Author: Zachary Bartsch

Relative Measures of Portfolio Performance

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This is the third and final installment of my series on portfolio performance measures among separate assets groups. First, I summarize the earlier posts, then I introduce relative performance measures. I start with the Markowitz cloud of possible portfolio weights, returns, and volatilities.

Absolute measures of performance contrast the realized portfolio performance with the performances that were possible simply by calculating the difference in, say, return or volatility. The drawback of this method is that different spreads of statistics can affect these differences apart from portfolio performance. That is, even if a portfolio of assets return was very high, some reference return can still be much higher and make the performance look poor.

Quasi-relative measures tackle this problem of different spreads by calculating the percentile of possible returns or volatilities. This allows us to compare portfolio returns to what was possible even among portfolios of different assets with Markowitz clouds of different volatility ranges. The drawback of quasi-relative measures is that the return at some percentile of possible returns is not the same as the return of the same percentile among possible portfolios. Said another way, each possible rate of return in the Markowitz cloud is not equally as likely. So, a low percentile among possible returns be due to a very high and unlikely return.

It should be obvious that returns and volatilities among possible portfolio weights are not equally likely. To help visualize the idea, see the below 3D quadratic for a simplified example that represents a portfolio of three assets. The x-axis represents returns and the z-axis represents standard deviation. The y-axis represents the weight on  the 3rd assets (returns and weights map directly to one another linearly). The set of possible portfolios lie on the surface of the quadratic function.

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Quasi-Relative Measures of Portfolio Performance

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Last week I discussed absolute measures of portfolio performance and management, specifically between two portfolios that are composed of different assets (utilities and tech). I began with comparing the basics of return, standard deviation, and Sharpe ratio to some other possible portfolio in the Markowitz cloud. But, simply comparing the difference between these possible portfolios can be sensitive to the spread of stats within a specific Markowitz cloud. In other words, it’s not scale independent. A larger spread of possible stats can make a portfolio look bad due to the spread return/standard deviation/Sharpe ratio alone.

In this post I introduce quasi-relative measures. Again, I lean on the Markowitz cloud. They’re pasted below (Utilities on the left, tech on the right).

If we can somehow express the returns, volatilities, and Sharpe ratios on a common scale that is independent of the level values, then we can make the realized portfolios more comparable. One thing that we can do is to express a stat as a weighted linear average between the maximum and minimum possible values. Conditional on the realized standard deviation, there exists a maximum and minimum of possible return. Something like the below. Rho is the weight on the maximum return. It’s also the proportion of possible conditional returns that are lower than the realized return.

The unconditional version is the same, but would be relative to the global maximum and minimum stats. We can represent the weigh on the maximum return and the percentile among possible returns as gamma.

A final quasi-relative measure of performance is the dissimilarity index between the realized portfolio weights and some reference portfolio weights. This provides a measure of how much the asset weights would need to change in order to adjust the portfolio.  If changing portfolio weights is costly, then it’s also a measure of the transaction cost of reallocation. It’s quasi-relative because it is independent of the spread of possible performance stats.

Below are the quasi-relative measures for each the utility and tech company portfolios.

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Absolute Measures of Portfolio Performance

Zachary Bartsch1 Comment

The basic idea is that we want to compare the performance of different portfolios or their managers. This is relatively easy as long as the portfolios contain the same assets. Then, the portfolios are simply characterized by the different weights among the different assets. But how do we compare the performance of portfolios whose assets are different? In finance, we usually assume that everyone can invest in everything. But there are plenty of cases in which that’s a bad assumption: when clients want exposure to particular industries, when there are statutory limitations on holding certain assets, or when an individual company is considering specific projects within the same company under conditions of scarce financing.

