MSNE Echoes PSNE

Let’s talk game theory. I’ve written in the past about Pure Strategy Nash Equilibria (PSNE). They identify possible equilibrium strategies, even if players are unlikely to adopt those strategies in real life. Students don’t like the implausibility of many PSNE strategies, and they sometimes struggle to limit their conclusions to the premises that yield PSNE. Students have a similar dissonance to Mixed Strategy Nash Equilibria (MSNE).

What is MSNE? A set of MSNE strategies allow a player to choose some strategies probabilistically – with probabilities that are less than 100%. That’s the feature of MSNE that distinguishes it from PSNE. In PSNE, a strategy is chosen with 0% or 100% probability.

Here’s an example to illustrate. Imagine that you are shopping at the grocery store with your shopping cart. You’re at one end of the aisle and another shopper is at the other end and your heading straight toward one another at a snail’s pace. Ideally, you’d not hit each other or awkwardly arrive in each other’s path. For simplicity, let’s say that each of you can walk on the right or the left side of the aisle only.* Below is a simultaneous normal form game with arbitrary payoffs.

There are two PSNE in the above game: each person walks on their right or their left side of the aisle. If you and the other person are both walking on your respective rights or lefts, then neither of you has an incentive to deviate. The alternative is that you are heading straight for one another and one of you must veer from their path or play an awkwardly low stakes game of chicken.

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One Up on Wall Street in the Meme Stock Era

Peter Lynch was one of the most successful investors of the 1970’s and 1980’s as the head of the Fidelity Magellan Fund. In 1989 he explained how he did it and why he thought retail investors could succeed with the same strategies in the bestselling book “One Up on Wall Street”. Given the meme stock exuberance of retail investors in the past few years, I thought the book might be due for a comeback.

Instead interest seems flat, and when I do hear Peter Lynch mentioned it is by institutional investors more than retail. But the book seems to me like it is still valuable, so I’ll share some highlights here. This one could easily have been written this year:

Where did the Dow close? I’m more interested in how many stocks went up versus how many went down. These so-called advance/decline numbers paint a more realistic picture. Never has this been truer than in the recent exclusive market, where a few stocks advance while the majority languish. Investors who buy “undervalued” small stocks or midsize stocks have been punished for their prudence. People are wondering: How can the S&P 500 be up 20 percent and my stocks are down? The answer is that a few big stocks in the S&P 500 are propping up the averages.

I see why the book hasn’t caught on with meme stock traders:

Nobody believes in long-term investing more passionately than I do… I think of day-trading as at-home casino care.

I’ve never bought a future nor an option in my entire investing career, and I can’t imagine buying one now. It’s hard enough to make money in regular stocks without getting distracted by these side bets, which I’m told are nearly impossible to win unless you’re a professional trader.

So where does he think retail investors have a chance to get “One Up on Wall Street”?

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The Crime Wave May Be Over

Crime of all forms certainly spiked in 2020 and 2021 in most of the US, and continued to remain high for a time after that. But recent data, especially homicide data compiled by AH Datalytics, suggest that crime is falling. When measured by homicide rates, the worst of crimes and the least likely to be underreported, homicide rates across 272 major cities in the US is down 17.6% in 2024 compared with the same period in 2023. And among the 20 cities with the most homicides in 2023, just one (Birmingham, the 20th on the list) saw an increase from 2023 to 2024.

But is this just coming down from a relative high? Are homicide rates still elevated from pre-pandemic? I went through the cities with the most homicides on the AH Datalytics list, and for those where I could find comparable data pre-pandemic, I created the following charts. As you will see, lots of these cities are down to or below pre-pandemic levels (for the period in 2024 that is comparable to prior years). Not every single city, of course, but most are close to 2019 or prior years.

How to (Almost) Double Your Investing Returns 2. Buy Deep in the Money Calls

Last week we described a simple way to achieve roughly double investing returns on some asset class like an S&P 500 stock basket, or a narrow class of stocks such as semiconductors, or on some commodity like gold or oil. That way is to buy one of the many exchange-traded funds (ETFs) which use sophisticated derivatives to achieve a 2X or even 3X daily movement in their share prices, relative to the underlying asset. For instance, if the S&P 500 stocks move up by 2% on a given day, the SSO ETF will rise by 4%.

