Investing

Is the Silver Bubble Bursting?

Scott Buchanan2 Comments

This is a five-year chart of the silver ETF SLV:

By most standards, this pattern looks like we entered a bubble a few months ago: speculative froth, unjustified by fundamentals. Economic history is replete with such madness of crowds. It is accepted wisdom on The Street that these parabolic price rises seldom end well. I lost a few pesos buying into the great gold bubble of 2011. All sorts of justifications were given at the time by the gold bugs on why gold prices ought to just keep on rising, or at least reach a “permanently high plateau” (in the famous words of Irving Fisher, just before the 1929 crash). Well, gold then proceeded to go down and down and down, losing some 60% of its value, until the price in 2015 matched the price in 2009, before the great bubble of 2010-2011.

Today, similar justifications are proffered as to why silver is going to the moon. There is a long-standing deficit in supply vs. demand; it takes ten years for a new silver mine to get productive; China has started restricting exports; Samsung announced a breakthrough lithium battery that can charge in six minutes, but requires a kilogram of silver; AI infrastructure is eating all the silver. These narratives seem to feed on each other. As the silver price moved higher in the past month, out came yet wilder stories that ricochet around the internet at high speeds: the commodities exchanges have run out of physical silver to back the paper trades; and the persistent claim that “they” (shadowy paper traders, central banks, commodity exchanges, the deep state, etc.) are “suppressing” silver and gold prices by means of shorting (which makes no sense). Given this popular shorting myth, it was with great glee that the blogosphere breathlessly spread the bogus story that some “systematically important bank” was in the process of being liquidated because it got squeezed on its silver short position.

The extreme price action at the very end of December (discussed below) was like rocket fuel for these rumors. Having bought a little SLV myself so as to not feel like a fool if the silver rally did have legs, I spent a number of hours as 2025 turned to 2026 trying to sort all this out. Here are some findings.

First, as to  the medium term supply/demand issues, I refer the reader to a recent article on Seeking Alpha by James Foord. He shows a chart showing that silver demand is increasing, but slowly:

He also notes that as silver price increases, there is motivation for more recycling and substitution, to compensate. He concludes that the current price surge is not driven by fundamentals, but by paper speculation.

The last ten days or so have been a wild ride, which merits some explanation. Here is the last 30 days of SLV price action:

Silver prices were rising rapidly throughout the month, but then really popped during Christmas week, reaching a crescendo on Friday, Dec 26 (blue arrow), amid rumors of physical shortages on the Shanghai exchange. To cool the speculative mania, the COMEX abruptly raised the margin requirements on silver contracts by some 30%,  from $25,000 to $32,500, effective Monday, Dec 29. I think the exchange was trying to ensure that speculators could make good on their commitment, and the raise in margin requirement would help do that. (Note, the exchange is liable if some market participant fails to deliver as promised and goes BK).

Anyway, this move forced long speculators to either post more collatoral or to liquidate their positions, on short notice. Blam, the price of silver dropped a near record amount in one day (red arrow). For me, a little minnow caught in the middle of all this shark tank action, the key part is what came after this forced decline. Was the bubble punctured for good? Should I hold or fold?

As shown above, the price has traded in a range for the past week, with violent daily moves. Zooming out to the a one-year view, it looks like the upward momentum has been halted for the moment, but it is unclear to me whether the bubble will deflate or continue for a while:

I sold about a quarter of my (small) SLV holding, hoping to buy back cheaper sometime in the coming year. Time will tell if that was a good move.

Usual disclaimer: Nothing here is advice to buy or sell any security.

P.S. Tuesday, Jan 6, 2025, after market close: I wrote this last night (Monday, Jan. 5) when silver was still rangebound. SLV was about $69, and spot silver about $76/oz. But silver ripped higher overnight, and kept going during the day, up nearly 7% at the close to new all time high. It looks like the bubble is alive and well, for now. Congrats to silver longs…

How Good Were 2025 Forecasts?

