I don’t want to torment you this week, but if you trust me play along and you’ll be paid off with some non-obvious lessons.
Imagine the wish you made on your 10-year-old birthday candles comes true. You are magically given $1,000,000. But there’s a catch. You must expose it to either of the following risks:
1) You must put it all on a single spin at the roulette wheel at the Cosmo. You can choose any type of bet you want. Sprinkle the wheel, pick a color, a lucky number, whatever you want.
2) You can put all the money in play on a roulette wheel that has 70% black spaces. Place any bet you want, but you must bet it all. And one more catch…you are required to play this roulette wheel 10x in a row. Your whole bankroll including gains each time.
Think about what you want to do and why. Even if you cannot formalize your reasoning, take note of your intuition. I’ll wait.
First of all, the correct answer for anyone without a private jet is #1. Just spread your million evenly, pay the Cosmo its $52,600 toll and try not to blow the rest of it before you get to McCarran. For many of you who computed the positive expected value of option #2 then you might feel torn.
Welcome to a constrained version of the St. Petersburg paradox.
The expected value of a single spin with a million dollars spread over the favorable blacks is $400,000 (.70 x $1,000,000 – .30 x $1,000,000). A giant 40% return.
But if you are forced to play the game 10x in a row, there is a 97% you will lose all your money (1-.70^10).
What’s going on?
This problem highlights the difference between arithmetic or simple average return vs a compounded return. If you made 100% in an investment over 10 years, the arithmetic average would be 10% per year while the compounded annual return would be 7.2%. I won’t demonstrate the math, but you can always ask me or just Google it. The mechanics are not the point. An understanding of the implications will be, so hang on.
In option #1, you will be in simple return land. In option #2, you are in compounded return land. Compounded returns are not intuitive, but they are much more important to your life. Let’s see why.
Sequencing and the geometric mean
- Compound returns govern quantities that are sequenced such as your net worth or portfolio. If you earn 10% this year, then lose 10% next year, you are net down 1%., right? While the arithmetic average return was 0% per year, your compound return is -.50% per year (.99^2 – 1).
- Let’s thicken the plot by increasing the volatility from 10% to 20%. If you win one and lose one, your arithmetic mean is 0, but now your compound return is -2% per year. Interesting.
- Let’s turn to Breaking The Market to see what happens when we tilt the odds in our favor and really ramp the vol higher. In his game, a win earns 50%, while a loss costs you 40%.
- The expected value of betting $1 on this game is 5%. But this is the arithmetic average. The geometric average is a loss of 5%!
- If you played his game 20x, your mean outcome is positive but relies on the very unlikely cases in which you have an almost impossible winning streak. You usually lose money.
- As BTM explains: Repeated games of chance have very different odds of success than single games. The odds of a series of bets – specifically a series of products (multiplication)- are driven by, and trend toward, the GEOMETRIC average. Single bets, or a group of simultaneous bets -specifically a series of sums (addition)-, are driven by the ARITHMETIC average.
The most important insights to remember!
- Arithmetic means are greater than geometric means; the disparity is a function of the volatility.
- Mean returns are greater than median and modal returns (Wikipedia pic). In other words, even in positive expected value games, if the volatility is high and you bet the bulk of your bankroll, your most likely outcomes are much worse than the mean.
Using this in real life
Recognize compounded returns when you see them. We have already seen them in the domain of betting and investing.
Consider these questions.
- I want to raise the price of my product by 60%, how many customers can I lose while maintaining current revenue?
- If CA experiences a net population outflow of 20% in the next 20 years, how much would it need to raise taxes on those that stayed behind to make up the shortfall?
- If muscle burns 2x as much calories at fat and I lose 40% of my muscle mass, how much less calories will I burn while at rest?
After groping around with those you may have found the general formula: X / (1-X)
If you lose 20%, you need to recover 25% to get back to even. Lose 50%, and you need 100% to get back to even. 100% volatility and you are certain to go broke. Look at the slope of that sucker as you pass 2/3.
In other words, negative volatility is a death spiral. Let the brutality of the math sink in.
