During an interview with Ted Seides, investor Andrew Tsai recounts an internship at the well-known trading firm Susquehanna in the mid-90s (disclosure: I worked there for 8 years after college). In particular, he remembers a company outing to a dog track that summer:
I’m sitting next to one of the partners and I’m looking at the sheet of all the races, and he’s like “How are you gonna bet?” I respond, “Well, I’ve never really done this before but this dog looks like he’s got a good track record and he’s been running strong lately.”
The guy looked at me like I was a complete idiot.
He’s like, “What are you talking about, ‘How is this dog doing?'”
Andrew is perplexed. Well, isn’t that kind of what we’re talking about.
The partner starts to explain, Look at the relative value of this dog and that dog.
The lightbulb went on for Andrew.
“We started talking about spread trading and trying to capture that basis and I’m like ‘These are my guys’. It was really this culture of dissection that I loved.”
Relative Value Goggles
One of my favorite Twitter follows is the anonymous account @econompic. He’s in my top 5 and you should follow him too (only about 15% of my followers follow him which is basically as stupid as a butterfly trading for a credit). Go for the finance stuff and stay for takes on breakfast cereal, Weezer, and the NBA. Oh and the polls. You see, Jake’s polls act like the Susquehanna partner while Andrew is the rest of #fintwit. They are cleverly designed to surface mispricings in how people think about risk or relative value.
His relative value instincts are well-tuned. It’s like he has goggles that allow him to filter the world through prices. It’s a lens that’s critical for trading. One of his recent tweets is a great example of this. I’ll withhold the full tweet for now since it has spoilers. Let’s start with this screenshot:
So which shot do you take?
(take note of your answer and reasoning before continuing)
The first thing that should leap off the screen is the gap between the free throw and the top-of-the-key 3. Using NBA dimensions, that’s a 15′ shot vs a 23’9″ shot. And you are rewarded 5X for it from the benevolent genie offering this bet. The reflex you need to hone is that:
Prices imply probabilities
Because of expected value. Expected value is the probability of payoff times its magnitude. Would you pay Best Buy $50/yr to insure a $1,000 TV? If there’s more than a 5% chance that it fails you might. If there was a $500 deductible then the benefit is cut by half and you need to think there’s at least a 10% chance the TV fails. And if you think you get more TV per buck every year thanks to innovation then purchasing insurance implies an even greater defect rate.
So when you weigh the cost of different choices (insure vs not insure, fix vs replace, cheaper product vs more durable product) you are implicitly weighing probabilities. Making that explicit can expose mispricings.
Let’s go back to basketball.
Dissecting the basketball shot.
Just to get a hang for the reasoning let’s start with a simplifying assumption. You are 100% to make the layup.
- Free Throws
How confident do you need to be from the free-throw line to forgo the certain $50,000 you’d make from a layup? At least 50% confident. If you can shoot a free throw with a better percentage than a coin flip the free throw “has more equity”. If you are a 60% free throw shooter than that option is worth $60,000.
- Top-of-the-key 3
$500k to make this shot. You only need to be 10% confident to justify forgoing the layup for a chance at some big money.
Ok, here is where the probabilities should really get your senses tingling. The free throw implies a 50% probability and the top-of-the-key 3 implies 10%. Are you 5x more likely to make a free throw than this 3-pointer?
Unless you are 7 and literally can’t heave a ball from the 3-point line, it’s hard to imagine your chance of making these shots to be so far apart. In fact, if the 7-year-old can’t reach the rim at all from long range, I have my doubts they can shoot consistently shoot 50% from the stripe in the first place. But I’m willing to concede that possibility. For an adult, that spread is too wide. You either can’t hit free throws with a .500 percentage or your chance of making a top-of-the-key 3 is greater than 10%.
To take an outside view, consider NBA players. Guys who shoot about 40% in games, can shoot between 65-75% in practice. HS coaches can tell you that a 30% 3-pt shooter can make about half their shots in practice. Since free throw percentages are bounded by 100% you are talking about no more than a spread of 2x between free throw and 3-pt percentage. Your margin of error on the spread could be 100% and you’d still only have a spread of 4x. These shots are priced at 5x!
An exactly 50% free throw shooter be a 12.5% 3-point-shooter using the most conservative estimates and this top-of-the key 3 is still “too cheap”. And remember, there is a conditional probability aspect to this since we are dealing in relative pricing. If you are certain you need a miracle to hit an uncontested 3-pointer there is almost no chance you are truly a 50% free throw shooter.
- The rest of the table
To amateurs the corner-3, without a view of the backboard or the chance for a lucky bank shot, is daunting. But are you really half as likely to hit a corner-3 vs the key-3? As we get into the low probability shots it’s reasonable for a person who really knows their habits to potentially parse these odds but it takes quite a bit of experience to know that you are really 100% better at top-of-the-key 3s then corner 3s. Without that conviction, I’d take the better implied odds in the corner-3.