James BaileyLeave a comment

Last January I shared a roundup of forecasts for the year from markets and professional economists. Were they any good? Here was their prediction for the US economy:

WSJ’s survey of economists reports that inflation expectations for 2025 were around 2% before the election, but are closer to 3% now. Their economists expect GDP growth slowing to 2%, unemployment ticking up slightly but staying in the low 4% range, with no recession. The basic message that 2025 will be a typical year for the US macroeconomy, but with inflation being slightly elevated, perhaps due to tariffs.

The verdicts (based on current data, which isn’t yet final for all of 2025):

Inflation: Nailed it exactly (2.7%)

GDP: We’re still waiting on Q4, but 2025 as a whole is on track to be a bit above the 2.0% forecast.

Unemployment: 4.6% as of November 2025, a bit above the 4.3% forecast

Recession: Didn’t happen, making the 22% chance forecast look fine

So the professional forecasters were probably a bit low on GDP and unemployment, but overall I’d say they had a good year. What about prediction markets?

For those who hope for DOGE to eliminate trillions in waste, or those who fear brutal austerity, the message from markets is that the huge deficits will continue, with the federal debt likely climbing to over $38 trillion by the end of the year. This is one reason markets see a 40% chance that the US credit rating gets downgraded this year.

While the US has only a 22% chance of a recession, China is currently at 48%, Britain at 80%, and Germany at 91%. The Fed probably cuts rates twice to around 4.0%.

Deficits: Nailed it, the federal debt is currently around $38.4 trillion.

US Credit Downgrade: It’s hard to score a prediction of a 40% chance of a binary event happening, but in any case Moodys downgraded the US’ credit rating in May, so that all three major agencies now rate it as not perfect.

The Fed: Cut rates a bit more than expected.

Foreign Recessions: China and Britain avoided recessions. Germany had a recession by the technical definition of Kalshi’s market, but not really in practice (FRED shows -0.2% Real GDP growth in Q2 followed by 0.00000% growth in Q3). Britain avoiding recession when markets showed an 80% chance was the biggest miss among the forecasts I highlighted.

Overall though, I’d say forecasters did fairly well in predicting how 2025 turned out, in spite of curveballs like the April tariff shock.

If you think the forecasters are no good and you can do better, you have more options than ever. Prediction markets are getting more questions and more liquidity if you’re up for putting your money where your mouth is; if you don’t want to put your own money at risk, there are forecasting contests with prizes for predicting 2026.

Investing: You Vs. All Possible Worlds

Zachary Bartsch1 Comment

This post illustrates a couple of things that I learned this year with an application in finance. I learned about the simplex when I was researching amino acids. I learned some nitty-gritty about portfolio theory. These combined with my pre-existing knowledge about game theory and mixed strategy solutions.

Specifically, I learned a way of visualizing all possible portfolio returns. This post narrowly focuses on 3 so that I can draw a picture. But the idea generalizes to many assets.

Say that I can choose to hold some combination of 3 assets (A, B, & C), each with unique returns of 0%, 20%, and 10%. Obviously, I can maximize my portfolio return by investing all of my value in asset B. But, of course, we rarely know our returns ex ante. So, we take a shot and create the portfolio reflected in the below table. Our ex post performance turns out to be a return of 15%.

That’s great! We feel good and successful. We clearly know what we’re doing and we’re ripe to take on the world of global finance. Hopefully, you suspect that something is amiss. It can’t be this straightforward. And it isn’t. At the very least, we need to know not just what our return was, but also what it could have been. Famously, a monkey throwing darts can choose stocks well. So, how did our portfolio perform relative to the luck of a random draw? Let’s ignore volatility or assume that it’s uncorrelated and equal among the assets.  

Visualizing Success with Two Assets

Say that we had only invested in assets A and B. We can visualize the weights and returns easily. The more weight we place on asset A, the closer our return would have been to zero. The more weight that we place on asset B, the closer our return would have been to 20%.

If we had invested 75% of our value in asset B and 25% in A, then we would have achieved the same return of 15%. In this two-asset case, it is clear to see that a return of 15% is better than the return earned by 75% of the possible portfolios. After all, possible weights are measures on the x-axis line, and the leftward 75% of that line would have earned lower returns.  Another way of saying the same thing is: “Choosing randomly, there was only a 25% that we could have earned a return greater than 15%.”  

