Measurement Not Prediction

Morgan tweeted a fun question today

It’s a provocative exercise to reduce investing to a single law. There are many thoughtful responses. It’s a healthy opportunity to consolidate your core beliefs about investing.

Twitter has exposed me to many ideas that I’ve re-mixed with my own experience. That process may uncover universal truths. It has helped form new beliefs, challenged old beliefs, and improved my understanding of some beliefs. I maintain a notebook of these beliefs.

There are 3 categories:

  1. Current Priors
    Examples: Long term bonds offer poor risk/reward or demographics are slow moving drivers of growth rates
  2. Beliefs Graveyard
    Examples: Profit margins should mean revert, printing money leads to inflation, more risk equals more reward
  3. Evergreen Beliefs
    I have publically shared these on my wiki.

I have recently added another to my “current priors”. It’s a belief that has underpinned my trading style for 20 years but I was only able to name it recently. Why? Because it only revealed itself in relief to what other people think investing is about. By observing how people think about investing, I’ve noticed my current prior is not widely obvious:

Investment edge is about measurement not prediction.

The distinction between prediction and measurement

Predicting means you make a probability-weighted opinion of future states of the world. This is typically built up from logic and reasoning.

Measurement is counting. Usually with an intent to rank. The idea is simple but not easy. Accounting is an entire field devoted to trying to create a meaningful picture of a business by normalizing quantities across industries.

These are of course not mutually exclusive activities. Measurement is done in current time and then projection leads to predictions. The prediction eventually becomes a point in the past, gets counted, and iteratively fed into the next projection.

Yet when I say edge is about measurement not prediction I am very much emphasizing the importance of measurement over prediction. There is just a single prediction underlying the entire framework:

Over time, cheap stuff outperforms expensive stuff.

But there is so much to measure. Once you accept this, your gaze shifts from the future to the present. The focus becomes finding the variables that matter, harvesting and cleaning data, and normalizing across markets. Every one of these steps leaves tremendous opportunity for creativity, discretion, and disagreement.

Why do I make this distinction?

There are many ways to skin a cat (don’t linger on that too long, it’s an expression better left unexamined).

Your cornerstone beliefs depend heavily on the path your career has taken. There are activist investors who use their rights to influence outcomes. Thematic investors concentrate their portfolio around a vision of the future (AI, crypto, cannabis, clean energy). They are the epitome of crystal ball predictors wagering their money and time. These strategies are high risk-reward with long feedback loops.

But I came into the investing world from the nearly the opposite angle. The market-making side. I see trading as the role of the house in a casino. That means diversifying over a wide range of discrete edges and not letting any single bet risk putting you out of business. Rapid feedback loops and large sample sizes. The focus is on operationalizing and measurement. In options, if you do not understand volatility time you cannot compare implied vols across markets. There’s only one major prediction: 1 If I measure cheap and expensive correctly it will pay off as surely as the casino gets paid.

If the distinction isn’t clear consider this story from Andrew Tsai about his internship at Susquehanna (disclosure: I worked there for 8 years after college). He recounts a company outing to a dog track:

I’m sitting next to one of the partners and I’m looking at the sheet of all the races, and he’s like “How are you gonna bet?” I respond, “Well, I’ve never really done this before but this dog looks like he’s got a good track record and he’s been running strong lately.”

The guy looked at me like I was a complete idiot.

He’s like, “What are you talking about, ‘How is this dog doing?’”

Andrew is perplexed. Well, isn’t that kind of what we’re talking about.

The partner starts to explain, Look at the relative value of this dog and that dog.

Tsai’s confusion was at the heart of prediction vs measurement. The partner didn’t have any opinion about the dog’s chances. He was just looking for mispriced lines. Which is another way of saying, “I don’t know what is going to happen but these prices are self-contradictory compared to the future state of the world.”

Yes, there’s some embedded prediction there, but the framing is very different from wondering which dog is feeling extra feisty.

Measurement — simple but not easy.

