Negative properties of continuously compounded returns
What you're missing is the continuous compounding computation doesn't work that way.  If you compound over n periods of time and a rate of return of r, the formula is e^(r*n), as you have to multiply the returns together with a mulitplicative base of 1.  Otherwise consider what 0 does to your formula.  If I get a zero return, I have a zero result which doesn't make sense.  However, in my formula I'd still get the 1 which is what I'm starting and thus the no effect is the intended result. Continuous compounding would give e^(-.20*12) = e^(-2.4) = .0907 which is a -91% return so for each $100 invested, the person ends up with $9.07 left at the end.  It may help to picture that the function e^(-x) does asymptotically approach zero as x tends to infinity, but that is as bad as it can get, so one doesn't cross into the negative unless one wants to do returns in a Complex number system with imaginary numbers in here somehow. For those wanting the usual compounding, here would be that computation which is more brutal actually: For your case it would be (1-.20)^12=(0.8)^12=0.068719476736 which is to say that someone ends up with 6.87% in the end.  For each $100 had in the beginning they would end with $6.87 in the end. Consider someone starting with $100 and take 20% off time and time again you'd see this as it would go down to $80 after the first month and then down to $64 the second month as the amount gets lower the amount taken off gets lower too.  This can be continued for all 12 terms.  Note that the second case isn't another $20 loss but only $16 though it is the same percentage overall. Some retail stores may do discounts on discounts so this can happen in reality.  Take 50% off of something already marked down 50% and it isn't free, it is down 75% in total.  Just to give a real world example where while you think a half and a half is a whole, taking half and then half of a half is only three fourths, sorry to say.  You could do this with an apple or a pizza if you want a food example to consider. Alternatively, consider the classic up and down case where an investment goes up 10% and down 10%.  On the surface, these should cancel and negate each other, right?  No, in fact the total return is down 1% as the computation would be (1.1)(.9)=.99 which is slightly less than 1. Continuous compounding may be a bit exotic from a Mathematical concept but the idea of handling geometric means and how compounding returns comes together is something that is rather practical for people to consider. You cannot use continuous compounding for returns less than or equal to 100% because a natural logarithm can only be taken for a positive amount.  This answer includes the accurate way to ascertain r, for which many people use an approximation. For example, using -20% monthly return for 12 months:- -0.2 -0.223144 0.0687195 Checking: 0.0687195 True Now trying -100% monthly return:- -1. Indeterminate Why?  Because a natural logarithm can only be taken for a positive amount.  So the latter calculation can not be done using (logarithmic) continuous compounding. Of course, the calculation can still be done using regular compounding.  For -100% the results go to zero in the first month, but -150% produces a more interesting result: -1.5 -11920.9 Well, one can easily have rates below -100%.  Suppose I start with $100, and end up with $9 after a year. What was my rate of return? It could be -91%, -181%, -218%, or -241%, or something else, depending on the compounding method.  We always have that the final amount equals the initial amount times a growth factor G, and we can express this using a rate r and a day count fraction T. In this case, we have T = 1, and B(T) = B(0) * 0.09, so: So, depending on how we compound, we have a rate of return of -91%, -181%, -218%, or -241%. This nicely illustrates that:
"Sorry in advance, but this will be long.  Also, it sounds like your friend is a tool. I hope this ""friend"" is not also your financial advisor... they would be encouraging you to make a very poor investment decision. They also don't know how to do financial math.  For what it's worth, I am not wrong. I have correctly answered a set of changing questions as you have asked them... Your friend is answering based on a third, completely different investment model, which you proposed in the edit to your last post. If that's what you meant all along, then you should have been more clear in the questions you were asking.  Please let me layout the following: How the previous questions//investment proposals were built How to analyze this current proposal What your other option is Why the other option is best in a 'real world' market The First Question My understanding of the initial proposal was to take out a $10,000 loan, invest the proceeds, and expect to not have any money of your own tied up in this. Because that OP did not specify that this is an interest-only loan (you still haven't in any of your questions), the bank will require you to make payments back to them each month that include principal and interest.  Your ""friend"" is talking about the total interest paid being the only cost of a loan. While that is (almost) true, regardless of what your friend says, significantly more cash is involved in making sure that all the payments are made on time---unless you set up an interest-only loan. But with the set up laid out in this post, and with the assumptions I specified there, the principal payments must be included because the borrower has to pay back the bank and isn't not tying up any of their own money.  