Visualizing Success with Three Assets

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Google’s TPU Chips Threaten Nvidia’s Dominance in AI Computing

Scott BuchananLeave a comment

Here is a three-year chart of stock prices for Nvidia (NVDA), Alphabet/Google (GOOG), and the generic QQQ tech stock composite:

NVDA has been spectacular. If you had $20k in NVDA three years ago, it would have turned into nearly $200k. Sweet. Meanwhile, GOOG poked along at the general pace of QQQ.  Until…around Sept 1 (yellow line), GOOG started to pull away from QQQ, and has not looked back.

And in the past two months, GOOG stock has stomped all over NVDA, as shown in the six-month chart below. The two stocks were neck and neck in early October, then GOOG has surged way ahead. In the past month, GOOG is up sharply (red arrow), while NVDA is down significantly:

What is going on? It seems that the market is buying the narrative that Google’s Tensor Processing Unit (TPU) chips are a competitive threat to Nvidia’s GPUs. Last week, we published a tutorial on the technical details here. Briefly, Google’s TPUs are hardwired to perform key AI calculations, whereas Nvidia’s GPUs are more general-purpose. For a range of AI processing, the TPUs are faster and much more energy-efficient than the GPUs.

The greater flexibility of the Nvidia GPUs, and the programming community’s familiarity with Nvidia’s CUDA programming language, still gives Nvidia a bit of an edge in the AI training phase. But much of that edge fades for the inference (application) usages for AI. For the past few years, the big AI wannabes have focused madly on model training. But there must be a shift to inference (practical implementation) soon, for AI models to actually make money.

All this is a big potential headache for Nvidia. Because of their quasi-monopoly on AI compute, they have been able to charge a huge 75% gross profit margin on their chips. Their customers are naturally not thrilled with this, and have been making some efforts to devise alternatives. But it seems like Google, thanks to a big head start in this area, and very deep pockets, has actually equaled or even beaten Nvidia at its own game.

This explains much of the recent disparity in stock movements. It should be noted, however, that for a quirky business reason, Google is unlikely in the near term to displace Nvidia as the main go-to for AI compute power. The reason is this: most AI compute power is implemented in huge data/cloud centers. And Google is one of the three main cloud vendors, along with Microsoft and Amazon, with IBM and Oracle trailing behind. So, for Google to supply Microsoft and Amazon with its chips and accompanying know-how would be to enable its competitors to compete more strongly.

Also, AI users like say OpenAI would be reluctant to commit to usage in a Google-owned facility using Google chips, since then the user would be somewhat locked in and held hostage, since it would be expensive to switch to a different data center if Google tried to raise prices. On contrast, a user can readily move to a different data center for a better deal, if all the centers are using Nvidia chips.

For the present, then, Google is using its TPU technology primarily in-house. The company has a huge suite of AI-adjacent business lines, so its TPU capability does give it genuine advantages there. Reportedly, soul-searching continues in the Google C-suite about how to more broadly monetize its TPUs. It seems likely that they will find a way. 

As usual, nothing here constitutes advice to buy or sell any security.

Visualizing the Sharpe Ratio

Zachary BartschLeave a comment

We all like high returns on our investments. We also like low volatility of those returns. Personally, I’d prefer to have a nice, steady 100% annual return year after year. But that is not the world we live in. Instead, there are a variety of returns with a diversity of volatilities. A general operating belief is that assets with higher returns tend to be associated with greater return volatility. The phrase ‘scared money don’t make money’ implies that higher returns are risky. The Sharpe ratio is a tool that helps us make sense of the risk-reward trade-off.

Let’s start with the definition.

By construction, the risk-free return is guaranteed over some time period and can be enjoyed without risk. Practically speaking, this is like holding a US treasury until maturity. We assume that the US government won’t default on its debt. Since there is no risk, the volatility of returns over the time period is zero.

Since an asset’s return doesn’t mean much in a vacuum, we subtract the risk-free return. The resulting ‘excess return’ or ‘risk premium’ tells us the return that’s associated with the risk of the asset. Clearly, it’s possible for this difference to be negative. That would be bad since assets bear a positive amount of risk and a negative excess return implies that there is no compensation for bearing that risk.

