Optimal Portfolio Weights

Zachary Bartsch6 Comments

All of us have assets. Together, they experience some average rate of return and the value of our assets changes over time. Maybe you have an idea of what assets you want to hold. But how much of your portfolio should be composed of each? As a matter of finance, we know that not only do the asset returns and volatilities differ, but that diversification can allow us to choose from a menu of risk & reward combinations. This post exemplifies the point.

1) Describe the Assets

I analyze 3 stocks from August 1, 2024 through August 1, 2025: SCHG (Schwab Growth ETF), XLU (Utility ETF), and BRK.B (Berkshire Hathaway). Over this period, each asset has an average return, a variance, and  co-variances of daily returns. The returns can be listed in their own matrix. The covariances are in a matrix with the variances on the diagonal.

The return of the portfolio that is composed of these three stocks is merely the weighted average of the returns. In particular, each return is weighted by the proportion of value that it initially composes in the portfolio. Since daily returns are somewhat correlated, the variance of the daily portfolio returns is not merely equal to the average weighted variances. Stock prices sometimes increase and decrease together, rather than independently.

Since the covariance matrix of returns and the covariance matrix are given, it’s just our job to determine the optimal weights. What does “optimal” mean? This is where financiers fall back onto the language of risk appetite. That’s hard to express in a vacuum. It’s easier, however, if we have a menu of options. Humans are pretty bad at identifying objective details about things. But we are really good at identifying differences between things. So, if we can create a menu of risk-reward combinations, then we’re better able to see how much a bit of reward costs us.

2) Create the Menu

In our simple example of three assets, we have three weights to determine. The weights must sum to one and we’ll limit ourselves to 1% increments. It turns out that this is a finite list. If our portfolio includes 0% SCHG, then the remaining two weights sum to 100%. There are 101 possible pairs that achieve that: (0%, 100%), (1%,99%), (2%,98%), etc. Then, we can increase the weight on SCHG to 1% for which there are 100 possible pairs of the remaining weights: (0%,99%), (1%, 98%), (2%, 97%), etc. We can iterate this process until the SCHG weight reaches 100%. The total number of weight combinations is 5,151. That means that there are 5,151 different possible portfolio returns and variances. The below figure plots each resulting variance-return pair in red.

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Economic Freedom Updates

James Bailey1 Comment

In September we covered the release of the Fraser Institute’s 2025 Economic Freedom of the World report. I said then:

The authors are doing great work and releasing it for free, so no complaints, but two additional things I’d like to see from them are a graphic showing which countries had the biggest changes in economic freedom since last year, and links to the underlying program used to create the above graphs so that readers could hover over each dot to identify the country

Well, now Matthew Mitchell of the Fraser Institute has done that:

I can only post a screenshot of a scatterplot here, but if you click through to the Fraser report you can hover over any dot to see which country it represents:

The Supreme Court Case on Trump’s Tariffs

Jeremy Horpedahl1 Comment

Today is a big day not only for Supreme Court watchers, but for everyone following economic policy: the Court will hear oral arguments for the case Learning Resources, Inc. v. Trump. The case concerns whether Trump’s tariffs imposed under the International Emergency Economic Powers Act are legal, which includes the famous “Liberation Day” tariffs from April 2025.

You should be able to livestream the arguments from the SCOTUS website starting at 10am ET (though it may start a little later). SCOTUS blog has a liveblog which should cover most of the legal arguments, but if you want to follow the economic arguments there are several people you can follow on Twitter, such as Scott Lincicome and Phil Magness (you can follow me too).

Is Tesla Stock Grossly Overpriced?

Scott Buchanan1 Comment

One of the more polarizing topics in investing is the valuation of Tesla stock. Its peers among the Magnificent 7 big tech leaders sport price/earnings ratios mainly in the 30s. Those are high numbers, but growth stocks deserve high P/Es. A way to normalize for expected growth of earnings is to look at the Price/Earnings/Growth (PEG) ratio. This number is usually 1.5-2.0 for a well-regarded company. Anything much over 2 is considered overvalued.