Of course, these leveraged funds will also go down two or three times as much. They also have a more subtle disadvantage, which is that when the markets go up and down a lot, they tend to lose value due to their daily reset mechanism.

In this post we describe a different way to achieve roughly double returns, which does not suffer from this volatility drag issue. This way is to buy long-dated deep in-the-money call options on a stock or a fund.

Say what? We have described how stock options work here and here. The reader who is unfamiliar with options should consult those prior articles.

A stock option is a contract to buy (if it is a call option) or to sell (if it is a put option) a given stock at some particular price (“strike price”), by some particular expiration date. Investors generally buy calls when they believe that the price of some stock or fund will go up.  For a call option with a strike price far below the current market price of a stock, the market price of the option will move up and down essentially 1:1 with the market price of the stock.

For instance, as I write this the market price of Apple is about $230. Suppose I think Apple is going to go up by say $40 in the next six months. One way for me to capture this gain is to invest $230 in buying Apple stock. The alternative propose here is to instead of buying the stock itself, buy, say, a call option with a strike price of $115 and an expiration date of January 17, 2025. The current market price of this option is about $119.

Other things being equal, we expect that the market value of this call option will go up by $40 if Apple itself goes up by $40. But we have invested only $119, rather than $230, so our return on our investment is roughly double with the option than by buying the stock itself.

There is a subtle cost to this approach. At a stock price of $230 and a strike price of $115, the intrinsic value of this call option is $115. But we pay an extra $3 of extrinsic value when we buy the option for $118. This extrinsic value will gradually decay to zero over the next six months.

Thus, if Apple went up by $40 within the next month or so, we could turn around and sell this call option for nearly $40 more than our purchase price. But if we wait for six months before selling it, we would only net $37 (i.e., $40 minus $3). This is still fine, but it illustrates that there is a steady cost of holding such options. This annualized cost is about equal to or slightly higher than the prevailing short term interest rate (5% /year). This option pricing makes sense, since an alternative way to control this many shares would be to borrow money at current interest rates (5%) and use those borrowed funds to buy Apple shares. Options and futures pricing is generally rational, to make things like this equivalent, or else there would be easy arbitrage profits available.

As a side comment, the reason I am focusing on deep in the money calls here is that the extrinsic premium you pay in buying the call gets lower the further away the strike price is (i.e. deeper in the money) from the current stock price. A deeper in the money call does cost you more up front, but net net its dollar movements up and down more closely track (1:1) the movements of the underlying stock. So, if I am not trying to guess right on any market timing, but simply want to get the equivalent of holding the underlying stock but tying up less money to do so, I find buying a call that is about 50% in the money generally works well.

How I Use Deep in the Money Call Options

I consider the technology-oriented stock fund QQQ to be a core holding in my portfolio, so I would like to stay exposed to its movements. But I might as well do this on a 2X basis, to make better use of my funds. I do hold some of the 2X ETF QLD. But if we experience a lot of market volatility, the price of QLD will suffer, as explained in our previous post.

As a more conservative approach here, I recently bought a deep in the money call on the QQQ ETF. As usual, I went for a call option with a strike price roughly half of the market price, with an expiration date 6-12 months away. When this gets close to expiration (May-June next year), I will “roll” it forward, by selling my existing call option, and buying a new one dated yet another 6-12 months further out. This takes little work and little decision making. I will pay the equivalent of about 5% annualized cost on the decay of the extrinsic option premium, but I come ahead as long as QQQ goes up more than 5% per year.

This is a little more work than just holding the 2X QLD ETF, but it gives me a bit more peace of mind, knowing I have done what I can to smooth out some of the risk there. Of course, if QQQ plunges along with the markets in general, I will be looking at double the losses. For that reason, I am taking some of the money I am saving by using these leveraged approaches, and stashing it in safe money market funds. In theory that should give me “dry powder” for buying more stocks after they drop. In practice, I may be too frozen with fear to make such clever purchases. But at any rate, I should not be appreciably worse off for having used these leveraged investments (2X funds or deep in the money calls).