If you normalize correctly then the cheap and expensive are just colors on a heatmap. In options, you must normalize vols, time, funding, and any parameter you believe drives pricing. Your feedback loops are punctuated by expiration cycles where the asset’s distribution needs to be settled. That’s the beauty of options. My friend Jeff explains this beautifully with respect to futures markets in his interview with Corey Hoffstein. Expirations force a convergence of truth. Manage risk, learn, repeat.

Exporting the logic to fundamental investing is outside my practical experience, but I can see the complexity of the problem. Consider stocks. As perpetual claims on future earnings, they have durations that vary with market discount factors. Today, fundamental investors feel like the ref (aka discount factor & interest rates) has tacked 90 minutes of extra time to the soccer match as earnings are amortized from ever further into the future.

I’m even more daunted when I consider wrangling fundamental parameters. A superficial process doesn’t distinguish between a value stock and a value trap. Even deeper, there’s a big difference between a stock whose cheapness reflects a low forward ROIC vs a stock that’s cheap because the market knows its inventory is overvalued.

From the outside, it appears that quant approaches to stock-picking come from the measuring camp. Yet commoditized approaches are likely frustrated by naïve measurement practices or practices that do not truly scale. So long as reading the fine-print in financial statements is labor-intensive and dependent on judgement, stock-picking will be an opportunity for analysts at the pinnacle of the art of measuring.

In addition to understanding business, there’s an adjacent skillset of understanding finance. In this thread by @10kdiver, we can see how stock based compensation can obscure the cap structure of a company. Any valuation estimate downstream from this analysis will be sensitive to how this impact is normalized across stocks’ dilution paths.

Selling a differentiated depth of knowledge in both finance and industry is easier than actually possessing it. So it makes sense to have a high bar when faced with someone claiming to have both.

A focus on measurement and agnosticm

I’m probably one of the few individuals who has ever leased on a seat on the Amex, NYSE, NYMEX, COMEX, and NYBOT/ICE (so every exchange in NYC. You are welcome to view this as a demerit as well).

By the end of my floor trading life, I leased (believe me I wish I owned) seats on all 3 exchanges in the NYMEX building at 1 North End Avenue. I was a ghost who would migrate from pit to pit based on where the action was. I learned the specific dynamics of several markets including oil, gas, precious metals, and soft commodities. Yet, I only dove into fundamentals deep enough to understand the major “gotchas” (usually squeeze mechanisms and bottlenecks) and seasonality trends. The Slovic study on horse bettors who were given more info than they needed showed that the extra data increased their confidence faster than their accuracy. And as you’ve heard, “It’s not what you don’t know that kills you, it’s what you know for sure that ain’t true.”

Find the variables that matter and ruthlessly tune. Normalization within and across these markets became the focus. I was a microcosm of the firms I had worked for. The specifics of a market needed to be learned to be fed up into a more bird’s eye view. Then it was a matter of sorting cheap and expensive. At no point did I have an opinion on underlying price. I’m a donkey. I just look for mispriced point spreads. Yes context matters, but as much as possible, I held an end goal of feeding normalized parameters into a framework to see what pops out.

Today, I remain skeptical of thematic and fundamental approaches. The former because it relies on prediction. For fundamental approaches, I’d like to see how a manager shortens the feedback loops. How they can map the variables they focus on to outcomes. Ideally variables that bounce around so there is trading opportunity and sample sizes. If this process lacks iteration it cannot improve. It just starts to look like thesis investing.

My bias (again, views are path-dependent and I am clearly giving my experience lots of weight) is towards agnostic, opportunist type of investors. Folks that are normalizing lines rigorously to find the mispriced horse. No allegiance to asset class or sector. When I talk to a manager I want to know what kind of metrics they look at, why those metrics matter, how they know if they still matter, and what short cycles they study to inform the metastability of their framework.

You are either a casino or a tourist. So if you claim you can count cards I need to be convinced they are countable. Can you measure?

Notes on Trading Volatility: Correlation, Term Structure, and Skew

Trading Volatility: Correlation, Term Structure, and Skew
Colin Bennett

The book is a broad reference on basic option theory, dispersion, and exotic options. It includes practical insight into managing a hedged book with a focus on correlation, term structure, and skew.

