Performance measurement, and how it is gamed

The subscription line effect on net IRR

A capital call facility secured by uncalled LP commitments lets a fund fund an investment before calling capital from its LPs. The investment's own cash flows do not change. The LP's cash flows move later, and because IRR is a function of timing and multiples are not, net IRR rises while DPI and TVPI fall by the cost of the borrowing.

Field	Value
Formula	Facility interest is added to the deferred call: call_t becomes call_t * (1 + c)^d at time t + d. IRR rises because d shortens the LP's exposure period; TVPI falls because PIC rises by the interest and distributions do not
Base case	No facility: paid-in 500.0000, LP distributions 848.0000, DPI 1.6960x, net IRR 11.9825 percent
Worked, 12-month facility at 6.00 percent	Paid-in 525.5000, of which 25.5000 is facility interest. LP distributions 853.1000, DPI 1.6234x, net IRR 13.5240 percent
The trade	Net IRR +154.2 basis points; DPI and TVPI -0.0726x. The LP paid 25.5000 of interest to make the headline number better
Worked, 24-month facility	Paid-in 552.5300, DPI 1.5538x, net IRR 15.9796 percent - a 399.7 basis point improvement bought with 52.5300 of interest
Unchanged throughout	The six investments, their dates, their proceeds, and the 2.2000x gross MOIC on invested capital

• This is the clearest case in the asset class where a headline metric and the underlying economics move in opposite directions. Nothing about it is improper - the facility genuinely shortens the period LP capital is at risk - but a net IRR comparison between a fund that uses a line and one that does not is not a comparison of anything.
• The GP gives something up too, and it is worth knowing what: on the reference fund the facility raises contributed capital, which raises the return-of-capital hurdle, so GP carry falls from 87.0000 to 81.9000. The GP trades 5.1000 of carry for 154 basis points of track record. That trade is only rational if the track record is worth more than the carry, which tells you what the facility is for.
• The right questions are numerical and short. Ask for net IRR computed as though every call had been made on the investment date; ask for the average number of days between investment and capital call; ask for total facility interest as a line in the expense schedule. The second number predicts the size of the first adjustment before anyone computes it.
• A facility long enough to span most of the investment period changes the character of the fund's reported return entirely. At 24 months the reference fund reports a 15.9796 percent net IRR on a portfolio that delivered 1.5538x to its LPs, against 11.9825 percent on 1.6960x with no facility. The better multiple is the worse-looking fund.

Source: ILPA guidance on subscription lines of credit recommends disclosure of net IRR both with and without the facility.

Why IRR and the multiple disagree, and when the identity holds

IRR is a rate; TVPI and MOIC are ratios. They are connected by a closed form only in the single-draw, single-return case. For any real call and distribution schedule there is no closed form, and the same multiple maps to a wide range of IRRs depending entirely on when cash moved.

Field	Value
Formula	Single draw and single return: IRR = MOIC^(1/n) - 1, equivalently MOIC = (1 + IRR)^n. For a stream, IRR solves 0 = sum of CF_t/(1 + IRR)^t and has no closed form
Worked, where the identity fails	The reference fund's net TVPI is 1.6960x over 10 years. The single-draw identity would give 1.6960^(1/10) - 1 = 5.4248 percent. The actual net IRR is 11.9825 percent
Why	Capital is called over five years and returned over six, so the average dollar is outstanding for far less than 10 years. The identity assumes it is outstanding for all of them
Gross against net	Gross IRR on deal-level cash flows is 17.2915 percent on a 2.2000x gross MOIC. Net IRR is 11.9825 percent on a 1.6960x net multiple. The 530.9 basis point gap is the fee and carry load expressed as a rate
The single-draw check	2.2000^(1/5) - 1 = 17.0805 percent, close to the 17.2915 percent gross IRR because five of the six investments were held exactly five years

• The gross-to-net gap is best read as two separate leaks. The multiple falls from 2.2000x to 1.6960x, of which 0.3300x is capital called that never bought an asset and 0.1740x is carry. The rate falls by 530.9 basis points, which mixes both leaks with the timing of the calls.
• IRR cannot be averaged across funds or added across periods. Pooling the cash flows and re-solving is the only correct aggregation, and it produces a different answer from an average of the individual IRRs in every case except a trivial one.
• Because IRR assumes interim distributions earn the IRR itself, a fund with an early realisation and a long tail reports a rate its LPs could not have earned on the cash they received. The reference fund distributes 110.0 in year 5 and is still calling capital in year 10; the 11.9825 percent rate assumes that 110.0 compounded at 11.9825 percent, which nobody guaranteed.

