Loan math problem
The price inflation isn't a percentage, it's a fixed amount. If the dealer adds $R to the price of both the trade-in and the purchased car, then everyone ends up with the right amount of money in their pockets. So your formula should be: D + T + R = 0.1 * (P + R) "Lachlan has $600 cash and a car worth $500. That's $1,100. The new car is priced at $21,800. Lachlan needs a loan for $20,700. However, the finance company insists that the buyer must pay a 10% deposit, which is $2,180. Lachlan only has $1,100, so no loan.  The car dealer wants to make a sale, so suggests some tricks. The car dealer could buy Lachlan's old banger for $1,500 instead of $500, and sell the new car for $22,800 instead of $21,800. Doesn't make a difference to the dealer, he gets the same amount of cash. Now Lachlan has $600 cash and $1,500 for his car or $2,100 in total. He needs 10% of $22,800 as deposit which is $2,280. That's not quite there but you see how the principle works. Lachlan is about $200 short. So the dealer adds $1,200 to both car prices. Lachlan has $600 cash and a car ""worth"" $1,700, total $2,300. The new car is sold for $23,000 requiring a $2,300 deposit which works out exactly.  How could we have found the right amount without guessing?  Lachlan had $1,100. The new car costs $21,800. The dealer increases both prices by x dollars. Lachlan has now $1,100 + x deposit. The car now costs $21,800 + x. The deposit should be 10%, so $1,100 + x = 10% of ($21,800 + x) = $2,180 + 0.1 x.  $1,100 + x = $2,180 + 0.1 x : Subtract $1,100 x = $1,080 + 0.1 x : Subtract 0.1 x 0.9 x = $1,080 : Divide by 0.9 x = $1,080 / 0.9 = $1,200 The dealer inflates the cost of the new car and the value of the old car by $1,200. Now that's the theory. In practice I don't know how the finance company feels about this, and if they would be happy if they found out."
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. Happy to help.   I always recommend you try and solve these types of problems by hand over financial calculators so that you can get a feel for how terms in the equation relate to one another. That's important while you're studying/learning.  But for work--by all means you should use the fastest method, which is probably excel. "For a personal finance forum, this is too complicated for sustained use and you should find a simpler solution. For a mathematical exercise, you are missing information required to do the split fairly.  You have to know who overlaps and when to know how to do the splits.   For an extreme example, take your dates given: Considering 100 days of calculation period,  If Roommate D was the only person present for the last 10 days, they should pay 100% of the grocery bill as they are the only one eating.   From your initial data set, you can't know who should be splitting the tab for any given day. To do this mathematically, you'd need:  But don't forget ""In Theory, Theory works.  In Practice, Practice works."" Good theory would say make a large, complicated spreadsheet as described above.   Good practice would be to split up the costs in a much, much simpler way." "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" "fair enough, but I wasn't sure if you actually get it or not because your question seemed to revolved around cash and transaction value.  You are really just said ""skewing"" is if you didn't add net debt the multiple wouldn't be a valid muliptle. If you understand it, why start out with this mental exercise of saying ""we shouldn't calculate multiples wrong, because then they would be wrong""?" Your calculations are correct. It is likely that the bank's software has a rounding error.  In effect, either your bank is overstating your interest by eight cents per month, or your bank is insisting that you prepay your principal by eight cents per month.  If the bank's ongoing interest calculations are correct, your final payment will be slightly smaller (because of the prepaid principal, and because of compound interest on those prepayments).  I have performed similar calculations for my mortgages over the years, and except upon early payoff in the middle of the month, I have always matched my banks' calculations to the penny.  Ironically, this means that my banks' formulas are a bit weird: After making these adjustments, even my calculations for the mid-month payoff matched to the penny. Using the facts in your comment: I use the pmt function in excel and the goal seek tool to determine an original balance of ~$24,241.33 Taking into account the statement in the questions that you have paid $6,072.26 in principal gives an estimated current balance of  which almost exactly matches your current balance statement of $18,168.56 a difference of 51 cents. Is it possible that during the time you were in college the accumulated interest caused the balance to grow from 20,800 to 24,241? This could happen if the loans were unsubsidized. The principal of the loan is the amount you borrow.  The capitalized interest is added to the principal of the loan, because you are not paying this interest as it accrues.  So when you begin payments, the principal of the loan is $5,500 + $436 = $5,936. Using the standard amortization formula (see this page for details), the per-month payment for a ten-year payment plan at 6.8% interest on principal of $5,936 is $68.31.  One hundred twenty payments (each month for ten years) totals $8,197.40. "&gt; My issue understanding this is I've been told that banks actually don't hold 10% of the cash and lend the other 90% but instead hold the full 100% in cash and lend 900%. Is this accurate?  