In addition its appendix includes the following topics and more:

  • a taxonomy of historical vol computations including and how they rank on “bias” and “efficiency”
  • shadow greeks
  • cap structure arbitrage theory

It’s an outstanding reference so I took notes. For public sharing I re-factored them by topic and tied some back to my own investment writing.

You can find these edited notes in my public Notion page. (Link)

Vol Premium [Partial] Justification

I’m about halfway through Colin Bennet’s terrific book Trading Volatility, Correlation, Term Structure and Skew (pdf).

Bennet is (or was) the Head of Quantitative Derivatives Strategy at Santandar. The book sits in a very sweet spot. It has lots of practical insights into managing vol portfolios and the mechanics of both vanilla and exotic options, var, and vol swaps. I’ll likely do a full post summarizing the takeaways I appreciated most, but in the meantime I thought to share this blurb about the oft-referred VRP (vol risk premium).

Just because implied vols trade over realized does not mean they are mispriced:

[To be fair the author asserts they still are. More importantly, you should read ch. 3 of the book to see how he decomposes the premium to systematic risk and pure vol demand premia.]

I wrote something similar a few weeks back:

Index options should be “overpriced”.

The question is how much premium do they deserve. If stocks warrant a risk premium over the RFR it’s because their systematic risk cannot be hedged. Index options must conceptually inherit this premium otherwise there would an arb in portfolio allocation.

An index option, held delta neutral, gets paid as correlations in the marketplace increase. It literally makes money when systematic risk embodies.

A standard for deciding if puts are expensive: Its price should have enough premium in it that by buying a put, if delta hedged, that you would actually have basis risk. In other words, it’s premium should make it uncertain that you would actually make money in a sell-off. If your argument is that it’s expensive in a vacuum (perhaps as a comparison to realized vol) then what if it was only 1% premium to realized? That sounds like a bargain for something that hedges the risk that, like, the whole world has. This isn’t news to most investors or anyone who understands portfolio construction and the beauty of neg correlations. It’s just another instance of my sun/rain example.

The ‘Volatility Is A Risk’ Strawman

In my short post Is Volatility A Risk?, I urged that any definition of risk:

should be evaluated by its usefulness. Any single definition is incomplete and insufficient for making an investment decision.

Here’s a specific case.

  • How The Sharpe Ratio Broke Investors’ Brains (Link)
    Institutional Investor

    This is a good overview the shortcomings of Sharpe ratio, most of which should be well-understood by anyone who has computed a standard deviation.

    I’ll expand on some of the less obvious points:

    • If you annualize Sharpes from monthlies you risk overstating it if the strategy returns are serially correlated.

      Why? Because you are understating the vol which you can no longer assume scales at the square root of time. This is a complicated issue because auto-correlation, while easy to compute, is itself subject to variation.

    • Pardon my yawn, but apparently option sellers game the Sharpe ratio fetish by selling nickels in front of a steam roller. If the image of straw allocators investing on the basis of a single measure keeps you up at night then, sure, sound the alarm. Skewness can hide within vol.

      A quick demo:

      a) Bet $1 on a fair coin
      b) Bet $.33 on heads on a coin that costs 9-1 if tails but has 90% of coming up heads (still a fair coin).

      These bets have the same vol ($.33 creates risk or vol parity weighting) but the payoff shape is materially different.

    • There are popular alt ratios like Sortino, Calmar, and Omega which try to correct for skewness by penalizing drawdowns and giving hall passes to upside volatility. These are not panaceas since they correlate strongly to Sharpes. This reinforces the idea that you can’t compress the nature of any strategy into a single number. (I feel the tendency to pretend that anybody evaluates investments so naively is a straw man drubbing of allocators signaling no deeper handle of the problem than an influencer who read Taleb on a cross-country flight. Like do you even know an allocator?)
    • A point the article didn’t mention: you can have high Sharpe strategies that cannot generate high returns. Like investing in T-bills. If the cost of levering the strategies is prohibitive then Sharpe would yet again not be the only number you can look at.

