Checking the math on a Truth-in-Lending Disclosure
As your question is written now, it looks like you have a typo.  Your stated APR is 5.542% = 0.05542, not 0.005542 as you've written.  I ran the numbers that you gave (accounting for the typo) through the formula at Wikipedia and got $849.2528 / month, which will round to $849.25 for most payments.  That doesn't match the number that you computed or the number on your TIL. (Maybe you also miskeyed the result of your calculation?) I agree that it's unlikely that this is just a calculation error by the mortgage company, although I wouldn't completely rule it out.  Are you paying anything else like a property tax escrow?  I didn't pull a blank TIL form to see what might go into the monthly payment line that you showed, but in many cases you do pay more than just principle and interest each month. (Not sure if that gets reflected at that point on the form though.)
I'd be interested in seeing a source regarding the transposing of signatures onto different documents.  That's blatant criminal fraud without even falling into a grey area.  About math, a study ran in the Economist showed that people who were bad at relatively basic math (understanding fractions, interest, percentages) were far more likely to have requested loans they could not afford. This is of course using 'new math'.  Namely, if I lend you $100 and you keep it for a month, I've lent you $100.  In fact, if I loan you $100 for a year I've loaned you $100.  But if I lend you $100 overnight, you pay me back in the morning, I lend you $100 overnight, you pay me back in the morning and we repeat that for a month, I've supposedly lent you $3,000.    That's some interesting math for sure. It's not an easy calculation so I made this demonstration that compares a fixed investment with a 100% mortgage, for a simple case.  (Obviously if you lessen the mortgage with a deposit it's somewhere between the extremes.)  The demonstrations shows some definite differences at higher interest rates.  You can probably decipher the calculations in the code if you're interested.  It's intended to be legible. http://demonstrations.wolfram.com/BuyOrRentInvestmentReturnCalculator/ Proof of funds for funding firms are used to determine if you have any money before they lend to you. (determine risk of lending to you if you defaulted) If you came in with 10 million dollars, the firms will turn a blind eye to any money laundering regulations and just lend to you. Interesting thing is not all of this analysis is typically done by the person you are speaking with re: your mortgage.  Likely there is an economic team (at some institution or another, as many many loans are sold/distributed in some fashion), finance team, and all of them are looking at various rates etc.  Typically what is offered is matching an offsetting liability somewhere else.  So there may be cases where the spread someone is looking to pick up might be tighter or looser within institutions or types of institutions vs others, even though the overall market is at one place.  (e.g. banks vs. insurance companies).    So you may have negotiating room at a certain term, but not another, or the reverse at another firm.  One firm might have lots of 10 year money, while another might be limited, so will be picky in what they choose.  Either more conservative loan terms/ltv, or a higher spread, for example.  So many things go into the ultimate rate someone can get, but in theory the blog did a decent job. Surely some borrowers and lenders make decisions about making and taking loans based on the actual interest rates on the actual loans? In which case it doesn't matter so much if the rates are calculated based on a fictional assumption about something. At the end of the day every borrower or lender in the market makes their own decision about which lending contracts they take part in. What you are looking here is the cost of capital, because that is what you are effectively giving up in order to invest in those loans. In a discounted cash-flow, it would be the *i* in the denominator (1+*i*). For instance, instead of purchasing those loans you could have lent your money at the risk-free rate (not 100% true, but typical assumption), and therefore you are taking a slight risk in those loans for a higher return.  There are several ways to compute that number, the one most often used would be the rate if the bank were to lend that money. In this way, it would be the Fed Funds rate plus some additional risk premium. "From the Op's comment below re: the ""audit""  http://www.gao.gov/new.items/d11696.pdf, the first paragraph states that loans outstanding peaked at just over 1 trillion in 2008.  So you'd look at the Fed's balance sheet of loans outstanding over time, and perhaps subtract amounts reported as more normal operations." The calculation can be made on the basis that the loan is equal to the sum of the repayments discounted to present value.  (For more information see Calculating the Present Value of an Ordinary Annuity.) With Deriving the loan formula from the simple discount summation.  As you can see, this is the same as the loan formula given here. In the UK and Europe APR is usually quoted as the effective interest rate while in the US it is quoted as a nominal rate.  (Also, in the US the effective APR is usually called the annual percentage yield, APY, not APR.) Using the effective interest rate finds the expected answer. The total repayment is £30.78 * n = £1108.08 Using a nominal interest rate does not give the expected answer. 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) I wouldn't say this is a lie.  Person 2 just made themselves the offer of 38k.  If it sells then they were outbid.  If it doesn't sell then they're stuck with the asset as if they had bid themselves.  They're paying themselves, or they're representing a party that is paying themselves.  That balances out to a $0 profit/loss + ownership of the asset.   I think something like this would be unethical:   Person 1: I'll buy your car for 30k  Person 2: Okay  Person 3:That's my car! "There is no equation. Only data that would help you come to the decision that's right for you.  Assuming the 401(k) is invested in a stock fund of one sort or another, the choice is nearly the same as if you had $5K cash to either invest or pay debt.  