NYSTRS and the Denominator Effect: What High-Frequency Fair Value Shows About the Pension Plan’s 2023 Sale of Private-Equity Funds

Abstract

The New York State Teachers’ Retirement System (NYSTRS) is a public pension plan that invests part of its assets in private-equity funds: funds that are run by outside managers and that own companies not traded on a stock exchange. This study covers 101 of the plan’s private-equity funds, not all of them, from January 2008 to January 2024; in any one week, between 45 and 75 of the 101 are active. We value each fund as a holding of the size NYSTRS committed to it, where a commitment is the total amount the plan promised to pay into the fund over the fund’s life. We state every value in US dollars and then multiply every dollar value by 3.9, which brings the combined value of our funds close to the total private-equity value that NYSTRS itself reported in 2021–2023.

The multiplication changes dollar amounts only. It applies equally to the value a fund’s manager reported and to our estimate of the fund’s fair value, meaning the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller, so the percentage gap between those two values is unchanged. The multiplication cannot make the 101 funds representative of the NYSTRS private-equity funds we left out.

What we did

We price 101 of the private-equity funds in which NYSTRS has invested, including 13 of the 34 funds NYSTRS sold, in whole or in part, on 29 December 2023 (Appendix B). The other 21 funds sold are not in our set, so nothing in this article measures them. We still had data at the close for 11 of the 13 sold funds; for the other 2, our data on the managers’ reported values and cash flows end in 2020 and 2021.

Sizing each fund takes three steps. The data we start from record each fund’s cash paid in, cash paid back and manager-reported values for a generic investor who promised $10 million to the fund, not for NYSTRS. First, we rescale those payments and reported values in proportion to the amount NYSTRS actually committed to the fund. Commitment figures come from Preqin’s record of NYSTRS’s investments. For the 13 funds in the sale, we used the original commitments listed in NYSTRS’s own Retirement Board minutes (NYSTRS 2024a). Where Preqin’s figure for another fund conflicted with NYSTRS’s own published records or with press reports of the commitment, we corrected it.

Second, we state every fund in US dollars. Of the 101 funds, 79 report in US dollars, 14 in euros, 7 in British pounds and 1 in Australian dollars. For each of the 22 funds that do not report in dollars, we convert every weekly estimate and every manager-reported value at the exchange rate of its own date, using the daily rates published by the Federal Reserve for the euro, the British pound and the Australian dollar (the last rate published on or before each date). Third, we multiply the dollar value of every one of the 101 funds by the same number, 3.9.

NYSTRS committed very different amounts to the 101 funds. Converted into dollars at the exchange rates of 29 December 2023, and before the multiplication by 3.9, its commitments add up to $10.13 billion. They range from $8.2 million to $300 million, and the median commitment, the one in the middle when all 101 are ranked by size, is $100 million.

What we found

A factor of about 3.9 is what it takes to bring our funds up to the size of the private-equity holdings NYSTRS reported. Divide the total private-equity value NYSTRS reported for a quarter by the combined value that the managers of our funds reported for the same quarter, in dollars, before any multiplication; the result is the factor that quarter would require. Table A-1 makes this calculation for the six quarters from the end of 2021 to March 2023, with the NYSTRS values taken from the plan’s own Retirement Board minutes: the row “Private Equity” of the “Market Value Summary” table in the Investment Committee Executive Summary, which states the value in thousands of dollars (NYSTRS 2023a, 2023b, 2023c). No single factor fits every quarter.

Source: The private-equity values of NYSTRS are the figures in the row “Private Equity” of the “Market Value Summary” table in the Investment Committee Executive Summary of the Retirement Board minutes named in the last column, where they are printed in thousands of dollars: $14,203,217 thousand for 31 December 2021, $14,561,238 thousand for 31 March 2022, $14,073,681 thousand for 30 June 2022, $13,859,720 thousand for 30 September 2022, $13,715,149 thousand for 31 December 2022 and $14,425,621 thousand for 31 March 2023. They are rounded here to the nearest $10 million. The second column of figures is the sum of the manager-reported values of the funds with a value in that quarter, each counted at the amount NYSTRS committed to it and converted into US dollars, before the multiplication by 3.9; a quarter with no published value is filled in as described above, and a fund its manager has written down to zero is left out. B denotes billions of US dollars.

The factor of 3.9 is the median (the middle value) of the six factors in Table A-1, which run from 3.74 to 4.08, rounded to one decimal; the median itself is 3.93. It is a middle value, not a precise fit for any one quarter. Every dollar amount quoted in this article for our funds combined is therefore approximate in the same way. The percentage gaps between our fair-value estimates and the managers’ reported values are not approximate in that way, because both values are multiplied by the same factor.

The values we fill in are a small part of the reported total on the two dates that matter most for this article, and they do not produce the gaps we report. On 30 September 2022, our assumed date for when the plan began looking to sell, the reported total covers 63 funds: 59 values that managers published and 4 that we filled in, which make up 4.5% of the total. On 31 December 2023, two days after the sale closed, the reported total covers 57 funds: 51 published values and 6 model estimates, which make up 7.5% of the total. Five of the 6 stand in for quarters with no published value; the sixth replaces one published value that we treat as a data error (Appendix B).

On 30 September 2022 the reported total covers one fewer fund than are active (63 of 64), because for one active fund our data hold neither a published value nor a model estimate of one. Our fair-value estimate for that fund is about $0.3 million, and leaving it out of our estimate as well leaves the gap at −4.3%. At the close the two totals cover the same 57 funds.

On 30 June 2023, the quarter end after the offering was first reported, the filled values are a larger part of the reported total: 11 of its 61 values, or 24.0% of it.

As a check, we repeated the comparison on published values only. On 30 September 2022 our fair-value estimate for the 59 funds with a published value was 4.3% below those published values, the same gap as for all the funds active on that date. At the close our fair-value estimate for the 51 funds whose published value we use was 7.5% above those published values, against 7.8% for all the funds active on that date. A 52nd fund also has a published value for that date; it is the value we treat as a data error, and Appendix B shows what the gap for all the active funds becomes if that value is left as published. On 30 June 2023 our estimate for the 50 funds with a published value was 9.2% above those published values, against 9.6% for all 61 funds active on that date.

Table A-2 gives, for each period and date the article discusses, how many of the 101 funds are active.

Source: Count of the funds, among the 101 NYSTRS private-equity funds we priced, that have a fair-value estimate in a given week (the active funds); a fund its manager has written down to zero is not counted from that quarter end on.

Exhibit A-1

Fund by fund through the 2008 financial crisis and the recovery, January 2008 to July 2011: another 32 funds

Each card shows one fund. The line is our weekly fair-value estimate (NAVcast™), and the markers are the manager’s quarterly reported values. Red shading marks the falling market and green the recovery. In each card, compare where the line turns upward with where the markers do. Exhibit 2 shows 12 funds; this exhibit shows another 32.

Source: Fair-value estimates and the values reported by the funds’ managers for individual NYSTRS private-equity funds. Values are in millions of US dollars. Each fund is counted at the size of NYSTRS’s own commitment to it, a fund that does not report in dollars is converted at the exchange rate of each date, and the value is then multiplied by 3.9. Where a manager published no value for a quarter, the marker is our model’s estimate of the value the manager would have reported. Each card has its own vertical scale, so the heights of the lines cannot be compared from one card to another.

Abstract

On 29 December 2023 the New York State Teachers’ Retirement System (NYSTRS), a public pension plan, sold all or part of its holdings in 34 private-equity funds to other investors. Private-equity funds are funds that are run by outside managers and that own companies not traded on a stock exchange. The plan sold in order to bring private equity back toward 9% of its total assets, the target it had set; on 30 September 2022 private equity had been 11.3% of the plan’s total assets. This appendix sets out the public record of that sale, with a source for each fact, beside our own valuations. The record gives the date of the sale and the list of funds sold, but not the price the buyers paid.

We priced 101 of the plan’s private-equity funds. For the funds active on a given date, we compare the values the funds’ managers reported, which managers publish once a quarter, with our weekly estimates of the funds’ fair value, meaning the price that would be representative of a transaction in a fund between two well-informed market participants when neither is a forced buyer or seller. On 30 September 2022, our assumed date for when the plan began looking to sell, our fair-value estimate for the 64 funds then active was 4.3% below the values their managers reported. On 29 December 2023, when the sale closed, our estimate for the 57 funds then active was 7.8% above the values their managers reported, or 0.1% above them if one published value that we treat as a data error is left as published.