The most primitive step is to compare the return and standard deviation of two different portfolios. However, higher risk investments tend to have higher returns in dynamic equilibrium. So, if we were to compare the returns of a tech company to a utility company, then we’d often see the tech companies performing better. But, if we compare the volatilities, then the utility companies would tend to perform better. Sharpe stepped in with a ratio to express the excess return (benefit) per standard deviation (the cost). This way, we can compare the price of volatilities between two portfolios. We’ll stick with just these basic 3 measures: return, standard deviation, and Sharpe ratio. (Others do exist)

Let’s put some meat on this with an example. Say that we have two portfolios, each composed of different assets. There’s a utility portfolio that’s composed of NEE, DUK, and SO. There’s also a tech portfolio that’s composed of AMD, MSFT, and NVDA. Both portfolios have weights of (0.33, 0.33, 0.34).  The results of the utility versus the tech portfolio are:

  • Returns: 14.2% vs 136.3%
  • Standard Deviation: 14.9% vs 32%
  • Sharpe: 0.684 vs 4.134

Goodness me! The tech portfolio returns much more in absolute terms and much more per unit of risk. It’s twice as volatile as the utility portfolio, but the returns are almost ten times as high. If you could, then many of us would choose the tech portfolio over the utility portfolio. But, what if, for one reason or another, you can only invest in one of the two industries? Or, what if you want to invest your money with a skilled manager, rather than a risky one?

One way to tackle this problem is to introduce the Markowitz cloud. Specifically, we can essentially list out all of the possible portfolios along with their return and standard deviations. Then, we can compare the actual performance to the entire menu of possible performances within each set of assets. Below are the possible performances for the utility (left) versus the tech (right) portfolio. The actual portfolios are marked with an X.

One way to evaluate the two portfolios is to compare their return, standard deviation, and Sharpe ratio to the other candidates that were achievable with the same assets. As we can see, conditional on the assets, neither portfolio minimized the volatility, maximized return, nor maximized the Sharpe ratio. Furthermore, assuming that the realized rate of return was the goal, neither portfolio minimized the conditional volatility. Assuming that the realized volatility was the goal, neither portfolio maximized the conditional return. Below are two tables that describe some candidate alternatives and how they differ from the realized portfolio.

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Breakfast Pudding

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When I was a kid, my family didn’t get JELLO puddings – or any puddings for that matter. As an adult, I realized that a lot of those are just sugar, cornstarch, and stabilizers. So, they became less appetizing.

You’re going to laugh at me.

A few years ago my wife and I went to a nice little breakfast restaurant for brunch in old town Fredericksburg, VA. I got this coconut milk chia seed parfait. I was blown away. That seems silly to say, but it was really nice.

For years we spoke longingly of that chia parfait and we’d speculate about when we might go there again. It was one of those conversations that married people have.

“Hey, remember that really good thing?”

“Yeah, it was really good.”

“We should try that again sometime.”

Then one day, while visiting Virginia, we noticed that the restaurant had closed. It wasn’t surprising because the restaurant had only been ‘fine’, except for the healthy and delectable layered treat from years past.

Now we have a handful of kids and we try different things periodically to make the morning routines go more smoothly. Having a responsible treat to entice juveniles from their room isn’t the worst thing that we’ve tried.

My wife, in her laudable creativity, refined a new creation that’s inspired by our now frustrated longing for a nice chia parfait.

Below is a recipe for peanut butter chocolate chia seed pudding. Basically, you mix it the night before and stir it again in the morning and it’s ready to go. It’s a crowd pleaser.

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Which Business Programs Require Economics?

Zachary Bartsch2 Comments

Disclaimer: This post might throw shade.

The vast majority of business majors across the US are required to take two or more Economics courses. You can look across the spectrum. All of the top 20 business schools require two or more econ classes. In fact, Wharton is the top-ranked business school and their business program is actually an *economics* program. They don’t have finance/accounting/business degrees. Instead, they have an Economics degree with the various business concentrations. Again – the top business school in the country is an Economics program.

