We are so skewed!

Note that I didn’t say “screwed” but skewed. Well, it wouldn’t have made a difference because today’s post is about how we get screwed by skewness. 

But I’m getting ahead of myself. The other day I asked myself why would anyone buy lottery tickets? The return profile is atrocious! The average payout is probably only about 50% of the money raised. In a hypothetical lottery with a one in a million chance for a $500,000 prize and a ticket price of $1.00, your expected return is -50% in one week, which means essentially -100% compounded over a year. The standard deviation is $500, so 50,000% relative to the $1 investment. And that’s on a weekly basis, which translates into over 360,000% annualized. What’s worse, that jackpot payout is usually stretched over many years or decades with a much lower lump-sum payment. And it’s subject to income taxes, so the after-tax return is even bleaker! If Vanguard or Fidelity or Schwab offered a mutual fund with return stats like that everybody involved would be facing federal indictments!

Then why not invest the lottery ticket money in stocks? No one can tell me that they’re afraid of equity risk (about 10-15% annualized) when they buy lottery tickets with 360,000% annualized risk. Nowadays you can buy stocks or equity mutual funds in very small amounts. Our 529 account has a $25 minimum investment and you can buy single stocks on Robinhood. Then what’s the appeal of a lottery? In one word: Skewness, see the Wikipedia definition. In particular, positive skewness!

Positive Skewness: higher probability of large positive outcomes (e.g. lottery). Negative skewness: higher probability of large negative outcomes (e.g. stock market). Source: Wikipedia

Positive Skewness means that the likelihood of large positive outliers is much higher than that of large negative outliers. Case in point, a lottery ticket: Your worst return is -$1, or whatever the price of the lottery ticket may be.  The largest positive outlier might be in the hundreds of millions. 

Side note: Just for the record, the guy on the title picture is not Dr. ERN. That’s Peter Sellers starring as Dr. Strangelove. And I don’t smoke either!

How popular are lottery tickets? According to The Atlantic and CNN, Americans spend about $70 billion on lottery tickets each year. This study was from 2015 when nominal GDP was around $18 trillion and disposable personal income was $12.4 trillion (according to the St. Louis Fed). So, lottery ticket sales are worth around almost 0.4% of annual GDP and just over 0.5% of disposable personal income. Not really huge numbers but if we dig a little bit deeper we find that poorer households spend a staggering 9% of their disposable income on lottery tickets, according to this study (unfortunately from back in 2010)! Why is there such a draw to buy lottery tickets? Blame evolution!

Humans are programmed to like positive skewness and dislike negative skewness!

For millions of years, humans and our ancestors lived a subsistence life – a life on the edge. There was no Social Security Administration, no disability benefits, no welfare, no charities and the like. One really bad draw in the lottery of life could be life-altering or even life-ending. This behavioral bias is still in us. By the same token, we also like positive skewness. There are many examples in the animal kingdom of the “winner-takes-all” principle, for examples alpha males and alpha females getting all the perks!

How is this relevant for personal finance?

So, skewness messing with our brains costs us billions of dollars in wasteful spending on lottery tickets. Who cares? Isn’t this mostly a problem of financially unsophisticated people? Far from it! Even sophisticated investors get screwed by skewness. Over shorter horizons, equities have more tail events on the downside (think Black Monday) and this feature makes equities unattractive to overly risk averse and myopic investors. As we pointed out in our post on option writing (part 1 and part 2), protecting the equity risk downside is excessively expensive. On the other hand, selling downside protection (i.e. selling put options) is quite profitable but creates even more negatively skewed return patterns than a plain vanilla equity investment (a skewness of about -2 for the put writing strategy, compared to roughly -0.5 for equities).

The problem is: negative skewness is where the juicy returns are. And positive skewness is where the sucker bets reside. I took the liberty of ranking different asset classes and return profiles by their skewness and expected returns and there seems to be a clear pattern of a tradeoff between skewness and returns, see below.

Skewness vs Return for different asset classes
Skewness vs. Return (or risk-adjusted Return) by Asset Class. These are just estimates and may not be to scale!

A few notes on this chart:

  • This is just my personal “guess-timate” and not written in stone! For example, I wasn’t entirely sure about the skewness of Real Estate investments and the expected returns of Venture Capital investments. This is up for discussion. So, please weigh in if you disagree with my view!
  • Short Put and Covered Call have the same return characteristics if the strike price is the same (Put-Call Parity). The reason I assign a more negative skewness to the Put Writing strategy is that I use out-of-the-money Puts (strike below current underlying) in my personal option strategy and most people who write covered calls do so using strikes at or above the current price of the underlying, see The Retirement Manifesto.
  • Government bonds and investment-grade bond funds have close to zero, even slightly positive skewness, though individual bonds with default risk can have negative skewness (limited upside potential, small chance of a large loss in default).