Disclaimer: As usual, nothing here should be considered advice to buy or sell any investment.

Maximizing winning versus minimizing losing

Spain just defeated England 2-1 to win the UEFA European Championship. I watched a fair amount of the last two European championships and the previous Men’s and Women’s World Cups, and one thing that stood out is the dichotomy in playing styles amongst the top 4-6 teams in their relative risk aversion. Putting aside the bottom teams whose relative talent level make it difficult to play anything but highly defensive soccer (bunkering in with 11 players behind the ball AKA parking the bus), it nonetheless remains shocking their seeming chasm in aggression between teams.

Watching England and Spain over the previous month, it is uncanny how much more willing to lose the Spanish sides were. Pressing hard, playing incisive but risky passes, each and every game they ran the risk of losing to a team that was fortunate enough to score off of an poorly placed pass or chaotic bounce. England, in point of contrast, played incredibly conservatively, themselves looking to win a close match by converting a small number of scoring opportunities at a higher rate through their superior talent and/or settling the match in a penalty shootout.

Playing conservatively is a valid strategy. Jose Mourinho bored countless millions of viewers while coaching superior teams to trophies that should have 1-0 etched onto them for eternity. But while watching these games and the those in the English Premiere League I’ve become aware of the compulsion English announcers have to for ascribing nearly every goal to an error made by the defense rather than an achievement of the offense. Spain scored two stunning goals today and the English commentator could not help but attribute at least (maybe both?) to the English team “falling asleep”.

This to me is another opportunity to ask yourself and others “What we are maximizing or minimizing?” Spain was maximizing the chance of winning the tournament. That meant focusing on nothing but the probability of winning each game. They were maximizing the mean. England, on the other hand, was minimizing the probability of losing to an inferior team. They were minimizing the variance. For two-thirds of the final, they continued to minimize the variance and eventually ended up down 1-0. They then subbed their obviously injured captain (who happens to be the best penalty kick taker in the world, maybe ever), put on a younger player, and were immediately more aggressive. They scored and then were scored upon. The bore the fruits and the costs of greater risk. I don’t know what would have happened if they had taken more risk the whole game, but I suspect their probability of winning would have increased.

When you see an organization that is minimizing variance that is often a good thing. It means they are valuing downside risk in a proper manner. In many sports contexts, however, over emphasis on minimizing downside risk is, to my eyes, an example of actors minimizing expected criticism. If England is looking for an explanation why they, the oft-proclaimed inventors of football, have not won a major international tournament in 58 years, the intense risk aversion within team managers seems a first order concern. Yes, I know, Italy has won plenty of tournaments being boring, but I would note that as offensive tactics have reduced the expected mean outcome of conservative play, they have themselves adapted, while England appears to be “fighting the last war”, so to speak.

I’m not English and have no particularly rooting interest, other than a hatred for boring sports (Audere est Facere). Boring, however, doesn’t offend me quite much as suboptimal outcome maximization and resource deployment. For that I must applaud the Spanish national team on their victory. May they stand as an example of the rewards for bravery, rational bravery, in any market.

Oster on Haidt and Screens

Emily Oster took on the Jonathan Haidt-related debate in her latest post “Screens & Social Media

Do screens harm mental health? Oster joins some other skeptics I know. She doesn’t fully back Haidt, and she does the economist thing by mentioning “tradeoffs.”

Oster, ever practical, makes a point that sometimes gets lost. Maybe social media doesn’t cause suicide. Maybe there is no causal relationship concerning diagnosed mental health conditions, as indicated by the data. That doesn’t mean that parents and teachers should not monitor and curtail screen time. Oster says that it’s obvious that kids should not have their phones in the classroom during school instruction.