Tesla’s forward P/E of about 270 is nearly ten times higher than peers. Its anticipated growth rate does not seem to justify this astronomical valuation, since its PEG of around 4-10 (depending on assumptions) is way higher than normal. This seems to be a case of the CEO’s personal charisma dazzling shareholders. There is always a new “story” coming out to keep the momentum going.

Tesla’s main actual business is selling cars, electric cars. It has done a pretty good job at this over the past decade, supported by massive government subsidies. With the phasing out of these subsidies by the U.S. and some other governments, and increasing competition from other electric carmakers, it seems unlikely that this business will grow exponentially. Ditto for its smallish ($10 billion revenue) business line of supplying large batteries for electric power storage. But to Tesla fans, that doesn’t really matter. Tesla is valued, not as a car company, but as an AI startup venture. Just over the horizon are driverless robo-taxis (whose full deployment keeps getting pushed back), and humanoid Optimus robots. The total addressable market numbers being bandied about for the robots are in the trillions of dollars.

Source: Wikipedia

From Musk’s latest conference call:

Optimus is Tesla’s bipedal humanoid robot that’s in development but not yet commercially deployed. Musk has previously said the robots will be so sophisticated that they can serve as factory workers or babysitters….“Optimus will be an incredible surgeon,” Musk said on Wednesday. He said that with Optimus and self driving, “you can actually create a world where there is no poverty, where everyone has access to the finest medical care.”

Given the state of Artificial General Intelligence, I remain skeptical that such a robot will be deployed in large numbers within the next five years. It is of course a mind-bending exercise to imagine a world where $50,000 robots could do anything humans can do. Would that be a world where there is “no poverty”, or a world where there is no wealth (apart from the robot owners)? Would there be a populist groundswell to nationalize the robots in order to socialize the android bounty? But I digress.

On the Seeking Alpha website, one can find various bearish articles with the self-explanatory titles of, for instance, Tesla: The Dream Factory On Wall Street, Tesla: Rallying On Robotaxi Hopium, and Tesla: Paying Software Multiples For A Car Business – Strong Sell . There are also bullish pieces, e.g. here,  here, and here.

Musk’s personal interaction with shares has propped up their value. He purchased about $1 billion in TSLA shares in September. This is chicken feed relative to its market cap and his net worth, but it apparently wowed TSLA fans, and popped the share price. What seems even more inexplicable is the favorable response to a proposed $1 trillion (!!) pay package for Elon. For him to be awarded this amount, Tesla under his watch would have to achieve hefty boosts both in physical production and in stock market capitalization. But… said package would be highly dilutive (like 12%) to existing shareholders, so, rationally they should give it thumbs down. However, it seems likely that said shareholders are so convinced of Musk’s value that they will approve this pay package on Nov 6, since he has hinted he might leave if he doesn’t get it.

Such is the Musk mystique that shareholders seem to feel that giving him an even greater stake in Tesla than he already has  will cause hundreds of billions of dollars of earnings appear from thin air. From the chatter I read from Wall Street professionals, they view all this as ridiculous magical thinking, yet they do not dare place bets against the Musk fanbase: the short interest in TSLA stock is only a modest 2.2%. Tesla is grossly overvalued, but it will likely remain that way as long as Elon remains and keeps spinning grand visions of the future.

Fresh observations of Americans working hard

Joy Buchanan1 Comment

It has been over 100 years since GK Chesterton visited America. I wrote about his observations on the American “enthusiasm for work” for Liberty Fund.

Henry Oliver commented on much the same thing this week in The American art of being busy.

The whole place was as busy as a hive. It went on and on. Everyone was cheerful. No-one fussed and bothered.

And what of the Americans who are not allowed to work because of the government shutdown? Here is a guy who has rapidly found a way to work in DC again and seems “cheerful”: This furloughed IRS lawyer is living out his dream of being a hot dog vendor

This restlessness and energy likely has something to do with us being currently in the lead for the race to AI. Ho hum… building God while still having the equivalent of the GDP of a small country to drop on Halloween trinkets.

