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.
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Negative properties of continuously compounded returns

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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.