The gross-to-net bridge

Gross figures are computed on cash flows between the fund and its investments. Net figures are computed on cash flows between the fund and its LPs. The bridge between them is the management fee, partnership expenses, carried interest, and the timing of capital calls.

Field	Value
Formula	Net multiple = (gross proceeds - carry) / (I + F + X). Each of the three deductions can be sized separately
Worked, the bridge	Gross MOIC 2.2000x on 425.0 of cost. Add 66.2 of fee and 8.8 of expenses to the denominator: 935.0/500.0 = 1.8700x. Deduct 87.0 of carry from the numerator: 848.0/500.0 = 1.6960x
Attribution of the 0.5040x	Fee and expenses cost 0.3300x of multiple; carry costs 0.1740x. Fees are the larger of the two on this fund
As a share of profit	Total fee and expenses 75.0 plus carry 87.0 = 162.0, against gross profit of 935.0 - 425.0 = 510.0. The manager takes 31.765 percent of gross profit
As a rate	17.2915 percent gross IRR against 11.9825 percent net IRR: 530.9 basis points

• Stating the load as a share of gross profit is the framing that survives comparison across strategies. A 162.0 deduction from 510.0 of gross profit is 31.765 percent whatever the multiple was; a 0.5040x deduction from a multiple means nothing without knowing the multiple.
• Fees cost more than carry on this fund, and that is the usual case at ordinary outcomes rather than the exception. Carry scales with success and fees do not, so the fee load dominates in exactly the outcomes that are most common.
• A gross IRR quoted without the corresponding gross MOIC is unreadable, because gross IRR on deal-level flows is silent about how long the fund took to deploy. On the reference fund the gross IRR of 17.2915 percent is essentially the five-year hold on a 2.2000x, which the multiple and the holding period would have told you directly.

TVPI, DPI and RVPI - read the split, not the sum

DPI is cash returned per unit of paid-in capital and is the only one of the three that cannot be asserted. RVPI is the manager's own mark per unit of paid-in. TVPI is their sum, which means two funds with identical TVPI can be in completely different positions.

Field	Value
Formula	TVPI = DPI + RVPI. Realisation share = DPI/TVPI
Worked, year 6	DPI 0.2872x + RVPI 1.2615x = TVPI 1.5487x. Realisation share 18.54 percent
Worked, year 8	DPI 1.3031x + RVPI 0.4531x = TVPI 1.7562x. Realisation share 74.20 percent
The trap	A fund at 1.5487x TVPI with 81.46 percent unrealised and a fund at 1.5487x TVPI fully realised report the same headline. Only one of them is a fact
Mark sensitivity at year 6	Marks 20 percent low would make the true TVPI 1.8010x; 20 percent high would make it 1.2964x. A 40-point range on one reported number

• The reference fund's TVPI peaks at 1.7562x in year 8 and falls to 1.6960x at termination. TVPI is not monotonic and a decline in it is not necessarily bad news - it can simply be marks converging to realisations, which is what happened here.
• DPI can be raised without a realisation by borrowing at the fund level against NAV and distributing the proceeds. The LP's DPI rises, its RVPI falls by less than the distribution, and it now sits behind a secured lender. Reconciling every distribution to a named realisation is the only check.
• RVPI should decline on a schedule set by the expected hold period. RVPI that persists well past the fund's stated term is a portfolio of assets that could not be sold at the marked price, whatever the mark says.

Source: ILPA Reporting Template defines DPI, RVPI and TVPI for LP reporting.

The J-curve

The characteristic path of a closed-end fund's reported return: negative early, because fees and expenses are charged against a portfolio still held at or near cost, then rising as marks and realisations arrive. The depth of the curve is set by the fee load and the speed of any early write-down; its length is set by deployment pace.

