# Computing Pooled IRR from the IRRs of parts

Suppose I have two cash flows:

• CF1: -10001001001100
• CF2: -20020301

I can compute now:

• IRR(CF1) = 10%
• IRR(CF2) =-55%
• IRR(CF1+CF2) = 4.46%

Is there a way to compute (or at least get a fair estimate) of the pooled IRR (i.e. 4.46%) by only knowing the two original IRRs along with, say, the initial investment values (or some additional partial metrics of the CFs)?

• I looked at this, but I don't think there is a solution. What it amounts to is: given a solution for each of two polynomial equations, can you find the solution of a polynomial which is the sum of the two polynomials. To my knowledge there is no mathematical theorem that allows you to do this. You have to solve the combined polynomial (i.e. find the combined IRR) all over again, there is no shortcut. – Alex C Dec 2 '15 at 23:37
• I didn't read all those comments and I bet other visitors would have done the same. Please add all the relevant information to the question or to the answers. Remember: All relevant information should be in the question or in the answer. – Bob Jansen Dec 3 '15 at 17:08
• Yes. I believe that is the case so far. – KT. Dec 3 '15 at 17:22

You could roughly estimate it by approximating the cash flows (which you do not know in full) using some "reasonable" simplified model. One example of such a model would be a cash flow of the form

$$-\mathrm{investment}, 0, 0, \dots{\small (n-1 \text{ zeros})}\dots, 0, \mathrm{returns}$$

Such a simple cash flow has IRR $r$ iff $$\mathrm{returns} = \mathrm{investment}\cdot(1+r)^n.$$

Now, if you only knew the IRRs $r_1$, $r_2$ and investment amounts $a_1$, $a_2$ for the two cash flows, you could approximate the PIRR by constructing the corresponding "model" cash flows and computing their pooled IRR:

$$\hat{\mathrm{PIRR}} = \left(\frac{a_1(1+r_1)^n + a_2(1+r_2)^n}{a_1 + a_2}\right)^{\frac{1}{n}} - 1$$

In your particular example, taking $n=3, a_1 = 1000, a_2 = 200, r_1 = 0.1, r_2 = -0.55$ you would obtain $\hat{\mathrm{PIRR}} = 3.98\%$.

Note that when $n=1$ this approximation corresponds to the weighted average of the two rates (with investment amounts being the weights). As $n\to \infty$ the approximation converges towards $\max(r_1, r_2)$.

The latter observation also illustrates why PIRR is not uniquely defined. In fact, if we used different real-valued lengths $n_1$, $n_2$ in the model representations of the two cash flows, we could have obtained almost any* desired resulting value of PIRR within $(\min(r_1, r_2), \max(r_1, r_2))$.

* It is always the case when both rates are positive. When one or both rates are negative this claim is not as obvious and might require proof.

• Could you elaborate what is the logic behind this approach? – KT. Dec 4 '15 at 0:34
• IRR / yield can be roughly considered as an annualized rate of return on investment. i.e. say you had 100 and you decided to invest for 3 years expecting to make 10% per year. how much cash you'll have after 3 years? 133~=100*(1+10%)^3. Now back to your CFs. let's estimate how much cash is left after 3 years: for investment 1 it would be 1331=1000*(1+10%)^3 and for investment 2 it would be 18=200*(1-55%)^3. so after 3 years you'be left with 1350 in total. so initially we had 1200 and after 3 years the value grows to 1350. what is the annualized rate? 4%=(1350/1200)^(1/3)-1. – Nicholas Dec 5 '15 at 10:52
• Cool. This is a good way to explain the logic behind a weighted average approximation as well (in that case you simply model the cash flows by their one-year approximations). – KT. Dec 5 '15 at 10:59
• @KT As soon as you make an assumption on the timing of the cashflows, the model will only be accurate if that assumption is met. It is essentially the same as saying "assume y1=y2 and give an example where y1=y2. Oh I get a perfect estimate!" – emcor Dec 5 '15 at 14:17
• @KT See example C1=(-1200,1200,0,0,100), C2=(-200,50,0,0,1). The IRRs are 7% and -64%. The PIRR is -3%. Your approach estimates y=3.61%, which is very wrong as it deviates by over 6%. My approach estimates y=0.361% which is much closer. Why? Because your model's assumptions on the cashflows are not met, while my model has no such assumption. – emcor Dec 6 '15 at 13:01

No there is no way since the calculated internal rate of return $r$ is by definition defined as:

$0 = \sum_{i=0}^{I} \frac{C_{i}}{(1+r)^{i}}$

You need to know the entire cash flow distribution and its timing if you want to compute the Pooled IRR.

One advantage of IRR is that it takes the irregular timings of cash flows into account, logically its disadvantage is that you need to know the timing and the associated amount.

