The Far-Right Tail Problem & Tail Tempering

The premise

Even though the probability for the return exceeding unrealistic values such as 1000x is extremely small its impact is very big. For this reason we made it sure that we are not overestimating our modeled returns because of a few unrealistic exits.

01: A poorly calibrated power-law model can make a fund strategy look more attractive than it really is

A pure Lomax distribution assigns a nonzero probability even to an extremely unlikely outcome such as a 1,000× return. The important issue is not only that the model puts some probability above 100×. It is what happens after a draw enters that region. A power law has no characteristic scale. Allowing such outcomes could distort the resulting mean and variability, so the maximum modeled return is tempered rather than allowed to extrapolate indefinitely.

The tail problem became obvious while analyzing the mixed pre-seed distribution. When the fitting parameter was low enough, individual values ballooned and an inflated standard deviation was reported—jumping from 7.78x in the tempered model to 14.08x in the uncapped model—while the mean was also overestimated from 3.17x to 3.69x (a 16.2% expected return inflation) because of wild exits such as 500×. The visible body of the distribution could still look reasonable. The problem was that a handful of outcomes, which were almost invisible on the plot, were controlling the numerical conclusions.

Figure 1Model output

Tempered Model Max (N=1M)

218× ($10.9B exit)

Untempered Model Max (N=1M)

5,800× ($290B exit)

Tempered Model Mean

3.17×

Untempered Model Mean

3.69×

Show the underlying values

Sample size (N)	Tempered max	Untempered max
500	58×	180×
2,000	82×	340×
10,000	124×	850×
50,000	142×	1,850×
200,000	174×	3,600×
1,000,000	218×	5,800×

While unicorn returns above 100× certainly exist and get quite the media coverage, they are extremely rare, of course. When we created our custom Mix distribution, we noticed that while the curve we fit to the return anchors matched the PitchBook report, the untempered probability of returns exceeding 100× was too large—sitting at 0.22% (1 in 464 deals).

A bigger problem than that is that physically impossible returns like 1,000× were actually not that impossible with such a fat-tailed distribution. In the untempered model, extreme draws reaching 1,000× to over 5,000× inflated the expected mean to 3.69× with an exploded standard deviation of 14.08×. Under the tempered model, maximum observed outcomes naturally level off around 150×–220×, yielding an expected mean of 3.17× and a standard deviation of 7.78×.

We implemented exponential tail tempering above 50.0× so that the probability density decays smoothly to zero rather than abruptly hitting an artificial hard cutoff.

Figure 2Model output

40× to 80×: Smooth tail separation at 50×80× to 150×: Continuous far-tail decay

Exponential Decay Threshold

T = 50.0×

Decay Constant (λ_decay)

30.0

P(MOIC > 50×)

0.45% (Tempered)

P(MOIC > 100×)

0.015% (Tempered)

Show the underlying values

Return threshold	Tempered probability	Untempered power law
Exceeding 50× ($2.5B exit at $50M entry)	0.45% (1 in 220)	0.91% (1 in 110)
Exceeding 80× ($4.0B exit at $50M entry)	0.045% (1 in 2,200)	0.38% (1 in 260)
Exceeding 100× ($5.0B exit at $50M entry)	0.015% (1 in 6,700)	0.22% (1 in 464)

That is why it is important to carefully validate every model one proposes. We implemented the exponential tempering of the tail. The $\lambda_{\text{decay}}$ parameter was chosen such that it fits empirical benchmarks from another VC investment return report—the Fenwick & West / Correlation Ventures venture return report—which reports that roughly 1% of financings fall between 20× and 50×, and approximately 0.5% exceed 50×.

Comparative analysis of calibrated results against historical benchmarks shows: 50×+ returns at 0.45%, aligning with a ~0.5% benchmark; 20× to 50× returns at 3.51%, consistent with PitchBook's seed-stage distribution; 100×+ returns at 0.015%; and 500×+ returns dropping to less than 0.000001%, effectively excluding unrealistic outliers.

Going back to the problem of the untempered distribution having P(>100×) at 0.22% (1 in 464 deals), comparing it with the tempered distribution highlights the difference: tempering brings this down to 0.015% (1 in 6,700 deals)—a roughly 14-fold reduction that keeps extreme exits in their realistic place without making them impossible.