Since stock returns are not fixed, but are a random distribution that somewhat resembles a bell curve, median about 10%, standard deviation about 14%. It's the age old question of ""getting a guaranteed X% (paying the debt) or a shot at 8-10% or so in the market."" This come up frequently in the decision to pre-pay mortgages at 4-5% versus invest. Many people will take the guaranteed 4% return vs the risk that comes with the market. For your decision, the 401(k) loan, note that the loan is due if you separate from the company for whatever reason. This adds an additional layer of risk and another data point to the mix. For your exact numbers, the savings is barely $50. I'd probably not do it. If the cards were 18%, I'd lean toward the loan, but only if I knew I could raise the cash to pay it back to not default." Sounds like he's just had the theory playing in his head since someone told him they did that instead of just purchasing it and going in to payments.  Perhaps I need to sit down and calculate it for my clueless self. "Wow, this turns out to be a much more difficult problem than I thought from first looking at it. Let's recast some of the variables to simplify the equations a bit. Let rb be the growth rate of money in your bank for one period. By ""growth rate"" I mean the amount you will have after one period. So if the interest rate is 3% per year paid monthly, then the interest for one month is 3/12 of 1% = .25%, so after one month you have 1.0025 times as much money as you started with. Similarly, let si be the growth rate of the investment. Then after you make a deposit the amount you have in the bank is pb = s. After another deposit you've collected interest on the first, so you have pb = s * rb + s. That is, the first deposit with one period's growth plus the second deposit. One more deposit and you have pb = ((s * rb) + s) * rb + s = s + s * rb + s * rb^2. Etc. So after n deposits you have pb = s + s * rb + s * rb^2 + s * rb^3 + ... + s * rb^(n-1). This simplifies to pb = s * (rb^n - 1)/(rb - 1). Similarly for the amount you would get by depositing to the investment, let's call that pi, except you must also subtract the amount of the broker fee, b. So you want to make deposits when pb>pi, or s*(ri^n-1)/(ri-1) - b > s*(rb^n-1)/(rb-1) Then just solve for n and you're done! Except ... maybe someone who's better at algebra than me could solve that for n, but I don't see how to do it. Further complicating this is that banks normally pay interest monthly, while stocks go up or down every day. If a calculation said to withdraw after 3.9 months, it might really be better to wait for 4.0 months to collect one additional month's interest. But let's see if we can approximate. If the growth rates and the number of periods are relatively small, the compounding of growth should also be relatively small. So an approximate solution would be when the difference between the interest rates, times the amount of each deposit, summed over the number of deposits, is greater than the fee. That is, say the investment pays 10% per month more than your bank account (wildly optimistic but just for example), the broker fee is $10, and the amount of each deposit is $200. Then if you delay making the investment by one month you're losing  10% of $200 = $20. This is more than the broker fee, so you should invest immediately. Okay, suppose more realistically that the investment pays 1% more per month than the bank account. Then the first month you're losing 1% of $200 = $2. The second month you have $400 in the bank, so you're losing $4, total loss for two months = $6. The third month you have $600 in the bank so you lose an additional $6, total loss = $12. Etc. So you should transfer the money to the investment about the third month. Compounding would mean that losses on transferring to the investment are a little higher than this, so you'd want to bias to transferring a little earlier. Or, you could set up a spreadsheet to do the compounding calculations month by month, and then just look down the column for when the investment total minus the bank total is greater than the broker fee. Sorry I'm not giving you a definitive answer, but maybe this helps." "For 3X, it's about 114, and for 4X, 144, which naturally, is twice 72.  These are close, back of napkin, results. With smart phone apps offering scientific calculators, you should get comfortable just taking the nth root of a number for a more precise answer.  Update in response to Brick's comment. The rule of 72 says that (n)(y)=72 to double your money. It answers both questions, how much time do I need, given a rate, and how much return do I need, given a time?  Logic tells me that if 72 is the number to double, 144 is the 4X. But I'm a math guy, and my logic might not be logical to OP. So -  Let's take the 20th root of 4.   This is the key to use. 4, (hit key) 20, equals. The result is 1.07177 or 7.177%. And this is the precise rate you'd need to quadruple your money in 20 years) Now (n)(y)= 20* 7.177 = 143.55 which rounds to 144. ""Rule of 144"" to quadruple your money.  This now answers OP's question, ""How to derive a Rule of X"" for a return other than doubling.  One more example? I want 10X my money. Of course I need the initial guess to enter one calculation. People like 8%, in general. It's a bit below the 10% long term S&P return, and a good round number. The Rule of 72 says 9 years to double, so, 18 years is 4X, and 36 years is 8X. For my initial calculation, I'll use 40 years. The 40th root of 10. I get 5.925% (Again the precise rate that gives 10 fold over 40 years) and multiplying this by 40, I get a ""Rule of 237"" which I'm tempted to round to 240.  At 6%, 237/6= 39.5 yrs,  1.06^39.5 = 9.99 At 6%, 240/6= 40.0 yrs,  1.06^40.0 = 10.29 You can see that you lose some accuracy for the sake of a number that's easier to remember, and manipulate. 72 to double is pretty darn accurate, so I'll stick with ""Rule of 237"" to get 10X my money.   To close, the purpose of these rules is to create the tool that lets you perform some otherwise tough calculations away from any electronic device. Of course I know how to use logs, and in real life I'm paid to explain them to students who are typically glad when that chapter is over. I've shown above how the ""Rule of X"" can be formulated with a power/root key, which, for most people, is simpler. Ironically, log calculations as @jkuz offered, force a continuous compounding which may not be desired at all. It would give a result of 230 for my 10X return example, and the following (using the first equation he offered) - At 6%, 230/6= 38.3 yrs,  1.06^38.3 = 9.31 which is further away from the desired 10X than my 237 or rounded 240."