That's the money multiplier effect being poorly described.  You take a loan out, but that loan eventually makes its way to other banks as cash deposits, which then are loaned out, and go to other banks, and loaned, etc., so that the economy is ""running"" on 10x cash, where 1x is in physical cash, and the other 9x is in this deposit-loan-deposit phenomenon.  &gt; The issue I see with it is that it becomes exponential growth that is uncapped.  Not true. If there is $1B outstanding ""physical"" cash (the money supply) with a 10% reserve, then the maximum amount of ""money"" flowing through is $1B / 10% = $10B. This assumes EVERYTHING legally possible is loaned out or saved in the banking system. As such, it represents a cap.   If you have an Excel spreadsheet handy, you can easily model this out in four columns.   Label the first row as follows: Deposit, Reserve, Cash Reserve, Loan Amount  A2 will be your money supply. For simplicity, put $100.   B2, your reserve column, will be 10%.  C2 should be =A2 * B2, which will be the cash reserve in the bank.  D2 should be A2 - C2, which is the new loan amount extended.   A3 should be = D2, as the loans extended from the last step become deposits in the next.   B3 = B2.  Now, drag the formulas down, say, 500 rows.  If you then sum the ""deposits"" column, it'll total $1,000. The cash reserve will total  $100, and the loan amount will be $900. Thus, there is a cap." From here The formula is M = P * ( J / (1 - (1 + J)^ -N)). M: monthly payment RESULT = 980.441... P: principal or amount of loan 63963 (71070 - 10% down * 71070) J: monthly interest; annual interest divided by 100, then divided by 12. .00275 (3.3% / 12) N: number of months of amortization, determined by length in years of loan. 72 months See this wikipedia page for the derivation of the formula Wow I honestly hope this is your first ever finance class. Anyway this isn't even finance, the only thing here finance related are the terms. It's really an algebra problem. Which is 1,312,500/x = 2 soooo find X.  X = 656250  so 131,250 in notes payable. If you don't know what to do know to find the quick ratio, then change majors This looks correct to me, for simple interest. If you are dealing with compound interest, the formula would be: So, A = 500000(1+0.036/365)^(30), or 501,481.57, or an interest of 1481.57, assuming the 3.6% is the annual nominal interest rate and it is compounded daily. Note that you are ignoring the depreciation and also ignoring the percentage of customers who will forfeit their debt in the 30 - 60 day period. "Something I found helpful when I learned this, is to just use Excel to expand the series.   So start with A1 as ""175"", then in A2 put ""=175+(A1*1.004308)"" and paste that same formula down for a few hundred rows.  You'll find your answer on A216.  Most non-math-centric people don't have an intuitive grasp of how exponents (aka ""compounding""... and never mind the natural logarithms this is all derived from) behave; but you can play with the numbers ""unrolled"" in Excel to get a better idea of how they work. That formula is just applying 0.4308% interest every period (row), and adding in a fresh $175." 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. "add the interest for the next 5 payments and divide that by how much you paid on the principal during that time Let's see - on a $200K 6% loan, the first 5 months is $4869. Principal reduction is $1127. I get 4.32 or 432%. But this is nonsense, you divide the interest over the mortgage balance, and get 6%. You only get those crazy numbers by dividing meaningless ratios.  The fact that early on in a mortgage most of the payment goes to interest is a simple fact of the the 30 year nature of amortizing. You are in control, just add extra principal to the payment, if you wish.  This idea sounds like the Money Merge Account peddled by UFirst. It's a scam if ever there was one. I wrote about it extensively on my site and have links to others as well. Once you get to this page, the first link is for a free spreadsheet to download, it beats MMA every time and shows how prepaying works, no smoke, no mirrors. The second link is a 65 page PDF that compiles nearly all my writing on this topic as I was one of the finance bloggers doing what I could to expose this scam. I admit it became a crusade, I went as far as buying key word ads on google to attract the search for ""money merge account"" only to help those looking to buy it find the truth. In the end, I spent a few hundred dollars but saved every visitor the $3500 loss of this program.  No agent who dialoged with me in public could answer my questions in full, as they fell back on ""you need to believe in it."" I have no issue with faith-based religion, it actually stands to reason, but mortgages are numbers and there's order to them. If you want my $3500, you should know how your system works. Not one does, or they would know it was a scam.  Nassim Taleb, author of ""The Black Swan"" offered up a wonderful quote, ""if you see fraud, and do not say 'fraud,' you are a fraud.""  The site you link to isn't selling a product, but a fraudulent idea. What's most disturbing to me is that the math to disprove his assertion is not complex, not beyond grade school arithmetic.  Update 2015 - The linked ""rule of thumb"" is still there. Still wrong of course. Another scam selling software to do this is now promoted by a spin off of UFirst, called Worth Unlimited. Same scam, new name."