Of those 57 funds, 11 were in the sale. For the funds that were sold, we price 13 of the 34 funds sold, and we still had data on both dates for 11 of those 13 (for the other 2, our data on the managers’ reported values and cash flows end in 2020 and 2021, well before the sale). For those 11 funds, our estimate was 4.6% below the managers’ reported values on 30 September 2022 and 4.7% above them at the close: the same pattern as all the active funds, with a smaller gap at the close.

What we did

We set out the sale fact by fact from public documents: the minutes of the plan’s Retirement Board and reports in the financial press (Table B-1). We then compared our fair-value estimate with the values the funds’ managers reported in three ways: for the funds active on each of four dates, 5 November 2021, 30 September 2022, 30 June 2023 and 29 December 2023 (Table B-2); for the funds that were sold (Table B-3); and at the end of every quarter from 2008 onward (Exhibit B-1). For the close of the sale on 29 December 2023, we set our estimate for the week of that date against the values the managers reported for 31 December 2023.

Table B-1 uses four terms. A fund’s net asset value (NAV) is the value the fund’s manager reports for it. A commitment is the total amount an investor has promised to pay into a fund, and an unfunded commitment is the part of that promise the investor has not yet paid. A discount is the amount by which the price a buyer pays falls short of the NAV the manager reported.

Source: the documents named in the right-hand column. The minutes are published by NYSTRS. The press reports of 23 June, 28 June and 28 September 2023 rest on unnamed sources.

What we found

Our fair-value estimate was below the managers’ reported values on 30 September 2022 and above them when the sale closed. Table B-2 gives both values on the four dates. Its dollar amounts are the values of our funds multiplied by 3.9, as Appendix A explains, not NYSTRS’s own reported figures; the percentages in the last column are the same with or without the multiplication.

Source: Fair-value estimates and manager-reported values for the funds active on each date, among the 101 NYSTRS private-equity funds we priced, with every dollar amount multiplied by 3.9. B denotes billions of US dollars. Where a manager published no value, the total uses our model’s estimate of the value the manager would have reported. Differences are computed from unrounded values. The last row replaces one fund’s published value for 31 December 2023 with our model’s estimate, as described below; with that value as published, the managers’ total is $14.93B and the difference is +0.1%.

On 30 September 2022 the managers’ reported values for the 64 active funds stood $0.58 billion above our fair-value estimate, which puts our estimate 4.3% below them. Public stock markets fell during 2022 and our weekly estimate fell with them, by 24% from its high on 5 November 2021, but the values the funds’ managers reported had not yet taken in that decline, so they made private equity look like a larger share of the plan’s total assets than fair value indicated. If the plan’s other private-equity funds showed the same gap, part of the excess over the 9% target that NYSTRS saw in that quarter was “phantom”: it came from reported values that had not caught up with the market, not from the holdings themselves.

By the close of the sale, public markets had recovered and the gap had reversed. From 30 September 2022 to the close, our fair-value estimate rose 16.6%, while the managers’ reported values rose 3.4%. At the close our fair-value estimate for the 57 active funds stood $1.09 billion, or 7.8%, above the values their managers reported. If the plan’s other private-equity funds showed the same gap, private equity was a larger share of the plan’s total assets than the plan’s own reports showed, and the sale was arguably more advisable than the managers’ reported values implied.

Of the 57 funds active at the close, 11 were in the sale and 46 were not. The funds that were sold follow.

The funds that were sold

Of the 34 funds NYSTRS sold, we price 13. We still had data on 30 September 2022 and at the close for 11 of them; for the other 2, our data on the managers’ reported values and cash flows end in 2020 and 2021, so they are in neither comparison. Table B-3 compares those 11 funds with all the active funds on both dates. The sold funds are shown at NYSTRS’s actual commitments, without the multiplication by 3.9.

Source: Fair-value estimates and manager-reported values. B denotes billions of US dollars. The first two rows are the funds active on each date among the 101 NYSTRS private-equity funds we priced. The last two rows are the 13 funds, of the 34 that NYSTRS sold on 29 December 2023, that we price, stated in US dollars. Two of the 13 are marked “Sold 50%” on the plan’s list; both are shown at the plan’s full commitment. With those two counted at half, our estimate is 4.1% below the managers’ reported values on 30 September 2022 ($0.92 billion against $0.96 billion) and 4.5% above them at the close ($1.40 billion against $1.34 billion). At the close, the managers published a value for 10 of the 11 sold funds, and the eleventh is our model’s estimate of the value its manager would have reported.

The sold funds show the same pattern as all the active funds. On 30 September 2022 our estimate is below the managers’ reported values by about the same percentage for both sets of funds: 4.6% for the sold funds and 4.3% for all active funds. At the close our estimate is above the managers’ reported values by less for the sold funds: 4.7%, against 7.8% for all active funds.

Two cautions apply to the sold-funds result. The first is coverage: we price 13 of the 34 funds sold, and by original commitment, with the funds marked “Sold 50%” counted at half, those 13 account for $1.40 billion of the $3.31 billion on the list, or 42%. The second is sample size: eleven funds are a small sample.

One published value that we treat as a data error

The 7.8% gap at the close depends on how we treat one value. For one fund, the value published for 31 December 2023 is about nine times the value published for the quarter before, although almost no cash was paid into or out of the fund between the two dates. We treat that published value as a data error and replace it with our model’s estimate of the value the manager would have reported, exactly as we fill in a quarter for which a manager published no value.

As published, that single value is 7.8% of the managers’ reported total for all the active funds: the total at the close is then $14.93 billion instead of $13.86 billion, and our fair-value estimate is 0.1% above the managers’ reported values instead of 7.8% above. The fund is not one of the funds that were sold, so the sold-funds comparison in Table B-3 is the same either way. The ordering of the two gaps is not the same either way: with the value as published, the 0.1% gap for all the active funds would be smaller than the 4.7% gap for the sold funds.

Exhibit B-1

How far our fair-value estimate for the funds active in each quarter stood from the managers’ reported values at the end of that quarter, 2008–2023

There is one bar for the last day of each quarter. The height of a bar is our fair-value estimate for the funds active at that date minus the combined value their managers reported, as a percentage of the managers’ reported value. A bar above zero means our estimate is the higher of the two: a plan relying on the managers’ reported values sees private equity as a smaller share of its total assets than it really is. A bar below zero means the managers’ reported values are the higher of the two: a plan relying on those values sees private equity as a larger share of its total assets than it really is. The bar labeled “Began looking to sell” is the quarter ending 30 September 2022, our assumed date for when the plan began looking to sell. The bar labeled “Sale closed” is the last bar, for the quarter ending 31 December 2023, two days after the sale closed on 29 December 2023. Red shading marks periods in which public stock markets were falling, and green shading periods in which they were rising.

Source: Estimates for the funds active in each quarter, among the 101 NYSTRS private-equity funds we priced (between 47 and 75 funds, depending on the quarter). Each bar uses our last weekly fair-value estimate (NAVcast™) on or before the quarterly date on which the managers reported values. The last bar replaces one fund’s published value for 31 December 2023 with our model’s estimate; with that value as published, the last bar would be +0.1%.

The managers’ reported values are below our fair-value estimate in most quarters, but not in all of them: of the 64 quarter ends in the exhibit, our estimate is above the managers’ reported values in 46 and below them in 18. Our estimate was furthest below the managers’ reported values at the end of March 2009, when our estimate for the 55 funds then active was 19.4% below them, and furthest above them at the end of December 2013, when our estimate for the 68 funds then active was 14.9% above them.

The 18 quarters in which our estimate is below the managers’ reported values are the ones that matter for a plan with a target for private equity. A plan that sold private-equity holdings in one of those quarters in order to return to its target would have been correcting, in part, a phantom excess: one that existed in the reported values and not in the holdings.

NYSTRS offered and sold its holdings in quarters of the opposite kind. At the end of each of the four quarters of 2023, our fair-value estimate was above the managers’ reported values: by 5.8% in March (62 funds active), 9.6% in June (61), 5.4% in September (61) and 7.8% in December (57), the last bar in the exhibit. Our conclusion that the NYSTRS sale was arguably more advisable than the managers’ reported values implied rests on those bars.