Here’s a personal story from this week. My son wants Roblox. The game says 12+, and I’ve told him that I’m sticking to that. No. He can’t have it now and he can’t start chatting with strangers online. We aren’t going to re-visit the conversation until he’s 12. Is he mad at me? Yes. You know what he does when he’s really bored at home? He starts vacuuming. I’ve driven him to madness, with these boundaries I set, or to vacuuming. (Recall he likes these books. Since hearing Harry Potter 1 as an audiobook in the car, he’s started tearing through the series himself via hardcover book.)

An innocent tablet game I let him play (when he’s allowed to have screen time) is Duck Life. Rated E for everyone.

Previously, I wrote “Video Games: Emily Oster, Are the kids alright?

And more recently, Tyler had “My contentious Conversation with Jonathan Haidt” Maybe Tyler should debate Emily Oster next about limiting phone use.

From Cubicles to Code – Evolving Investment Priorities from 1990 to 2022

I’ve written before about how we can afford about 50% more consumption now that we could in 1990. But it’s not all bread and circuses. We can also afford more capital. In fact, adding to our capital stock helps us produce the abundant consumption that we enjoy today. In order to explore this idea I’m using the BEA Saving and Investment accounts. The population data is from FRED.

The tricky thing about investment spending is that we need to differentiate between gross investment and net investment. Gross investment includes spending on the maintenance of current capital. Net investment is the change in the capital stock after depreciation – it’s investment in additional capital not just new capital.  Below are two pie charts that illustrate how the composition of our *gross investment* spending has changed over the past 30 years. Residential investment costs us about the same proportion of our investment budget as it did historically. A smaller proportion of our investment budget is going toward commercial structures and equipment (I’ve omitted the change in inventories). The big mover is the proportion of our investment that goes toward intellectual property, which has almost doubled.

It’s easiest for us to think about the quantities of investment that we can afford in 2022 as a proportion of 1990. Below are the inflation-adjusted quantities of investment per capita. On a per-person basis, we invest more in all capital types in 2022 than we did in 1990. Intellectual property investment has risen more than 600% over the past 30 years. The investment that produces the most value has moved toward digital products, including software. We also invest 250% more in equipment per person than we did in 1990. The average worker has far more productive tools at their disposal – both physical and digital. Overall real private investment is 3.5 times higher than it was 30 years ago.

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“Cheapflation”: Inflation Really Does Hit the Bottom Harder

During the peak of the Covid inflation in 2022 I speculated that food inflation was worst for the cheapest products:

typical McDouble now costs well over $2 in most of the US, while a typical Big Mac is still well under $6. You used to be able to get 4-5 McDoubles for the price of a Big Mac; now you typically get less than 3 and sometimes, as in Keene, less than 2.

What’s going on here? First, the McDouble was always absurdly cheap. Second, prices rise most quickly where demand is inelastic, and demand is less elastic for goods that are cheaper and goods that are more like “necessities” than “luxuries”.

That post was just based on a couple anecdotes from my personal experience, but a new NBER working paper by Alberto Cavallo and Oleksiy Kryvtsov confirms that this really was a general trend:

We use micro price data for food products sold by 91 large multi-channel retailers in ten countries between 2018 and 2024. Measuring unit prices within narrowly defined product categories, we analyze two key sources of variation in prices within a store: temporary price discounts and differences across similar products. Price changes associated with discounts grew at a much lower average rate than regular prices, helping to mitigate the inflation burden. By contrast, cheapflation—a faster rise in prices of cheaper goods relative to prices of more expensive varieties of the same good—exacerbated it. Using Canadian Homescan Panel Data, we estimate that spending on discounts reduced the change in the average unit price by 4.1 percentage points, but expenditure switching to cheaper brands raised it by 2.8 percentage points….

The prices of cheaper brands grew between 1.3 to 1.9 times faster than the prices of more expensive brands—and only when inflation surged, not before or after.

