returns - 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)?

Answer

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.}