Possible Insight

I privately received a couple of interesting comments on my diversification post:

One of RSCM‘s angel advisors wrote, “I would think most smart people get it intellectually, but many are stuck in the mindset that they have a particular talent to pick winners.”

One of my Facebook friends commented, “VC seems to be a game of getting a reputation as a professional die thrower.”

I pretty much agree with both of these statements. However, even if you believe someone has mad skillz at die-rolling, you may still be better off backing an unskilled roller. Diversification is that powerful! To illustrate, consider another question:

Suppose I offered you a choice between the following two options:

(a) You give me $1M today and I give you somewhere between $3M and $3.67M with 99.99% certainty in 4 years.

(b) You give me $1M today and a “professional” rolls a standard six-sided die.  If it comes up a 6, I give you $20M in 4 years. Otherwise, you lose the $1M. But this guy is so good, he never rolls a 1 or 2.

The professional’s chance of rolling a 6 is 25% because of his skill at avoiding 1s and 2s. So option (b) has an expected value of $5M. Option (a) only has an expected value of $3.33M. Therefore, the professional has a 50% edge. But he still has a 75% chance of losing all your money.

I’m pretty sure that if half their wealth were on the line, even the richest players would chose (a).  Those of you who read the original post probably realize that option (a) is actually an unskilled roller making 10,000 rolls.  Therefore:

Diversifying across unskilled rolls can be more attractive than betting once on a skilled roller.

Of course, 1 roll versus 10,000 hardly seems fair.  I just wanted to establish the fact that diversification can be more attractive than skill in principle.  Now we can move on to understanding the tradeoff.

To visualize diversification versus skill, I’ve prepared two graphs (using an enhanced version of my diversification spreadsheet).  Each graph presents three scenarios: (1) an unskilled roller with a standard 1 in 6 chance of rolling a 6, (2) a somewhat skilled roller who can avoid 1s so has a 1 in 5 chance of rolling a 6, and (3) our very skilled roller who can avoid 1s and 2s so has a 1 in 4 chance of rolling a 6.

First, let’s look at how the chance of at least getting your money back varies by the number of rolls and the skill of the roller:

The way to interpret this chart is to focus on one of the horizontal gray lines representing a particular probability of winning your money back and see how fast the three curves shift right.  So at the 0.9 “confidence level”, the very skilled roller has to make 8 rolls, the somewhat skilled roller has to make 11, and the unskilled roller has to make 13.

From the perspective of getting your money back, being very skilled “saves” you about 5 rolls at the 0.9 confidence level. Furthermore, I’m quite confident that most people would strongly prefer a 97% chance of at least getting their money back with an unskilled roller making 20 rolls to the 44% chance of getting their money back with a very skilled roller making 2 rolls, even though their expected value is higher with the skilled roller.

Now let’s look at the chance of winning 2.5X your money:

The sawtooth pattern stems from the fact that each win provides a 20X quantum of payoff.  So as the number of rolls increases, it periodically reaches a threshold where you need one more win, which drops the probability down suddenly.

Let’s look at the 0.8 confidence level.  The somewhat skilled roller has a 2 to 5 roll advantage over the unskilled roller, depending on which sawtooth we pick.  The very skilled roller has a 3 roll advantage over the unskilled roller initially, then completely dominates after 12 rolls. Similarly, the very skilled roller has a 2 to 5 roll advantage over the somewhat skilled roller, dominating after about 30 rolls.

Even here, I think a lot of people would prefer the 76% chance of achieving a 2.5X return resulting from the unskilled roller making 30 rolls to the 58% chance resulting from the very skilled roller making 3 rolls.

But how does this toy model generalize to startup investing? Here’s my scorecard comparison:

• Number of Investments. When Rob Wiltbank gathered the AIPP data set on angel investing, he reported that 121 angel investors made 1,038 investments. So the mean number of investments in an angel’s portfolio was between 8 and 9. This sample is probably skewed high due to the fact that it was mostly from angels in groups, who tend to be more active (at least before the advent of tools like AngelList).  Therefore, looking at 1 to 30 trials seems about right.
• “Win” Probability. When I analyzed the subset of AIPP investments that appeared to be seed-stage, capital-efficient technology companies (a sample I generated using the methodology described in this post), I found that the top 5% of outcomes accounted for 57% of the payout. That’s substantially more skewed than a 1 in 6 chance of winning 20X.  My public analysis of simulated angel investment and an internal resampling analysis of AIPP investments bear this out. You want 100s of investments to achieve reasonable confidence levels. Therefore, our toy model probably underestimates the power of diversification in this context.
• Degree of Skill. Now, you may think that there are so many inexperienced angels out there that someone could get a 50% edge. But remember that the angels who do well are the ones that will keep investing and angels who make lots of investments will be more organized. So there will be a selection effect towards experienced angels. Also, remember that we’re talking about the seed stage where the uncertainty is the highest. I’ve written before about how it’s unlikely one could have much skill here. If you don’t believe me, just read chapters 21 and 22 of Kahneman’s Thinking Fast and Slow. Seed stage investment is precisely the kind of environment where expert judgement does poorly. At best, I could believe a 20% edge, which corresponds to our somewhat skilled roller.

