Author: Zachary Bartsch

Portfolio Efficient Frontier Parabolics

Zachary Bartsch5 Comments

Previously, I plotted the possible portfolio variances and returns that can result from different asset weights. I also plotted the efficient frontier, which is the set of possible portfolios that minimize the variance for each portfolio return.* In this post, I elaborate more on the efficient frontier (EF).

To begin, recall from the previous post the possible portfolio returns and variances.

From the above the definitions we can see that the portfolio return depends on the asset weights linearly and that the variance depends on the asset weights quadratically because the two w terms are multiplied. Since the portfolio return can be expressed as a function of the weights, this implies that the variance is also a quadratic function of returns. Therefore, every possible portfolio return-variance pair lies on a parabola. So, it follows that every pair along the efficient frontier also lies on a parabola. Not every pair lies on the same parabola, however – the efficient frontier can be composed on multiple parabolas!

I’ll use the same 3 possible assets from the previous post, below is the image denoting the possible pairs, the EF set, and the variance-minimizing point.

One way to find the EF is to calculate every possible portfolio variance-return pair and then note the greatest return at each variance. That’s a discrete iterative process and it definitely works. One drawback is that as the number of assets can increase the number of possible weight combinations to an intractable number that makes iterative calculations too time consuming. So, we can instead just calculate the frontier parabolas directly. Below is the equation for a frontier parabola and the corresponding graph.

Notice that the above efficient frontier doesn’t appear quite right. First, most obviously, the portion below the variance-minimizing return is inapplicable – I’ve left it to better illustrate the parabola. Near the variance-minimizing point, the frontier fits very nicely. But once the return increases beyond a certain level, the frontier departs from the set of possible portfolio pairs. What gives? The answer is that the parabola is unconstrained by the weights summing to zero. After all, a parabola exists at the entire domain, not just the ones that are feasible for a portfolio. The implication is that the blue curve that extends beyond the possible set includes negative weights for one or more of the assets. What to do?

As we deduced earlier, each pair corresponds to a parabola. So, we just need to find the other parabolas on the frontier. The parabola that we found above includes the covariance matrix of all three assets, even when their weights are negative. The remaining possible parabolas include the covariance matrices of each pair of assets, exhausting the non-singular asset portfolios. The result is a total of four parabolas, pictured below.

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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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Looking Ahead: Post-Powell Interest Rates

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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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Does Trump Weaken the US Dollar?

Zachary Bartsch2 Comments

Talk to some economists and they’ll tell you that exchange rates aren’t economically important. They say that exchange rates between countries are a reflection of supply and demand for one another’s stuff. So, at the macro, it’s a result and not a determinant of transnational economic activity.

For individual firms at the micro level, it’s the opposite. They don’t affect the exchange rate by their lonesome and are instead affected by it. If you have operations in a foreign country, then sudden changes to the exchange rate can cause your costs to be much higher or lower than you had anticipated. The same is true when you sell in a foreign country, but for revenues. This type of risk is called ‘exchange rate risk’ since it’s possible that none of the prices in either country changed and yet your investment returns change merely because of an appreciated currency.

Supply & Demand

Exchange rates are determined by supply and demand for currencies. Demand is driven by what people can do with a currency. If a country’s goods become more attractive, then demand for those goods rise and demand for the currency rises. After all, most retailers and wholesalers in the US require that you pay using US dollars. Importantly,  it’s not just manufacturing goods that drive demand for currency. Demand for services, real estate, and financial assets can also affect the supply and demand for currency. In fact, many foreigners  are specifically interested in stocks, bonds, US treasuries, and other investments. The more attractive all of those things are, the more demand there is for them.

Of course, the market for currency also includes suppliers. Who does that? Answer: Anyone who holds dollars and might buy something. Indeed, all buyers of goods or financial products are suppliers of their medium of exchange. In the US, we pay in dollars. Especially since 1972, suppliers have also included other central banks and governments. They treat the US currency as if it’s a reserve of value, such as gold, that can be depended upon if they need a valuable asset (hence the name, “Federal Reserve”). This is where the term ‘reserve currency’ comes from – not from the dollar-denominated prices of some internationally traded commodities. Though, that’s come to be an adopted meaning.  