Limits

• The date on which the plan began looking to sell is our assumption. We place it at 30 September 2022, the date for which the board minutes show private equity at 11.3% of the plan’s total assets. No public document says when the plan decided to sell: the first public report of the offering is dated 23 June 2023, and the date on which the plan engaged its adviser is not public. The board minutes show 10.8% for 30 June 2022.
• No price or discount is on record for this sale, so nothing in this appendix measures what the sale cost or gained the plan.
• The comparison for all active funds is not a comparison for the funds sold: the plan kept 46 of the 57 funds active at the close. The comparison for the sold funds covers 13 of the 34 funds sold, and we still had data at the close for 11 of them (for the other 2, our data on the managers’ reported values and cash flows end in 2020 and 2021, well before the sale).
• We do not know whether NYSTRS adjusts the values its fund managers report before it reports its own private-equity value. We assume that it does not. If the plan does adjust those values, the gaps between fair value and the plan’s own figures would differ from the gaps in Table B-2.

Abstract

A private-equity fund is a fund that a professional manager runs on behalf of investors and that owns companies not traded on a stock exchange, so the fund has no market price and its value has to be estimated. Any method of estimating that value should expect to be cross-examined, as a witness is in a courtroom, and the first question will be what data the method uses and what it assumes. This appendix answers that question for our method with one worked example: one of the 101 private-equity funds of the New York State Teachers’ Retirement System (NYSTRS), a public pension plan, that we priced for this article.

We value the fund three ways from the cash flows and manager-reported values that its investor observes: two simple ways, each of which falls short, and our own weekly estimate of the fund’s fair value, meaning the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller. We call that estimate a NAVcast™. For this fund, which reports in euros, the NAVcast™ reaches a high of 153.9 million in the fund’s own currency, whereas the highest value the manager ever reported is 133.7 million. One fund illustrates how the method works; it is not evidence that the method is accurate.

What we did

The example fund is a euro fund, followed from 2006 to April 2022. All amounts in this appendix are for a holding of 150 million in the fund’s own currency, the euro: the commitment figure a data vendor recorded for NYSTRS in this fund. A commitment is the total amount an investor promises to pay into a fund; the manager “calls” that money in installments as it is needed.

Unlike the figures for the 101 funds combined elsewhere in this article (Appendix A), the amounts in this appendix are not converted into US dollars at the exchange rate of each date and are not multiplied by 3.9.

An investor in a private-equity fund observes only three things: the cash the investor pays into the fund, the cash the fund pays back, and the value the fund’s manager reports once a quarter, which is called the fund’s net asset value (NAV), or its “mark.” We build up from those three observations to our estimate in four steps. Each of the first three steps leaves a problem unsolved, and the next step takes that problem on.

1. Net cash paid in. Add up everything the investor has paid into the fund and subtract everything the fund has paid back. The result is a poor measure of what the fund is worth, because it ignores the return earned on the money while it is invested.
2. The accounting rule. A fund’s value this week equals its value last week, grown by that week’s investment return, plus any cash paid in and minus any cash paid out. The rule itself is not in question, but nobody observes the weekly return of a private-equity fund, and the rule needs one for every week.
3. The “beta-naive” shortcut. This common shortcut sets the fund’s value equal to the manager’s reported value each quarter, and between those quarterly dates moves the value up and down with the public stock market. Beta is the standard name for how much a fund’s value moves when the public stock market moves; the shortcut is called beta-naive because, between the manager’s quarterly reports, it relies on beta and on nothing else. The shortcut tries to answer two separate questions with one number, what value the manager would report today and what the fund is actually worth today, and it answers neither question well.
4. Our estimate, the NAVcast™. We apply the accounting rule week by week, using a weekly return made of the four parts described below, and we do not reset the fund’s value to the manager’s reported value each quarter. The NAVcast™ is built to answer only the second question: what the fund is actually worth today.

The weekly return in our estimate has four parts.

1. Sensitivity to the public market (beta). How much the fund’s value moves when the public stock market moves.
2. Steady out- or under-performance (alpha). The return the fund earns, period after period, above or below what its sensitivity to the public market alone would produce.
3. The link to a comparable traded asset. A comparable asset is a publicly traded investment that resembles what the fund holds. From the comparable asset’s return we use only what is left after we take out the movement that the comparable asset shares with the broad stock market, which the first part of the weekly return already accounts for.
4. The fund’s own unexplained movement. Changes in the fund’s value that none of the first three parts accounts for. The typical size of those changes, which we call the fund’s own volatility, varies over time. We include this part because the value of every fund moves for reasons of its own, as the price of every public stock does.

We estimate these four parts from three kinds of data.

• The values the manager reports. Managers smooth the values they report, which means the reported values change more gradually than the fund’s underlying worth. The reported values also lag behind the public markets: they take time to reflect a rise or a fall that has already happened. Before we draw any information from the reported values, we correct them for the smoothing, for errors that fall at random on either side of the fund’s worth, and for any consistent tendency to report too high or too low.
• The fund’s cash flows. Our model treats each payout as a fraction of the fund’s value, so the size and timing of a fund’s payouts are evidence of what the fund is worth. The payouts also reveal the manager’s liquidity preference, meaning how quickly the manager chooses to return cash to investors (Appendix D).
• The returns of comparable traded assets. We use the week-by-week returns of the comparable traded assets described above.

Exhibit C-1

One private-equity fund valued three ways from the same cash flows and manager-reported values

Top panel: the fund’s value in millions, at NYSTRS’s commitment and without conversion into US dollars; the fund is a euro fund. The markers are the values the fund’s manager reported each quarter. “Net cash paid in” is the cash the investor paid into the fund minus the cash the fund has paid back. “Beta-naive” is the shortcut that sets the fund’s value equal to each value the manager reported and, between those reports, moves the value with the public stock market; the line jumps each time a new reported value arrives. “NAVcast™” is our weekly fair-value estimate; it is grown week by week without being reset to the manager’s reported value, and its highest point is above the highest value the manager reported. Bottom panel: cash the investor paid into the fund (bars pointing up) and cash the fund paid out to the investor (bars pointing down).

Source: Estimates for one NYSTRS private-equity fund, for a holding of 150 million, the commitment figure in our data. The data we start from record this fund’s cash flows and values for an investor who committed 10 million, so every figure in the exhibit is the figure in those data multiplied by 15. Before the manager’s first reported value, the NAVcast™ and beta-naive lines are drawn on the net-cash-paid-in line. The exhibit does not plot the weekly return of the comparable traded asset, or the week-by-week estimates of the fund’s beta, alpha and own volatility.

The three approaches give visibly different pictures of the same fund. Net cash paid in ends the period at minus 90.0 million, because the fund paid back more cash than it called; that figure says nothing about the value of what remains in the fund. The beta-naive line jumps each time a new manager-reported value arrives. Our estimate, which is never reset to the manager’s reported value, reaches its highest point, 153.9 million, on 1 February 2013; the highest value the manager ever reported for the fund is 133.7 million, for a date a month earlier.

Our model also estimates each part of the fund’s weekly return and revises those estimates every week as new data arrive, so over the life of this fund each estimate moves within a range. The estimate of beta ranges from 0.90 to 1.14 and ends at 0.94; a beta of 1.00 would mean the fund’s value moves one for one with the market.

The estimate of alpha ranges from −10.0% to +3.8% a year and ends at +2.3% a year.

The estimate of the fund’s own volatility ranges from 3.5% to 61.4% a year and ends at 26.7% a year. We report these ranges to show what the model estimates.

Exhibit C-2

The same fund in real time: our estimate and the model’s estimates of beta, alpha and σ_F, stopped at the same week

Press play, or drag the slider, to stop the chart at any week. Everything drawn is what the model knew in that week: our weekly fair-value estimate (NAVcast™) up to that week, the values the manager had reported by then, and, as faint lines behind them, the model’s estimates of the fund’s beta, alpha and σ_F as they stood each week. Beta is the fund’s exposure to the market; alpha is the fund’s return above or below what that exposure explains; σ_F is the volatility of the fund’s own return. The faint lines show how each estimate moved, not its level; the line under the chart gives the values for the week shown. The bars at the bottom are the cash paid into and out of the fund up to that week. Red shading marks the 2008 financial crisis, from 12 October 2007 to 6 March 2009, and green shading the recovery that followed, up to 29 April 2011, the week our estimate reached its highest point in 2010 and 2011.

▶ Play

Source: Estimates for the same fund as Exhibit C-1, in millions in the fund’s own currency, without conversion into US dollars. Each week’s estimates use only the data available through that week. The model’s estimates of beta and alpha begin on 4 August 2006, and its estimate of σ_F on 14 December 2007.