Field	Value
Formula	TVPI at year t = (cumulative distributions + NAV_t) / cumulative paid-in. The trough occurs where the marginal fee call exceeds the marginal increase in value
Worked, the trough	TVPI 0.8801x at year 1 and 0.9356x at year 2, crossing 1.0000x in year 3 at 1.0432x
Why year 1 is 0.8801x	119.30 of paid-in against 105.0 of NAV. The 14.30 of fee and expenses called in year 1 bought no asset and there is no gain yet to offset it
Worked, IRR path	-12.4057 percent at year 2, +4.1519 percent at year 3, peaking at +14.3984 percent in year 6, then declining to +11.9825 percent at termination
Year 1	Not defined. The only cash flows are a 119.30 call and a 105.0 NAV struck on the same date, which has no internal rate of return

• The since-inception IRR peaks in year 6 and then falls for four straight years while the fund distributes 708.0 of cash. That is not deterioration; it is the arithmetic of a rate declining as the same profit is spread over a longer period. Reading a falling IRR in a harvesting fund as bad news is a common and expensive error.
• A fund that reports no J-curve is worth a question. The two usual explanations are a subscription facility deferring the calls that would have created it, or early marks written up above cost, and the two have very different implications.
• The trough depth is roughly the year-one fee and expense call divided by commitments called - 14.30 on 119.30 here, or 11.99 percent. A fund that calls fees on committed capital while deploying slowly has a deeper and longer trough for reasons entirely unrelated to its investments.

Unrealised marks and the reported multiple

Until a fund is fully realised, part of every reported multiple is an estimate produced by the party being measured. The share that is an estimate is exactly RVPI/TVPI, and it is computable from the report itself.

Field	Value
Formula	Share of TVPI that is unrealised = RVPI/TVPI. Sensitivity of TVPI to a uniform mark error of e = e * RVPI
Worked, year 6	RVPI/TVPI = 1.2615/1.5487 = 81.46 percent of the reported value is a mark
Sensitivity	A 10 percent mark error moves TVPI by 0.10 * 1.2615 = 0.1262x. A 30 percent error moves it by 0.3784x
Worked, what actually happened	Investment B was marked at 45.0 in year 4 against a cost of 60.0 and realised 30.0 in year 6. Investment C was marked at 280.0 in year 6 and realised 310.0 in year 7
Realisation test	Across the six investments, the last mark before exit totalled 842.0 against 935.0 realised - the portfolio was marked 9.95 percent below where it sold

• The most useful single statistic a manager can publish is the historical ratio of realisation proceeds to the last mark before exit, by investment. It converts the credibility of the marks from a matter of opinion into a distribution.
• Mark error is not symmetric in its consequences. An overstated mark inflates a fee charged on NAV, inflates an accrued carry that has not been earned, and supports a fundraise; an understated one does none of those things. Ask which direction the incentive points before assuming the errors cancel.
• A mark held flat at cost for several years is a decision, not an absence of one. Investment B sat at 60.0 for two years before being written down, and the write-down when it came was 50 percent of cost.

Kaplan-Schoar PME

A wealth ratio rather than a rate. Every fund cash flow is restated into a common date using the index, and the ratio of restated inflows plus terminal NAV to restated outflows is taken. Above 1.00 the fund produced more terminal wealth than the same cash invested in the index on the same dates.

Field	Value
Formula	KS-PME = (sum of D_t * I_T/I_t + NAV_T) / (sum of C_t * I_T/I_t)
Worked	(1100.9628 + 0.0)/914.3383 = 1.2041
Reading	The fund produced 20.41 percent more terminal wealth than the index would have on the same cash flow schedule
In excess currency	1100.9628 - 914.3383 = 186.6246 of extra terminal wealth on 500.0 of capital called
Discounting instead of compounding	The same ratio results if every flow is discounted to time zero by the index: 5.131779/4.147128 = 1.2374 against a constant 7.00 percent index, and identically 1.2041 against the series used here

• KS-PME is the most robust of the four methods because it is a ratio and therefore always defined. It cannot produce the negative intermediate value that breaks Long-Nickels, and it does not depend on solving for a rate.
• Its weakness is the same as TVPI's: it says nothing about time. A KS-PME of 1.2041 over ten years and over four years are very different results reported identically, which is why it is normally read alongside Direct Alpha.
• The choice of index dominates the answer and is not a technical detail. A PME against a broad equity index and a PME against a levered small-cap index on the same fund can land on opposite sides of 1.00, and neither is wrong - they answer different questions about the LP's alternative.

Source: Kaplan and Schoar, Private Equity Performance: Returns, Persistence, and Capital Flows, Journal of Finance, 2005.

