Public market equivalent (PME)

Key takeaways

• A fund IRR is money-weighted and an index return is time-weighted, so the two cannot be compared directly even over identical dates.
• The Long-Nickels PME buys and sells an index shadow portfolio with the fund's cash flows and compares the fund IRR with the IRR of that shadow, but the shadow NAV can turn negative for successful funds.
• The Kaplan-Schoar PME divides the index-compounded value of distributions plus NAV by the index-compounded value of contributions; above 1.0 the fund beat the index, below 1.0 it lagged.
• PME+ scales the fund's distributions so the shadow portfolio ends exactly at the fund's NAV, which avoids a negative shadow balance at the cost of altering the distribution stream.
• Direct Alpha is the IRR of the fund's cash flows after each has been compounded by the index to a common date, giving an annualized excess return that is consistent with the Kaplan-Schoar PME.
• Every PME is only as good as the index chosen: in Fund E the same cash flows give a Kaplan-Schoar PME from 1.28 down to 0.95 depending on the benchmark.

Why an IRR cannot be set against an index return

An index return is a time-weighted return (TWR). It measures what one dollar held in the index from the first date to the last would have grown to, and it is unaffected by when anyone added or withdrew money. A fund IRR is a money-weighted return (MWR). It weights each period by how much of the LP's capital was in the fund at the time, and the GP decides that timing. Put a 13 percent fund IRR next to a 6 percent index return and the gap mixes two things: the fund's actual outperformance and the difference between the two methods of measurement.

The distortion can run either way. If the index rose sharply while the fund held little capital, the comparison is unfair to the fund; if it fell then, the comparison flatters the fund. The earlier topic on time-weighted and money-weighted returns shows identical underlying performance producing a 9.1 point apparent gap purely from the difference in method.

There is also a horizon problem. A '10-year index return' runs between two fixed dates, while a fund's capital was invested for only part of any such window. The only fair benchmark is one that invests and divests on exactly the fund's dates in exactly the fund's amounts.

The example: Fund E and its index

Fund E is a hypothetical fund observed over six years with annual cash flows, which keeps every step visible. The LP base contributes $100m in years 0 to 3 and receives $120m of distributions in years 3 to 6, and the fund still holds a NAV of $40m at the end of year 6. In year 3 the fund both calls $15m and distributes $10m, so the net flow that year is an outflow of $5m.

The benchmark index starts at 100 and ends at 142. Its path is deliberately uneven: it rises 12.0 percent in year 1, falls 6.25 percent in year 2, then rises, dips and rises again. Over six years it compounds at 6.0 percent a year. Fund E's own IRR on its net cash flows, with the ending NAV treated as a final inflow, is 13.0 percent. Its TVPI is 1.60x and its DPI is 1.20x.

The naive reading is that Fund E beat the index by 7.0 points a year (13.0 minus 6.0). That happens to be close to what the methods find for this index path, by coincidence rather than by construction; with other paths the naive gap and the PME answer diverge.

Long-Nickels PME: the index shadow portfolio

The first PME method, proposed by Austin Long and Craig Nickels, builds a shadow portfolio. Each time the fund calls capital, the same amount is imagined invested in the index. Each time the fund distributes, the same amount is withdrawn from the index holding. Between cash flows the shadow holding grows or shrinks with the index. At the report date the shadow holding has a value, the shadow NAV, which replaces the fund's actual NAV.

The Long-Nickels PME is then the IRR of the fund's actual contributions and distributions with the shadow NAV as the terminal value. That IRR is the money-weighted return the index would have delivered on the fund's schedule. The comparison is fund IRR minus Long-Nickels IRR, a spread in percentage points. Because both IRRs use the same dates and the same interim cash flows, the timing distortion described above cancels out.

For Fund E the shadow portfolio ends with $3.0m, against the fund's actual $40m NAV. The index, fed the same cash, would have run almost dry, because the fund's distributions were large relative to what the index earned. The Long-Nickels IRR is 6.0 percent, so the fund outperformed by 7.0 points on a like-for-like basis.

When the shadow NAV turns negative

The weakness of the Long-Nickels method appears when a fund is very successful early. Its distributions are then larger than anything the index holding could have grown to, so the shadow portfolio is asked to pay out more than it holds. Arithmetically the shadow NAV goes negative, which means the method is implicitly shorting the index to fund the distributions. From that point the shadow gains when the index falls and loses when it rises, which is not a sensible benchmark for a long-only allocation.