What about at the other end of the spectrum? I live in Florida. Every single Florida state school requires both Micro and Macroeconomics for business majors. These schools include everything from Florida State University to the local Florida state college down the road. I didn’t look at other state-run higher education systems in other states. There are a lot of states…

I teach at a private Catholic university. We’re listed in something called ‘The Newman Guide’ which recommends 17 Catholic schools. Many of these are liberal arts schools, but the list also includes Catholic University of America, which is an R1. Most of these schools also require two or more Economics classes in their Business major programs. The only exception is University of Dallas, which has Economics in the core curriculum.*

So, overwhelmingly undergraduate business programs across the country require two economics courses. But, why? The students are often not happy to be there, and I’ve even heard business professors demean the math as performatively rigorous and superfluous. They argue that plenty of people get rich or are otherwise successful without all of the quantitative skills that economics leverages.

I think that the fear of math is both a red herring and a scapegoat. Rather, Economics confronts students with the liberal arts – whether they like it or not. Be careful. Liberal Arts are not the same as Humanities. They include argumentation, the ability to write and communicate, clear and consistent logic, and, yes, even math. Accounting can tell you how to keep track of the money, but it doesn’t include a theory for when you should produce more or less in contrast to your competitors. Finance does better since it has the time value of money and ‘with vs without’ analysis. That’s closer to marginal thinking. But finance lacks a theory of markets outside of portfolio theory and arbitrage.**

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How do Income Tax Brackets Work?

Zachary Bartsch1 Comment

I was listening to an episode of The Deduction, a podcast by the Tax Foundation. As if that first sentence isn’t evident enough, I was reminded of how confusing taxes are – period. Even experts disagree and see grey areas. As I was listening, I thought “man, they need a graph”. So, here we are.

Income Tax Vocabulary

The money that you are paid by your employer is your gross income. Not all of it is taxable. You can deduct money from your gross income to get your taxable income. Most people subtract the ‘standard deduction’ from their gross income, which is how I’ll proceed in this post. Since the standard deduction for 2026 is $16,100 for a single earner, that means that your taxable income is $16,100 less than your gross income. By following a formula, one can calculate the amount of money that they must pay the government. These payments can be all at once, throughout the year, or even directly from your paycheck. The total that’s due to the government by April 15 is called the total tax liability. Finally, the money that the government doesn’t take, and that you get to keep, is called your net income. It’s your income net of taxes.

If you’ve had a job, then you are probably most familiar with your gross income, what your employer pays you, and your net income, what you get to take home. The steps in between might include some hand-waving.

Marginal Tax Rates

One of the most confusing pieces of the income tax code is marginal income taxes. Below are the brackets for 2026.

Marginal Tax rates work like this: Every dollar that you earn faces a tax rate. If your taxable income would be below zero, then you pay zero in taxes. But if your taxable income is $5k, then it gets taxed at a rate of 10%. That part should be pretty straightforward. But what if your taxable income is $15k? According to the table, you face a tax rate of 10% for dollars earned up to $12,400. That would be a tax liability of $1,240. But the remainder of your $15k in taxable income exists in the next tax bracket. That portion of your taxable income faces a tax rate of 12%. Sticking with the example, $2,600 is in the 12% tax bracket, so the tax liability for that portion of your taxable income is $312 (=$2.6k*0.12). Therefore, your total tax liability would be the sum of your tax liabilities across all applicable tax brackets: $1,552 (=$1,240+$312).

There are some features of marginal tax rates that are worth mentioning. Since the tax rates on the lower taxable income brackets don’t change, earning more gross income never reduces your net income unless the tax rate exceeds 100% (which it doesn’t here). So, when someone says that their taxable income is in the 35% tax rate bracket, they probably just mean that their last dollar earned is there. They’re only paying 35% on the taxable income that’s above $256,225. They’re not paying 35% of all earned dollars to the Internal Revenue Service (IRS).

Below is a graph that details the different marginal tax rates with shaded areas. The blue line is the average tax rate. It’s calculated by dividing the tax liability by the gross income. Even though one might earn an income that’s greater than $257k where the marginal tax rate is 35% or greater, the average tax rate remains lower, topping out at about 30% in this figure. The average tax rate is lower than an earner’s top marginal tax rate because the income in those lower brackets never disappears or get taxed at a higher rate.