Looking Ahead: Post-Powell Interest Rates

Zachary BartschLeave a comment

Jerome Powell’s term as Fed Chair ends in late May 2026. President Trump has said that he will nominate a new chair and the US senate will confirm them. It may take multiple nominations, but that’s the process. The new chair doesn’t govern monetary and interest rate policy all by their lonesome, however. They have to get most of the FOMC on board in order to make interest rate decisions. We all know that the president wants lower interest rates and there is uncertainty about the political independence of the next chair. What will actually happen once Jerome is out and his replacement is in?

The treasury markets can give us a hint. The yields on government debt tend to follow the federal funds rate closely (see below). So, we can use some simple logic to forecast the currently expected rates during the new Fed Chair’s first several months.

https://fred.stlouisfed.org/graph/?g=1NwVQ

Here’s the logic. As of October 16, the yield on the 6-month treasury was 3.79% and the yield on the 1-year treasury was 3.54%. If the market expectations are accurate, then holding the 1-year treasury to maturity should yield the same as the 6-month treasury purchased today and then another one purchased six months from now. The below diagram and equation provide the intuition and math.

Since the federal funds rate and US treasury yields closely track one another, we can deduce that the interest rates are expected to fall after 6 months. Specifically, rates will fall by the difference in the 6-month rates, or about 49.9 basis points (0.499%).  This cut is an expected value of course. Given that the cut is between a half and a zero percent, we can back out the market expectation of for a 0.5% vs 0.0% cut where α is the probability of the half-point cut.* Formally:

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The Art of Spending Money

James Bailey1 Comment

The author of The Psychology of Money, Morgan Housel, has a new book “The Art of Spending Money” out this month. Its main point is that people tend to be happier spending money on things they value for their own sake- rather than things they buy to impress others, or piling up money as a yardstick to measure themselves against others (this is repeated with many variations).

Overall it is well-written at the level of sentences and paragraphs with well-chosen stories and quotes, but I’m not sure what it all adds up to. The main points seem obvious to me, though maybe that’s my fault for reading a book titled this when I’m already fairly happy with how I spend money. I think I err a bit on the frugal side, but I just don’t see many opportunities to turn money into happiness by spending it- I was maybe hoping for ideas on that front but I got none from the book. After reading it I don’t plan to do anything differently and don’t find myself thinking about spending differently.

Still, some highlights. The book is full of well-chosen quotes from others:

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Bad Claims About Food Stamps (SNAP)

Jeremy Horpedahl11 Comments

One of the likely effects of the federal government shutdown is that recipients of SNAP benefits (what used to be officially called “food stamps,” a term still used by the general public, especially those that dislike the program) may lose their benefits next month. This would obviously be a hardship for those that depend on this program, but it has also led to bad claims being made about the program, from both supporters and opponents of the program.

Let’s start from the political right: Matt Walsh makes the claim that by subsidizing food consumption “obviously drives up the cost” of groceries.

As with all bad claims, there is a nugget of truth baked into them. If the government subsidizes anything, we would expect demand to increase, and thus unless supply is perfectly elastic, there will be some effect on prices. However, we need to think more carefully about the nature of the subsidy.

The way SNAP works is that beneficiaries receive an electronic voucher to spend at the grocery store, which is about $300 per month on average for a household. That $300 must be spent on groceries. However, if that household had already planned to spend $300 or more on groceries, it is unlikely they will spend all of the additional $300 on food. In the limit, it’s entirely possible they will spend no additional money on groceries, merely reducing their out-of-pocket spending on groceries by $300. They will then effectively have $300 more to spend on other goods. More likely is that they will spend some of the additional $300 on groceries, and some of it on other goods.

Many studies have tried to look at the extent to which SNAP benefits affect household spending, but these were mostly observational studies. There was no treatment and control group. But a 2009 paper titled “Consumption Responses to In-Kind Transfers: Evidence from the Introduction of the Food Stamp Program” has a better approach to studying the question. Since the original Food Stamp program was slowly rolled out across the country over more than a decade, you can compare counties that entered the program first to counties that entered it later. By doing so, Hilary Hoynes and Diane Schanzenbach find out some first interesting things about the causal effects of SNAP benefits.