Ok, I’m done suspending my disbelief that anyone uses a a single metric in isolation to decide anything of importance. The post is worthy reading for new investors who just discovered Sharpe before they run out and impale themselves on it. I hope my additions made it a touch more interesting for the initiated.

40% Of Your Assets In…OTM Options?

The treasury issues EE Bonds that yield 3.5% guaranteed if held for 20 years. In the interim, they are totally illiquid.

Meanwhile 20-year US treasuries yield 1.5% if held to maturity. However these are liquid, so you can sell them anytime.

Is it worth giving up 2% per year for the liquidity?

In How Much Extra Return Should You Demand For Illiquidity I explore this question and what it depends on. There are other examples of how other investments are priced based on their liquidity. I provide 2 frameworks to consider as you try to price liquidity.
Applying the logic to the current environment
Putting your money in a lockbox for 20 years to earn 3.5% per year might sound attractive if you decide liquidity isn’t worth much to you. Especially when the equivalent liquid treasury only yields 1.5%.

But as @econompic shows, there is no period in the last 75 years that if you looked back 20 years at stocks did you only earn 3.5% per year.

It’s reasonable to point out that stocks are not bonds so the comparison is made of straw. But the counter to the counter is that if you are putting the money in a box and throwing away the key for 20 years, then the comparison is not crazy. A significant benefit of bonds comes from the ability to rebalance. But with a 3.5% bond trapped in a box you lose the option to rebalance.

So we are stuck with that 1.5% bond. It’s nearly cash. Let’s not sugarcoat this. Bonds at current pricing are just an option on deflation. And the premium is all extrinsic value since they have negative real returns. Since they are now an option that you pay for in real terms, they are no longer an investment but an insurance policy. Once you see it like that, you have to wonder if their appropriate allocation size should be more commensurate with that line of thinking. Would you put 40% of your portfolio in option hedges? I didn’t think so.

Is anyone still putting 40% of their portfolio in bonds? Asking for an industry.

How Much Extra Return Should You Demand For Illiquidity?

In some corners of asset management, marketers are offering to lock up your money to “save you from yourself”. These Samaritans don’t want you to succumb to behavioral biases and overtrading. I’m fine if private funds want to argue that the best opportunities are illiquid (I don’t have to believe them but I’m ok with them making this argument). But don’t tell me your lockups are doing me a favor. Don’t act like you shouldn’t be giving me a discount for tying up my money.

Should You Care About Liquidity Even If You Are Talking About Money You Don’t Need For A Long Time

Perhaps you are one of these people who doesn’t want to put their hands on the wheel. You are self-aware enough to know you’d chop yourself to pieces in the market. First, I commend this level of self-awareness but you still deserve a discount all else equal (I know it never is).


First of all, your needs or preferences don’t set the marginal price. Don’t be so vain, not everything is about you 🎶. The price of illiquid investments are set by those who do care about liquidity even if you don’t. You inherit that discount the same way you get power windows for free nowadays. You get that even if you think you’d be better off with the physical exercise of cranking your own windows.

The second reason why illiquidity deserves a discount or liquidity deserves a premium is liquidity itself is an option. Any argument that says liquidity is bad, whether for behavioral or any other reason, needs to address the value of that option.

Uh oh. Are we going to need to price some abstract option?

Fortunately, no. We can build the intuition from option theory to demonstrate that liquidity is not only valuable, but quantifiably so. It gets better. We can also point to another approach that demonstrates the measurable value of liquidity without pricing options. The best part is its driven by the same underlying logic that makes the option approach work.

Before we start thinking about the value of liquidity, let me start with why I started thinking about this question.

3 things I’ve come across recently have made wonder about how big a liquidity premium is warranted.

1. “Networks of confidence

I was listening to @Jesse_Livermore on the Invest Like The Best podcast.

I’ll paraphrase:

High valuations are increasingly dependent on liquidity or what he terms “networks of confidence”. He refers back to prior work that shows how you’d need a healthy discount to intrinsic to buy an asset you couldn’t sell.

On Twitter, he later posed a cool thought experiment where you price an asset that has no fundamental risk but unpredictable, perhaps zero liquidity in the future. The only thing you can rely on is its unchanging (even in real terms) dividend.