In that case, my initial analysis is correct--your breakeven is in the low teens for an annual required return.  The Second Proposal Your second proposal... before any edits... refined things a little bit, to try to capture the any possible returns by not selling something. As I indicated there, (with what was an exaggerating assumption), the lack of clarity makes for an outlandish required return.  The Second Proposal...with edits, or the one proposed above I will get to the one proposed above in a second, but first let me highlight a few problems with your friend's analysis.  Simple interest: the only place (in the US at least) that will lend with simple interest is student loans. Any loan that you actually take out will be compound interest.  Not an interest only loan: your ""friend"" is not calculating interest correctly. Since this isn't an interest-only loan, the principal balance will reduce every time you make a payment, by ~$320-$340 each month. This substantially reduces the total interest paid, to $272.79 over the total 24 months.  ""Returns"": I don't know what country, or what business your friend works in, but ""returns"" are a very ambiguous concept. Investopedia defines returns as gains or losses. (I wish I could inhabit the lala land that your friend lives in when returns are always positive). TheFreeDictionary.com defines a return for finance as ""The change in the value of a portfolio over an evaluation period, including any distributions made from the portfolio during that period.""  When you have not made it clear that any other money is being used in this investment plan (as was the case in scheme #1 and scheme #2a,) the loan still has to be paid. So, clearly the principal must be included in the return calculations.  How to evaluate this proposed investment scheme Key dimensions: Loan ($8,000 ... 24 months ... 0.27% monthly rate... monthly compounding... no loan origination fees) Monthly payment (PMT in Excel yields $344.70). Investment capital (starting = $8,000) Monthly Return (Investment yields... we hope it's positive!)  Your monthly contribution from your salary Taxes = 10%. Transaction Fees = $20  Go and lookup how to build an amortization table for a loan in Excel. Your life will be infinitely better for it.  Now, you get this loan set up and invested into something... (it costs $20 to buy the assets).  So you've got $7980 chugging away earning interest. I calculate that your break even, with you paying in $344.70 of your own money each month is 1.81% annually, or 3.42% over the 24 month life of this scheme. That is using monthly compound interest for the payments, because that's what the real world would use, and using monthly compounding of the investments' returns.  Your total interest expense would be $272.79.  This seems feasible. But let's talk about what your other option is, given that you're ready to spend $344.70 each month on an investment.  Your other option  I understand the appeal of getting $8,000... right away... to invest in something. But the risk behind this is that if the market goes down (and markets do) you're stuck paying a fixed amount for your loan that is now worth less money.  Your other option is to take your $344.70, and invest it step-by-step. (You would want to skip a month or two buying assets in the market, so that you can lessen transaction costs). This has two advantages: (1) you save yourself $272 in interest. (2) When the market goes down, you still win.  With this strategy, you still win when the market goes down because of what is commonly called ""dollar cost averaging"". When the market is up, your investments are also up. When the market goes down, your previous investments decrease in value but you can invest new money at the lower rates.  Why the step-by-step, invest your own money strategy is better At low rates (when you're looking for your break-even), the step-by-step model outperforms the loan. At higher rates of return (~4% + per year), you get the benefit of having the borrowed money earning more gains.  In fact, for every continuous (meaning set... not changing month-to-month) interest rate that you can dream up that is greater than about 4% per year, the borrowed money earns more.  At 10% per year, the borrowed money will earn about $500 more over the 2 years than your step by step investment would.  BUT I recognize that you might feel like the market will always go up. That's what everyone thinks. And that's alright.  But have one really bad month, or a couple of just-not-great-months, and your fixed 'loan' portfolio will underperform. Have a few really bad months, and your portfolio could be substantially reduced in value... but you would still be paying the same amount for it each month.  And if that happened (say your assets declined -3% in 3 of the 24 months...) You'd be losing money relative to the step-by-step portfolio." Let's say I have $10,000, and I invest said monies in mutual fund   XXXXX at $100/share, effectively giving me 100 shares. Now, let's   assume at the end of the year I have a 5% return. My $10,000 is now   $10,500. At what point does my investment benefit from compounded interest?   