• Comments are not for extended discussion; this conversation has been moved to chat. – Bob Jansen Dec 3 '15 at 17:06
• The answer is not true, you can express the solution based on the IRRs. It includes the cashflows, but it depends on the IRRs. – emcor Dec 3 '15 at 18:42
• @emcor I'm just answering the question : No there is no way to compute the pooled IRR by only knowing the two IRRs. In your answer you assume you know the full cash flow. – Malick Dec 3 '15 at 23:05

Exact solution:

Assume we agree that for $y_1:=IRR(CF1)$, $y_2:=IRR(CF2)$, $y:=IRR(CF1+CF2)$, the following equations hold by definition:

$$-1000+\frac{100}{1+y_1}+\frac{100}{(1+y_1)^2}+\frac{1100}{(1+y_1)^3}=0$$ $$-200+\frac{20}{1+y_2}+\frac{30}{(1+y_2)^2}+\frac{1}{(1+y_2)^3}=0$$ $$-1200+\frac{120}{1+y}+\frac{130}{(1+y)^2}+\frac{1001}{(1+y)^3}=0$$

These equations represent the definition of IRR and must hold. It is known that a system of equations can be solved by adding the equations, so it follows: $$-1000-200+\frac{100}{1+y_1}+\frac{20}{1+y_2}+\frac{100}{(1+y_1)^2}+\frac{30}{(1+y_2)^2}+\frac{1100}{(1+y_1)^3}+\frac{1}{(1+y_2)^3}\stackrel{!}{=}-1200+\frac{120}{1+y}+\frac{130}{(1+y)^2}+\frac{1001}{(1+y)^3}$$ So we need to solve $$\frac{100}{1+y_1}+\frac{20}{1+y_2}+\frac{100}{(1+y_1)^2}+\frac{30}{(1+y_2)^2}+\frac{1100}{(1+y_1)^3}+\frac{1}{(1+y_2)^3}=\frac{120}{1+y}+\frac{130}{(1+y)^2}+\frac{1001}{(1+y)^3}$$ Call the left hand side "$X$" and multiply the equation by $(1+y)^3$: $$X(1+y)^3=120(1+y)^2+130(1+y)+1001$$ This is a cubic equation: $$X(1+y)^3-120(1+y)^2-130(1+y)-1001=0$$ The real-valued solution to such equation given by here is: $$1+y= - \frac{1}{3a}\left(b\ +\ C\ +\ \frac{\Delta_0}{C}\right)$$ where $C = \sqrt[3]{\frac{\Delta_1 + \sqrt{\Delta_1^2 - 4 \Delta_0^3}}{2}}$, $\Delta_0 = b^2-3 a c$, $\Delta_1 = 2 b^3-9 a b c+27 a^2 d$.

It follows: $$y= - \frac{1}{3a}\left(b\ +\ C\ +\ \frac{\Delta_0}{C}\right)-1$$ with $a=X$, $b=-120$, $c=-130$, $d=1001$.

This solution incorporates the two IRRs $y_1,y_2$. It happens to have the same expression as when calculating the "normal" IRR based on the cashflows since $X=1200$ by definition. Note that this must be the case, because we are looking for the same $y$ that solves both equations.

As requested by the OP and moderator, I will write out this solution:

$$y= - \frac{1}{3X}\left(-120\ +\ \sqrt[3]{\frac{\Delta_1 + \sqrt{\Delta_1^2 - 4 \Delta_0^3}}{2}}\ +\ \frac{\Delta_0}{\sqrt[3]{\frac{\Delta_1 + \sqrt{\Delta_1^2 - 4 \Delta_0^3}}{2}}}\right)-1$$ $$= - \frac{1}{3\left(\frac{100}{1+y_1}+\frac{20}{1+y_2}+\frac{100}{(1+y_1)^2}+\frac{30}{(1+y_2)^2}+\frac{1100}{(1+y_1)^3}+\frac{1}{(1+y_2)^3}\right)}\left(-120\ +\ \sqrt[3]{\frac{2 b^3-9 a b c+27 a^2 d + \sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2 - 4 \Delta_0^3}}{2}}\ +\ \frac{\Delta_0}{\sqrt[3]{\frac{2 b^3-9 a b c+27 a^2 d + \sqrt{\Delta_1^2 - 4 \Delta_0^3}}{2}}}\right)-1$$

I stop writing out the solution here as it would not add more information. This equation infers the PIRR from the IRRs, all other values are constants. It is the exact solution.

However, it shows that the solution must incorporate the cashflows, it is not possible to separate out the solution to depend only on $y_1,y_2$. The third root term in the equation immediately implies that the cashflows cannot be canceled out.

Approximate Solution:

The IRR $y$ of the sum of the cashflows must lie withing the range $[y_1,y_2]$ where $y_1\leq y_2$. If $y$ is higher than $y_1$ and $y_2$, then the above equation has no solution since all cash flows on the right were discounted at a higher rate than the left such that the equation could not hold. By the same argument it follows that $y$ must be larger than the minimum of $y_1,y_2$.

A natural guess would be to take the average of the two: $$y\approx (y_1+y_2)/2=-0.225$$ Since we have a sum of CF1 and CF2, we may further weight the average by the absolute sum of cash flows to get: $$y\approx \frac{2200}{2200+251}0.1+\frac{251}{2200+251}\cdot(-0.55)=0.03345$$ This estimate is very close to the exact value of $0.046$.

• Comments are not for extended discussion; this conversation has been moved to chat. – Bob Jansen Dec 3 '15 at 17:05