The standard deviation of an asset’s returns are a measure of risk. An asset might have a higher or lower value at sale or maturity. Since the future returns are unknown and can end up having any one of many values, this encapsulates the idea of risk. Risk can result in either higher or lower returns than average!

Putting all the pieces together, the excess return per risk is a measure of how much an asset compensates an investor for the riskiness of the returns. That’s the informational content of the Sharpe ratio, which we can calculate for each asset using historical information and forecasts. Once we’ve boiled down the risk and reward down to a single number, we can start to make comparisons across assets with a more critical eye.

Sometimes friends or students will discuss their great investment returns. They achieve the higher returns by adopting some amount of risk. That’s to be expected. But, invariably, they’ve adopted more risk than return! That means that their success is somewhat of a happy accident. The returns could easily have been much different, given the volatility that they bore.

Let’s get graphical.

Consider a graph in (standard deviation, return) space. In this space we can plot the ordered pair for some portfolios. The risk-free return occurs on the vertical intercept where the return is positive and the standard deviation is zero. Say that a student was thrilled with asset A’s 23.5% return and that it’s standard deviation of returns was 16%. Meanwhile, another student was happy with asset B’s 13.5% return and 5% standard deviation. With a risk-free rate of 3.5%, the Sharpe ratios are 1.25 & 2 respectively. We can plot the set of standard deviation and return pairs that would share the same constant Sharpe ratio (dotted lines). Solving for the asset return:

The above is simply a linear function relating the return and standard deviation. In particular, it says that for any constant Sharpe ratio, there is a linear relationship between possible asset returns and standard deviations. The below graph plots the two functions that are associated with the two asset Sharpe ratios. The line between the risk-free coordinate and the asset coordinate identifies all of the return-standard deviation combinations that share the same Sharpe ratio. This line is known as the iso-Sharpe Line.

With this tool in hand, we can better interpret the two student asset performances. There are a couple of ways to think about it. If asset A’s 23.5% return had been achieved with an asset that shared the Sharpe ratio of asset B, then it would have had risk that was associated with a standard deviation of only 10%. Similarly, if asset A’s volatility remained constant but enjoyed the returns of asset B’s Sharpe ratio, then its return would have been 35.5% rather than 23.5%. In short, a higher Sharpe ratio – and a steeper iso-Sharpe line – imply a bigger benefit for each unity of risk. The only problem is that a such an nice asset may not exist.

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Are Your Portfolio Weights Right?

Zachary BartschLeave a comment

What do portfolio managers even get paid for? The claim that they don’t beat the market is usually qualified by “once you deduct the cost of management fees”. So, managers are doing something and you pay them for it. One thing that a manager does is determine the value-weights of the assets in your portfolio. They’re deciding whether you should carry a bit more or less exposure to this or that. This post doesn’t help you predict the future. But it does help you to evaluate your portfolio’s past performance (whether due to your decisions or the portfolio manager).

Imagine that you had access to all of the same assets in your portfolio, but that you had changed your value-weights or exposures differently. Maybe you killed it in the market – but what was the alternative? That’s what this post measures. It identifies how your portfolio could have performed better and by how much.

I’ve posted several times recently about portfolio efficient frontiers (here, here, & here). It’s a bit complicated, but we’d like to compare our portfolio to a similar portfolio that we could have adopted instead. Specifically, we want to maximize our return given a constant variance, minimize our variance given a constant return or, if there are reallocation frictions, we’d like to identify the smallest change in our asset weights that would have improved our portfolio’s risk-to-variance mix.

I’ll use a python function from github to help. Below is the command and the result of analyzing a 3-asset portfolio and comparing it to what ‘could have been’.

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Michael Burry’s New Venture Is Substack “Cassandra Unchained”: Set Free to Prophesy All-Out Doom on AI Investing

Scott BuchananLeave a comment

This is a quick follow-up to last week’s post on “Big Short” Michael Burry closing down his Scion Asset Management hedge fund. Burry had teased on X that he would announce his next big thing on Nov 25. It seems he is now a day or two early: Sunday night he launched a paid-subscription “Cassandra Unchained” Substack. There he claims that:

Cassandra Unchained is now Dr. Michael Burry’s sole focus as he gives you a front row seat to his analytical efforts and projections for stocks, markets, and bubbles, often with an eye to history and its remarkably timeless patterns.