Look through the thread and you will not be able to unsee how little thought we put into pricing liquidity.

2. Illiquid vs Liquid Bonds

I came across this article about US Government EE bonds which showed how a feature of these treasury-issued bonds is, if held for 20 years, you are guaranteed 2x your money back. That means your worst case is you earn a 3.5% CAGR nominally. The catch: they don’t trade in an open market. Compare that to liquid 20 year US treasuries at about 1.5% yield. [Let’s set aside the fact that one can only buy $10k worth of EE bonds per year.]

There is a 2% per year difference in yield if you held both to maturity! That sounds big. But is it? This comparison is a perfect example of why we’d really like to be able to quantify the value of liquidity.

3. Insurance products

I know someone who is considering jamming a bunch of savings into an insurance product that “guarantees” around 3.5% per year if held for about 25 years. I don’t want to turn this into a post about insurance, I have enough brain damage from the email threads I’m privy to. The larger point is there are products where you can earn more yield for sacrificing liquidity even after after adjusting for the credit risk and the actuarial features of these products. (PSA: If you are interested in getting technical about insurance my buddy @RajivRebello covers it from the institutional side. In other words, he understands the math and levers in ways you cannot pry out of retail brokers.)

Hopefully I have convinced you that a) liquidity is worth a premium and b) we are faced with real-life comparisons that beg us to price it.

How big should the liquidity premium be?

As I alluded earlier, there are 2 frameworks in which I have started to think about this. Let’s start with the approach I found personally more intuitive (although I suspect most of you will find the second approach more natural).

The Liquidity-Is-An-Option Replication Approach

First, what’s the obvious advantage of liquidity?

You can cut risk.

The fact that a market is willing to show you a bid for your investment at all times has a real theoretical value. That may sound abstract but the entire options market is actually built upon that idea. Let’s see how.

Go back to the untradeable EE bond vs the 20 year treasury. The nominal EE bond has a nominally guaranteed CAGR of 3.5% if held to maturity. But it’s real return is not guaranteed. In real terms, you can technically lose 100%. In contrast, the liquid treasury bond can be sold. If you placed a stop order on it, you can create one of those hockey stick payoff diagrams where the most you can lose is your stop price.

So…you have created an option. This was the entire basis of portfolio insurance.

[reader recoils 🤮]

I know, I know.

1987 ruined the term portfolio insurance. But the reality is some version of it is done every time a market maker sells an option and delta hedges it. The market maker is trying to dynamically replicate the option they have sold. They are “manufacturing” a long option. The market maker hopes the accumulation of losses due to negative gamma (buying high and selling low) is less than the premium they collected up front for the option.

The key here is to recognize that the ability to “manufacture” an option by trading is possible because of liquidity.

Sure, there are caveats. The “manufactured” option fails in the presence of gaps. It’s not as valuable as “hard” or contractual optionality. 1987, in fact, makes my point…the constraint on theory is liquidity. Liquidity is valuable in itself because it sustains options. And options are good.

[aside: options are valuable because they allow you to fine tune risk. Slice & dice the expressions of your desired exposure or lack of exposure. Equity is an option. Capital structures allocate options according to what shape of risk people are willing to take. Some investors require insurance. Some investors only equity, while others may prefer debt. And there are some debt holders who want CDS, another type of option that can be relative-value arbitraged against vanilla options]

Back to the bonds…

So we can think of a liquid bond as having an option to sell that the EE bond does not.

Liquid bond = EE bond + Option

What’s that option worth? Pricing the option (if we assume the market is continuous) will be an exercise in portfolio insurance-esque replication.

The recipe will look something like this:

1. Pick some theoretical strike price (ie maybe a desired stop price)

2. Estimate what option is worth

3. Add it to the cost of an EE bond that guarantees 3.5% for 20 yrs.

Compare the portfolio comprising a 3.5% EE bond + this theoretical option  to a portfolio which simply holds the 1.5% treasury and you are taking a big step toward quantifying the value of liquidity!