Long-Nickels PME

Constructs a hypothetical index portfolio into which the fund's contributions are invested and out of which its distributions are withdrawn, then reports the IRR of the fund's own cash flows using the index portfolio's terminal value in place of the fund's NAV. The result is the rate the index would have delivered on this cash flow schedule.

Field	Value
Formula	PME NAV_T = sum of C_t * I_T/I_t - sum of D_t * I_T/I_t. LN-PME IRR solves 0 = sum of (-C_t + D_t)/(1 + r)^t + PME NAV_T/(1 + r)^T
Worked	PME NAV = 914.3383 - 1100.9628 = -186.6246. LN-PME IRR = 7.2267 percent
Spread	Fund net IRR 11.9825 percent less 7.2267 percent = 475.6 basis points of outperformance
Not the index CAGR	The index compounded at 8.3504 percent over the ten years. LN-PME reports 7.2267 percent because it weights the index's returns by when the fund's cash was actually invested
The known failure mode	The PME NAV is negative, meaning the index portfolio was short by 186.6246 at the end. The IRR still solves here, but the construct has no economic interpretation once the position goes short

• The negative PME NAV is the standard criticism of the method and it is not hypothetical - it happens whenever a fund distributes more, earlier, than the index would have supported, which is to say whenever the fund substantially outperforms. The method degrades exactly where the answer is most interesting.
• There is a special case worth knowing: if the index return is constant, LN-PME IRR equals that constant exactly. Against a flat 7.00 percent index the reference fund's LN-PME is 7.0000 percent to four decimal places. Any variation in the reported LN-PME comes entirely from the interaction between the index's path and the fund's cash flow timing.
• The 112 basis point gap between the index's 8.3504 percent CAGR and the 7.2267 percent LN-PME is real information: the fund called capital heavily in years 1 to 5, which included the index's -8.00 percent year, and this schedule earned less from the index than a lump sum would have.

Source: Long and Nickels, A Private Investment Benchmark, 1996.

PME+

A modification of Long-Nickels that scales all distributions by a single factor chosen so the hypothetical index portfolio ends with exactly the fund's own terminal NAV. This removes the possibility of a negative index position at the cost of misstating the individual distributions.

Field	Value
Formula	lambda = (sum of C_t * I_T/I_t - NAV_T) / (sum of D_t * I_T/I_t). PME+ IRR solves 0 = sum of (-C_t + lambda*D_t)/(1 + r)^t + NAV_T/(1 + r)^T
Worked	lambda = (914.3383 - 0.0)/1100.9628 = 0.830490
Check	914.3383 - 0.830490 * 1100.9628 = 0.0000, exactly the fund's terminal NAV
Result	PME+ IRR = 7.6266 percent; spread against the fund's 11.9825 percent net IRR = 435.6 basis points
Against Long-Nickels	LN-PME 7.2267 percent, PME+ 7.6266 percent - a 40.0 basis point difference in the same benchmark from the choice of adjustment method alone

• The 40 basis point gap between LN-PME and PME+ on identical inputs is the reason a PME figure without its method named is not a number. Two managers using different PME conventions on the same fund and the same index will report different outperformance.
• PME+ scales every distribution by the same factor, which is defensible as a construct and false as a description: it says the fund distributed 83.05 percent of what it actually distributed. The scaling is a device to force the terminal condition, not a claim about the cash flows.
• PME+ is at its most useful for a fund with substantial remaining NAV, which is exactly the case where the LN construct is most likely to go short. On a fully realised fund the two methods converge in purpose and PME+ retains only its terminal-condition advantage.

Direct Alpha

Restates every fund cash flow into terminal-date money using the index, then solves for the IRR of the restated series. Because the index return has been stripped out of every flow, the resulting rate is the annualised excess return directly, not a spread between two separately computed rates.