Another major supplier of currency is the US central bank. It has the advantage of being able to print US dollars. But it doesn’t have an exchange rate policy. So, it’s not targeting a particular price of the US dollar versus any other currency. The Fed does engage in some international reserve lending, but it’s not for the purpose of supplying currency to foreign exchange markets.  

The US Exchange Rate in 2025

One of the reasons that the US has such popular financial assets is that we have highly developed financial markets and the rule of law. People trust that, regardless of the individual performance of an asset, the rules of the game are mostly known and evenly applied. For example, we have a process to follow when bond issuers default. So, our popularity is not merely because our assets have higher returns. Rather, US investment returns have dependably avoided political risk – relative to other countries anyway.

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Are Imports Bad for GDP?

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A periodically recurring conversation on social media is whether imports are bad for GDP. Everyone thinks they are clearly right, and then they lazily defer to brief dismissal of the opposing view. Some of this might be due to media format. Something just a tiny bit more thorough could help to resolve the painfully unproductive online interactions… And just maybe improve understanding.  

It starts with the GDP expenditure identity:

The initial assertion is that imports reduce GDP. After all, M enters the equation negatively. So, all else constant, an increase in M reduces Y. It’s plain and simple.

Many economists reply that the equation is an accounting identity and not a theory about how the world works and that the above logic is simply confusing these two things. This reply 1) allows its employers to feel smart, 2) doesn’t address the assertion, & 3) doesn’t resolve anything. In fact, this reply erects a wall of academic distinction that prevents a resolution. What a missed opportunity to perform the literal job of “public intellectual”.

How are Imports Bad/Good/Irrelevant for GDP?

Let’s add a small but important detail to the above equation to distinguish between consumption of goods produced domestically and those produced elsewhere.

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All of the Prices

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Today I’m just sharing a truly awe-inspiring resource. The University of Missouri has what is essentially a central clearinghouse for prices and wages. If you want the price of anything, then they should be your first stop.

See the screenshot at the bottom. The website links to the original sources for household consumption prices, occupation wages, etc. They make it easy to cut the data by date, industry, location, etc. Because they cite their sources, you can see some data series that are not even available on FRED – without having to perform the painful sleuthing on a government website.

I especially like this site for its historical data. One of the challenges of historical US data is that individual cities may not have prices that are representative of the national levels or trends. Lower levels of market integration make representative samples even more important than in modern data. But really, that was more of a concern for 20th century researchers. Now, we love our panel data. So, the historically less integrated markets of the US provide ‘toy economies’ that include greater regionalism and local shocks.

Although David Jacks has loads of tabulated data, he doesn’t have it all. The Missouri library site links to PDFs of original statistical publications which, while digitized, have never been tabulated into useable data fit for modern researchers.

Go have a look around. You won’t regret it.

https://libraryguides.missouri.edu/pricesandwages/1870-1879

Children Don’t Die Like They Used To

Zachary Bartsch2 Comments

Academics generally agree on the changing patterns of mortality over time. Centuries ago, people died of many things. Most of those deaths were among children and they were often related to water-borne illness. A lot of that was resolved with sanitation infrastructure and water treatment. Then, communicable diseases were next. Vaccines, mostly introduced in the first half of the 20th century, prevented a lot of deaths.

Similarly, food borne illness killed a lot of people before refrigeration was popular. The milkman would deliver milk to a hatch on the side of your house and swap out the empty glass bottles with new ones full of milk. For clarity, it was not a refrigerated cavity. It was just a hole in the wall with a door on both the inside and outside of the house. A lot of babies died from drinking spoiled milk. 

Now, in higher income countries, we die of things that kill old people. These include cancer, falls that lead to infections, and the various diseases related to obesity. We’re able to die of these things because we won the battles against the big threats to children. 

What prompts such a dreary topic?

I was perusing the 1870 Census schedules and I stumbled upon some ‘Schedule 2s’. Most of us are familiar with schedule 1, which asks details about the residents living in a household. But schedule 2 asked about the deaths in the household over the past year.  Below is a scan from St. Paul, Minnesota.

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Now Published: Prohibition and Percolation

Zachary Bartsch2 Comments

My new article, “Prohibition and Percolation: The Roaring Success of Coffee During US Alcohol Prohibition”, is now published in Southern Economic Journal. It’s the first statistical analysis of coffee imports and salience during prohibition. Other authors had speculated that coffee substituted alcohol after the 18th amendment, but I did the work of running the stats, creating indices, and checking for robustness.

