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EDIT, 10/31/19:

I’m withdrawing this piece. I’ve become convinced that since the Kelly Criterion is not directly applicable to venture, using the spirit of it to justify follow-ons is cheating. I still think I’m right, but if I want to make the argument I think I have to do the math.

EDIT, 6/20/17:

Kelly justified betting a larger amount of your bankroll when you have a larger informational advantage. But the Kelly Criterion itself imagines a specific scenario: sequential parlay betting when you have an edge. Since venture investing is never really sequential (the ‘bet’ is not immediately resolved) and only angel investors really parlay, it can’t be literally applied to determine exact portfolio allocation. This post was not meant to suggest that it should be. This post makes a single claim: the Kelly Criterion suggests that because in venture capital edge tends to increase faster than odds as more information becomes available, it usually makes sense to increase the amount invested in a portfolio company–to follow on.

If you try to allocate via the Kelly Criterion alone, you’ll quickly find that you run out of money to invest due to unresolved bets.

I’ve made a couple of edits below to clarify this.

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Should you follow-on in later rounds?

The pros:

1. You want to show support for the founders you’ve backed (signaling.)
2. After working with the company for a year or two, you have a better idea of value than the new investors, so if they’ve underpriced relative to actual value, you’re buying at a bargain (the asymmetrical information argument.)

The cons:

1. If the company is succeeding and each round is an up-round, then the earliest rounds have the highest multiple at exit. Investing in later rounds always averages down your exit multiple (the bragging rights argument.)
2. If you sometimes follow-on and sometimes don’t, then not following-on is a signal to the new investors that they’re making a mistake (signaling again.)
3. The more companies you invest in, the better chance you have of a massive exit, so better to spread the money around (diversification.)

These are good arguments, but they’re secondary considerations to the underlying question. Before all of the strategic pros and cons, how should your portfolio be allocated to maximize your expected return?

There is a way to compute an optimal investment size in a series of investments, it’s called the Kelly Criterion. John Kelly, Jr. was a researcher at Bell Labs in the 1950s and a devotee of fellow Bell Labs researcher Claude Shannon’s information theory. He solved the problem in his 1956 paper A New Interpretation of Information Rate¹.

Kelly asked: if you have a private stream of information related to a bet, not known to others and so not reflected in the odds, you have an edge; if you have an edge, how much should you wager? If you only have one shot at a positive expectation bet, you wager everything you can afford. But what if you have a stream of these bets and can wager previous winnings²? If you wager everything, you’ll eventually lose it all and walk away with nothing. But what amount between nothing and all?

Kelly determined that the fraction of your bankroll to wager is equal to \[\frac{edge}{odds}\].

The odds are the multiple of your wager your bankroll increases by if you win. When you roll a die, fair odds are 5 to 1: if you roll your point you win $5 (and keep the dollar you’ve bet), if you don’t you lose $1. In this case the odds are 5 to 1, or just 5.

If it’s a fair die, the probability of rolling your point is one-sixth, and you have no edge. Your expected value is getting your wager back. But imagine it’s a loaded die and it rolls 2 one-fifth of the time and something else four-fifths of the time. No one else knows this, so the odds remain the same. You would bet the 2, of course, because you have an edge.

For every dollar you bet on the 2, you can expect to win $5 one-fifth of the time and lose $1 four-fifths of the time. Your edge is

\[\frac{$5}{5} – \frac{$1*4}{5} = $0.20\].

Your edge per dollar is 0.2. According to Kelly you should bet 0.2/5 of your bankroll, 4%, on each roll of the die.

\[f^{*}\], the fraction of your bankroll to bet, is

\[f^{*} = \frac{bp – q}{b}\],

where \[p\] is the probability of winning, \[q\] is the probability of losing, and \[b\] are the odds.

Since \[q=1-p\], we only need two numbers to figure out \[f^{*}\]: the probability of winning, \[p\], and the multiple of your wager you get if you win, \[b\]. Note that the Kelly Criterion assumes a binary outcome, you either win or lose.

How does Kelly fare? Below is a chart of a simulation of betting on the crooked die. The blue line is your bankroll using Kelly and the green line is your bankroll betting a fixed amount equal to 10% of your initial bankroll on each roll. The bankroll starts at $10. Kelly obviously grows much more quickly (the bankroll is on a log scale; the edge here is huge, btw, that’s why the bankroll grows so fast.)

Before we apply this to venture, two simplifying assumptions. First, each investment is independent of the others. Second, you are aiming for a certain fund IRR. If you are aiming for 20% IRR per year, and you think you will hold each investment on average five years, then your fund should return 1.2⁵=2.5x. Valuing each investment to achieve this overall return creates your edge (and accounts for the time value of money.)

Here’s an example. You’re running a $50 million seed fund that you expect to have a 20% IRR and hold investments on average five years, a 2.5x return overall. You invest in a company you think has a 1 in 50 (2%) shot at a billion dollar exit. The value of the company now is $1 billion x 2% / 2.5 = $8 million.

