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 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.
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!
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.
Summary
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!