Reportedly the subscription cost is $39 a month, or $379 annually, and there are 26,000 subscribers already. Click the abacus and…that comes to a cool $ 9.9 million a year in subscription fees. Not bad compensation for sharing your musings on line.

Michael Burry was dubbed “Cassandra” by Warren Buffett in recognition of his prescient warnings about the 2008 housing market collapse, a prophecy that was initially ignored, much like the mythological Cassandra who was fated to deliver true prophecies that were never believed. Burry embraced this nickname, adopting “Cassandra” as his online moniker on social media platforms, symbolizing his role as a lone voice warning of impending financial disaster. On the About page of his new Substack, he wrote that managing clients’ money in a hedge fund like Scion came with restrictions that “muzzled” him, such that he could only share “cryptic fragments” publicly, whereas now he is “unchained.”

Of his first two posts on the new Substack, one was a retrospective on his days as a practicing doctor (resident in neurology at Stanford Hospital) in 1999-2000. He had done a lot of on-line posting on investing topics, focusing on valuations, and finally left medicine to start a hedge fund. As he tells it, he called the dot.com bubble before it popped.

The Business Insider summarizes Burry’s second post, which attacks the central premise of those who claim the current AI boom is fundamentally different from the 1990s dot.com boom:

The second post aims straight at the heart of the AI boom, which he calls a “glorious folly” that will require investigation over several posts to break down.

Burry goes on to address a common argument about the difference between the dot-com bubble and AI boom — that the tech companies leading the charge 25 years ago were largely unprofitable, while the current crop are money-printing machines.

At the turn of this century, Burry writes, the Nasdaq was driven by “highly profitable large caps, among which were the so-called ‘Four Horsemen’ of the era — Microsoft, Intel, Dell, and Cisco.”

He writes that a key issue with the dot-com bubble was “catastrophically overbuilt supply and nowhere near enough demand,” before adding that it’s “just not so different this time, try as so many might do to make it so.”

Burry calls out the “five public horsemen of today’s AI boom — Microsoft, Google, Meta, Amazon and Oracle” along with “several adolescent startups” including Sam Altman’s OpenAI.

Those companies have pledged to invest well over $1 trillion into microchips, data centers, and other infrastructure over the next few years to power an AI revolution. They’ve forecasted enormous growth, exciting investors and igniting their stock prices.

Shares of Nvidia, a key supplier of AI microchips, have surged 12-fold since the start of 2023, making it the world’s most valuable public company with a $4.4 trillion market capitalization.

“And once again there is a Cisco at the center of it all, with the picks and shovels for all and the expansive vision to go with it,” Burry writes, after noting the internet-networking giant’s stock plunged by over 75% during the dot-com crash. “Its name is Nvidia.”

Tell us how you really feel, Michael. Cassandra, indeed.

My amateur opinion here: I think there is a modest but significant chance that the hyperscalers will not all be able to make enough fresh money to cover their ginormous investments in AI capabilities 2024-2028. What happens then? For Google and Meta and Amazon, they may need to write down hundreds of millions of dollars on their balance sheets, which would show as ginormous hits to GAAP earnings for a number of quarters. But then life would go on just fine for these cash machines, and the market may soon forgive and forget this massive misallocation of old cash, as long as operating cash keeps rolling in as usual. Stocks are, after all, priced on forward earnings. If the AI boom busts, all tech stock prices would sag, but I think the biggest operating impact would be on suppliers of chips (like Nvidia) and of data centers (like Oracle). So, Burry’s comparison of 2025 Nvidia to 1999 Cisco seems apt.

What Tariffs Mean For Your Finances

James BaileyLeave a comment

That’s the title of a talk I’ll be giving Saturday at the Financial Capability Conference at Rhode Island College. Registration for the conference, which also features personal finance speakers and top Rhode Island politicians, is free here.

A preview: after many changes, the average tariff on the goods Americans import has settled in the 15-20% range:

If the tariffs stay in place, which is far from certain, this will represent roughly a 2% increase in overall costs for Americans (a ~17% tax on imports which are ~14% of the economy predicts a 2.4% increase, but a bit of that will be paid by foreign producers lowering prices).