Monthly? Quarter? Yearly? Does it even benefit? Daily would be my answer as your investment, unless you are selling shares or not re-investing distributions is getting the following day's change that impacts the overall return. Consider how if your fund went up 2% one day and then 2% another day from that $10,000 initial investment. The first gain brings it up to $10,200 and then the second makes it $10,402 where the extra $2 is from the compounding. The key though is that these are generally small movements that have to be multiplied together. Note also that if your fund goes up and down, you may end up down overall given how the returns compound. Consider that your $10,000 goes up 10% to $11,000 and then down 10% to result in $9,900 as the return for up x% and down x% is (1+x)(1-x)=1-x^2 which in this case is 1% as 10% of 10% is 1%. The key is how long do you keep all the money in there so that the next day is applied to that amount rather than resetting back to the initial investment. I have done something similar to this myself.  What you are suggesting is a sound theory and it works.  The issues are (which is why it's the reason not everyone does it) : The initial cost is great, many people in their 20s or 30s cannot afford their own home, let alone buy second properties.  The time to build up a portfolio is very long term and is best for a pension investment.  it's often not best for diversification - you've heard not putting all your eggs in one basket? With property deposits, you need to put a lot of eggs in to make it work and this can leave you vulnerable. there can be lots of work involved. Renovating is a huge pain and cost and you've already mentioned tennants not paying!  unlike a bank account or bonds/shares etc.  You cannot get to your savings/investments quickly if you need to (or find an opportunity)  But after considering these and deciding the plunge is worth it, I would say go for it, be a good landlord, with good quality property and you'll have a great nest egg.  If you try just one and see how it goes, with population increase, in a safe (respectable) location, the value of the investment should continue to rise (which it doesn't in a bank) and you can expect a 5%+ rental return (very hard to find in cash account!)  Hope it goes well! A stopped clock is right two times a day.   We may get a market crash similar to the financial crisis or the dot com crash or we may not. What if over the next 10 years rates rise very sluggishly, low inflation, and  low growth more or less continues with maybe one brief and shallow recession. In that case I bet US stocks produce 4-5% returns per year and US bonds produce maybe 1-2% per year.  European and emerging market stocks should have higher returns because they are in an earlier part of the cycle.  I think your baseline has to look something like that.  The last two crashes were caused by the tech bubble and the housing bubble - where is the bubble today? US stocks are expensive, but probably not in bubble territory. Bonds worldwide are unattractive with low or negative yields - negative yields maybe a bubble, but central banks will be the most hurt by negative rates and they are in a strong position to take the pain.   There could be a crash I just don't see how we get there yet - maybe china? A stopped clock is right two times a day.   We may get a market crash similar to the financial crisis or the dot com crash or we may not. What if over the next 10 years rates rise very sluggishly, low inflation, and  low growth more or less continues with maybe one brief and shallow recession. In that case I bet US stocks produce 4-5% returns per year and US bonds produce maybe 1-2% per year.  European and emerging market stocks should have higher returns because they are in an earlier part of the cycle.  I think your baseline has to look something like that.  The last two crashes were caused by the tech bubble and the housing bubble - where is the bubble today? US stocks are expensive, but probably not in bubble territory. Bonds worldwide are unattractive with low or negative yields - negative yields maybe a bubble, but central banks will be the most hurt by negative rates and they are in a strong position to take the pain.   There could be a crash I just don't see how we get there yet - maybe china? "IMO: If you look long and hard enough at your data you're going to bump into these kinds of ""patterns"". There is no ""curse of 7"".  “Looking at the chart makes you want to hide under your desk when you see it, but remember this is only a sample size of 11, and the average performance is greatly skewed by big drops late in 1907, 1937, 1957, and 1987. Also, consider that 1907, 1937, and 1957 were all recessionary years, and equities had run up 40% for the year in August 1987, so a pullback was not surprising."" [Quote](https://lplresearch.com/2017/08/11/the-curse-of-years-ending-in-7/)" "You're supposed to be filling form 433-A. Vehicles are on line 18. You will fill there the current fair value of the car and the current balance on the loans. The last column is ""equity"", which in your case will indeed be a negative number. The ""value"" is what the car is worth. The ""equity"" is what the car is worth to you. IRS uses the ""equity"" value to calculate your solvency. Any time you fill a form to the IRS - read the instructions carefully, for each line and line. If in doubt - talk to a professional licensed in your state. I'm not a professional, and this is not a tax advice." I think the definition of overcollateralization on investopedia will answer this question for you.  Namely this part: For example, in the case of a mortgage backed security, the principal amount of an issue may be $100 million while the principal value of the mortgages underlying the issue may be equal to $120 million. The bond is packed with more mortgages than the face value indicates.  It's effectively sold at a discount to underlying value.