Exhibit C-3

The same fund from its three sources of evidence: the manager’s smoothed values, a comparable traded asset and the fund’s payouts

Press play, or drag the slider, to stop the chart at any week. Top panel: the gold line joins the values the manager reported each quarter with straight lines from one report to the next, and the magenta line is our weekly fair-value estimate (NAVcast™). As the chart plays, our estimate unravels the gold line week by week and leaves it behind as a faint trace. The gold line is drawn ahead of the estimate only to show what is being unravelled: in real time the manager’s later values are not yet known, and our estimate for each week uses none of them. Below it: the return of a broad index of public stock prices over the previous 13 weeks, red when the market was falling and green when it was rising, and the weekly return of the comparable traded asset. Bottom panel: the payouts the fund made to the investor and, from 2011, the payout our model predicted at each payout date. Red shading marks the 2008 financial crisis and green shading the recovery, up to 29 April 2011.

▶ Play

Source: Estimates for the same fund as Exhibit C-1, in millions in the fund’s own currency, without conversion into US dollars. Each predicted payout is the model’s weekly predictions added up since the previous payout; the prediction for each week uses only data up to the week before.

Managers smooth the values they report and report them late, so the gold line changes gradually and trails the market. Our estimate does not copy those values. It takes from them the information they carry once the smoothing, delay, noise and bias are unravelled, which is why it moves week by week while the gold line drifts from one report to the next.

The comparable traded asset is a publicly traded investment that resembles what the fund holds. Its returns, in the third panel, are observed every week, and our estimate uses the part of their movement that the broad stock market does not explain.

The fund’s payouts are the third source of evidence; Exhibit C-4 looks at them more closely.

Exhibit C-4

The payouts our model predicted for the same fund against the payouts the fund actually made, 40 dates from 2011 to 2022

Amounts are in millions, without conversion into US dollars, as in Exhibit C-1. Each date has two bars side by side: the payout the fund actually made to the investor that week, and the payout our model predicted.

Source: Estimates for the same fund as Exhibit C-1. The prediction for each week uses only data up to the week before. The model produces a prediction every week, whereas the fund pays out only occasionally, so the weekly predictions are added up between one actual payout and the next.

Each week the model predicts what the fund would pay out if it made a payout that week, and Exhibit C-4 sets those predictions beside the payouts the fund actually made. The predictions matter because the model treats each payout as a share of the fund’s value: every payout the fund makes tells the model something about what the fund is worth, and the model uses that evidence alongside the manager’s reported values and the returns of comparable traded assets.

A daily extension

The same model can produce a value for every weekday, not only for every week, and each day’s value uses only the information available on that day. Exhibit C-5 shows the daily NAVcast™ for the example fund.

Exhibit C-5

The example fund, day by day: the daily NAVcast™

Left: the whole period, from 2006 to April 2022. Right: a closer look at the first half of 2020, when stock markets fell and recovered within weeks. The magenta line is the daily NAVcast™, the open circles in the right panel are the weekly NAVcast™ and the gold dots are the values the manager published.

2006 to April 2022

27 January to 30 June 2020

Source: Estimates for the same fund as Exhibit C-1, in millions of euros for a commitment of 150 million. The daily series are those in the model’s own report for this fund, each day’s value made with only the data available on that day.

Exhibit C-6 summarises the method: four kinds of data go in, the weekly NAVcast™ comes out, and the model separates the fund’s signal from the noise in the inputs.

Exhibit C-6

From inputs to NAVcast™: what goes in, what comes out, and what is filtered out

The same structure applies to every fund we price.

Abstract

A private-equity fund, meaning a fund that a professional manager runs on behalf of investors and that owns companies not traded on a stock exchange, pays cash back to its investors only occasionally and in irregular amounts. Even so, those payouts carry information about what the fund is worth, and our valuation model uses them. This appendix shows how, with one example fund that makes 10 payouts.

There are two findings. First, the model estimates every week how large a payout would be if the fund made one that week, and it puts a range around that estimate; of the fund’s 10 actual payouts, 7 fall inside the model’s range in the week they are paid. Second, the pace at which a manager chooses to pay out, which we call the manager’s “liquidity preference,” decides how much of the investors’ money stays in the fund and how much comes back as cash.

The example fund is an illustration: it is not one of the 101 private-equity funds of the New York State Teachers’ Retirement System (NYSTRS) that we price in the rest of this article, and nothing in this appendix is a result about NYSTRS. Its values are in millions, scaled so that the fund is worth about 1 million in its first week; the method works the same at any scale.

What we did

Our model treats each payout as a fraction of the fund’s value at the moment the payout is made. The manager’s liquidity preference is the size of that fraction: a manager with a stronger preference for returning cash pays out a larger fraction of the fund’s value.

We ask two questions of the example fund. First, how fast does this manager return the fund’s value to its investors as cash? Second, given that actual payouts are irregular, can the model still estimate, each week, what a payout would be if the fund made one that week? An estimate of this kind, which describes the present before the data for it have arrived, is called a nowcast.

The example fund runs for 328 weeks and makes 10 payouts. To answer the first question we run a what-if comparison. We take the fund manager’s liquidity preference as the model estimated it from this fund’s data, and recompute the fund’s value twice more: once as if the manager paid out 0.40 times that liquidity preference, and once as if the manager paid out 2.50 times that liquidity preference.

What we found

Exhibit D-1

Ten irregular payouts by one fund against the model’s weekly estimate of what a payout would be

Each bar is a payout the fund actually made. The line is the model’s estimate, for each week, of how large a payout would be if the fund made one in that week. The shaded band around the line is the range in which the model expects an actual payout to fall. The model learns the width of that band from the fund’s data, and we call that width the payout noise. A bar that ends inside the band is a payout that fell within the model’s range.

Source: Estimates for one example fund. The horizontal axis counts weeks since the fund began. The payout noise is a percentage of the estimated payout: the band runs from the estimated payout reduced by that percentage to the estimated payout increased by the same percentage. Values are in millions, at the scale described above.

The fund pays out only 10 times, at irregular intervals and in irregular amounts, yet the model’s range follows those payouts closely.

Exhibit D-2

The value of the same fund under three payout speeds

Each line is the fund’s value, week by week, under one payout speed. The solid line uses the fund manager’s liquidity preference as the model estimated it. The other two lines use 0.40 times and 2.50 times that fraction. The more slowly the manager pays out, the more value stays in the fund; the faster the manager pays out, the sooner the fund’s value is returned to investors as cash. The table gives the fund’s value at the end of the period and the total cash paid out under each speed.

Payout speed, as a multiple of the fund manager’s liquidity preference	Fund value at the end of the period (millions)	Total cash paid out over the period (millions)
0.40×	12.55	4.34
1.00× (the manager’s liquidity preference)	6.81	8.33
2.50×	1.22	11.50

Source: Estimates for the same example fund as Exhibit D-1, recomputed at three payout speeds. Values in the chart and the table are in millions, at the same scale.

The pace of payouts decides how much of the investors’ money is still in the fund and how much has come back as cash. A manager who pays out at 0.40 times the manager’s liquidity preference returns 4.34 million in cash to investors and leaves 12.55 million of value in the fund at the end of the period; one who pays out at 2.50 times the manager’s liquidity preference returns 11.50 million in cash and leaves only 1.22 million in the fund. Because the outcome depends so heavily on the manager’s liquidity preference, our model estimates that preference from the fund’s data instead of assuming it. The table is best read as a comparison across the three payout speeds.

Abstract

A private-equity fund owns companies that are not traded on a stock exchange, so the fund has no market price and nobody knows its true value. An estimate of a real fund’s value therefore cannot be checked against the truth. A simulation removes that obstacle: we generate artificial funds by computer, so we know each artificial fund’s true value in every week and can measure how far any estimate falls from that true value.

We score two estimates. The first is our own weekly estimate of a fund’s fair value, which we call a NAVcast™; fair value means the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller. The second is a simple alternative that we call beta-naive. Beta-naive sets a fund’s value equal to the value the fund’s manager reports each quarter (the manager’s “mark”) and, between those reports, moves the value with the public stock market.

Our estimate comes about twice as close to the true value as beta-naive does. On average across six simulated settings, and for a portfolio of the artificial funds scaled so that the commitments to its funds (the total amounts investors promised to pay into them) add up to $1 billion, our estimate misses the portfolio’s true value by about $4 million at each comparison, made every week, and beta-naive misses it by about $8 million. This is a preview of our simulation testing, not the full study.