The entire payoff schedule suggests that you should either take a layup or a corner 3 as you are being offered very cheap relative pricing on those options. You can check out the rest of the tweet for the comments and replies. (Link)
What If You’re Broke?
If you read the thread there’s mention about how being broke can push you towards the layup even if the expected value of another choice is higher. This is a great opportunity to bring ideas like “risk aversion” or “diminishing marginal utility of wealth” into practical consideration.
The expected value framework above is an optimal case. It assumes every dollar has equivalent value to the player. The fancy term for this is “risk neutral”. If you have $5,000 and making another $5,000 has a “happiness value” that is equal and opposite to the “sadness value” that you experience if you lose $5,000 then you are risk-neutral. Since you are not a robot and need to eat, you are not risk-neutral. You would not bet all your money on a 50/50 coin flip. And you probably wouldn’t do it if you had a 60% of winning the flip. You are “risk-averse”.
A related concept is the diminishing value of additional wealth. This is pretty obvious. Jeff Bezos’ first million probably felt good. Today, it would be an imperceptible amount on his Mint dashboard.
Without knowing the lingo we all understand the intuition. If you are a broke college kid you might always opt for the layup. A sure $50k might mean getting out from under that 15% credit card APR, while $100k is ‘nice to have’, not ‘need to have’. That first $50k can be life-changing by getting you off the wrong path.
Likewise, the rich gal with a vacation house in Malibu is not so constrained. She can rely on the optimal pure expected value prescription. Just as a trading firm with a huge bankroll is willing to bet large sums on small edges. They will optimize for EV when the bet sizes are small relative to capital.
Our intuition moves us in the right direction. It tells us that the college student will be more conservative in choosing which shot to take. By mixing in a simple concept like “utility of wealth”, we can actually re-price all the probabilities implied by the shot payoffs.
Adjusting Probabilities For Risk Aversion
Linear vs Concave Utility
- Risk-neutral utility curves are linear.
If you are the risk-neutral robot every dollar you make is worth exactly the same to you. Your second million is as sweet as the first. That’s a linear utility function. Those are the curves embedded in any expected value proposition which simply spits out “pick the highest one”. I presumed such a framework in the prior table that said: “Min Probability To Accept Shot”.
- Risk-averse utility curves are concave
If you are risk-averse, every additional dollar is not worth quite as much as the one before it. And every extra dollar you lose hurts just a tad more than the one before it. Losing your rent money hurts more than losing your Ferrari money. So instead of a linear function, we need a function that:
1. Is always increasing to reflect that more money is always better than less money (‘Mo Problems and other first-world complaints notwithstanding).
2. Slope starts out faster than the linear model then flattens as we make more money.
Luckily, there is a simple function that does exactly that. The log or natural log function. People who study “risk-aversion” and diminishing marginal utility of wealth don’t think about it linearly. They don’t presume $5,000,000 is twice as “useful” as $2,500,000. They might say it’s only 1.75 as “useful” ( ln 5 / ln 2.5 = 1.75).
Re-computing Minimum Probabilities As A Function Of Starting Wealth
- 25 year old with $10,000 to his name.
The guaranteed layup increases his wealth by 6x and log wealth by 2.8x.
The free throw increases his wealth by 11x but his log wealth by only 3.4x!
Look how much it raises the minimum probabilities for him to accept various shots if he has a log wealth utility preference. He needs to shoot 3s as well as a good [contested] NBA shooter to gamble on the big money instead of the layup!
- Give that guy a $10,000,000 bank account, and he’ll choose according to Spock-like expected value prescriptions.
- Finally, check out the implied minimum shot probabilities for various levels of wealth. The larger your bankroll the more you can rely on probabilities imputed simply by expected value. If you are fabulously rich, you aren’t paying up for life insurance, home insurance, and so forth. You’ll deal with those bills as they come. For most of us, calamities mean financial ruin.
How we decide depends not just on the expected value but on our own situations. The more secure we are (on the flatter section of the log wealth curve) the more we can afford to act optimally.
(There is quite a bit of fuel for liberal policymakers here. They will realize that this is another example of Matthew effect or accumulated advantage. Richer people can avoid negative EV trades like insurance. Another thought. The inflection point on the so-called Laffer curve is probably much further to the right if we re-scale the axis in terms of log wealth suggesting we may tolerate much steeper graduated tax brackets. I’m not making a political opinion so don’t @ me. I’m just observing things that I’m sure have been discussed elsewhere.)
Prices impute probabilities. By taking the extra effort to make this explicit we can de-fog our relative value goggles. This improves our decision making in trading and life.
Since we are not “risk-neutral” robots the correct decisions are often theoretical. Translating the prescription to your own situation is an extra step that we typically leave to our intuition. This is quite reasonable. At the end of the day, we aren’t going to define our own wealth functions in Excel (log wealth is just one example of a non-linear function that seems to accommodate our intuition but the actual slopes and smoothness can vary quite a bit from person to person).
I recommend following Jake. His polls will help you tune your intuition.