate some of the table mechanics, but I assumed that if you're here asking you would have gotten stuck running some of those.  Here's the spreadsheet: https://docs.google.com/spreadsheet/ccc?key=0Ah0weE0QX65vdHpCNVpwUzlfYjlTY2VrNllXOS1CWUE#gid=1" Incredible article, tons of data. Thank you!   It does answer the above posters question if you're willing to read through. It provides data with and without 'revolving debt'.  Side note; interesting to see how age and income trend. Debt increasing during the family-middle aged years, and during the peak income earning years.  I'd say you want these credit card debt lower overall and on average; but with the distribution it may be sustainable. There is one basic principle to apply here:  to compare money paid at different times, all the amounts must be compounded or discounted to the same point in time. In this case, the moment of the initial  $225,000 loan is convenient. At that moment, you get $225,000 You then make 30 payments on the 40% mortgage.  The amount of these payments has to be calculated;  they're paying off a $90,000 mortgage with 30 monthly payments at a monthly rate of 0.5% Finally, you make 30 payments of an amount X, starting one month after the 40% mortgage ends. So far we've just listed the amount and time of all the payments back and forth. A time-line type diagram is a huge help here. Finally, use compound interest and annuity formulas to bring all the payments to the starting point, using an interest rate of 1% a month! Equate money in with money out and solve for X