How Much To Bet On A Deal?

Say you are managing a corpus of $100 and intend to invest it across a portfolio of ‘N’ bets, how should you determine the size of each bet?

The Kelly Formula, as outlined by the famous math professor, investor, and gambler Ed Thorp, shows us a path.

One question that I have been studying for a while now is how to most optimally bet on a given deal? Public market investors call this ‘position sizing’ – say you are managing a corpus of $100 and intend to invest it across a portfolio of ‘N’ bets, how should you determine the size of each bet?

A. Nuances for every strategy

Based on studying how some of the best public market and venture investors approach position sizing, it’s clear to me that, like most things in life, it’s part art and part science.

Every investor will have their own nuanced perspective on this topic based on their individual ‘strategy’. This includes the following aspects:

1/ Asset class

Publicly listed stocks are liquid and play in a mostly-efficient, no information-asymmetry market. On the other hand, venture capital is perhaps the most inefficient market, governed by intense power laws and extreme loss ratios.

In a totally different game, real assets generate cash flows and are, therefore, conducive to debt financing that improves levered-equity returns.

2/ Beliefs and personality

Warren and Charlie believe in buying extraordinary businesses at fair prices. Joel Greenblatt believes in special situations. YC and 500Startups believe in the ‘Moneyball’ style of venture investing. Benchmark and Kleiner Perkins believe in the classical, craftsperson style of venture capital. Brookfield believes in buying high-quality real assets on a value basis.

3/ Circle of competence

Often called an ‘edge’ or ‘competitive advantage’. Peter Thiel and Vinod Khosla understand revolutionary technologies better than others. Li Lu gets China more than Western fund managers. Berkshire is unique in its understanding of insurance businesses.

4/ Selection criteria

Don Valentine, Founder of Sequoia, famously said that “great markets make great companies”. Keith Rabois of Founders Fund has a founder-driven investing style where he looks to figure out whether this founding team can build an iconic company that changes the world.

Public market OG Chuck Akre’s investment criteria are captured in the ‘three-legged stool’ – (1) extraordinary business, (2) talented management, and (3) great reinvestment opportunities and histories.

5/ Portfolio construction

On the public market side, Charlie Munger’s Daily Journal Corp has a super-concentrated portfolio of 4 stocks (~40% Wells Fargo, ~40% Bank of America, ~15% Alibaba Group, and the rest is U.S. Bancorp). Bill Ackman of Pershing Square has a classical, concentrated ’10×10′ portfolio that presently includes 8 stocks, with each position being 10-20% of the portfolio. Seth Klarman of Baupost Group is comfortable with a bit more diversification, owning 28 stocks at present with the largest holding at ~15% portfolio, and a bunch of positions in the single digit % range.

In venture capital, given its high-risk profile, the importance of diversification is generally well-understood. Yet, firms exhibit significant variance in their approaches to portfolio construction. While the likes of Brad Feld and Mark Suster believe in taking 30-40 shots even from reasonably large $300Mn+ funds, Mike Maples Jr. of Floodgate believes that a typical venture portfolio becomes statistically diversified at 12 companies, and beyond 25, there is no incremental value from excess diversification. Miriam Rivera of Ulu Ventures has studied data from LPs and concluded that even the best VCs have ~4.5% picking skills and therefore, a portfolio of 70-100 shots at goal is needed. Finally, an accelerator like YC funded 229 startups in just one Summer 2023 batch.

So, as we can see, position sizing approaches can vary dramatically based on the investing context and strategy being followed. But are there any broad rules and heuristics that can be useful for any investor out there?

B. The Kelly Formula

John L. Kelly was a researcher at Bell Labs in the 1950s. He developed a mathematical theory on how to bet most-optimally from a finite bankroll, in favorable gambling games.

Without going into the mathematical details of it*, here’s the basic idea behind the theory as explained by Rob Vinal of RV Capital in his H1 2023 Investor Letter:

The basic idea is that the greater the upside relative to the downside, the more an investor should bet. However, if there is a probability of a total loss, the bet size should be zero as the product of any series of numbers with a zero in it is zero.