A simulation cannot by itself settle how the method performs on real funds, whose true value is never observed. The second half of this appendix closes that gap with a net-present-value test: a fund’s value today should match the cash it later pays out, discounted back to today at a rate that reflects the fund’s risk. On the simulated funds, the version of the test that needs no true value moves closely with the version that does. On 319 NYSTRS funds in four asset classes, our model’s error is smaller than beta-naive’s, for all the funds together and for each asset class. A second check on the same real funds scores a separate series of our model, its estimate of the values managers will report, against the values they then published; added up over 96 quarters, it misses them by less than half as much as beta-naive does.

What we did

Each simulated fund is a synthetic clone of a real fund. We start from 314 real funds of four kinds: 223 private-equity funds, 14 venture-capital funds, 57 real-estate funds and 20 private-debt funds. Each clone keeps its real fund’s calendar and the money its investors paid in, but its value from week to week is generated by us from settings we fix in advance, so the clone’s true value is known in every week. We generate every clone in six different settings, which we call “worlds.” Each world sets the following characteristics.

• Excess return (alpha). The return a fund earns each year above what its sensitivity to the public stock market alone would produce.
• Beta. How much a fund’s value moves when the public stock market moves. A beta of 1.00 means the fund moves one for one with the market, a beta of 0.70 means it moves less than the market, and a beta of 1.40 means it moves more.
• Volatility. The typical size of the swings in a fund’s value over a year: a simulated fund with a volatility of 20% a year swings about twice as widely as one with a volatility of 10% a year.
• Comparable link. How strongly a simulated fund’s value moves with its comparable traded asset, a publicly traded investment that resembles what the fund holds; a higher number means the fund’s value moves more strongly with that asset. The comparable link is specified only in the fifth world, at 0.1, and in the sixth, at 0.7.

The settings differ by asset class. Table E-1 lists them; worlds 5 and 6 differ only in how strongly each clone moves with its comparable traded asset.

Exhibit E-1 shows how a clone is made, for one real private-equity fund. We take the fund’s characteristics, meaning its calendar, the dates on which its manager reported and the money its investors paid in, and change its performance and its noise. Because the clone is simulated, we can see its true value in every week. The exhibit shows two clones of the same fund that differ only in their beta, 0.70 in one and 1.40 in the other: the same fund, the same money paid in, the same calendar, and two different truths.

Exhibit E-1

One real fund and two of its synthetic clones

Top left: the real fund, with the values its manager published and the money paid in and out each quarter. Top right and bottom left: two clones of that fund, generated with an excess return of 8% a year and a volatility of 20% a year, one with beta 0.70 and one with beta 1.40. Each clone keeps the real fund’s calendar, its reporting dates and the money its investors paid in; its published values, its payouts and its true value are generated. Bottom right: the two clones’ true values, with the real fund’s own published values and cash flows on top.

The real fund

Clone: beta 0.70

Clone: beta 1.40

Both clones’ true values (dotted: beta 0.70; solid: beta 1.40), with the real fund on top

Source: One real private-equity fund and two of its simulated clones; millions of US dollars for an investor who committed $10 million.

In each world we add all 314 clones, of all four kinds, together into one simulated portfolio and compare two estimates of that portfolio’s value: our NAVcast™, and the single alternative we test, beta-naive, applied to the values the simulated managers report each quarter.

We score both estimates every week, from January 1996 to March 2024, by comparing each with the true value of the whole simulated portfolio; we call each of these comparisons a reading. At each reading, the miss is the size of the difference between the estimate and the true value, whether the estimate is too high or too low. The score is the running total of that miss: each new reading’s miss is added to the misses of all the readings before it. We express the running total in dollars for a hypothetical portfolio scaled so that the commitments to its funds, meaning the total amounts investors promised to pay into them, add up to $1 billion. The value of that portfolio is not fixed at $1 billion; it changes from week to week.

What we found

Exhibit E-2

A simulated portfolio of funds whose true value is known: the full history, and closer looks at three periods

The value of the whole simulated portfolio, with all the simulated funds added together, in billions of US dollars. The dashed line is the true value, which is known because we generated it. The markers are the simulated versions of the values that fund managers report each quarter. “NAVcast™” is our weekly fair-value estimate. “Beta-naive” is the alternative that sets the portfolio’s value equal to each of those simulated reported values and, between them, moves the value with the public stock market. Compare how closely the NAVcast™ line and the beta-naive line each track the dashed true value, in the full history and in the three closer looks: the market fall of 2007 to 2009, the period from late 2017 to the end of 2019, and 2021 to 2022.

1995 to March 2024

June 2007 to December 2009

October 2017 to December 2019

January 2021 to December 2022

Source: Simulation of one portfolio of simulated funds, generated in the second of the six worlds listed above, ending 29 March 2024. The full history is plotted every fourth week; the three closer looks are plotted every week. The dollar scale on this exhibit is the simulated portfolio’s own scale. It is not the scale of Exhibits E-3 and E-4, which restate each simulated portfolio so that the commitments to its funds add up to $1 billion, so read this exhibit for the shape of the lines and the percentage gaps between them, not for the dollar amounts.

Our estimate tracks the true value closely but not perfectly, which is about the best any estimate of a value that cannot be observed can do.

Exhibit E-3

The running total of each estimate’s miss against the true value, averaged over the six simulated worlds

The vertical axis shows the running total of the miss, in billions of US dollars, for a hypothetical portfolio whose funds’ commitments add up to $1 billion. At each weekly reading, the miss is the size of the difference between an estimate and the true value of the simulated portfolio. Each line adds up the miss reading after reading, which is why the running total can exceed $1 billion. We plot each running total below zero, so both lines run downward as the misses add up: a running total of −$6.18 billion is $6.18 billion of accumulated misses. The line that stays closer to zero belongs to the more accurate estimate.

Source: Simulations. Each line is the average of the six simulated worlds, with one reading every week, ending 29 March 2024. The beta-naive line leaves out the weeks in which beta-naive’s values were most extreme.

Beta-naive misses the true value by about twice as much as our estimate does. For the hypothetical portfolio with $1 billion of commitments, our estimate misses the true value by about $4 million on average at each weekly reading. Beta-naive misses it by about $8 million.

Exhibit E-3 shows the same result as running totals. Added up over all 1,474 weekly readings, the misses come to $6.18 billion for the NAVcast™ and $12.37 billion for beta-naive, which the exhibit plots as −$6.18 billion and −$12.37 billion.

The NAVcast™’s advantage is consistent over time. The running total of the miss is smaller for the NAVcast™ than for beta-naive at 1,463 of the 1,474 weekly readings, which is 99% of the time.

Exhibit E-4

The running total of the miss in each of the six simulated worlds

Each card shows one simulated world, with the same two lines as Exhibit E-3: the running total of the miss for our estimate (NAVcast™) and for beta-naive. All six cards share one vertical scale, so they can be compared directly. The final value of each line is printed under its card. In every world the NAVcast™ line ends closer to zero than the beta-naive line.

Source: Simulations of six worlds. Values are in billions of US dollars for a hypothetical portfolio whose funds’ commitments add up to $1 billion.

The result does not depend on one favorable setting. In each of the six worlds, taken separately, the NAVcast™ ends with a smaller running total of the miss than beta-naive does.

The worlds above add all four asset classes together. Within a single asset class, two worlds can also be compared directly, because they are matched on excess return, volatility and comparable link and differ only in beta. Exhibit E-5 shows all the private-equity clones at beta 0.70 and at beta 1.40, and all the real-estate clones at beta 0.50 and at beta 1.20. The blue line grows the same cash flows at the return of a 70/30 stock-bond market. By the end of the period the low-beta truth sits below that line and the high-beta truth above it, and in both the NAVcast™ stays on the true value.

Exhibit E-5

The same asset class at two generating betas: the NAVcast™ follows the true value in both

Each panel adds up all the clones of one asset class in one world. The two worlds in each row are the same except for beta. The dashed line is the true value, the magenta line is the NAVcast™ and the blue line is the same cash flows grown at the return of a 70/30 stock-bond market.

Private equity, beta 0.70

Private equity, beta 1.40

Real estate, beta 0.50

Real estate, beta 1.20

Source: Simulations: all 223 private-equity clones and all 57 real-estate clones, each counted at $10 million committed and added up; billions of US dollars. Weekly, to the first quarter of 2024.

