# IRR of IRR or weighted average of IRR

I have a list of investments with their expected IRR(Internal Rate of Return). I'm confused about which is the right metric to depict for my population: IRR of IRR or weighted average of IRR. It's tough to understand which is needed when. Can someone help? I'm not even sure what to term the IRR of all my individual IRRs?

• Say, if we decide to use an average IRR, how will it be meaningful given that (I assume) investments have different cash flow sizes? That is, a large investment with a lower IRR can make you much richer than a small investment with a higher IRR. – Nik Tuzov Jul 28 '16 at 18:50
• This isn't really a statistical question. I think this is off topic here. However, I suspect you could get a good answer on the Quantitative Finance SE site. (You could also flag your Q & ask the moderators to migrate it for you.) – gung Jul 28 '16 at 19:29

You could add the cash flows (CF) by period and then compute the IRR for that accumulated cash flow (CCF). The resultant IRR will be like a weighted IRR. That will be different from IRR weighted by NPVs or Initial investment of each investment.

Let say you have two investments with this CFs:

       [0]  [1]  [2]  [3]  [4]  [5]  [6]
Inv1 -1560  200 1300  800   NA   NA   NA
Inv2 -1560  200 1300  800   NA   NA 1600


With a 25% discount rate you will have the NPVs and IRRs:

          NPVs      IRRs
Inv1 -158.4000 0.1902671
Inv2  261.0304 0.3164824


The accumulated cash flow will be:

       [0]   [1]   [2]   [3]   [4]   [5]   [6]
CCF  -3120   400  2600  1600     0     0  1600


You can verify that the total NPV is equal to the NPV of CCF at the same discount rate: 102.6304. Nothing new, this is one of the properties of NPV rule.

You could also compute the IRR of the CCF: 0.2652826, that will be the IRR of your investments.

That rate is different from the mean of IRRs (0.2533748), and from the weighted IRR using as weights the NPVs (0.5112835) or the initial investments (0.2533748).

Lets change the investment 1 to:

       [0]  [1]  [2]  [3]  [4]  [5]  [6]
Inv1 -2560  200 3300 1800   NA   NA   NA
Inv2 -1560  200 1300  800   NA   NA 1600


Then the NPVs and IRRs

         NPVs      IRRs
Inv1 633.6000 0.3806551
Inv2 261.0304 0.316482


The CCF

       [0]   [1]   [2]   [3]   [4]   [5]   [6]
CCF  -4120   400  4600  2600     0     0  1600


Then the IRR of the CCF: 0.3528677. Again different from the mean of IRRs (0.3485688), and from the weighted IRR using as weights the NPVs (0.3619311) or the initial investments (0.3563567).

Each IRR is the [or rather a] solution of a polynomial equation, whose coefficients are the cash flows of the specific projects.

If all the projects are carried out at the same time, the cash flows from combining the projects form the coefficients of a new polynomial.

This overall polynomial equation must be solved from scratch to find the IRR of the overall project. There are AFAIK no "shortcuts" such as as adding or weighing the solutions (IRRs) of the subprojects. It is a whole new IRR problem that must be solved from scratch.