For the claim by Walsh in his Tweet, the most relevant result from the paper is that food stamps impact household spending similarly to a cash transfer. Yes, the program increases household spending on groceries, but it also increases spending on other goods and services. And it does so almost identically to how cash transfers impact household spending. In other words, while pitching the program as assistance for buying groceries may make it more politically palatable, SNAP benefits are no different from a similarly-sized cash transfer for the average recipient. If they do cause any inflation, they do so in the same way as a cash transfer would, and thus there is no specific impact on food inflation.

A second bad claim about SNAP comes from the political left, in this case Minnesota Governor Tim Walz:

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WW II Key Initiatives 3: Kurt Tank Gives Germany a Superior Fighter Plane, the Focke-Wulf 190

Scott BuchananLeave a comment

This is the third in a series of occasional blog posts on individual initiatives that made a strategic (not just tactical) difference in the course of the second world war.

World War II was not only the biggest, bloodiest conflict, in human history. It played a definitive role in giving us the world we have today. Everyone can find something to complain about in the current state of affairs, but think for a moment what the world would be like if the Axis powers had prevailed.

Having control of the air became crucial in the second world war. It meant you could drop bombs on enemy soldiers, ships, tanks, cities, factories, etc., etc. The Germans showed early on how important that can be. Their terror bombing of the Dutch city of Rotterdam compelled the Netherlands to surrender to spare other cities from being likewise bombed, even though the Dutch armed forces could have held out for some time longer. The German breakthrough in their invasion of France in 1940 was facilitated by a concentrated Stuka dive bombing attack on a key sector of the French front lines. The 1940 Battle of Britain was an air war, where the Germans hoped to whittle the British Air Force capability down enough to permit them to invade across the English Channel. And so on.

The main German fighter plane at first was the Messerschmidt Me 109. It was a good plane, although by 1941 the British Spitfire had become a match for it. Both the Me 109 and the Spitfire were designed around in-line engines, where the cylinders were arranged in two long rows in the engine block. That gave a narrow engine, and hence a skinny profile to the airplane, which tended to reduce wind resistance and make for higher speeds. A weak point of all in-line engines is that they need to have a circulating coolant system, going through a radiator, to cool down the engine block from the heat of the internal combustion. This makes for more complicated maintenance and is very vulnerable to being damaged by enemy fire,

Just when the Brits were starting to wrest air superiority back from the Germans, the FW 190 appeared in the skies over France. Allied pilots were shocked. The new German fighter could out-climb, out-roll, and in many cases out-fight the current Spitfire models. This so-called “butcher bird” gave air superiority back to the Germans.

Its remarkable performance was the result of one man’s engineering philosophy and persistence: Kurt Tank, chief designer at the German aircraft manufacturer Focke-Wulf. Tank was a pilot as well as an engineer, with long and varied prior military experience. He chose a radial engine for his plane, to make it more rugged and easy to maintain. With a radial engine, the individual cylinders all stick out from a central crankcase; airflow past the fins on the cylinders cools the engine. Hence, no vulnerable cooling system and radiator. The conventional thinking was that a radial engine was so fat that an airplane using it would have a wide, draggy profile. His ingenious design features allowed him to make a fast, agile plane. However, was an uphill job for Tank to sell his concept to the German military establishment. Eventually, his results spoke for themselves and the Fw 190 was produced. With its critical spots armored, the Fw 190 was hard to kill. Tank deliberately gave a wide stance and long travel to the landing gear, to allow deployment in rough frontline airfields.

The Fw 190 was a superb low-medium altitude fighter, and was also widely pressed into service (due to its rugged design) as a precision bomber on the front lines. Around 20,000 Fw 190s were produced. They shot down many thousands of Allied planes, killed untold thousands of Allied airmen and soldiers, and destroyed thousands of Allied vehicles, mainly on the Eastern Front. It was not enough to change the ultimate outcome of the war, but Tank stretched it out appreciably, by (largely single-handedly) giving the Germans such a versatile and deadly weapon.

Sports gambling has a problem other prediction markets don’t

mdmakowsky1 Comment

Sports gambling is entering it’s first series of major crises since widespread legalization. While there is the typical handwringing around the intersection of vice and broad entertainment, there is also the added dimension of the role that insider information can and should play within any speculative market. Those arguments, conducted earnestly, are of course completely valid, but I think they are not giving enough attention to a key distinction in the online incarnation of sports gambling.

