Takahashi-Alexander model: forecasting the cash flows of private markets funds

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If you invest in closed-end funds, you commit money that the manager only calls gradually and pays back years later. The Takahashi-Alexander model, also called the Yale model after its origin, uses five parameters to predict when capital calls and distributions will come and how the fund's value (NAV) will develop. This article explains the formulas, works through a fund year by year, shows how to estimate the parameters from data and uses them to plan commitments across many vintages until a target allocation is reached.

Origin: the Yale endowment's model

Dean Takahashi and Seth Alexander worked in the Investments Office of Yale University, whose endowment follows the endowment model and holds a large share of illiquid assets. That investment approach is also often called the Yale model; here, the term refers only to the cash flow model. Takahashi and Alexander (2002) published their model in the Journal of Portfolio Management. It is meant to help institutional investors project future fund values and cash flows for venture capital, leveraged buyouts, real estate and natural resources partnerships. The historical figures of the investor's own funds serve as the starting point, and scenarios with different commitments, call and distribution assumptions and returns can be run quickly.

Every investor in private markets knows the underlying problem: the capital commitment is not the same as the investment. The manager calls the money when it is needed and pays it back when holdings are sold. According to Kaplan and Strömberg (2009), a buyout fund usually has a life of ten years, extendable by up to three. If you want to hold a fixed allocation and remain able to pay at all times, you have to forecast these cash flows. The model does this in a deliberately simple way: deterministically, with a few parameters that are easy to explain.

The formulas

The calculation runs in years t = 1, 2, … from the commitment, with all cash flows at year end. Each year the NAV grows at G, part of it is distributed, and new calls are added:

Calls shrink geometrically because a fixed share of the remainder is called every year. The distribution rate, by contrast, rises with the fund's age: (t / L)^B is small at the start and reaches exactly 100% in year L, when the entire remaining NAV is distributed. As long as (t / L)^B is below Y, the minimum distribution Y applies; it stands for current income such as rents, interest or dividends. In the form used here, the fund calls nothing more from year L onward, and whatever has not been called by then lapses.

What the parameters do

The bow factor B has the greatest influence on the pattern. With B = 1, the distribution rate rises evenly by 1 / L per year, and in the sixth of twelve years half the NAV flows back. With B = 2.5, it is only 17.68% in year 6 and 48.71% in year 9; with B = 5, the rate only exceeds the minimum distribution in year 7 (6.8%). The larger B is, the longer the money stays in the fund, the higher the NAV and the more the distributions bunch up at the end. Venture capital funds with a few late exits tend to fit a large B, while real estate or credit funds with current income fit a higher Y.

Worked example, year by year

A commitment of 100 to a fund with a twelve-year life, a rate of contribution of 25% in every year, a bow factor of 2.5, growth of 10% and a minimum distribution of 5%, i.e. the assumptions from the worked example on the Yale model plus growth. The first three years in detail:

Calibration: where the parameters come from

The parameters are not constants of nature. You estimate them from the cash flows of your own or comparable funds, separately by strategy, because buyout, venture capital, real estate and private debt funds call and distribute capital very differently. Each parameter has an observable counterpart:

• Rate of contribution: RC_t = C_t / unfunded commitment at the start of the year, averaged across funds of the same age. A fund that has called 57.81% after three years fits 25% a year: 1 − 0.75³.
• Bow factor and minimum distribution: from the observed distribution rate D_t / (NAV_t−1 × (1 + G)) for each fund age; Y corresponds to the rate in the first years, and B follows from RD_t = (t / L)^B, i.e. B = ln RD_t / ln(t / L).
• Growth: the fund's expected return after fees. Because G in the model is the IRR (see below), the assumption should match the return expectation for the strategy.
• Fund life: from the fund agreement, including the usual extensions.

For an existing portfolio, you do not start the model at zero but at each fund's current NAV and current unfunded commitment, and you continue calculating from its current age. This is what Takahashi and Alexander (2002) did: they combine the actual history with the forecast and compare forecast and reality every year.