Zoning Matters for Rising Housing Costs, Especially After 1980

From a new working paper “The Price of Housing in the United States, 1890-2006” by Ronan C. Lyons, Allison Shertzer, Rowena Gray & David N. Agorastos (emphasis added):

“Zoning was adopted by almost every city in our sample during the 1920s. We see a slightly steeper gradient over the next two periods (coefficients of .48 and .29, respectively). In these periods it is possible both that the existing zoning regimes were causing higher price growth and that home price appreciation was incentivizing cities to adopt even more restrictive measures, particularly by the 1970s (Fischel, 2015; Molloy et al., 2020). The gradient in the final period (1980-2006) is even steeper, however (coefficient of .67), suggesting a closer relationship between zoning and home price appreciation towards the end of the 20th century.”

The authors acknowledge that they cannot establish causality with their data, but this is consistent with existing research, such as a paper by Gyourko and Krimmel that I previously discussed.

How to Roughly Double Your Investing Returns 1. 2X (or 3X) Leveraged Funds

Most years, stocks go up, by something like 9%. Wouldn’t it be nice to invest in a fund that went up double those amounts? Such funds exist. They use futures or other derivatives to move up (or down!) by double, or even triple, the percentage that the underlying stock or index moves, on a daily basis.

For instance, a common unleveraged fund (ETF) is SPY that roughly tracks the S&P 500 index of large U.S. stocks is SPY. SSO is a 2X fund, which gives double the returns of SPY, on a daily basis. UPRO is a 3X fund, giving triple the returns. 2X funds exist for many different asset classes, including semiconductor stocks, treasury bill, and crude oil – see here. And similarly for 3X funds.

Since all the action in stocks these days seems to be in large tech companies, I will focus on the NASDAQ 100 index universe. The leading unleveraged fund there is QQQ. The 2X version is QLD, and the 3X is TQQQ. Let’s look at how these three funds performed over the past twelve months:

QQQ is up a respectable 36%, but QLD is up by 70%, and TQQQ by a mouth-watering 106%. You could have doubled your money in the past twelve months simply by investing in a 3X fund instead of holding boring 1X QQQ. 

These leveraged funds can be utilized in more than one way. One approach is to just put the monies you have allocated for stocks into such funds, and hope for higher returns. Another approach is to put, say half of your speculative funds into a 2X fund (to get roughly the same stock exposure as putting all of it into a 1X fund), and then use the remaining half to put into other investments, or to keep as dry powder to give you the option to buy more equities if the market crashes.

What’s not to like about these funds? It turns out that a year of daily doubling of returns does not necessarily add up to doubling of yearly returns. There is “volatility drag” associated with all the exaggerated moves up and down. As an illustration of how this works, suppose you held a stock that went down by 50% one day, say from a price of $100 to $50. The next day, it went back up by 50%. But this would only get you back to $75, not $100.

It turns out that with these leveraged funds, as long as stocks are generally going up, the yearly returns can match or even exceed the 2X or 3X targets. But in a period with a lot of volatility, the yearly returns can fall far short. And in a down year, the combination of the leverage and the volatility drag lead to truly horrific losses. For instance, here is what 2022 looked like for these funds:

QQQ was down by 31%, which is bad enough. But imagine your $10,000 in TQQQ melting down to $3,300 that year.

And here is the chart from January 2022 to the present:

QQQ is up 27% in the past 2.5 years, 2X QLD is up only 16%, while 3X TQQQ is actually down by 6%, as it could not recovery from 2022.

This was a kind of a worst-case scenario, since 2022 was an exceptionally bad year for QQQ, coming off a fabulous 2021. A chart of the past five years, which includes the 2020 Covid crash and recovery, and the 2022 crash and subsequent recovery still shows the leveraged funds coming out ahead over the long term:

The net returns on QLD (321%) were about double QQQ (158%), while the more volatile TQQQ return (386%) was plenty high, but fell well short of three times QQQ.

In my personal investing, I hold some QLD as a means to free up funds for other investments I like. But if I smell major market trouble coming, I plan to swap back into plain QQQ until the storm clouds pass.

There are some other ways to get roughly double returns, which suffer less from volatility drag than these 2X funds. I will address those in subsequent posts.

Disclaimer: As usual, nothing here should be considered advice to buy or sell any investment.