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The Arithmetic of Family Punctuality

Zachary Bartsch4 Comments

My children are getting more capable. They get more responsibility that comes with the independence that capability implies. Specifically, when getting ready in the morning they like to leave so that they arrive at school just barely on time. Except, when something comes up, they are rushed, flustered, short-tempered, and tardy. They lament that “if only the unforeseeable event X hadn’t happened, I would have been on time”.

It doesn’t matter what X is. Maybe they forgot to pack a lunch, or set out their clothes, or they have a flat tire on their bikes, or… whatever. The specific time-consuming event is unforeseeable. But, that *any* time-consuming event will occur is very foreseeable. What’s a Bayesian to do?

Before we even start the analysis, let’s acknowledge that being perfectly on time for some event usually involves stress and a lack of preparedness. Yes, you were ‘on time’, but given the probability of heavier traffic, difficulty finding a parking spot, or whatever, we know that tardiness is just one unforeseen event away.

Individual Punctuality

How long does it take to get somewhere? It takes both travel time and time preparing to depart. Let’s just generally call this ‘preparation’ time. Let’s assume that you complete everything that you would complete. That means that you aren’t forgoing a shower or breakfast or whatever lower priority you might choose to forgo to arrive at some obligation punctually.

Random events can occur either as you travel to work or as you prepare to depart, but let’s place the random travel events to the side and focus on what one can do to get out of the house ‘on time’. In my personal case, my children have a 30min interval during which they can arrive at school. They almost never arrive in the first 15min of that interval. That’s more of a policy choice than an accident. They don’t want to sit in a cold gymnasium for 20min if it’s avoidable. So, their planned arrival time has an effective 15min window.

Here is the problem. A time-consuming random event, X, is a right-skewed random variable. Discretely, the modal day includes X=0min. Though the most common delays are greater than 0min. See the distribution below. A 0min random event occurs 35% of the time. But, a time-consuming event happens 65% of the time. So, if you try to arrive exactly on time to your obligation, then you will be punctual 35% of the time and you will be tardy 65% of the time. That’s not a good look and not a good reputation to build – and that’s apart from building a habit of imprudence and the material consequence of not being ready for the task at hand.

Someone with just enough insight to be dangerous might say ‘Ah! Instead, leave with enough time to accommodate the expected unforeseen event’. Mathematically, that’s the weighted average. In this case, that’s six minutes. So, if you plan to arrive 6min early, then you will be punctual – on average. But even that’s not really what we’re after. We’d like to be on time for a preponderance of the days. Building in a 6-minute buffer does two things. 1) Every time that there is a 0min or 5min unforeseen event, you get to your destination 6min or 1min early. That’s good for your nerves, performance, and reputation. But, that also means that you’re late whenever there is a 10min, 15min, or 20min unforeseen event – and those occur 35% of the time!

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Price Level: Noise vs Signal

Zachary Bartsch2 Comments

My university recently hosted a guest speaker. Among their content, they included some nominal macroeconomic values from pre-2020, back in the era when inflation was very low. That roughly includes the years 2012-2019. Truly, inflation stayed below 2% through February of 2021, but I think that we can all agree that the economy was different in a few ways beginning in 2020.

I asked the speaker why not express the nominal values in real terms. They were emphatic that the low rates of inflation at the time implied that the signal-to-noise ratio was too low. Therefore, the ‘real’ inflation adjusted values would not be more precise because excessive noise would be introduced into the series during a period when not much deflating was necessary in the first place.

My answer to this is a firm ‘maybe’. It makes sense and it’s plausible (Jeremy has written about error and revisions in the past). We can think about the noise in price indices in a few ways.

1) It may be information is incomplete and becomes more complete as time passes. This sort of noise only exists in the short-run and is resolved as more information becomes available later in time. Revisions tend to happen each month for prior months, as well as each year for prior years. There are also big revisions after methodological, consumption weight, and data source changes.

2) Another type of noise is due to incomplete information that is never resolved. After all, the government statisticians can’t see literally all of the transactions. Those unobserved transactions will never make it into the official inflation measures and we’ll never get a perfect picture.  

