Return and Risk Methodology

Measurement issues arise for nonpublicly traded assets for a host of reasons, for example, relative illiquidity, stale pricing, and irregularly timed cash flows, which confound attempts to compute standard time-weighted returns typical of publicly traded assets. Dollar-weighted measures, such as IRR, are most frequently reported but are not without their problems. For example, cash flows are not necessarily reinvested over the life of the fund at the measured internal rate of return as implicitly assumed. Neither can IRR be compared with time-weighted return, nor does IRR measure the opportunity cost of capital. We follow up on these criticisms in the discussion further on.

Public market equivalents are directed at measuring the opportunity cost of capital. These are mimicking portfolios that invest private equity cash flows in a publicly traded benchmark (for example, the S&P 500). Total dollars that would have accumulated in the benchmark are then compared to the dollars earned in the private equity investment. Indexing dollar distributions at time t by D_i and drawdowns, or dollar capital calls, by C_j and assuming the fund lives T periods, then with continuous compounding, PME is the ratio of the value of the private equity investment (distributions are reinvested in the benchmark) to the opportunity cost of funds:

Private equity's excess return over the benchmark—its alpha gross of risk—is the scalar adjustment to the benchmark return in the denominator of PME necessary to equilibrate the future values of capital calls to distributions, that is, to make PME equal to 1. (This is why it is referred to in the literature as excess IRR.) That is, α solves the following but is still unadjusted for risk:

The Long and Nickels methodology combines IRR with a public market equivalent and compares the IRRs on two competing cash flows—those from the private equity investment and a matching set of cash flows in the public benchmark. For example, suppose an equivalent amount of capital invested with the general partner (GP) was invested in the public benchmark. When the GP makes a distribution, that dollar amount is also harvested from the public benchmark, that is, it shows up as a cash flow distribution for that period in the IRR calculation for the public benchmark. The remaining funds in the benchmark continue to grow at the market rate of return. In effect, the benchmark portfolio's cash flow size and timing is determined by the GP. When the GP liquidates the fund, the IRRs are computed for both the private equity and the benchmark investments. The difference between the two IRRs is the private equity alpha. This alpha is also gross of risk.

Timing can be everything. To see why, consider an illustrative example given in Table 19.1, in which we make a single investment of $100, which pays $200 at some point in the 10-year life of the fund. (All cash flows occur at the end of the year.) With no loss of generality, we assume a zero residual value at year 10. Assume the benchmark return is 7.2 percent so that it also returns $200 on the 10-year $100 investment. In this deterministic setting, a fund manager knows therefore that his multiple (distributions divided by capital calls) is equal to two no matter which year he distributes the $200, and his opportunity cost is the constant 7.2 percent benchmark return.

Table 19.1 IRR, ICM-α, and IRR Example.

Let's begin with the last column in which the distribution occurred in year 10. The IRR and the benchmark return are identical at 7.2 percent per year, thereby generating a Long and Nickels alpha of zero. The PME is unity, since the future value of the distribution is equal to the future value of the capital call and this, in turn, implies that the PME alpha is zero as well.

Keeping the fund's life constant at 10 years, let's now incrementally shorten the duration of the $200 distribution. Note that the multiple and the benchmark return stay constant at 2 and 7.2 percent, respectively. The PME grows from 1.0 to 1.87 and the PME alpha (not adjusted for risk) grows to 6.9 percent. The IRR, on the other hand, grows exponentially to 100 percent and the Long and Nickels alpha grows likewise to 92.8 percent. Keep in mind that these are annualized figures and that even though the investment returned 100 percent in the first year, it returned zero in the last nine years. Yet, the annualized IRR does not change. Note the impact of time—when the $200 distribution moved from year two to year one, alpha tripled but the PME rose only from 1.74 to 1.87, meaning that, instead of $1.74, we would have had to invest $1.87 in the benchmark to generate a return equal to a $1 investment in the private equity fund. Thus, while the Long and Nickels alpha tripled from 34.2 percent to an annualized 92.8 percent, the PME alpha rose by only 79 cents. Clearly, IRR-based performance measures tell a much different story in this example.

This exercise illustrates the huge range of these performance measures. Private equity managers often refer to their multiples but should know that multiples are essentially meaningless, since any given multiple is consistent with an unbounded range of IRRs as well as Long and Nickels alphas. PME is much less sensitive due in large part to it being a function of the benchmark return, which, in general, is much lower than computed IRRs.

Again, none of the preceding measures are adjusted for risk. To get at risk, Driessen et al. estimate the fund's beta in a CAPM context (along with alpha) using the generalized method of moments (GMM). They also extend the single-factor CAPM to the three-factor Fama-French model to identify funds’ style factors.

