Do you think proportionally?

Yale Finance professor Kelly Shue explains non-proportional thinking and the mistakes it can cause you to make in arenas as disparate as trading stocks or picking medical treatments

Every winter, I have the pleasure of (re)teaching a Wharton MBA course that covers classic psychology research on decision biases and heuristics. I’ve been teaching this material for fifteen years but it still feels fresh and fun because the takeaways are so clear and practical. In a nutshell, I teach my students that humans are mediocre intuitive statisticians who tend to make predictable (and avoidable) mistakes as a result. I’m always looking for interesting new examples of how much this matters, and a few years ago I stumbled across a truly terrific paper by a Yale Finance professor demonstrating how one classic decision bias can lead investors to make major errors. An interview about that paper is the focus of this month’s newsletter, and I hope you’ll find it intriguing!

This Month’s Recommended Listens and Reads

• What Reliably Boosts Progress on Long-Term Goals?: If you set yearly or even monthly goals, check out this Scientific American op-ed where my co-authors and I share our research on a simple and often overlooked tactic for boosting your progress.

• An Economist’s Research Shows How the Distant Past Shapes our Present: This terrific interview with economist Daron Acemoglu on the People I (Mostly) Admire podcast covers questions as disparate as how colonialism still affects us today and who benefits from new technologies.

• Look Again is a fantastic new book out today from the incredible Tali Sharot and Cass Sunstein that will help you refresh your point of view. It reveals why it’s easy to be lulled into complacency about anything and how to prevent falling into this trap.

• A Conversation with Cass Sunstein about the Past, Present and Future of Behavioral Public Policy: Last month, along with Penn psychology professor Angela Duckworth, I had the pleasure of interviewing the legendary Harvard law professor Cass Sunstein at Penn, and our conversation is now available on YouTube.

---

Q&A: Non-Proportional Thinking

In this Q&A from Choiceology, Yale Finance professor Kelly Shue discusses her research on non-proportional thinking and the mistakes it can cause you to make when trading stocks or picking medical treatments.

Me: I’m really excited to talk to you about non-proportional thinking. What is this bias and how have you studied its effects on investors?

Kelly: Non-proportional thinking is when investors think about stock price reactions to news in dollars rather than percentages. For example, if there's good news that a skilled new CEO has joined, investors should think this firm's value will go up maybe one percent. The wrong way to think about it, and that would be non-proportional thinking, is, ‘well, other times when firms had similar good news, they went up about a dollar. So, this firm should also go up in value by a dollar per share.’

This is the wrong way to think about it because share prices are basically meaningless. You could take the same firm, divide it into a different number of shares, and have a different share price. Suppose you had two identical firms, same size, one is trading at $20 per share and the other is trading at $40, but the $40 per share firm has a higher share price simply because it's divided into half the number of shares. It would be wrong to think that due to a new CEO arriving, both share firms should go up by $1. That's non-proportional thinking. Because if you think that way, then the $20 per share firm has gone up 5% and the $40 per share firm has only gone up 2.5%. When they should have the same percentage return to this good news.

Me: That’s really clear! I’d love to talk about the underlying psychology. You've argued that this is closely related to denominator neglect or ratio bias. Can you explain what those biases are?

Kelly: Denominator neglect or ratio bias is the tendency of people to focus too much on the numerator and not enough on the denominator when they're thinking about things that are ratios or percentages.

I think the most interesting and important example is in the medical context. There's a paper by Yamagishi in 1997 that finds that when cancer is described as killing 1,286 people out of 10,000 (a 12.8% kill rate), people think that's riskier than when a cancer kills 24 out of 100 people. Even though that translates to a higher mortality rate. People should be thinking about 12% versus 24%. But, instead, they're drawn to the numerators: 1,286 people dying or 24 people dying.

Me: That’s so interesting. Please tell me more about the research you’ve done on the implications of this bias for the stock market.

Kelly: Well in finance and in the stock market, people should be thinking about returns. And returns are basically ratios — it's the dollar change divided by the share price as of the previous day. But if people neglect the denominator and they only focus on the numerator, which is the dollar change, then they might react the same to stocks that are otherwise identical, but trading at different share prices.

In particular, low-priced stocks might have greater return reactions to news. Say we think stocks should move up by a dollar based on some news. That $1 divided by a lower share price is going to translate into a greater return movement. We tested this in financial markets and focus on what happens before and after stock splits.

Me: A stock split is when, if you own two shares of a stock, suddenly that would become four shares because everything's being divided in half. If you own two shares worth $40 each, now you own four shares worth $20. Am I summarizing that adequately?

Kelly: Yes, that is exactly what a stock split is. Nothing about the firm has changed but the number of shares doubles and so the share price is half of what it used to be. So, with our scenario where investors think a stock should move up by $1, that same news should move up the share price by only 50 cents after the stock split. But suppose investors think the same news should continue to move up the share price by $1 after the stock split. This will translate to much bigger return swings. In other words, bigger return volatility after the stock split. And that's exactly what we find in the data. Even for a very large, publicly traded firms, what we find is share price volatility goes up by about 20 to 30% after a stock split and it remains elevated persistently after that for up to a year.

Me: So, the share price bounces around more than it used to because people are overreacting to news more now that individual shares are half as expensive as they used to be (even though the company’s total value is the same)?

Kelly: Exactly. And the firm is continuously getting news. Some of the news is good, so the share price will move up. Some of the news is bad, so the share price will go down.

Me: Why do you think it is that people are overreacting like this after stock splits?

Kelly: I think we can go back to how information is displayed to investors. So, the Wall Street Journal used to only publish dollar price changes for stocks. You can figure out the percentage change, but you have to do that work on your own. And that's still true, even for modern applications and television shows. The CNBC ticker shows how much a stock has moved in dollar units relative to the closing price yesterday. I got frustrated looking at the CNBC ticker, and my coauthor, Rick Townsend and I, were also talking about how silly it was that people said the Dow index had the largest point drop in history when the Dow index is at a much higher level than it was 10, 20, or 50 years ago. So the fact that it had a very large point drop is actually not a big deal. It had far larger percentage drops in its previous history.

Me: I love that origin story for this research, and certainly the way information is displayed to us shapes the way we think about it. If we abstract away from thinking about buying and selling stocks, what are the takeaways from understanding non-proportional thinking and denominator neglect?

Kelly: People have to be very careful thinking about problems in the right units. When you're thinking about the mortality risk of a cancer or any type of disease, you want to be thinking in percentages, not just how many people died.

There's also a famous example by Tversky and Kahneman that shows that people are more willing to travel 20 minutes to a different store to save $5 on a $15 calculator than they are to save $5 on a $125 calculator. And this is a mistake because what you should be comparing is your cost of traveling to a different store with your savings, which is $5. You should not think about $5 divided by $15 or $5 divided by $125. So I think people should try to think very carefully. Should I be thinking about a ratio or should I just be thinking about the raw dollar amount that I'm saving or getting?

Me: Oh, I love that. That’s a really good example. Thank you, Kelly.

This interview has been edited for clarity and length.

To learn more about Kelly’s work on using non-proportional thinking and denominator neglect, listen to the episode of Choiceology where we dig into the topic.

That’s all for this month’s newsletter. See you in March!

Katy Milkman, PhD

Professor at Wharton, Host of Choiceology, an original podcast from Charles Schwab, and Bestselling Author of How to Change

P.S. Join my community of ~100,000 followers on social media, where I shares ideas, research, and more: LinkedIn / Twitter / Instagram