My contributions include:

  • National and state indices for coffee and coffee shops from major and local newspapers.
  • A textual index of the same from book mentions.
  • I uncover that prohibition is when modern coffee shops became popular.
  • The surge in coffee imports was likely not related to trade policy or the end of World War I
  • Both demand for coffee and supply increased as part of an intentional industry effort to replace alcohol and saloons.
  • An easy to follow application of time series structural break tests.
  • An easy to follow application of a modern differences in differences method for state dry laws and coffee newspaper mentions.
  • Evidence from a variety of sources including patents, newspapers, trade data, Ngrams, naval conflicts, & Wholesale prices.

Generally, the empirical evidence and the main theory is straightforward. I learned several new empirical methods for this paper and the economic logic in the robustness section was a blast to puzzle-out. Finally, it was an easy article to be excited about since people are generally passionate about their coffee.

Bartsch, Zachary. 2025. “Prohibition and Percolation: The Roaring Success of Coffee During US Alcohol Prohibition.” Southern Economic Journal, ahead of print, September 22. https://doi.org/10.1002/soej.12794.

What’s the Best Major to Prepare for Law School?

Zachary Bartsch4 Comments
  • This is post coauthored with Jack Cavanaugh, Ave Maria University Graduate of 2025.

Say that you want to become a successful lawyer. What does that mean? One possible meaning is that you are well-compensated. Money is not everything, but it does give people more options for how to spend their time and resources. Law degrees are a type of graduate degree. So, what bachelor’s degree major should one choose in preparation for law school? We lack rich administrative data on college majors and LSAT scores.

Luckily, the 2023 American Community Survey (ACS) comes to the rescue. It has all of the typical demographic covariates, income, occupation, and college major. So, if we make the small leap that well-prepared law school students become high-performing lawyers who are ultimately paid more, then what college major puts you on the right path? What should your major be?

We don’t look at an exhaustive list. We place several occupations into bins and examine only a few alternative majors. Any unlisted major falls under ‘other’. Below are the raw average incomes by occupational category and college major. Note two majors in particular. First, Pre-law literally has the word ‘law’ in the name and is marketed as preparation for law school. However, it is the undergraduate major associated with the lowest paid lawyers. For that matter, Pre-law majors have the lowest pay no matter what their occupation is. Second, Economics majors are the most highly paid in all of the occupations.

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What’s Wrong with Sales Tax Holidays?

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Tax holidays are when some set of goods are tax-free for a period of time. These might be back-to-school supplies for a week or a weekend, or hurricane supplies for several months. These policies tend to be popular among non-economists.

There are practical reasons for anyone to decry tax holidays. Usually, there is a particular type of good that qualify for tax-free status. These are often selected politically rather than by an informed and reasoned way with tradeoffs in mind. Sometimes, there is a subpopulation that is intended to benefit. However, the entire population gets the tax holiday and those with the most resources, who often have higher incomes, are best able to adjust their consumption allocations and enjoy the biggest benefits. A tax holiday weekend is no good to a single-mom who can’t get off work during that time.

Getting more economic logic, these holidays also concentrate shopping on the tax-free days, causing traffic and long lines that eat away at people’s valuable time – even if they aren’t purchasing the tax-free items. Furthermore, retailers must comply with the law. This means ensuring that all items are taxed correctly, making neither mistakes in over-taxing or under-taxing. Given the variety of goods and services out there, this is a large cost for individual firms.

Finally, as economists know, there is a deadweight loss anytime that there is a tax. As a consequence, you might think that economists would love anytime that taxes are low. But, holding total tax revenue constant, a tax break on a tax holiday implies that there must be greater tax revenues on the other non-holidays. In particular, economists also know that losses in welfare increase quadratically with changes in tax rates. Therefore, higher tax rates on some days and lower rates on other days causes more welfare loss than if the tax rate had been uniform the entire time. In the current context, such welfare loss manifests as forgone beneficial transactions. These non-transactions are hard for non-economists to understand because we can’t see purchases that don’t happen, but would have happened in the absence of poor policy.

Let’s look at some graphs.

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