The basic GMM model draws on the PME setup. Since a detailed discussion can be found in Driessen, Lin, and Phalippou, I stick to a high-level presentation here. The objective is to define moment conditions that will provide estimates of alpha and beta in the CAPM setting. To do that, recall that each fund i invests C_{i ,tj} at time _tj and distributes D_{i ,tk} at time _tk. The first step is to future-value each cash flow to the liquidation date (l_i) of the fund and then average these over n_i, the number of investments (capital calls) for each fund:

The second step collects these into portfolios with N_p funds per portfolio. A portfolio might consist of the funds within a subclass of private equity such as venture, or a portfolio might consist of all funds originating in a particular vintage year. Averaging again, this time across the N_p funds in the portfolio, generates the desired moment conditions:

Finally, iterate on (α,β) to minimize the quadratic loss function given by:

There are as many moment conditions for GMM estimates as funds. Aggregating cash flows by vintage year would generate as many moment conditions as vintage years. Cash flow aggregation by vintage year tends to reduce idiosyncratic risk, which, in turn, will improve the precision of GMM. However, there are cases in which vintage year aggregations can confound bootstrapping procedures when the distribution of funds across vintage years is either too thin or highly variable, as it is in our case. Bootstrapped standard errors are estimated by randomly drawing N funds with a replacement, where N is the total number of funds in the group, and using GMM to estimate the model parameters of the bootstrap sample. This sampling procedure is repeated N times. The z scores are computed using the standard deviations of the parameter estimates in the N bootstrap samples.

The Long and Nickels methodology is an interesting one; it forces the benchmark to match the private equity cash flows. In effect, it asks whether a given cash flow sequence that is determined by the GP would have earned more or less on an IRR basis in the public market. It does not ask whether the opportunity cost of the investments to private equity over, say, a 10-year horizon is too high. That is a different question and one that sometimes yields a different answer. Consider a $1 two-year investment for which private equity and the public benchmark earn 75 percent and 0 percent, respectively, in year one and 0 percent and 100 percent in year two. Assume the GP harvests dividends of $0.75 after year one and the public benchmark must match this policy. Then the GP has cash flows of –$1, $0.75, and $1.00 for an IRR of 44 percent while the public benchmark has –$1, $0.75, and $0.50 for an IRR of 18 percent. But left alone, the benchmark would have doubled its investment to $2 in two years versus $1.75 for the GP. This measures the true opportunity cost of fund, not the IRR.

If the limited partner (LP) wants to know whether a particular sequence of cash flows would have earned more in the public market, then the Long and Nickels alpha would seem to provide a useful solution methodology. If, on the other hand, the LP wants to know about opportunity cost, then PME is more direct. And extending PME to a CAPM structure and using GMM provides risk-adjusted performance. In any case, we tend to get conflicting results, depending on the method used.

To prove this point, let's set up a CAPM model with the Russell 3000 returns to simulate private equity returns as a test of Long and Nickels. We have quarterly returns to the benchmark on 77 quarters, spanning June 1989 to June 2008. We simulated private equity return series using the CAPM, setting alpha and beta before the event. For example, if we set alpha equal to zero and beta equal to two, we created a return series with twice the risk of the benchmark. Returns were subsequently permitted to accumulate over time in both the benchmark and the simulated private equity series. We then select a set of random distribution dates with random distribution amounts (not to exceed available funds) from the simulated return series and match these distribution amounts in the benchmark portfolio as prescribed in the Long and Nickels methodology. The IRRs to the benchmark and the simulated series were then computed and differenced to estimate the Long and Nickels alpha.

As a baseline, we first simulated a series using (α,β) = (0,1) to generate two identical cash flows which, by definition, produce zero alpha. In a deterministic single-factor model like this one, the simulated return r_i is linear in the benchmark r_m according to:

The expected excess return is:

Now, the average annualized return to the Russell 3000 over this time was about 10.35 percent, which, for simplicity, we'll round to 10 percent. Thus, beta equal to 1.0 implies zero excess return, beta equal to 1.5 implies an excess return of 5 percent, beta of 2 implies 10 percent, and so forth. Note that alpha is zero by design—excess return is from risk and not skill. Then we estimated Long and Nickels alpha. It was zero in the first case as it should be, but as beta rose, the Long and Nickels alpha began to increase exponentially, for example, beta of 1.5 generated an average Long and Nickels alpha in 1,000 trials of 17.5 percent (standard deviation of 0.78 percent). The Long and Nickels alpha for beta equal to 2 was 34.6 percent (standard deviation of 1.75 percent)—almost three times as high as what the expected excess return would indicate. When beta is set to 3, the Long and Nickels alpha was a whopping 62 percent (with standard deviation of 2.2 percent).

It is possible that this exaggerated outperformance is due to cash flow matching. Private equity distributions will generally be larger in a high beta environment and the public benchmark must match the dividend policy of a much riskier investment. This is clearly favorable to the private equity manager if his performance is gauged relative to the less risky public benchmark. In effect, cash flow matching alters the production function of the public benchmark to the extent that its capital has been depleted by a rule that acts as a constraint on its opportunity set. If, for example, the investment horizon were two years, the benchmark would have returned more than the private equity investment on an annualized basis. On the other hand, cash matching would have crushed the benchmark because the benchmark portfolio would have been required to harvest most of its first-year capital. Thus, it is possible that Long and Nickels, in addition to not adjusting for risk, may bias private equity performance in favor of the GP.