This is bad for US consumers, but not as bad as the Covid-era inflation, and likely not as bad as our upcoming problems with debt and plans to weaken the dollar. It is more valuable for most people to make sure they are getting the personal finance basics right than to think about how to avoid tariffs, though they may want to consider investments that hold their value with a weakening dollar.

“Big Short” Michael Burry Closes Scion Hedge Fund: “Value” Approach Ceased to Add Value?

Scott Buchanan1 Comment

Michael Burry is famed for being among the first to both discern and heavily trade on the ridiculousness of subprime mortgages circa 2007.  He is a quirky guy: brilliant, but probably Asperger‘s. That comes through in his portrayal in the 2015 movie based on the book, The Big Short.

He called it right with mortgages in 2007, but was early on his call, and for many months lost money on the bold trading positions he had put on in his hedge fund, Scion Capital. Investors in his fund rebelled, though he eventually prevailed. Reportedly he made $100 million himself, and another 700 million for his investors, but in the wake of this turmoil, he shut down Scion Capital.

In 2013 he reopened his hedge fund under the name Scion Asset Management. He has generated headlines in the past several years, criticizing high valuations of big tech companies. Disclosure of his short positions on Nvidia and Palantir may have contributed to a short-term decline in those stocks. He has called out big tech companies in general for stretching out the schedule of depreciation of their AI data center investments, to make their earnings look bigger than they really are.

Burry is something of an investing legend, but people always like to take pot shots at such legends. Burry has been rather a permabear, and of course they are right on occasion. For instance, I ran across the following OP at Reddit:

Michael burry is a clown who got lucky once

I am getting sick and tired of seeing a new headline or YouTube video about Michael burry betting against the market or shorting this or that.

First of all the guy is been betting against the market all his career and happened to get lucky once. Even a broken clock is right twice in a day. He is one of these goons who reads and understands academia economics and tries to apply them to real world which is they don’t work %99 of the time. In fact guys like him with heavy focus on academia economic approach don’t make it to far in this industry and if burry didn’t get so lucky with his CDS trade he would be most likely ended up teaching some bs economic class in some mid level university.

Teaching econ at some mid-level university, ouch.  (But a reader fired back at this OP: OP eating hot pockets in his moms basement criticizing a dude who has made hundreds of millions of dollars and started from scratch.)

Anyway, Burry raised eyebrows at the end of October, when he announced that he was shutting down his Scion Asset Management hedge fund. This Oct 27 announcement was accompanied by verbiage to the effect that he has not read the markets correctly in recent years:

With a heavy heart, I will liquidate the funds and return capital—minus a small audit and tax holdback—by year’s end. My estimation of value in securities is not now, and has not been for some time, in sync with the markets.

Photo

To me, all this suggested that Burry’s traditional Graham-Dodd value-oriented approach had gotten run over by the raging tech bull market of the past eight years. I am sensitive to this, because I, too, have a gut bias towards value, which has not served me well in recent years. (A year ago I finally saw the light and publicly recanted value investing and embraced the bull, here on EWED).

Out of curiosity, therefore, I did some very shallow digging to try to find out how his Scion fund has performed in the last several years. I did not find the actual returns that investors would have seen. There are several sites that analyze the public filings of various hedge funds, and then calculate the returns on those stocks in those portfolio percentages. This is an imperfect process, since it will miss out on the actual buying and selling prices for the fund during the quarter, and may totally miss the effects of shorting and options and convertible warrants, etc., etc. But it suggests that Scion’s performance has not been amazing recently. Funds are nearly always shut down because of underperformance, not overperformance.

Pawing through sites like HedgeFollow (here and here) , Stockcircle, and Tipranks, my takeaway is that Burry probably beat the S&P 500 over the past three years, but roughly tied the NASDAQ (e.g. fund QQQ). This performance would naturally have his fund investors asking why they should be paying huge fees to someone who can’t beat QQQ.

What’s next for Burry? In a couple of tweets on X, Burry has teased that he will reveal some plans on November 25. The speculation is that he will refocus on some personal asset management fund, where he will not be bothered by whiny outside investors. We shall see.

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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