<prompt> Human: I read this a lot: for any given dollar, just compare the interest rate of your loan to the estimated rate of return of your investment to figure out where to put it. Return is never guaranteed, so unless the interest rate on the loan is low, its better to pay off the loan. This doesnt feel right. I have no math background at all, so here goes my back of the envelope try at it--Id be interested to read your responses.Lets say you get 1000.Case 1a: you have no debt, and invest at a guaranteed 7 rate of return for ten years. You end up with 1967.15 a net of 967.15.Case 2a: you have a 1000 debt at 7, and pay it off immediately and do nothing else. Immediately, and at the end of ten years, youre out 1000, but you also avoided paying 393.30 in interest. So you effectively net 393.30, no?Now factor in the fact that you have debt in both cases, and you have to pay it off regardless of investing that extra 1000 or not. Case 1b: 10,000 loan at 7 for ten years. Pay off 1000 of it 393.30, and then the rest of the loan -12,540 This nets -12,146.7.Case 2b: 10,000 loan at 7 for ten years. Invest 1000 967.15, then pay off the loan -13,933 This nets -12,965.85.This results in a difference of 819.15 between the two strategies, advantage to paying off the loan rather than investing.Now factor in a longer time horizon, beyond the end of the loan.Case 1c: Case 1b, but after ten years, you have nothing invested. So, you invest what you would have paid toward the 10,000 loan each month 116 for ten years, yielding 19,842 or a net, after 20 years, of 7,695.3 Case 1c - Case 1b.Case 2c: Case 2b, and invest monthly what you were paying toward the loan 116 for another ten years. The 1000 grows to 3869.68 and the monthly investing yields 19,842, for a total of 23,711.68, or a net, after 20 years, of 10,745.83 Case 2c - Case 2b.Now this suggests the opposite: advantage to investing rather than paying off the loan. And now I get bogged down in factoring in the fact that investment returns arent guaranteed.Id love a final answer as to what to do, but doesnt this show that it isnt as simple as comparing rates?This is what I get for procrastinating while studying for the bar exam...  <response>: Your math is incorrect because case 2a assumes you pay off the 1000 over time. But in fact, you pay off the 1000 immediately, and save 70 on interest every year. And that 70 goes on to compound as well, which will end up with the same savings as case 1a.In other words, imagine I lend you 1000 at 7 interest. You have 1000, and have two options. However, Im feeling generous, so your loan of 1000 has no deadline -- you may choose to pay whenever, although anything you dont pay will generate interest for me. This is very similar to how credit cards work, except that credit cards monthly payment is usually not 0.You have two options:1 You may invest the 1000, and completely ignore my loan. 2 You may pay off my loan.You have 1000.In the first case, you choose to invest in something that generates 7 returns, and net 967.15. However, the loan I gave you also generates 967.15 in interest, which means that you failed to make or lose any money. I, on the other hand, gained 967.15 I made an investment that generated 7 returns every year for 10 years -- the loan I gave you was my investment.In case 2, you pay off the loan and have no money left. You also make no money, but lose no money.---This equivalency becomes apparent in my example, because I myself loaned you the 1000 at 7 interest. Loans are a zero-sum game, meaning that every dollar of interest I gain comes at an equivalent loss for you. I can only make X in interest if you lose X in interest to me. Thus, from my perspective, I made an investment that yields 7 return per year. This must mean that, by the equivalency noted earlier, you must have lost an equivalent amount.Put another way, imagine you invested the 1000 you owe by lending the money to someone else at 7 interest. So you owe me 1000 at 7 interest, but someone else owes you 1000 at 7 interest. Hopefully, its obvious that you are not going to make or lose money. 