3) Methodological artifacts may also include known biases. This type of noise doesn’t get corrected except after major changes to the series. If those changes never happen, then we just sort of live with imprecision. Luckily, so long as the bias is consistent, then percent change in the price indices will approximate the underlying true levels. However, if there are non-random biases in the percent change, then it can cause some trouble.

One way to get an idea for the amount of noise in the data is to observe the magnitude of revisions. Of course, this only helps us with the first type of noise above that eventually gets resolved with more information. It’s much harder to get a handle on the imprecision that is not identifiable. The Philadelphia Federal Reserve Bank provides an easy-to-use database that puts all of the archival and revised numbers for many macro series in a single place: the Real-Time Data Set (RTDS). It includes every historical PCE price index value for each publication month. Let’s limit our sample to the 21st century.

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Take Your Kids to the Movies

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Economists talk about the ‘Covid Shock’ in 2020 because it was a mostly unpredictable event that had big, measurable effects. People spent a lot less time being in close quarters. Especially hit hard were movie theaters, and other events spaces.

In the several years prior to Covid, “Recreational Service” industry sales had been chugging along, growing at healthy annual rate of 3.4% (inflation adjusted). This category of services includes clubs, sports centers, theaters, and museums. In the blink of an eye, the covid shock drastically reduced spending in that category by more than 60%. See the graph below.

Unfortunately, we don’t have disaggregated series for the components of “Recreational Services”. But we do know that movie theaters were already well past their hay-day. Theaters had been closing and consolidating for more than a decade and ticket sales were down. Many give credit to the popularity of streaming video services and other digital media alternatives. Covid added insult to injury.

Now, going to a movie theater is exceptional. As a teenager in the early naughts, I’d go to the theater easily half a dozen times per year. Now, I don’t think that I’ve gone six times in the last five years. Real growth in the entire recreational service category has grown annually by an anemic 1.8% since 2019. It’s not dead, but that’s also the total industry. I’ve heard the news stories of sports events making a big comeback. I’ve not heard anything like that for movie theaters.

I went to the movies recently and it is not what you remember.

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A Rant about Long Run Problems and Passe Solutions

Zachary Bartsch5 Comments

If you listen to or read major economists discussing what they think are big-picture problems, then their list usually includes three topics: Fertility, Culture, & the Fiscal Health.  On the wonkier side, you’ll also hear that housing scarcity and affordability is a problem, but let’s stick with the first three.

Fertility

People are deciding to have fewer children for a variety of reasons. In no particular order, the reasons include greater access to financial institutions, more popular female education, higher female wages, lower infant mortality, and falling religiosity. Some also speculate that housing affordability, safety regulations, and social safety nets contribute too.

What’s wrong with lower fertility? In an objective sense, there is nothing wrong. But, in the sense that people value similar things, we are in somewhat uncharted territory. Realized fertility is dropping across the globe. We know that economies of scale increase productivity and real wages. We also know that technological innovation comes from having more minds engaged with economic problems. It’s possible that labor productivity rises faster than the productivity that we lose with smaller scale, but it’s an open question. What happens to the liberal societies and polities when the liberals fail to persist? These are big geopolitical concerns.

Culture

People seem to be more fragmented religiously and culturally. Social scientists used to discuss Judeo-Christian norms more often. Sometimes you’d hear about English or Roman legal tradition or enlightenment values. But now, there seems to be very little in terms of common social cohesion. In the USA, the general common culture seems to be ‘smile and be nice’. That’s not the worst common rule, but it’s not enough to hang our hat on for a capable liberal state.

The lack of cultural cohesion isn’t my own particular concern – public intellectuals in economics and elsewhere feel like there is a problem. There is a mix of reasoning behind the concern. Some people are worried about transmitting values to the next generation, some are worried about how people behave when no one’s watching, and still others are worried about simply lacking a Schelling  point that coordinates large scale economic cooperation.

Fiscal Health

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