That identity one more time:

Liquid bond = EE bond + Option

What is the main driver of the option’s value?


The premium we are willing to pay for liquidity depends on volatility. The higher the volatility the more the liquidity option is worth and the larger the gap should be between a liquid and illiquid price. 

It’s interesting to consider in light of recent valuations. The more volatile the future is, the bigger the discount we should ascribe to illiquid assets. Today, with implied vols relatively elevated, private investing should be worth less if all else is equal. (I expect private managers to claim that all the alpha is in private markets which is an argument they are entitled to)

To restate the main point of the liquidity-as-an-option replication approach:

Increased volatility raises the value of liquidity because it raises the value of the option embedded in the ability to trade.

I mentioned there’s a second framework for valuing the premium we can ascribe to liquidity.

Rebalancing premium

The ability to rebalance your portfolio is valuable.

Here’s an intuitive demonstration:

Markets take a dive. Pretend 1/2 your wealth was in stocks and 1/2 in your home. If homes were down more than stocks, you could sell stocks & upgrade your home while restoring a 50/50 allocation.

In Lessons From Coin Flip Investing, I showed how rebalancing between a coin flipping investment and a positive expectancy investment enhances performance. Specifically, rebalancing pushes your geometric return up towards the expected arithmetic return (remember geometric returns are lower than arithmetic because of volatility drain). You earn a “premium” for rebalancing.

There are 2 main drivers of the rebalancing premium.

  • Volatility 

    The size of  the premium is a direct function of volatility since the drain is half the variance. This should be satisfying — the option replication framework also said that volatility increases the value of liquidity.

  • Correlation

    A full explanation would be out of scope here. Instead, I direct you to
    @breakingthemark and his blog. His recent post, The Great Age of Rebalancing Begins, shows how lower fees/spreads provide unprecedented opportunity to capture “Shannon’s Demon” — the underlying concept behind rebalancing premium first identified by Claude Shannon. 

Key Takeaways

  • Liquidity-As-An-Option and rebalancing premium are 2 ways to price the value of liquidity
  • Both methods agree — the greater the volatility, the more liquidity is worth.
  • You should get a better deal for accepting less liquidity

Path: How Compounding Alters Return Distributions

Compounded returns experience “variance drain”. This idea captures the fact that typical result of compounded returns is lower than if you compute arithmetic returns even though the expected value is the same. We mostly care about compounded returns. This describes the situation in which your bet size or allocation is a fixed percent of your wealth, savings, or bankroll.

This is in contrast to keeping your bet size fixed (ie if you invested $10,000 in the stock market every year regardless of your wealth).

The distinction is critical because as humans we experience the path of our investments so we care about the distribution of returns in addition to the expected value.

Let’s back up for moment.

Recapping Intuition

  • What land are we in?
    • Compounding Land

      If you bet 1% of your wealth on a coin flip and win then lose, you are net down money. This is symmetrical. If you lose, then win, still down money.

      1.01 * .99 = .99 * 1.01

    • Additive Land

      In additive or non-compounding land we bet a fixed dollar amount regardless of wealth.

      So if I start with $100 and win a flip, then bet $1 again and lose the flip I’m back to $100. The obvious reason is the $1 I bet when my bankroll increased to $101 is less than 1% of my bankroll.

  • The order of win then lose, or lose then win leaves you in the same place in both worlds.

    The order does not matter if we are consistent about how we size the bet (so long as we are consistent to the style whether it’s fixed dollar or fixed percentage).

So is fixed percentage somehow “bad” in that it opens you up to volatility or variance “drag”? 

Well in the last example we used an alternating paths. Win then lose or vice versa. Let’s look at the case where instead of alternating wins and losses, we trend. Win-win or lose-lose.

  • In the additive case, we are either up 2% or down 2%
  • In the compounded case we are up 2.01% or down 1.99%

Wait a minute. In the compounded case, we are better off both ways! So the compounded case is not always worse.

The compounded case is better when we trend and worse when we “chop”.

If bet a fixed percent of our bankroll fair coin toss game we are in compound return land.