How to trick my brain into liking negative skewness

Apart from the fact that a lot of investments with negative skewness pay a generous expected return, I rationalize my preference for negative skewness with (at least) two mental tricks:

1: Negatively skewed return distributions have higher probabilities of positive outcomes, all else equal. That sounds counter-intuitive, but let’s look at the chart below. It displays two distributions with mean zero and standard deviation of one. One is a nice symmetric distribution and looks almost like a Normal Distribution (albeit truncated and on discrete points between -2 and +2). It has zero skewness. The other has a negative skew, so a much larger probability of a -2 outcome than a +2 outcome. But we can’t just take away probability weight from the +2 outcome and shift it to the -2. That would alter the mean and variance. By putting more weight into the very bad outcome we have to compensate with much more weight on the moderately positive outcome of +1 to keep the mean and variance the same. And that implies that the probability of positive outcomes just went up. Not all is bad with negatively skewed distributions!

Skewness pdf
Two distributions with mean zero and a standard deviation of one. A distribution with negative skewness has a higher probability of big losses but also a higher probability of moderately positive outcomes.

A great example of a return pattern with negative skewness is that of an option writing strategy (see here how I implement this). The upside is limited while the downside is unlimited. It sounds like the opposite of what everybody desires, but that’s why it’s so profitable! I looked at my personal records and found that 94% (!) of my trades were profitable, even though they are very short-term, usually a week or shorter. 85% of daily returns are positive, much better than for equities. That makes the occasional large loss a lot more palatable.

2: Time-diversification! Negative skewness becomes less of a problem when averaging over many different independent bets. Here are the skewness stats for the Big ERN option trading portfolio at different horizons:

  • Daily: -5.05 (ouch! The S&P500 has a negative skewness, too, but usually only around -0.5)
  • Weekly: -4.87. Still very nasty at the average holding period of my short puts.
  • Monthly: -1.91. Still very negative, but already a lot better than at daily frequency!
  • Quarterly: +0.14. That’s not a typo! The skewness is now essentially zero at a quarterly frequency. The occasional large drawdowns (e.g. August 2015 and January 2016) are averaged out over enough weeks and months that the occasional sharp short-term drawdown doesn’t even register in the quarterly return series.

What’s the explanation? Say thanks to basic mathematics and statistics, in particular, the “Central Limit Theorem.” It states that the average outcome from independent draws from even very non-Normal distributions (e.g. very negatively skewed distributions) will look more and more “Normal” and thus unskewed.


Whatever happened to the finance rule “higher risk = higher return”? It can’t be universally true. One way to see that is to note that an option seller and buyer both face the same risk measured as variance or standard deviation. But option buyers, especially put option buyers to protect the downside have very poor expected returns. Expected returns are tied to risk, but mostly the downside risk, of course. Not only will we not get compensated much from exposure to the upside but we’d likely have to pay a premium for a lottery-style payoff. Getting over the aversion to downside risk and negative skewness took me some time and getting used to. But then I stopped worrying and learned to love the bomb negative skewness, to borrow from the movie. Let’s just hope our risk model works out better than for Dr. Strangelove. 😉

I hope I didn’t bore you with my ramblings about mathematics. Feel free to leave your own ramblings below! We are still traveling in Europe this week and might be slower than usual to respond to comments! Have a great rest of the week!



30 thoughts on “We are so skewed!

  1. WOW, great analysis! You’re right though, there’s no way there’s universal truth to higher risk = higher reward. And history has proven that over and over again. Sure, sometimes it works out, but the likelihood is actually low, such in the case of lottery tickets.

  2. Big ERN,
    Having a bad day at the ranch (so to speak). Thanks for another great post that helped brighten my day.

    If you look at negative skewness through the lens of Prospect Theory and Loss Aversion (Kahneman & Tversky), the psychological impact can be rather challenging to overcome. I really like your use of framing effects to help overcome the limitations of the wiring in our grey matter.

    Something tells me you have something up your sleeve wrt to skewness’ miscreant cousin kurtosis for a future blog post.

    1. My mind went right to loss aversion as well as I read this article. It’s an innate tendency of humans. In general my approach is just to automate as much as possible to take me out of the equation. Thats why I’m not trading options in any way. It’s not necessarily that I disagree I can make a better return doing it right. It’s that I know I’ll be tempted to step beyond or react to my own behavioral biases. As always I appreciate the mathematical ramblings.

      1. Yup, loss aversion is a very important topic. Stay tuned for a future post on that.
        I have streamlined the option trading so much, it doesn’t take much time any more. Maybe 5 minutes 3x a week. But I can see how the passive crowd just wants to go with the passive index funds.

    2. Exactly! Loss aversion has the same origin! Definitely worth its own post because it’s such an important behavioral bias!
      Oh my! Kurtosis is a tough one. Thinking about how to write something interesting on that one! 🙂
      Thanks for stopping by!!!

