# 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.

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