# Private Equity: Direct Alpha vs Excess IRR

I'm trying to understand the advantages and disadvantages of using Direct Alpha versus Excess IRR for computing excess returns over a market index for private assets.

Wikipedia references a highly informative paper that compares the Direct Alpha against PME, PME+, mPME, and KS-PME and discusses the limitations of these as well as analyzes the correlations between them.

I'm looking for a similar resource to that compares the two in my title.

## 1 Answer

They are "essentially" the same thing. IPP (or excess IRR) is the excess return over the annualized benchmark such that the adjusted PME is 1:

$$\text{PME} = 1 = \frac{\sum_{i=1}^n d_i \left(1 + \dfrac{b_{T_i, T_N}}{q} + \dfrac{r}{q}\right)^{q(T_N-T_i)} }{\sum_{j=1}^m c_i \left(1 + \dfrac{b_{T_j, T_N}}{q} + \dfrac{r}{q}\right)^{q(T_N-T_j)}},$$

where $c_i$ and $d_i$ are the contribution and distribution at time $t_i$, respectively, $b$ is the annualized public market benchmark return over the relevant periods, $r$ is the annualized excess IRR/IPP, and $q$ is the compounding frequency (typically $q=1$ in the IPP setting).

If you let $q\to\infty$ (i.e., continuous compounding) and with some redefinition (e.g., $e^\alpha = 1+a$), it is easy to show that excess IRR/IPP converges to direct alpha.

So simply put, direct alpha is just the limiting case of excess IRR/IPP, but they're conceptually the same thing. As such, they have the same advantages and disadvantages. IMO, the derivation of IPP/excess IRR is more direct (resembling the way OAS is defined in the fixed income market) and the use of discrete compounding makes the output more comparable to other reported returns.