A model that values a fund well must also recover the fund’s risk. Exhibit E-6 measures how far each method’s estimate of a fund’s beta, made with the data available at each age of the fund, falls from the beta found after the fact, by a standard regression over the fund’s whole simulated life; no method running in real time could use that benchmark. Averaged over every age, our model’s beta error is 16.0%, against 63.4% for beta-naive, and it is lower in every asset class: 18.4% against 61.5% in private equity, 13.9% against 69.8% in venture capital, 19.2% against 72.8% in real estate and 13.2% against 71.2% in private debt.

Exhibit E-6

Recovering beta: each method’s beta error at each age of the fund

Lower is better. The beta error is the gap between a method’s estimate of the fund’s beta and the after-the-fact fit, as a percentage of that fit, averaged across the simulated funds of the same age. Top: all 1,884 simulated fund-scenarios. Below: each asset class.

Private equity, 1,338 fund-scenarios

Venture capital, 84 fund-scenarios

Real estate, 342 fund-scenarios

Private debt, 120 fund-scenarios

Source: Simulations: the 314 clones in six worlds. One reading every four weeks of fund age.

From simulated funds to real funds: the net-present-value test

A real fund has no observable true value, so the comparison above cannot be repeated on real funds. A different test can. Whatever a fund is worth today should match the cash it will later pay out to its investors, net of the cash it will still call from them, discounted back to today at a rate that reflects the fund’s risk. This is a net-present-value (NPV) test. Because it needs only the fund’s cash flows, which are known once they have happened, it can be run on real funds as well as simulated ones. We run it first on the simulated funds, where we can also check it against the true value, and then on 319 NYSTRS funds.

The test splits each fund’s life at a date, which we call τ (tau), and works in three steps.

1. Estimate, using only the data up to τ. Each method estimates the fund’s beta, and in a second version its alpha as well, from the fund’s cash flows, reported values and market returns between the start of the fund and τ. Nothing that happens after τ is used.
2. Discount the cash flows after τ. We take the cash the fund actually paid out, net of the cash it actually called, in every week after τ to the end of the fund’s life, and discount it back to τ. Each week’s discount is the public stock market’s return in that week multiplied by the estimated beta, plus the estimated alpha in the second version. Whatever value is left in the fund at the end of our data counts as one final payment. The result is the fund’s net present value at τ.
3. Compare. We compare that net present value with a reference value for the fund at τ. On the simulated funds the reference is the true value, which we generated. On real funds, which have no true value, the reference is each method’s own value estimate at τ; this is the only part of the test that changes between simulated and real funds, and we show below that the change does not alter the result: the two versions of the test are highly correlated, and visually very similar. The NPV error is the gap between the net present value and the reference, as a percentage of the reference.

We repeat the three steps at every date τ in each fund’s life. For our model, alpha and beta are the model’s own estimates. For beta-naive, the shortcut described above, they come from a standard regression of its own values on the public stock market.

We summarize the error two ways: its average across funds at each date, and its running average over every fund and every date up to that point. For the simulated funds together, we first average the six worlds within each asset class and then average the four asset classes.

Exhibit E-7

The net-present-value test on simulated funds: how far each method’s discounted cash flows fall from the true value

Lower is better: a lower line means a smaller error. Each panel shows the NPV error as a percentage of the true value, on a scale on which each step up multiplies the error by ten. Left: the average across funds at each date. Right: the running average over every fund and every date up to that point. Top row: each method’s beta only. Bottom row: alpha and beta.

Beta only, at each date

Beta only, running average

Alpha and beta, at each date

Alpha and beta, running average

Source: Simulations: 314 clones of real funds (223 private equity, 14 venture capital, 57 real estate and 20 private debt), each generated in six worlds; within each asset class the six worlds are averaged, then the four classes are averaged. One reading every week, through December 2024.

Simulated funds: the net present value against the generated true value. Because we generated each simulated fund, we know its true value at every date τ, so the cleanest comparison is the fund’s net present value at τ against that generated true value. On that comparison, our model’s estimates discount the funds’ cash flows most accurately. Averaged over every fund and every date through 2024, the NPV error is 9.9% with our model’s beta, against 112.2% for beta-naive. With alpha and beta it is 45.2%, against 171.0%.

Real funds: the net present value against the estimate. A real fund has no generated true value, so the reference has to change. We replace it with each method’s own value estimate at τ: the NAVcast™ for our model, and beta-naive’s own value for beta-naive. The test then asks whether a method’s value at τ agrees with the cash the fund actually paid after τ, discounted with that same method’s own risk estimates.

Does the switch change the answer? The correlation. On the simulated funds we can run both versions of the test on the same funds and compare them (Exhibit E-8): the net present value against the generated true value, and the net present value against the estimate. Date by date, the two error lines move closely together: with alpha and beta, the correlation between them is 0.93 for our model and 0.94 for beta-naive, on a scale on which 1 means that they move in perfect step. The version that needs no true value therefore tells the same story as the version that does, which is what lets us run it on real funds.

Exhibit E-8

The same simulated funds scored two ways: against the true value, and against each method’s own value estimate

Lower is better. Alpha and beta. The average NPV error across funds at each date, as a percentage of the reference, on a scale on which each step up multiplies the error by ten. Left: the reference is the true value. Right: the reference is each method’s own value estimate, the only reference available for a real fund.

Reference: the true value

Reference: each method’s own value estimate

Source: Simulations, as in Exhibit E-7. The correlation is measured date by date, through December 2024, on the logarithms of the two lines.

We then run the test on 319 NYSTRS funds: 225 private-equity funds, 15 venture-capital funds, 59 real-estate funds and 20 private-debt funds (Exhibit E-9). We did not have NYSTRS’s commitment amounts for these funds, so each is counted as if $10 million had been committed; the NPV error is a percentage, so this does not change it. The results for all the funds together pool every fund. With alpha and beta, the NPV error averaged over every fund and every date through 2024 is 64.7% for our model, against 328.3% for beta-naive. With beta alone it is 54.6%, against 188.5%. Our model’s error is the smaller of the two in each of the four asset classes.

Exhibit E-9

The net-present-value test on 319 NYSTRS funds: all asset classes together, and each class separately

Lower is better. Alpha and beta; the reference is each method’s own value estimate. The NPV error is a percentage of that reference, on a scale on which each step up multiplies the error by ten. Top: all 319 funds, at each date (left) and as a running average (right). Bottom: the running average for each asset class.

All 319 funds, at each date

All 319 funds, running average

Private equity, 225 funds, running average

Venture capital, 15 funds, running average

Real estate, 59 funds, running average

Private debt, 20 funds, running average

Source: Estimates for 319 NYSTRS funds; we did not have NYSTRS’s commitment amounts for them. One reading every week, through December 2024.

A second check on real funds: modeling the values managers report

Our model is multivariate. Alongside the NAVcast™, its estimate of what a fund is worth, it produces a second, separate series: its estimate of the value the fund’s manager will report, which we call the GP-NAV estimate. The two are easily confused, so we keep them apart. The NAVcast™ is the fair-value estimate used everywhere else in this article. The GP-NAV estimate reproduces the way a manager smooths the values it reports and reports them late.

The two series come from the same model. To reach the NAVcast™, the model unravels the smoothing, delay, noise and bias in the values managers report; the GP-NAV estimate is the same model with that ravelling put back in. Unlike a fund’s true value, the values managers report are observed, so the GP-NAV estimate can be checked directly on real funds, against each value a manager later publishes. A model that predicts the published values well has captured the ravelling it must undo to reach fair value.

Exhibit E-10

One real fund: the GP-NAV estimate follows the values the manager publishes, while the NAVcast™ follows the market

The dots are the values the fund’s manager published. The gold line is the GP-NAV estimate, the model’s estimate of the value the manager will publish; the magenta line is the NAVcast™, the model’s fair-value estimate. Both lines come from the same model. Left: the fund’s whole life. Right: a closer look at the 2008 financial crisis and the recovery that followed.

The fund’s whole life

July 2007 to March 2011

Source: Estimates for one of the 319 NYSTRS funds, in millions of US dollars for an investor who committed $10 million; we did not have NYSTRS’s commitment amount for this fund. Weekly, through December 2024.

When the stock market fell in 2008, the values this manager published barely moved, and the GP-NAV estimate moved with them. The NAVcast™ fell with the market. The same model produced both lines: one describes what the manager would report, the other what the fund was worth.