  3. You shouldn’t apologize for your “mathematical ramblings” — that’s exactly what I and I’m sure many others come here for. 🙂 Enjoy your travels!

  4. Thank you for this. I find these kind of analyses really fascinating.

    I think we need to update the saying from “higher risk = higher returns” to “higher perceived risk = higher returns.” The market determines returns. The market is made up of people. People have the biases that you laid out above. Like you, I try to look at the numbers and base my decisions on probabilities when investing. Once you force yourself to incorporate probabilities into decision-making, you eliminate a lot of the natural biases in our intuitive thought process.

  5. Oh man, my head hurts. I have a more simpler exercise, and that is to try and practice predicting the future and make benefits before everyone else catches on. Sometimes it works, sometimes it’s so obvious it’s nuts!

    1. Haha, thanks for stopping by! It got a little bit technical today, true!
      I don’t try to beat anyone in the finance game. I’m pretty sure stocks and option selling will continue to perform well because they pick up a risk and skewness premium. But I like to find out more about your investment strategies. Please keep us posted on your excellent blog! 🙂

  6. Big ERN – great stuff as usual (We are so skewed and your options posts). What are your thoughts on this CBOE paper and on the PUTW Etf (which I’m contemplating investing as it seems to be a decent alternative to directly selling options each week/month)? This research seems to imply selling monthly put options on the S&P may actually be preferable to weeklies (weeklies had lower max drawdown and vol, but monthly had higher returns / sharpe ratio over time period used). https://www.cboe.com/micro/buywrite/put-oleg.pdf. Also, what is your thoughts on volatility – to me it is an imprecise measure of risk as the only large movement that concerns me is downward (not upward).

    1. Hi Steve,
      Yes, the CBOE paper is spot on. I still prefer the weekly options the way I implement the strategy (with leverage and out-of-the-money puts). A lot more can go wrong in a month than in a week when you use leverage.
      The PUTW ETF is a nice vehicle, but it’s no competition to implementing it yourself. The disadvantage of the ETF(aside from the expense ratio!) is that it tries to replicate the CBOE put writing index, which specifies holding the margin cash in very short-term investments (e.g. money market fund). But it’s more efficient to hold the margin funds in slightly higher-yielding bonds (e.g. Muni Bonds) with additional yield and diversification benefit.
      As you indicate, volatility is only a problem on the downside. There are other measures, e.g., downside vol to account for that. Or people use the Sortino Ratio (https://en.wikipedia.org/wiki/Sortino_ratio) to account for shortfall risk.

      1. Thanks Big Ern! Appreciate it and I got it now- the OTM weeklies would have a much smaller draw down risk, which is essential when using leverage. Also, thanks for insight on the cash component of the Put write index.

  7. ERN,

    Nice post, I enjoyed it. As the $VIX drops, how much of a premium reduction are you seeing? I was eyeballing the equity curve in your other blog post. Taking out the draw downs the gains are more or less linear. If you had been reinvesting the gains, an exponential rise is expected had the premium stayed the same.

    I sell puts on stocks I’d like to own from time to time but never at anything close to your size. They usually expire worthless (small delta). Only recently had I been defending by rolling forward. These days I have an extreme bullish outlook, so I’ve been making highly directional bets.

    Good luck trading,

    1. Thanks for stopping by, NR!
      Yes, I have to admit, right now it’s slim pickings in the put selling world. Vol is too low! But then again, the market is so range-bound right now, there seems to be little potential for large drawdowns. I probably make about a quarter less in premiums than what I’m used to, but then again, this year I have kept close to 80% of the options premiums. Smaller income but it’s very steady so far this year.
      If you want to own the stocks, why not sell puts with a larger (to be precise more negative) delta? Why not at-the-money (delta=-0.50)? Pickup the stocks on the dips and then enjoy the rally. Just a thought. 🙂
      Good luck trading, to you as well!

      1. ERN,

        I should have been clearer. It’s stocks that I’d like to own at much lower prices. I tend to sell puts at delta=-0.2 with 4-6 wks to expiration with the principle aim of having them expire OTM. The fact I don’t mind owning them is a protection for getting exercised. Someone who specializes in premium selling will just turn around and write calls but I use it as a means to build a position at an attractive valuation.

        More recently I have been making directional bets in tech stocks which is where the action will be for the next 12-18 months. The volatility is much higher but days like yesterday and today make it worthwhile.


  8. Interesting stuff ERN!
    I wonder if you’ve calculated really big down events in your strategy. I’m sure you did. It’s just that your strategy is kind of the opposite of Nissim Taleb.

    1. Yes. I did some backtests during 2008 and even 1987 and the strategy didn’t blow up for moderate levels of leverage.
      All to be taken with a bit of caution because who knows how an investor would have behaved back then. And there is the issue of having to make some assumptions about some of the option price data because I don’t have the full dataset of option data.

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