Why has nearly every real estate developer you know went bust at some point? Because they are in the most cyclical business in the world and love leverage. Leverage amplifies the volatility of their returns by multiples. Compounded returns are negatively skewed. Mercifully for them, zero (aka bankruptcy) is an absorbing barrier.
- Diversify your bets. In the earlier casino example, if you could divide your million dollars into 10 100k bets you would now have a basket of uncorrelated bets. If you could bet 1/10th of your bankroll on 10 such wheels you’d expect to make 400k in profit (7 wins out of 10 spins). With a standard deviation of 1.45 you now have a 95% chance of getting at least 5 heads and breaking even on the bet instead of a 97% to go bust in the version where you bet everything serially.
- When a bet is very volatile, reduce your bet size. If you put 100% of your net worth into a 20% down payment on a home you lose half your net worth if housing prices ease 10%. In investing applications, variations of Kelly criterion are good starting points for bet sizing.
- Remember that for parallel bets to not be exposed to disastrous volatility, your investments must not be highly correlated. Having a lot of investment in the stock market and high beta SF real estate simultaneously is an illusion of diversification. Likewise, if you own 10 businesses, you will likely want them in separate LLCs. For those in finance, you will immediately recognize the divergence in interests between a portfolio manager of a multi strat fund and the gp of the fund. Izzy Englander wants his strategies to diversify each other while he gets paid on the assets, while the individual PM wants to take maximum risk. Izzy risks his net worth, the PM just her job. If you take one thing away from this paragraph: a basket of options is worth more than an option on a basket.
- Insurance is by necessity a negative expected value purchase. You buy it because it ensures financial survival. In arithmetic return land it’s a bad deal, but if the insurance avoids ruin, it may have a profoundly positive effect on compounded returns which is what we actually care about.
- Finally, the power of portfolio rebalancing. If you hold several uncorrelated assets, by rebalancing periodically you narrow the gap between the median and mean expected returns. This is more apparent if there is wide differences in the volatilities of your assets.
- I ran a bunch of Monte Carlo sims on “coin flip assets” with positive drift. Some takeaways were a bit surprising.
- If the volatility of your portfolio is about 9% per year, median returns are about 90% of the mean returns. At this level of volatility, rebalancing has little effect.
- If the volatility of your portfolio is about 15% per year, median returns are about 50% of the mean returns if you rebalance.
- Rebalancing actually lowers your mean returns when the volatility of the portfolio is high even though it raises the median. My intuition is by taking profits in the higher volatility assets it truncates the chance of compounding at insane rates, but it also cuts the volatility by so much that it provides a much more stable compounded return. The higher the volatility the more of the mean return is driven by highly unlikely right upside moves.
- The impact of high volatility is stark. It is extremely destructive to compounded returns.
For finance folk and the curious
- Compounded returns are negatively skewed. Black-Scholes option models use a lognormal distribution to incorporate that insight. The higher the volatility, the greater the distance between the mean and mode of the investment. Example pic from Quora.
- A recollection from the dot com bubble. Market watchers like to say the market was inefficient. The options market would disagree. Stock prices and volatilities were extremely high reflecting the fact that nobody understood the ramifications of the internet. Had you looked at the option-implied distributions is was not uncommon to see that a $250 stock had a modal implied price of $50. To be hand-wavey about it, the market was saying something like “AMZN has a 10% of being $2050 and a 90% chance of being worth $50.” In other words, if you bought AMZN there was a 90% chance you were going to lose 80% of your money. If you are itching to get technical on the topic Corey Hoffstein’s paper explores how risk-neutral probabilities relate to real-world probabilities.
- For option wonks, (assuming no carry costs) you’ll recall the concept of variance drain. The median expected stock price is S – .5 * variance. The mode is S – 1.5*variance. The higher the variance, the lower the median and mode! The distribution gets “squished to the left” as the probability the stock declines increases in exchange for a longer right tail like we saw during the dotcom days.
- The expensive skew embedded in SPX option prices reflects 2 realities. First, the average stock in the index will see its volatility increase but more critically the cross-correlation of the basket will increase. Since index option variance is average stock variance x correlation, there is a multiplicative effect of increasing either parameter. The extra rocket fuel comes from the parameters themselves being positively correlated to each other.