Across all 319 NYSTRS funds, we score the GP-NAV estimate at every quarter in which managers published values. For each published value, we take the estimate from the week before that value arrived, so the estimate does not know the value it is scored against. At each quarter we add up the estimates and the published values across the funds and measure the gap between the two totals; Exhibit E-11 adds those gaps up, quarter after quarter, from March 2000 to the first quarter of 2024. Over those 96 quarters the GP-NAV estimate misses the published values by $1.31 billion in all, against $3.34 billion for beta-naive, for funds each counted at $10 million committed, because we did not have NYSTRS’s commitment amounts for them.

Exhibit E-11

The GP-NAV estimate against the values the managers of 319 NYSTRS funds published: the miss, added up quarter by quarter

Closer to zero is better. At each quarter, the miss is the gap between the published values, added up across the funds, and each method’s estimates from the week before those values arrived, added up the same way. Each line adds up the miss quarter after quarter and is plotted below zero, so a line that stays closer to zero belongs to the more accurate method. The GP-NAV estimate is a separate series of our model, not the NAVcast™.

Source: Estimates for 319 NYSTRS funds, each counted at $10 million committed because we did not have NYSTRS’s commitment amounts for them; billions of US dollars. One reading at each of the 96 quarters with published values, March 2000 to the first quarter of 2024. Published values more than ten times, or less than a tenth of, the estimate are left out. Beta-naive is shown without the weeks in which its values were most extreme.

Abstract

A private-equity fund is run by a professional manager on behalf of investors and owns companies not traded on a stock exchange. Because such a fund has no market price, it is normally valued by its own manager, about once a quarter. Sometimes someone other than the manager also values the fund; we call that a third-party valuation. The NAVcast™, our weekly estimate of a fund’s fair value (the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller), can take a third-party valuation into account, with a weight that the user of our model sets: a “degree of belief,” from none (our estimate ignores the valuation) to full (our estimate moves almost all the way to the valuation).

We show how the degree of belief works on one simulated fund, that is, a fund whose history we generated by computer. In week 52 of the simulated fund’s life, a third party values the fund at about $670 million. Our estimate for that week is about $660 million with full belief, lower with no belief, and between those two with “some belief,” a setting between none and full (Table F-1). This first look shows only that more belief moves our estimate closer to the third-party valuation; how much belief any particular third-party valuation deserves needs its own study.

What we did

The simulated fund starts at a value of $100 million and runs for 200 weeks. Its manager reports a value every 13 weeks, as a real manager does about once a quarter, and a third party values the fund twice, in week 52 and in week 120.

We calculate our weekly estimate of the simulated fund’s fair value three times from the same 200-week history, changing only the degree of belief given to the two third-party valuations: none, some or full.

What we found

Exhibit F-1

Our weekly estimate of one simulated fund’s value, calculated three times: with no belief, some belief and full belief in the same two third-party valuations

The vertical axis is the fund’s value in US$ millions, and the horizontal axis counts weeks since the fund began. First panel: weeks 38 to 72, around the third-party valuation made in week 52. Second panel: the full 200 weeks. The three lines are our weekly fair-value estimate (the NAVcast™) with no belief, some belief and full belief; the round markers are the values the fund’s manager reported every 13 weeks, and the diamonds are the third-party valuations. Look at the diamond at week 52: the full-belief line passes almost exactly through it, the no-belief line passes well below it, and the some-belief line runs between those two lines.

Weeks 38 to 72

From the fund’s start to week 200

Source: Simulation of one fund that starts at a value of $100 million; all values are in US dollars. Unlike the other exhibits in this article, which show each week’s estimate as it stood in that week, the three NAVcast™ lines here are revised estimates, made looking back at the end of the simulation with all 200 weeks of the fund’s history known. We show them this way because revising its estimates of the past as more information comes to light is another capability of the model.

The more belief the user sets, the closer our estimate moves to a third-party valuation in the weeks around that valuation. In week 52 the third party values the simulated fund at about $670 million, well above the $560 million that the fund’s own manager reported for that week; our estimate is about $660 million with full belief, $600 million with some belief and $580 million with no belief (all rounded to the nearest $10 million).

In this example the degree of belief matters little by the last week of the simulation. In week 200 the three estimates are within about $16 million of each other (computed on unrounded values) and are no longer in order of belief: the some-belief estimate, at $1,426 million, is the lowest of the three.

For a pension plan that uses our model, the practical point is that a third-party valuation does not force an all-or-nothing choice between adopting the valuation and ignoring it. A plan that trusts the valuation can set full belief, and our estimate then moves almost all the way to the valuation (about $660 million against about $670 million in week 52); a plan that has doubts can set a lower degree of belief, and the valuation then moves our estimate by less.

Abstract

A private-equity fund’s manager reports a value for the fund about once a quarter; that reported value is often called a “mark.” Occasionally the reported value jumps or drops by far more than movements in the public stock market or the fund’s cash flows (the cash that investors pay into the fund and the cash that the fund pays out to them) can explain. This appendix shows what the NAVcast™, our weekly estimate of a fund’s fair value (the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller), does when that happens. The example is one of the 101 private-equity funds of the New York State Teachers’ Retirement System (NYSTRS), a public pension plan, that we priced for this article.

What we did

We feed the fund’s history into our model in the order in which an investor in the fund would have learned it, one payment or manager-reported value at a time, and each week’s estimate uses only the information available up to that week.

The fund is a US-dollar fund into which NYSTRS promised to pay a total of $160 million over the fund’s life (that promised total is called a commitment); every dollar figure below is stated for NYSTRS’s own holding in the fund, scaled to that $160 million. We follow the fund to early January 2024.

What we found

Exhibit G-1

Two sudden steps in the value one fund’s manager reported, and how our weekly estimate responded

Top panel: the fund’s value in US$ millions; the line with markers is the value the fund’s manager reported each quarter, and the other line is our weekly fair-value estimate (the NAVcast™). Bottom panel: cash that investors paid into the fund (bars up) and cash that the fund paid out to them (bars down). The two circles mark the manager-reported values that jumped or dropped suddenly: the first is 35% above, and the second 25% below, the value the manager reported a quarter earlier. The dashed vertical lines A and B mark the weeks in which the model flagged a shift in the manager’s reporting. Look at our estimate beside each circle: it rises 3.9% in the week of the first circled value and falls 13.2% in the week of the second.

Source: Estimates for one NYSTRS private-equity fund. Our data state the fund’s cash flows and manager-reported values per $10 million that an investor promised to pay into the fund; NYSTRS promised $160 million, so we multiply every figure by 16. The dates and sizes of the two shifts in the manager’s reporting are the model’s own estimates, made with only the data available at the time.

The first shift. The manager’s reported value rose 35.3% in one quarter, from $228.3 million on 2 July 2021 to $308.9 million on 1 October 2021. Neither the market nor the fund’s cash flows explain a rise that large: over that quarter the public stock market fell 0.9%, comparable traded assets fell 0.6%, and the fund paid out $27 million while calling only $1.6 million. In the week the new value arrived, the model flagged the jump as a shift in how the fund’s value was being reported, a one-step move of 26% in the reported values, not a gain in what the fund was worth. Our estimate rose 3.9% that week, against the 35% jump in the reported value.

The second shift. Three quarters later the reported value fell 24.7% in one quarter, from $311.8 million to $234.8 million, while the public stock market fell 11.5%. The model flagged this drop as a second shift in reporting and estimated a one-step move of 15% in the reported values.

What the model does with a flagged shift. A step this large means either that the manager changed the way it values or reports the fund and caught up in a single step, or that one of the model’s inputs is wrong, such as a missing payment into the fund. Either way, the model treats the step as a change in how the value is reported, not as a change in what the fund is worth, so our estimate does not simply copy the step.

The model that estimates a fund’s value each week can also project the fund forward. This appendix projects the example fund of Appendix C forward from 1 February 2013, the week in which its NAVcast™ was at its highest, with the scenario-forecast engine of our platform. The reader chooses the scenario: how the stock market behaves, how strongly the fund moves with it, and the fund’s payouts and capital calls.

How a scenario forecast works

Each scenario is one possible path for the fund, week by week. In each week the stock market moves by a random amount around the expected return set below, with the volatility set below. The fund’s return is its alpha plus its beta times the market’s return, plus a movement of its own. Payouts reduce the fund’s value and capital calls add to it. Alongside the NAVcast™, the engine projects the value the fund’s manager would report, the GP-NAV forecast, which follows the fund’s value with the lag and smoothing the model estimated for this manager. Every input starts at the fund’s own estimate in the week of 1 February 2013, from the same model run as Appendix C, and the charts show the range across all the scenarios.

