Asymmetric Risk

Suppose historical stock returns yield 10% per year. In a decade of HODL, you'd get:

  • Investment A: \( \alpha = 1.1^{10} =  2.59\) times capital returned (or 159% gains).

Now suppose you're gambling on some startups, and there's a .5% chance to get a 518X capital returned over the next decade (the other 99.5% is a total loss). You'd get:

  • Investment B: \(E[\alpha]= .005 \times 518 = 2.59\) times capital returned (same returns)

Which is a better investment?

On the outset, it seems like only an insane person would take 0.5% at success, but the reward is so big that the expected value is the same as if you were to conservatively put all your eggs in the cheap index funds basket.

But let's look closer on the risks involved here.

Investment A

In Investment A, you're risking all of your capital to be able to multiply that money by 2.59 times at the end of the decade. What happens if the market crashes right as you need that money? Maybe at the end of the decade, you'd really like to spend that money traveling and enjoying life before you get too old, or maybe you just need it to pay off hefty medical bills. While you'll probably get away with investment A maybe 75% of the time, the 25% of the time when your timing is bad, you'll be in hot water (assuming we're 25% of the time in a recession).

Investment B

Downside Calculation

In Investment B, you can achieve the same returns as investment A by dividing up your money into 200 chunks. With each chunk of money, you just bet on things that have a .5% of returning you 518X capital returned. But how likely is it even for you to hit anything if you only have a .5% chance of success?

Let's answer the reverse question first: how likely is it for you to not hit a single 0.5% jackpot? It's \((1-.005)^{200} = 36.7%\).

In other words,

  • Investment A there's a 25% chance you'll have significant less than your original capital while
  • in investment B there's a 36.7% you lose everything.

Upside Calculation

So what's the upside of investment B? There's a 63.3% that you'll at least perform as well as investment A's average performance. But actually, how many times would you be able to win twice, or 3 times?

  • Win 2 times (P(Lose 198 times)*P(Win 2 times)*All possible combos you in 2 times): \((1-.005)^{198} \times .005^{2} \times 200C2=.1844\) or 18.44%.
  • Win 3 times: \((1-.005)^{197} \times .005^{3} \times 200C3=.0611\), or 6.11%.

...And the probability drops off from there. This makes sense because the cumulative probability for you to win should add up to 63.3%, and you can win up to 200 times.

With a similar calculation, it turns out in order to win exactly once is 36.88%!

\[(1-.005)^{199} \times .005^{1} \times 200C1=.3688=36.88\%\]

So your winning probabilities actually look like this:

  • 36.7% you lose everything
  • 36.88% you break even with average performance of investment A
  • 26.42% you do better than investment A.

So which is better?

It depends. Investment A has a lot more downside protection because it's very unlikely that you'll experience a total loss with an index fund (unless you're very unlucky and it's a hyperinflation scenario like we saw at the end of WWI and the entire currency you're in is obliterated). Investment B has a 36.7% of losing everything which is exxtremely risky. However, investment B can "win twice" or "win thrice" a good 24.55% of the time, which would be significantly better than investment A (since they'd yield 5.18X and 7.77X returns on capital, respectively).

Thus, investment B is great for those trying to maximize upside, and investment A is great for those trying to preserve capital. If your risk tolerance is somewhere in between, you may choose to have a blend of them. Examples of things that behave like (but with likely different characteristics) investment B are:

  • Call options
  • Risky cryptocurrency coins
  • Flipping NFTs
  • Angel investing

The hard part about investment B is assigning an appropriate expected multiplier return and what the probability of success is. Another hard part with investment B is that you have to do a lot of bets for the numbers to work out well. In my example above, you need to make 200 bets. But if you had .5% of winning and only made 30 bets, you are much better off with investment A for example.

Personally, I like doing investment A with some of money money and then using the rest of the money (i.e. money I am OK with losing) to gamble with on investment B.

Exercise for the reader

  1. What happens if you could make 800 bets (instead of 200 bets) of .5% probability of winning 518X?
  2. What conclusions can you draw about making a ton of Investment B bets?
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