Compounding is not “bad”, it just alters the distribution of our terminal wealth

Your net compounded return in the coin-flipping game is negative more often than it’s positive, even though the game has zero expectancy.

So why is the median outcome negative?

It goes back to the trend vs the chop. Compounding likes trending and hates chopping as we saw earlier.

  •  Chopping happens more 𝐨𝐟𝐭𝐞𝐧 so you get a negative median
  • …but this is balanced by a larger trending bonus due to compounding.

Let’s illustrate.

2 Coin Flips

There’s 4 actual scenarios:

2u (trend)
1u, 1d (chop)
1d, 1u (chop)
2d (trend)

Zoom in on “compounding bonus/drag”:


  • Chop and trend happen equally.
  • The magnitude of the boost/drag is also equal.

3 Coin Flips

There’s 8 total outcomes, but again order doesn’t matter. So there’s really just 4 outcomes.

The “chops” are bolded. They represent compounding “drag”


  • You drag 75% of the time!
  • The larger positive boost magnitudes make up for the frequency.

Now that you have the gist, let’s do 10 flips.

10 Coin Flips

  • 65% of the results are chop giving you compounding drag.
  • The times you trend though crush your performance if you only bet fixed dollar!

Visualizing “The Chop”

Let’s take a look visually at paths where N=10  to see the “chop”.

Pascal’s Triangle is a quick way to to get the coefficients of a binomial tree. The coefficients represent combinations which are weighted by the probabilities in the binomial expansion.

I enclosed the “chop” or drag paths

100 Coin Flips

  • The negative median now becomes very apparent in the “cumulative probability” column.
  • The chop occurs in 68% of paths. The median return is -.50% after 100 flips though the expectancy is still zero.
  • In additive world if you win 50 $1 bets and lose 50 $1 bets your p/l is zero.
  • In compounding world, where you bet 1% each time you are down 50 bps in that scenario.
  • The negative median associated with compounding is balanced by better outcomes in the extremes.

Both the maximum and minimum returns in simulations are better than the fixed bet case. This simulation by Justin Czyszczewski (thread) shows just how substantial the improvement is in those less probably trending cases:

Lessons From Compounding Coin Flips

  • Your overall expectancy is zero because the common chop balances the rare but heavily compounding trends.
  • Paths affect distribution of p/l even if they don’t affect expectancy.

Since we actually experience “path” and all its attendant emotions, it pays to think about the composition of expectancy and returns.

Making Property Taxes Apples to Apples

You will be working from home more often. Not all of you but many of you. That means browser tabs devoted to Zillow searches in Austin, Nashville, Vegas, Denver, and Miami. Geo-arbitrage won’t be as dramatic as software devs had hoped since the big companies will cut your pay when you leave, but in some of these places you could sustain a 20% pay cut and still be better off (at least if you’re leaving SF).

One of the biggest inputs into cost-of-living comparisons are so-called SALT (state and local) taxes. Since 2018, SALT deductions are limited to $10,000. They were previously uncapped. This has created even larger disparities in cost-of-living between states. CA, IL, NJ, and NY have income taxes that get a bit handsy with their residents.

Beyond state income taxes, one needs to consider property taxes for a more complete picture. Texans enjoy zero state income tax but hefty property taxes. NJ residents are assaulted from both ends — above average state income taxes and punitive property taxes. How about CA? The state income tax, gas tax and the cost of renewing a vehicle registration are nothing short of sunny weather ransoms.

But what about CA property taxes? The answer to this is sneaky and can be used to understand the impact of property taxes in general. But I’d go further and say that if you have not walked through the math the way we are about to, then you may be walking around with some very mistaken impressions about the cost of housing.

Property Taxes: Apples to Apples

The effect of property taxes depends on 2 core variables. The property tax rate and the assessed value. If you are weighing a house in CA to a house in NJ you want to make an apples-to-apples comparison. How do you do that when the rates are different and the methods of assessing value are different?

Let’s isolate each effect.