Field	Value
Formula	Direct Alpha = IRR of the series {-C_t * I_T/I_t, +D_t * I_T/I_t, +NAV_T}
Worked	Direct Alpha = 4.0641 percent per annum on the reference fund
Against a constant index	With a flat 7.00 percent index, Direct Alpha is 4.6565 percent and (1.07)*(1.046565) - 1 = 11.9825 percent, exactly the fund's net IRR. The decomposition is exact
Against a varying index	(1 + 0.072267)*(1 + 0.040641) - 1 = 11.5845 percent against the fund's actual 11.9825 percent. The exact decomposition does not survive a varying index
Against the spread methods	Direct Alpha 4.0641 percent; the LN-PME spread 475.6 basis points; the PME+ spread 435.6 basis points. Three answers to the same question, 69 basis points apart

• Direct Alpha is the only one of the four that produces an excess return without subtracting one IRR from another, which matters because IRRs are not additive. A spread between two IRRs is not a rate of anything; Direct Alpha is.
• The exact multiplicative decomposition holds only when the index return is constant. Against a real index series it is an approximation, and on the reference fund it is 39.8 basis points off. Anyone presenting Direct Alpha as an exact attribution should be asked which index path makes that true.
• Direct Alpha and KS-PME are complements, not alternatives: KS-PME gives the size of the outperformance as a wealth ratio and Direct Alpha gives its rate. Reported together they are enough. Reported alone, either can be made to look better by choosing the other's blind spot.

Source: Gredil, Griffiths and Stucke, Benchmarking Private Equity: The Direct Alpha Method, 2014.

Horizon IRR against since-inception IRR

A since-inception IRR uses every cash flow from the fund's first call. A horizon IRR uses only a recent window, treating the NAV at the start of the window as a purchase and the NAV at the end as a sale. They answer different questions and can differ by more than the entire return.

Field	Value
Formula	Horizon IRR over the window (T-n, T] solves 0 = -NAV_(T-n) + sum over t in the window of (-C_t + D_t)/(1 + r)^(t-(T-n)) + NAV_T/(1 + r)^n
Worked, one year	Opening NAV 72.0 at end of year 9, then a 1.10 call, an 80.0 distribution and a zero closing NAV: -12.3333 percent
Worked, three years	Opening NAV 395.0 at end of year 7: -0.6386 percent
Worked, seven years	Opening NAV 355.0 at end of year 3: 13.6730 percent, which is higher than the since-inception 11.9825 percent
The point	The same fund reports -12.3333 percent, -0.6386 percent, 10.3387 percent, 13.6730 percent and 11.9825 percent over five different windows, all ending on the same day

• A horizon IRR on a fund in wind-down is close to meaningless, because the opening NAV dominates the window and the return consists of that NAV being paid out. The reference fund's -12.3333 percent one-year horizon IRR describes an entirely successful final year in which 80.0 was distributed against a 72.0 opening mark.
• The seven-year horizon IRR exceeds the since-inception figure because it excludes the J-curve. That is not a distortion; it is the definition of the window. It does mean that a manager choosing a horizon is choosing an answer.
• Horizon IRRs are the standard basis for benchmark comparisons across managers, and they are the metric most sensitive to the vintage mix of the portfolio being compared. A programme dominated by young funds and one dominated by old funds cannot be compared on a three-year horizon IRR at all.

IRR cannot be averaged, added, or carried across periods

IRR is the root of a polynomial in the cash flows. Roots do not add. An average of two funds' IRRs is not the IRR of the two funds together, and a return computed over two consecutive windows cannot be chained to give the return over the combined window.

Field	Value
Formula	The only correct aggregation is to pool the cash flows and re-solve: IRR_pooled solves 0 = sum over all funds and all t of CF_(j,t)/(1 + r)^t
Worked, chaining fails	The reference fund's first seven years produce a 14.2780 percent since-inception IRR and its last three a -0.6386 percent horizon IRR. Chaining gives (1.142780^0.7)*(0.993614^0.3) - 1 = 9.5820 percent against the actual 11.9825 percent
Worked, size weighting	A 100.0 fund at 20 percent and a 900.0 fund at 5 percent do not average to 12.5 percent, and they do not average to 6.5 percent by size either, because IRR weights by time as well as by amount
What can be averaged	Multiples can, if the denominators match. A pooled TVPI is the sum of all distributions and NAV over the sum of all paid-in, and that arithmetic is exact

• This is the practical reason a fund-of-funds or an LP programme reports pooled cash flow IRRs rather than average fund IRRs, and why the two figures for the same portfolio can differ by hundreds of basis points.
• The same non-additivity is why a since-inception IRR is not recoverable from a series of published annual returns, and why an LP that only receives horizon IRRs cannot reconstruct its own experience without the underlying cash flows.
• The workaround, where cash flows are unavailable, is to switch metric rather than to fix the arithmetic. Pooled TVPI and pooled DPI aggregate correctly and require only the totals.