I've been trying to wrap my head around a downside of using IRR - the reinvestment assumption. I don't really understand the concept on how IRR assumes that we reinvest at the IRR, if IRR is for discounting backwards. Would be great if someone had an analogy or example so I can understand the actual concepts. Also, some people say that it's a myth or fallacy, and the IRR does not actually assume anything, which makes me even more confused. There's barely any information available, and I've tried to read all the research papers yet I still do not understand. Please help.
The idea behind the IRR is that there is some discount rate that makes the discounted cash flows add up to zero. That is, given a set of cash flows with amount a_i at time t_$, the sum over all i of a_i*d^(t_i) is zero (this is including all money put into the investment, including the initial principal, as negative). The IRR is then 1/d-1, or (1-d)/d. In other words, we find some number that when substituted in results in zero, and we take that as the associated return. (There are some situations where it can have more than one solution, though.)
This analysis does require that the total discount at any particular time be given by some constant raised to the time of the discount. So if we take A1 dollars out Y1 years after our initial investment, and A2 out Y2 after, we're treating A2 as being less valuable than A1, by a factor of d^(Y2-Y1). Equivalently, we're treating A1 as being more valuable by a factor of d^(Y1-Y2), or by a factor of (1+IRR)^(Y2-Y1). This implies that we're assuming that between receiving A1 and receiving A2, we were able to invest A1 somewhere that received a return of IRR, i.e., we reinvested all our returns at the IRR.
For example, suppose we invest $100k initially, and we get $60k back the first year, and $70k the second year. To calculate the IRR, we assume there is some r such that the $70k is the result of getting a return of r for two years, while the $60k is the result of getting r for one year. If we calculated everything in terms of current dollars, we divide the $60k by 1+r, and the $70k by (1+r)^2, while the $100k stays the same:
-$100k + $60k/(1+r) + $70k(1+r)^2 = 0
If we calculate everything in terms of final dollars, the $70k stays the same, the $60k is multiplied by 1+r, and the $100k is multiplied by (1+r)^2:
-$100k(1+r)^2 + $60k(1+r) +$70k = 0
Note that these two equations are equivalent, as the second can be obtained by the first by multiplying by (1+r)^2.
So going through the second equation, at the end of the second year, we have $70k that is just right there, we have $60k that we're assuming we were able to invest at a rate of r, and then there's $100k that has an opportunity cost (and thus is negative) that we could have invested and gotten (1+r)^2, and so we lost out on $100k(1+r)^2 by putting it in this investment instead.
Basically, IRR is asking "If there were some constant rate of return that allowed us to compare all our cash flows, what would it be?" There's a time value for the $60k being received a year before the $70k, and for us to be using a single rate of return that applies to not just time value, but to all the other returns, we assume that we reinvested the $60k at the IRR.