PME, Direct Alpha, & Replication

8/24/2025

Benchmarking private equity to public markets sounds tidy: could I have just bought the index instead? This post gives you a simple mental model and a hands-on playground to explore three sibling answers:

Kaplan-Schoar PME: a clean wealth ratio.
Direct Alpha: an IRR after you first divide each cash flow by the index.
Replication view: track how many index units a cash-matching strategy would hold.

All three share one move from finance 101: a change of numeraire. Measure dollars in units of a public index, then compare paths on that common scale.

The setup

Let $C_t$ be contributions (cash you pay in) and $D_t$ be distributions (cash you receive). Let $I_t$ be the level of a chosen public index at time $t$ . Let $T$ be the final time in our toy examples. To keep the story focused on timing, assume no residual NAV at $T$ . We handle NAV in the caveats.

Use the controls

We keep one shared set of parameters for all plots. Change them here and watch everything update.

Controls

Years

Commitment

Contrib years

Dist start

Dist years

Exit multiple

Index r (annual, %)

Shock on

Shock year

Shock drop %

1) See the cash flows next to the index

This first plot shows contributions and distributions as bars, with the index as a line. If you toggle a shock, a light band marks the shock year.

Why it matters. All three metrics care about when cash moves relative to where the index is. Big distributions when the index is high help the fund look good versus the index. The same dollars in a drawdown do not.

2) Kaplan-Schoar PME

Kaplan-Schoar PME compares terminal wealth after you scale each cash flow by the index growth from its date to $T$ :

\text{PME}_{\text{KS}} = \frac{\sum_{t=0}^{T} D_t \cdot \tfrac{I_T}{I_t}}{\sum_{t=0}^{T} C_t \cdot \tfrac{I_T}{I_t}}.

Read it like this: pretend you invested each contribution into the index that day and let it grow to $T$ . Do the same for each distribution. If scaled dists exceed scaled contribs, $\text{PME} > 1$ and the fund beat the index in a wealth sense.

How to eyeball it from the first plot. If distributions cluster when the index is high, the numerator grows and PME rises. If contributions are made at high index levels and distributions land at low levels, PME falls.

Long-Nickels PME (ICM): the original PME

Idea. Build a hypothetical public index investment that mirrors the fund’s cash flows. When the fund calls $C_t$ , the replicator buys the index; when the fund distributes $D_t$ , it sells the index. The remaining index holding at the end, valued at $I_T$ , is the terminal value of that public replication. The LN-PME number is the IRR of that hypothetical public-market series (same calls and dists as the fund, plus the replication’s terminal value at $T$ ). Many LP decks compare Fund IRR to LN-PME IRR as a spread.

Algebra. Work in index units first. One index unit costs $I_t$ dollars at time $t$ . The replication’s unit inventory through time is $\text{Units}_t=\sum_{s\le t}\frac{C_s}{I_s}-\sum_{s\le t}\frac{D_s}{I_s}.$

Value those units at $T$ : $\text{NAV}_{PME}=I_T\cdot \text{Units}_T=\sum_{t=0}^{T} C_t\frac{I_T}{I_t}-\sum_{t=0}^{T} D_t\frac{I_T}{I_t}=FV(C)-FV(D).$

Define LN-PME IRR as the IRR of the public-replication cash-flow series with flows $-C_t + D_t$ each period and a terminal payoff $+\text{NAV}_{PME}$ at $T$ . That IRR is the market’s “answer” to the fund’s timing, on the same cash schedule.

Link to KS-PME. Because $\text{NAV}_{PME}=FV(C)-FV(D)$ , we have $\text{KS-PME}=\frac{FV(D)}{FV(C)}=1-\frac{\text{NAV}_{PME}}{FV(C)}.$ So LN-PME and KS-PME are two views of the same replication accounting: LN-PME returns a rate for the index-matched strategy; KS-PME returns a wealth multiple for the fund versus that same index growth.

Limitation. If the replication goes negative (i.e., would require shorting the index to finance early distributions), the LN-PME terminal value can be negative and its IRR unstable or undefined. This is exactly the shorting problem that PME+ addresses later by scaling distributions to enforce a long-only constraint.

3) Direct Alpha

Direct Alpha first deflates each cash flow by the index level at its date, then computes an IRR on that deflated series:

\sum_{t=0}^{T} \frac{-C_t + D_t}{I_t} \cdot \frac{1}{(1+\alpha)^t} = 0.

$\alpha$ is an annualized rate. You can read it as a dollar weighted excess return over the index. Positive $\alpha$ means the fund outperformed after accounting for timing.

Link back to the first plot. If distributions line up with high index levels, the deflated distributions are smaller, which can reduce $\alpha$ unless the raw dollars are strong. The trick is that we are no longer in dollars; we are in index units.