A negative terminal shadow NAV also turns the final cash flow of the LN series into an outflow. The series then changes sign more than once, and, as the IRR topic explains, such a series can have several IRRs or none that means anything. The example below produces two mathematical roots, neither of which is a plausible return for an index that rose over the period.

Kaplan-Schoar PME: a ratio of index-adjusted values

Steven Kaplan and Antoinette Schoar proposed a ratio rather than a spread. Compound every contribution and every distribution forward to the final date at the index's return from its own date. Add the fund's ending NAV to the compounded distributions. Divide by the compounded contributions. The result, the Kaplan-Schoar PME (KS-PME), is the fund's total value per dollar of contributions, where both sides are measured in index terms. Compounding everything to the final date gives exactly the same ratio as discounting everything back to the first date, which is how the formula appears in the previous topic, because the same index factor cancels.

The interpretation is direct. A KS-PME of 1.0 means the LP would have ended with the same wealth by putting the same cash into the index on the same dates. Above 1.0, the fund did better: a KS-PME of 1.28 means the fund produced 28 percent more ending value than the index would have on its schedule. Below 1.0, the index would have been the better choice. The ratio is cumulative, not annual, so a 1.28 over six years and a 1.28 over twelve years are different degrees of outperformance.

The KS-PME never needs a shadow balance, so it cannot break the way the Long-Nickels method does. And when the index is flat it collapses exactly to TVPI: it is TVPI with every dollar restated in index-equivalent terms.

PME+: rescaling distributions

PME+, developed at Capital Dynamics, keeps the Long-Nickels idea of an IRR comparison but repairs its failure mode. Instead of letting the shadow portfolio absorb the fund's actual distributions, it multiplies every distribution by one constant, lambda, chosen so that the shadow portfolio ends exactly at the fund's actual NAV. The shadow is never forced short by construction, because its terminal value is pinned to a real number.

The PME+ IRR is the IRR of the fund's contributions, lambda times its distributions, and the fund's ending NAV. It is the money-weighted return the index would have earned if it had paid out the same proportional pattern of distributions and ended with the same residual value. The comparison is fund IRR minus PME+ IRR. For Fund E, lambda is 0.7134: the index could have afforded only about 71 percent of the fund's distributions while ending at a $40m NAV.

The cost of the fix is that PME+ no longer uses the fund's actual distribution amounts, so the benchmark's cash flow pattern differs from the LP's. When the Long-Nickels shadow stays positive, as it does for Fund E, the two spreads tend to be close.

Direct Alpha: an annualized excess return

Direct Alpha, set out by Oleg Gredil, Barry Griffiths and Rüdiger Stucke, combines the robustness of the Kaplan-Schoar ratio with the annualized form of an IRR. Compound every fund cash flow to a common date using the index, exactly as in the KS-PME, and treat the ending NAV as a flow on the final date. Then compute the IRR of those index-adjusted cash flows. Because the index return has been stripped out of every flow, the IRR that remains is the fund's return in excess of the index, per year.

The authors present the result both as the discrete annual rate, a, and in continuously compounded form, ln(1 + a). The choice of common date does not change the answer, because compounding every flow by the same additional factor leaves the IRR unchanged. Direct Alpha is consistent with the KS-PME: the KS-PME is above 1.0 exactly when Direct Alpha is above zero.

For Fund E the index-adjusted IRR is 6.6 percent a year, not the 7.0 point spread from the IRR-based methods. A spread is a difference of two money-weighted returns; Direct Alpha is one return computed on excess performance, so the two usually differ modestly.

Choosing a method and choosing an index

The methods agree on direction whenever they work: a fund that beats the index on one beats it on the others. They differ in the form of the answer and in robustness. A committee that thinks in multiples will find the KS-PME natural. One that thinks in annual returns will prefer Direct Alpha or a PME+ spread. The Long-Nickels spread is intuitive but should be used only after checking that the shadow NAV stayed positive.

The larger source of disagreement is not the method but the index. A buyout fund benchmarked against a large-cap index, a small-cap index and a sector index can pass or fail depending on the choice. Leverage matters too: buyout equity carries debt, so an unlevered index understates the risk the LP took. Some analysts adjust the index for leverage or beta; the adjustment itself requires assumptions and should be disclosed. Currency matters as well: the index and the fund should be measured in the LP's base currency, or both in the fund's currency, never one of each.

The table below runs Fund E's identical cash flows against four hypothetical index paths. The fund does not change. The verdict does.