Commitment pacing: commitments across many vintages

If you want to hold a permanent allocation to private markets, you make new commitments every year, vintage by vintage. This planning is called commitment pacing. Because the full commitment is never invested at once, and early vintages are already distributing while new ones are still calling capital, you have to commit more than the target allocation implies. The sum of NAV and unfunded commitments then exceeds the target: this is overcommitment. The model tells you how much overcommitment is needed and how much liquidity has to be kept available for it.

The example works with assets of 100 that are initially fully liquid and earn 5% a year there. At the start of each year, the investor commits a fixed share of assets; each vintage follows the worked example above. Calls are paid from the liquid portion, and distributions flow back into it. The goal is the constant commitment rate at which the private markets allocation reaches 15% over the long run:

Two consequences matter. First, building up the allocation takes almost as long as a fund's life: with the solved rate, the target is only reached in the tenth year; if you want to get there faster, you commit more at the beginning and reduce later. Second, the unfunded commitments, 8.99% of assets in the example, must be available to be called at any time. In the steady state, distributions from the old vintages largely fund the calls of the new ones; in a crisis this can break down if distributions dry up while calls continue. On top of that comes the denominator effect: if the liquid portion falls, the allocation rises without a single new commitment.

Advanced: properties and extensions

• IRR equals G: dividing NAV_t = NAV_t−1 × (1 + G) + C_t − D_t by (1 + G)^t gives NAV_t / (1 + G)^t − NAV_t−1 / (1 + G)^(t−1) = (C_t − D_t) / (1 + G)^t. Summed up, the present value of all cash flows including the residual value at the rate G is zero. The since-inception IRR is therefore G in every year, and the TVPI depends only on how long the money stays in the fund.
• Linear in the commitment: all cash flows are proportional to CC. A program of many vintages is the sum of the individual paths, weighted by the commitments; mathematically, a convolution. With commitments as a fixed share of assets, the allocation settles at a fixed value that a solver can target.
• Stochastic models: Buchner et al. (2010) model the contribution rate as a mean-reverting square-root process and distributions as a Brownian motion with time-dependent drift, estimated with conditional least squares on European funds. Their model explains up to 99.6% of the variation in average cumulative net cash flows and, unlike the Yale model, delivers a distribution instead of a single path.
• Dynamic commitments: de Zwart et al. (2012) develop a strategy that sets anew each year how much to commit, depending on distributions received, unfunded commitments and the current allocation compared with the target. The Yale model provides the expected cash flows of the existing funds for this.
• Measuring instead of planning: the model does not show whether a fund actually performed well. For that, there is the PME of Kaplan and Schoar (2005) and direct alpha of Gredil et al. (2023), which compare the actual cash flows with an index.

Limitations and common mistakes

• Deterministic: the model delivers an expected path. Real funds call capital irregularly, distribute in waves and vary widely from one another.
• No market dependence: in reality, calls and distributions depend on the market. In crises, exits and distributions stall while calls continue; scenarios with a lower G and later distributions therefore belong in the analysis.
• No risk in the NAV: the NAV grows smoothly at G. It is unsuitable for volatility and correlation; for those you need desmoothed series or listed proxies.
• Fees and recallable distributions: the basic model knows nothing of management fees, recallable distributions, credit lines or fund life extensions. They shift cash flows and belong in the parameters or in separate adjustments.
• Old parameters for new funds: calibrated on a different interest rate environment or strategy, pace and shape will be off. Compare forecasts with actual cash flows regularly.

How Portfolio Lab calculates

The Private markets tool in the Plan section calculates exactly the formulas above in the Forecast tab, year by year with cash flows at year end. Rates of contribution can be entered for each year, after which the last value applies; from year L onward, the fund calls nothing more, and the rest of the commitment lapses. The Plan commitments tab adds a vintage with the same model every year, with a fixed commitment or as a percentage of assets at the start of the year, grows the liquid portion at a fixed return and optionally deducts an annual withdrawal. A solver searches for the constant commitment rate at which the allocation in the steady state hits the target (the average of the last L years of a calculation over 200 years), and returns no solution if the liquid portion would turn negative. The Analyze a fund tab measures actual funds with IRR, TVPI, DPI, PME and desmoothing, as in the article measuring private markets. The details are in the private markets methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.