4) Replication picture and PME+

Plain English idea. Try to shadow the fund using only the index:

When the fund calls capital $C_t$ , you buy the index.
When the fund pays a distribution $D_t$ , you sell the index to raise cash.

To compare apples to apples, convert dollars to index units. One index unit costs $I_t$ dollars at time $t$ , so:

units bought at $t = C_t / I_t$
units sold at $t = D_t / I_t$

Your inventory of index units over time is $\text{Units}_t$ :

\text{Units}_t = \sum_{s\le t}\frac{C_s}{I_s} - \sum_{s\le t}\frac{D_s}{I_s}.

If $\text{Units}_t$ dips below zero, your shadow strategy would need to short the index to fund distributions. Some allocators do not allow shorting. PME+ asks a stricter, long only question:

What constant fraction of the fund distributions could a long only index strategy finance, using the same contributions?

We answer that by shrinking every distribution by a single factor $\gamma^* \in (0,1]$ so the inventory never goes negative:

\gamma^* = \min_{t:\;\sum_{s\le t}\tfrac{D_s}{I_s} > 0} \frac{\sum_{s\le t}\tfrac{C_s}{I_s}}{\sum_{s\le t}\tfrac{D_s}{I_s}}.

Intuition: at every time, cumulative units sold must be less than or equal to cumulative units bought. The smallest of those ratios across time is the tightest no short constraint. Scale distributions by that number and you avoid dipping below zero.

Then compute PME+ by plugging the scaled distributions $\gamma^* D_t$ into the KS ratio:

\text{PME+} = \frac{\sum_t \gamma^* D_t \cdot \tfrac{I_T}{I_t}}{\sum_t C_t \cdot \tfrac{I_T}{I_t}} = \gamma^* \cdot \text{PME}_{\text{KS}}.

Because $\gamma^* \leq 1$ , PME+ is always less than or equal to KS-PME. It is deliberately conservative: it asks how much of those distributions a long only index could have financed.

How to read this plot. The filled curve shows $\text{Units}_t$ for the unscaled cash flows.

If the curve stays above zero, a long only replication is feasible and $\gamma^* = 1$ , so PME+ equals KS-PME.
If the curve goes below zero, the fund timing would force the replicator to short the index. PME+ fixes that by shrinking all distributions by the same factor $\gamma^*$ . The plot itself is not rescaled; it is a diagnostic that shows whether shorting would be required.

5) How results change with exit multiple

Sweep the exit multiple while you keep the timing pattern fixed. This shows how each metric responds to stronger or weaker outcomes.

Reading tips.

The dotted lines mark $\text{PME} = 1$ and $\alpha = 0$ .
KS-PME and PME+ typically rise with the exit multiple. The slope depends on whether distributions land at high or low index levels.
Direct Alpha is often smooth under steady index paths and develops kinks when shocks hit cash heavy periods.

6) Path dependence: slide the shock

Hold your exit multiple steady and move a one time index shock across the fund life.

What to notice.

A shock during contribution heavy years can make both PME and $\alpha$ look better. You bought the index cheaper.
A shock during distribution heavy years can make both look worse. You sold into weakness.
PME and Direct Alpha can disagree on magnitude because they summarize timing differently.

Caveats

Residual NAV. If there is terminal NAV, add it to the KS numerator as NAV_T, and add $NAV_T / I_T$ as a final positive deflated flow in Direct Alpha.
Index choice. Pick $I_t$ with intent. A sector or size tilt can move results more than timing does.
PME+ reframes the question. By forbidding shorting, PME+ asks what multiple a long only index replication would achieve.

Connections and broader context

This is the same mathematical move as discounting in fixed income. There you discount by a curve. Here you rescale by an index.
The replication lens echoes option replication: trade a state variable (the index) against cash to match flows. Shorting shows up for the same structural reason.
Replace $I_t$ with a factor basket to step toward replication portfolios. KS-PME can be decomposed by factor, and Direct Alpha becomes excess return relative to that basket.
Direct Alpha is an IRR on a deflated series. It rhymes with log excess returns and, in an SDF frame, with changing the pricing kernel.

Translator’s note: unpacking the jargon

Wrapping up

Kaplan-Schoar PME, Direct Alpha, and PME+ are three windows on the same landscape. One shows the wealth multiple, one gives you a rate of outperformance, and one shows whether a simple index replication could keep up. None is the “true” answer alone, but together they sketch the shape of timing and scale.

The point isn’t to crown a winner, but to see how they complement each other. PME shows cumulative wealth, Direct Alpha reframes it as a return rate, and PME+ asks whether a long-only replicator could realistically match the flows. Used together, they help you understand not just what outperformance happened, but how it came about.