Speculating on sports outcomes produces the same elicitation and aggregation of information as a more traditional speculative market, such as commodities futures or stock equity markets. Information, acted upon through purchase, reveal each individual’s beliefs about the true value of a contract paid upon the conclusion of a sporting event or the price of agricultural commodity at a given date and time. The market exchange of these contracts aggregates these beliefs into a collective piece of information in the form of a market price. Some contract holders get richer, some poorer, and the broader world benefits from the distillation of private information into public prices. The problems within sports gambling stem from the second channel through which entertainment is provided and paid for: random outcome generation.

Sports match outcomes are something you speculate on. Random outcomes are something you gamble on. Yes, there is random chaos in sports the same way there is random weather in agriculture. There is no speculating on a roulette wheel, however, that’s a pure gamble. I believe the major sports leagues and the online gambling companies they partner with have made a grievous error allowing their sites to offer (nearly) pure gambles.

Think about how much casinos invest in the integrity of their games as pure and fair gambles. Dice are rigorously inspected and routinely replaced. Roulette wheels are engineered with astounding tolerances. Card games occur under multiple layers of scrutinous observation. Manipulation under such conditions is sufficiently costly such that it is almost never worth undertaking.

How do you go about making similar investments in monitoring 6 inches of horizontal manipulation of the first pitch of a baseball game? Of a marginal player taking himself out of a game injured a few minutes early? The answer is you largely can’t. So now you have human roulette wheels who can decide what number they land on. Which brings us to the second, closely related problem in the new regime of sports gambling: inframarginal game outcomes. Once a game is probabalistically decided before its official conclusion, teams will often play their substitutes to finish out the formality in order to rest their main players and protect them from injury. These players typically earn smaller salaries, often over far shorter careers, with less scrutiny over their quality of play. These are the exact players for whom a couple hundred thousand dollars may be worth incurring a small amount of risk. The product of their play in terms of success (i.e. scoring, hitting, etc) is still highly conditional on their ability relative to their opposition, but the play itself (i.e. shooting, swinging, pitching choices, fouling, etc) is entirely within their control. It may be less purely random, but it is nonetheless sold to gambling customers as fair.

Whether the outcomes in question are quasi-random outcomes or merely inframarginal, what matters is that they are not joint products of competition. To significantly manipulate these outcomes does not require the explicit or implicit coordination of multiple individuals across competing teams. Yes, one player can tilt the odds, but if you are looking to make significant money manipulating sports gambling, you can’t just tilt the odds a few percentage points. There’s a reason the Black Sox Scandal of 1919 involved eight players (seven if you consider Buck Weaver innocent, which I do).

As I love to point out, coordination across individuals is very difficult. Crimes involving coordination are, in turn, far easier to monitor. Online gambling massively reduced the transaction costs in sports gambling, opening the door for orders of magnitude increases in the number and variety of bets that could be taken. There’s obviously demand for pure gambling alongside outcome speculation, and that demand could now be met through random and inframarginal in-game player outcomes.

The danger, of course, is that few of these events are truly inframarginal. Every pitch and available player counts towards the outcome. Enough manipulation by enough players will graze away the integrity of the core product. The subsidy of lower end players through gambling will change how they approach their careers and how management approaches their employment. Fans will react accordingly as well, adjusting how they view outcomes. We’re already seemingly hardwired to view everything as causal and conspiratorial, overestimating bias in refereeing and player preferences. This will only stoke those fires further.

Organized crime famously offered a “numbers game” prior to state lotteries. Desperate for a credibly random outcome, a common mechanism was to use the middle three digits of the number of shares traded on the NYSE as the winning number. There are no shortage of lotteries now, but there obviously remains latent demand, and customers clearly enjoy bundling gambling with a product far more entertaining to consume than scratching off a ticket. Pro sports was unable to deny the profits from exactly such a bundling, but the cross contamination with their core product may prove to be of greater cost.

I’m not businessman, just a lowly economist and sports fan, but if I were running a $11.3 billion per year firm, I would be far more risk averse.