The conclusion I think you should draw is that even if you think you or someone you know has some skill in picking seed stage technology investments, you’re probably still better at focusing on diversification first.  Then try to figure out how to scale up the application of skill.

And be warned, just because someone has a bunch of successful angel investments, don’t be too sure he has the magic touch. According to the Center for Venture Research, there were 318,000 active angels in the US last year. If that many people rolled a die 10 times, you’d expect over 2,000 to achieve at least a 50% hit rate purely due to chance! And you can bet that those will be the people you hear about, not the 50,000 with a 0% hit rate, also purely due to chance.

In science, there isn’t really any such thing as a “fact”.  Just different degrees of how strongly the evidence supports a theory. But diversification is about as close as we get. Closer even than evolution or gravity. In “fact”, neither evolution or gravity would work if diversification didn’t.

So I’ve been puzzled at some people’s reaction to RSCM‘s startup investing strategy.  They don’t seem to truly believe in diversification. I can’t tell if they believe it intellectually but not emotionally or rather they think there is some substantial uncertainty about whether it works.

In either case, here’s my attempt  at making the truth of diversification viscerally clear.  It starts with a question:

Suppose I offered you a choice between the following two options:

(a) You give me $1M today and I give you $3M with certainty in 4 years.

(b) You give me $1M today and we roll a standard six-sided die.  If it comes up a 6, I give you $20M in 4 years. Otherwise, you lose the $1M.

Option (b) has a slightly higher expected value of $3.33M, but an 83.33% chance of total loss. Given the literature on risk preference and loss aversion (again, I highly recommend Kahneman’s book as an introduction), I’m quite sure the vast majority of people will chose (a).  There may be some individuals, enterprises, or funds who are wealthy enough that a $1M loss doesn’t bother them.  In those cases, I would restate the offer.  Instead of $1M, use $X where $X = 50% of total wealth. Faced with an 83.33% chance of losing 50% of their wealth, even the richest player will almost certainly chose (a).

Moreover, if I took (a) off the table and offered (b) or nothing, I’m reasonably certain that almost everyone would choose nothing. There just aren’t very many people willing to risk a substantial chance of losing half their wealth. On the other hand, if I walked up to people and credibly guaranteed I’d triple their money in 4 years, almost everyone with any spare wealth would jump at the deal.

Through diversification, you can turn option (b) into option (a).

This “trick” doesn’t require fancy math.  I’ve seen people object to diversification because it relies on Modern Portfolio Theory or assumes rational actors.  Not true.  There is no fancy math and no questionable assumptions. In fact, any high school algebra student with a working knowledge of Excel can easily  demonstrate the results.

Avoiding Total Loss

Let’s start with the goal of avoiding a total loss. As Kahneman and Tversky showed, people really don’t like the prospect of losing large amounts. If you roll the die once, your chance of total loss is (5/6) = .83.  If you roll it twice, it’s (5/6)^2 = .69.  Roll it ten times, it’s (5/6)^10 = .16. The following graph shows how the chance of total loss rapidly approaches zero as the number of rolls increases.

By the time you get to 50 rolls, the chance of total loss is about 1 in 10,000.  By 100 rolls, it’s about 1 in 100,000,000.  For comparison, the chance of being struck by lightning during those same four years is approximately 1 in 200,000 (based on the NOAA’s estimate of an annual probability of 1 in 775,000).

Tripling Your Money

Avoiding a total loss is a great step, but our ultimate question is how close can you get to a guaranteed tripling of your money.  Luckily, there’s an easy way to calculate the probability of getting at least a certain number of 6s using the Binomial Theorem (which has been understood for hundreds of years).  One of many online calculator’s is here. I used the BINOMDIST function of Excel in my spreadsheet.

The next graph shows the probability of getting back at least 3x your money for different numbers of rolls.  The horizontal axis is logarithmic, with each tick representing 1/4 of a power of 10.

As you can see,  diversification can make tripling your money a near certainty. At 1,000 rolls, your probability of at least tripling up is 93%. And with that many rolls, Excel can’t even calculate the probability of getting back less than your original investment. It’s too small. At 10,000 rolls, the probability of less than tripling your money is 1 in 365,000.

So if you have the opportunity to make legitimate high-risk, high-return investments, your first question should be how to diversify. All other concerns are very secondary.

Now, I will admit that this explanation is not the last word. Our model assumes independent, identical bets with zero transaction costs. If I have time and there’s interest, I’ll address these issues in future posts. But I’m not sweeping them under the rug. I’m truly not aware of any argument that their practical effect would be significant with regards to startup investments.