Rob Vinal

*For those who are mathematically inclined, check out a couple of old must-reads by the famous math professor, investor, and gambler Ed Thorp – The Mathematics of Gambling and The Kelly Criterion and the Stock Market.

Based on studying the Kelly system, including commentary on it from the likes of Ed Thorp and Rob Vinal, here are some key rules that any investor should be aware of while position sizing for any strategy:

1/ Play only when you have an advantage

Here’s how Ed Thorp describes it:

The Kelly system calls for no bet unless you have the advantage. Therefore, it would tell you to avoid games such as craps and slot machines. However, if you have the knowledge and skill to gain an edge in blackjack, you can use the Kelly system to optimize your rate of gain.

Ed Thorp

Warren Buffet’s ‘Circle of Competence’ rule is also a play on this idea. To have the best odds of winning, choose a game you have an edge in and choose to play at the table with weaker players.

TLDR: focus on identifying your edge before thinking through bet sizing.

2/ Avoid the risk of ruin

In repeated games (say a coin toss) where there are some odds of a total loss (eg. heads you win, tails you lose), if you bet everything in each turn knowing that you have an edge in each turn (say you have odds of 0.52 for getting heads in each turn), as the number of turns ‘N’ increases, the probability that you will be ruined tends to 1 or certainty.

In the Kelly system, you never bet everything in a single turn so the chance of ruin is zero.

3/ Bet more when asymmetricity is high

The Kelly formula tells us to bet large where there is a big asymmetry between upside and downside. Conversely, it shows that if the risk of loss is too high on a single bet (eg. in Roulette), it’s too dangerous to bet a large fraction of your bankroll.

The former scenario is the method that top-value investors follow – betting a big proportion of the fund (10-20%+) on each high-conviction, high-quality business with a large margin of safety. Case in point: Berkshire has ~50% of its publicly traded portfolio in Apple.

We don’t put the most money into things that are going to give us 7-10x returns. We put the most in positions where we will never lose money.

Joel Greenblatt

Conversely, the latter ‘Roulette’ insight is the method venture capital investors follow while investing in extremely high-risk startups. They play on the right side of the power law curve – assembling an optimally diversified portfolio of high-risk, high-reward bets; and deploying enough capital in each bet so as to ensure enough ownership per company such that if and when it wins, it wins big enough to compensate for all the other losses in the portfolio.

4/ Value of holding cash

By using concepts like bankroll, betting small portions of it at a time, and not going broke, the Kelly formula also subliminally suggests the value of always holding cash in the portfolio.

Berkshire is famous for holding significant amounts of cash ($100Bn+ in recent years) on its balance sheet at all times.

We believe in always having cash. There have been few times in history where if you don’t have it, you don’t get to play the next day.

Cash is like oxygen. It’s there all the time but if it disappears for a few minutes, it’s all over.

Warren Buffet

Holding cash also helps in going on offense when unforeseen crises like the dotcom crash or GFC occur. As asset prices crash, these become once-in-a-lifetime opportunities to deploy capital.

C. Incorporating special considerations

While following the above heuristics from the Kelly criterion, it’s also important to keep some room to account for special considerations in your personal context. Eg. Rob Vinal keeps some buffer in case LPs want to withdraw money on short notice:

RV Capital H1 2023 Investor Letter

D. Closing Thoughts

Position sizing is both an art and a science. Having a well-defined view on it that is congruent with your overall investment strategy is crucial for any investor.

As you think through its nuances, it’s useful to keep in mind the guardrails that the Kelly formula tells us. The Kelly heuristics guide us towards the most optimal, risk-adjusted path for generating returns in probabilistic games like investing while avoiding the risk of total ruin.


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Author: Soumitra Sharma

Operator-Angel I Product Leader I US-India corridor I Believer in Power Laws I Love building & learning

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