On May 5, 12:42*am, "joeu2004" wrote:
[1] AFAIK, IRR is only defined as the rate that causes the NPV to be zero..
Someone else in this forum claims that there are multiple definitions of
IRRs. *IMHO, he is misusing the term IRR per se. *The "other definitions"
are, in fact, different definitions of rate of return. *To paraphrase a
familiar syllogism: *"IRR is a rate of return, but not all rates of return
are an IRR".

If that someone else you referred to is me then let me explain what I
mean when I say IRR may be defined in more than one way.

A) At an interest rate that equals the IRR internal rate of return,
the NPV or net present value of the cash flows is ZERO

B) If the NPV of cash flows is ZERO, then it follows that NFV or net
future value is also ZERO.

This is so as NFV is nothing more than the product of NPV and future
value interest factor that compounds the net present value to reflect
its future worth at i% interest rate for N periods.

C) NPV may be defined as the difference of discounted costs from
discounted benefits (B-C=0) thus we can rearrange this fact to define
profitability index such that PI is the ratio of discounted benefits
over discounted costs (B/C=1) given that C<0. This holds true as long
as the costs are not ZERO at which point the function is undefined due
to division by zero

NPV = discounted_benfits - discounted_costs
1) At IRR, NPV = 0
discounted_benfits - discounted_costs = 0

NFV = NPV x FVIF(i%,N)
2) At IRR, NFV = 0
NFV = NPV x (1+i)^N
NPV x (1+i)^N = 0

PI = discounted_benfits / discounted_costs
This follows from definition of NPV
NPV = discounted_benfits - discounted_costs
discounted_benfits = discounted_costs
discounted_benfits / discounted_costs = 1 [given that discounted_costs
< 0]

3) At IRR, PI = 1
discounted_benfits / discounted_costs = 1
[discounted_benfits / discounted_costs] - 1 = 0