Every scenario pays the fund out over its remaining life. By the platform’s wind-down rule, the size of each payout starts at three times the fund’s own typical size (each payout up to February 2013 as a share of the fund’s value the week before, scaled to the fund’s age), and the payouts are spread at random over the fund’s remaining life of about five and a quarter years: for this fund, 40 payouts. Capital calls start at the €18.75 million of the commitment still to be called in February 2013, spread evenly over the next three years. The three buttons are presets of the platform, and they change only the stock market. Status quo keeps the stock market’s average return and volatility from 2006 to February 2013, 7.6% and 16.0% a year. Market stress sets the expected return to zero and doubles the volatility. Market rally sets the expected return to 1.5 times the larger of the average and 8%, and lowers the volatility by a quarter.

Exhibit H-1

The example fund, projected forward: the range of the NAVcast™ and of the reported value across the scenarios, with the fund’s cash flows

Choose a preset and the exhibit is recalculated at once; the settings below the buttons show the values that preset uses. The magenta line is the NAVcast™ to 1 February 2013 and, after it, the middle of the scenarios; the shaded bands hold the middle half and the middle 80% of the scenarios. The gold dashed line is the middle of the scenarios for the value the manager would report. The lower panel shows the fund’s payouts and capital calls: the actual ones before 1 February 2013 and, after it, the average across the scenarios in each quarter.

Source: The platform’s scenario-forecast engine, run in this page with the same calculation, from the example fund’s estimates in the week of 1 February 2013 in the same model run as Appendix C; millions of euros for a commitment of 150 million. The random paths are drawn in the page.

Active fund

A fund for which we have a fair-value estimate in a given week, which is not the case before the fund’s first cash flow, after it has wound down or after our data for it end. A fund counts as wound down from the first quarter end at which its manager reports a value of zero and never again reports a positive value.

Allocation

The share of an investor’s total assets held in one kind of asset; for example, private equity’s share of a pension plan’s total assets.

Alpha (α)

The part of a fund’s yearly return that is left over after removing the part explained by movements in the public stock market.

Beta (β)

How much a fund’s value moves when the public stock market moves; a beta of 1.0 means the fund moves one for one with the market.

Beta-naive

A simple way of valuing a fund that resets the value to the manager’s reported value each quarter and, between reported values, moves the value up or down with the public stock market.

Capital call (contribution)

Cash that an investor pays into a fund when the fund’s manager asks for it, as part of the total the investor has promised to pay.

Cash flow

Cash that moves between a fund and its investors: cash the investors pay into the fund or cash the fund pays out to them.

Commitment

The total amount an investor has promised to pay into a fund over the fund’s life, which the fund’s manager asks for in installments.

Comparable traded asset

A publicly traded investment, or a blend of several, chosen because it resembles what a private-equity fund holds, so that its price movements give a clue to how the fund’s value is moving.

Data error (suspect value)

A figure in our data that we judge to be wrong and replace with the model’s estimate of the value the manager would have reported; this article treats one published value in that way (see the text after Table 2).

Degree of belief

The weight the user of our model tells the model to give a valuation made by someone other than the fund’s manager, from none (ignore the valuation) to full (move the estimate almost all the way to the valuation).

Denominator effect

A rise or fall in private equity’s share of a pension plan’s total assets that happens only because the plan’s publicly traded holdings are repriced every day and its private-equity funds about once a quarter: when public markets fall quickly, the plan’s total assets (the denominator of the share) shrink at once, the reported private-equity values lag, and private equity’s share rises although the plan has bought nothing.

Discount to NAV

The percentage by which the price a seller receives for its holding in a private-equity fund falls short of the value the fund’s manager last reported for that holding. Caution is needed in treating all transactions as the same, since forced-buyer or forced-seller dynamics alter a sale’s price away from fair value.

Evergreen fund

A fund that holds private investments and has no fixed end date, which lets investors put money in and take money out at set intervals at the value the fund’s manager reports.

Excess return

In our simulations, the return a simulated fund earns each year over and above the return explained by how much the fund moves with the public stock market (the fund’s beta).

Exchange-rate conversion

Restating in US dollars a value recorded in another currency, by multiplying it by the number of US dollars that one unit of that currency bought on the value’s date.

Factor of 3.9

The number by which we multiply the value of every one of the 101 private-equity funds we price, so that their combined dollar value is roughly as large as the private-equity value NYSTRS reported; the multiplication changes dollar amounts and leaves percentage gaps unchanged.

Fair value

Fair market value: the price that would be representative of a transaction in the fund between two well-informed market participants when neither is a forced buyer or seller; because private-equity funds are rarely traded, it cannot be observed and has to be estimated. The NAVcast™ is our estimate of it, made every week.

Filled-in manager value

The model’s estimate of the value a fund’s manager would have reported for a quarter in which our data hold no published value for the fund; it is an estimate of the reported value, not of fair value.

Forced seller

An investor that has to sell private-equity holdings because private equity’s share of its total assets has risen above the highest level its own policy allows, not because it judges the sale attractive.

General partner (GP)

The firm that manages a private-equity fund and reports its value; in this article, “the fund’s manager”.

Intermediary (adviser)

A firm that a seller hires to find buyers for the holdings the seller is offering and to run the sale.

Liquidation

The word NYSTRS’s board minutes use for the plan selling all or part of its holding in a fund; it does not mean that the fund itself was wound up.

Liquidity preference

How quickly a fund’s manager returns the fund’s value to its investors as cash; a manager with a high liquidity preference pays out a larger fraction of the fund’s value at each payout.

Mark

The value that a fund’s manager reports for the fund as of a given date, usually once a quarter.

Net asset value (NAV)

The value of a fund’s holdings as reported by the fund’s manager; in this article “NAV,” “mark” and “manager-reported value” all mean this same reported number.

NAVcast™

Our name for the estimate of a fund’s fair market value that our model produces each week.

Net-present-value (NPV) test

A check of a valuation method on real funds: a fund’s value at a date should match the cash it later pays out, net of the cash it later calls, discounted back to that date at a rate that reflects the fund’s risk (Appendix E).

Nowcast

An estimate of what something is worth, or how large it is, right now, made before the figures for the current period have been reported.

NYSTRS

The New York State Teachers’ Retirement System, a public pension plan for teachers in New York State.

Over-allocation, under-allocation

Holding more (over-allocation) or less (under-allocation) of one kind of asset, measured as a share of the investor’s total assets, than the target share the investor has set for that kind of asset.

Payout (distribution)

Cash that a fund pays out to its investors.

Phantom over- or under-allocation

An over-allocation or under-allocation that appears only because the values the funds’ managers reported are out of date, and that would not appear if the same funds were measured at fair value.

Policy band and trigger

The policy band is the range around a plan’s target within which the plan allows one kind of asset’s share of its total assets to move; a trigger is the share at which the plan’s policy requires the plan to act, selling above a sell trigger or buying below a buy trigger.

Private equity

Ownership of companies whose shares are not traded on a stock exchange, usually held through funds that a professional manager runs on behalf of investors such as pension plans.

Public market index

A single number that tracks the combined price level of a broad set of publicly traded stocks through time.

Reporting lag

The delay between the date a manager’s reported value refers to and the later date on which the investor actually receives that value.

Running total of the miss

The gap between an estimate of a fund’s value and the fund’s true value, added up period after period, which our simulations use to score how far an estimate has strayed in total.

Sale close

The date on which a sale is completed, which for NYSTRS’s 2023 sale of holdings in private-equity funds was 29 December 2023.

Scenario forecast

A projection of a fund’s value and cash flows forward under chosen assumptions about the stock market and the fund, repeated over many random paths so that the range of outcomes can be read (Appendix H).

Secondary sale

The sale by an investor of its existing holding in a private-equity fund to another investor.

Shift in the manager’s reporting

A one-time step up or down in the values a fund’s manager reports, after which the reported values continue from the new level.

Simulated fund

A fund that does not exist but is generated by computer simulation, so that its true value is known and an estimate can be checked against it.

Target

The share of its total assets that a plan has decided it wants to hold in one kind of asset, such as private equity.

Third-party valuation

A valuation of a fund made by someone other than the fund’s own manager.

Unfunded commitment

The part of the total an investor has promised to pay into a fund that the investor has not yet paid in.

Volatility

How much a value moves up and down over time, usually stated as a percentage per year.

Wound-down fund

A fund whose manager has reported a value of zero and never again a positive value; from that quarter end on we leave the fund out of both totals, the managers’ and ours, and no longer count it as active.