If the company is successful your investment will return $1 billion/$8 million = 125x. This is the multiple, \[m\]. The odds are one less than the multiple because the multiple includes the original investment, \[b=m-1\], so \[b=124\] and

\[f^{*} = \frac{bp – q}{b}=\frac{124×2\%-98\%}{124}=\frac{1.5}{124}=1.21\%\].

1.21% of a $50 million fund is $605 thousand. This is what Kelly says you should invest in this company. (Edit: While conceptually true, the practical issue here is: what is the fund size when you make an investment? It has to include the current value of whatever companies you’ve already invested in, although those are hard to determine. Also, they are illiquid: you may find the Kelly Criterion suggesting you up a bet when all of your cash is already in other companies. This is because the Kelly Criterion was built on a scenario where all bets pay off immediately.)

(The $8 million value is not the price you invest at, it doesn’t account for dilution. If you assume that after your round the company will increase its issued shares by 50% because of later rounds and options issuance, you invest at $8 million/1.5 = $5.3 million. You will own $605,000/$5,333,333 = 11.34% of the company. This doesn’t really matter here because your expected multiple remains the post-dilution multiple, I just wanted to point it out. This way of getting to price is called the Venture Capital Method of valuation³.)

Note that the numerator of \[f^{*}\] is your expected fund multiple, \[e\], less one. This is because \[bp-q=(m-1)p-(1-p)=mp-p+p-1=mp-1\] and \[mp = e\]. So,

\[f^{*} = \frac{e-1}{b}\].

The Kelly Criterion is well known to investment managers. But venture capital has a twist most investors don’t have: investments are staged, there are multiple rounds at different prices and risk profiles. VCs have to figure out how their optimal allocation changes as new information comes in.

Imagine another investor decides to invest in this company a year later. The company performed well and mitigated some risk. The ultimate exit value if the company is successful remains the same, but the probability of success has increased. There is now a 10% chance of the company reaching $1 billion. The new fund has a 2.1x goal (because time value of money: it’s a year later and 1.2⁴=2.1). They value the company at $1 billion x 10% / 2.1 = $48 million (post-dilution but, again, we don’t care about that here.) Their multiple is 21, so the odds are 20. For them

\[f^{*} = \frac{e-1}{b}=\frac{2.1-1}{20}=5.5\%\]

The new investor should invest 5.5% of their fund in the company.

The funny thing is, so should you. The additional information they have, a year later, is information you now also have. Your calculation changes, and is identical to theirs. To remain at the Kelly optimum you should increase the percentage of your fund invested in the company from 1.21% ($605 thousand) to 5.5% ($2.75 million.) You should invest $2.14 million in this round⁴. (Edit: This assumes your bankroll is still worth $50 million. It may in fact be more–the company in question, at least, is worth more. But other bets in your portfolio may pull that down and it is worth less. If the portfolio has grown in value in that time, then Kelly indicates you should invest at least that amount. Note that this amount is probably much more than your pro-rata anyway: if the company is raising $10 million in the new round, 20%-ish, then your pro-rata is 11.34% of that, or $1.134 million.)

In most up-rounds, especially from the early stages, Kelly recommends increasing your position. Based on an initial investment made with a success probability of 5% and an eight year time to exit, the table below shows the increase in allocation based on the increase in probability of success and decrease in time to exit. In this case, Kelly suggests you should increase your position unless there is a minimal increase in the probability of success a long time after your investment (the red numbers). While these are technically “up” rounds, that company is going sideways. And, of course, if you think the probability of success is only 10% but exit is imminent it’s probably your thought process you should worry about, not your portfolio allocation.

You may also have strategic considerations (signaling, what you’ve promised your LPs, etc.) that have to be set against the Kelly optimum. Do the calculations each time and use them as inputs for your thinking. But the math says you should almost always follow-on.

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1. Originally titled “Information and Gambling” until AT&T nixed that, wanting to distance themselves from gamblers. An entertaining, if non-mathematical, telling of the story is in Poundstone, W. (2006). Fortune’s Formula. http://amzn.to/2sHEPtu ↩

2. The Kelly Criterion assumes bettors parlay: winnings are recycled into the bankroll. While some venture funds do limited recycling, and angel investors obviously recycle, it’s a conceptual issue for the non-recycled part of the winnings. The underlying observation that information that increases your edge should increase your bet size remains true though. ↩

3. See https://ocw.mit.edu/courses/sloan-school-of-management/15-431-entrepreneurial-finance-spring-2011/lecture-notes/MIT15_431S11_lec01.pdf if you want a more in-depth explanation of the Venture Capital Method. ↩

4. Your pro-rata in this example is almost certainly less than this, but you should take as much as you can get, up to $2.14 million. ↩