[Obviously the cost to buy a home has many factors that can mostly be tucked under the headings of supply and demand. Yet the effect of property taxes is significant so it’s worth isolating. It’s also worth noting that since a primary residence is most people’s largest asset, a property tax is a defacto, albeit incomplete, wealth tax. Economically it’s passed-thru to renters so it hits everyone]

Assessed Value Effect

Property taxes are waged on assessed value. In NJ, assessed value resets whenever a home trades. So if you buy a $1,000,000 home and the property tax rate is 1% you owe $10,000 per year in property tax. As the estimated market value of your home changes, your assessed value changes. So if your market value jumps 15% in one year you can expect a big increase in your tax bill. It may lag the full market return but the idea is the assessed value tracks the value of the home. Downturns in prices require homeowners to plead their case that the home’s value has declined if they want relief on their taxes.

Like NJ, CA assessed value resets to the purchase price after a transaction. But then CA diverges from other states. A month before I was born, in June 1978, CA passed Prop 13, a ballot proposition that has created distortions in wealth that few could have foreseen. Prop 13 froze assessed values at 1976 levels for homes which have not since traded. It also limits increases in assessed value to a cap of 2% per year.

Combined with a NIMBY attitude to permitting new construction, CA features a lopsided sight to behold — multi-million dollar homes with single-digit thousand tax bills. Nice for those owners but not socially desirable.


  • The flipside of having seniors be able to stay in their homes is that it limits worker mobility by poorly allocating big homes to people who don’t need them. It basically keeps rooms off the market. If you are a senior citizen on a fixed income you are not going to sell the home you’ve outgrown to buy a condo with much higher property tax than the big house you leave behind. And that’s after you pay a huge cap gains bill.
  • Prop 13 starves the state of tax revenue that needs to come from somewhere. So the state income tax can be seen as a wealth transfer from young, working Californians to older, entrenched Californians.

In a state that has seen generational wealth built on a loop of buying real estate, and cash-out refis it’s easy to see how Prop 13 has contributed to the party. Let’s pretend you buy a home in CA and NJ.


  • Each home costs $1,000,000
  • Each has a property tax of 2.5%. We are isolating the assessed value effect so need to hold the tax rate constant.
  • Each home has a real (inflation-adjusted) return of 2% per year.
  • The only difference is the CA home is assessed only when you buy it, but the NJ home is assessed each year.

The CA home’s IRR will be .14% after-tax while the NJ home’s IRR is -.52%. The CA home outperformed the NJ home by .66% per year over 30 years. On a $1mm home that’s over $275,000 simply because the NJ home is re-assessed every year.

It gets crazier. The effect actually explodes with higher appreciation rates. If we double the appreciation rate to 4% per year, the CA homes nets you $700,000 more than the NJ home. Remember that the tax rates are the same! We are just isolating the impact of fixing the assessed value at the purchase price.

The main takeaway is Prop 13 is a call option on inflation. Your home is much less of an inflation hedge than you think if its assessed value increases in-step with the market value.

[This year Prop 15 is on the CA ballot. Prop 15 would repeal Prop 13 for commercial properties only. Based on the examples above, it’s obviously something RE investors are highly concerned about.]

Rate Effect

What if you wanted to compare the price of homes in 2 places with different property tax rates? Let’s pretend CA no longer had Prop 13. Like NJ, it’s property taxes were re-assessed annually. This allows us to simply isolate the impact of differing tax rates.

Let’s assume:

  • Each home costs $1,000,000
  • CA tax rate is 1%
  • NJ tax rate is 2.5%.
  • The homes do not appreciate over 30 years (just to keep it simple)

Let’s explore 2 methods of comparison:

The Mortgage Method

If the homes do not appreciate then their assessed value remains fixed at $1mm. This makes it easy — the CA home owes $10,000/yr in taxes and the NJ home owes $25,000. On a monthly basis, the NJ home costs an extra $1,250. If mortgage rates are 3% we can find that a $300,000 30-year mortgage corresponds to a $1,250 monthly payment. So we can say that a $1mm house in CA costs the same as a $700,000 house in NJ since the $700,000 plus an additional $300,000 mortgage would equate to the cost of the CA home.

The IRR Method

The IRR on your home’s value will approximately differ by the spread in the tax rates. In the table below, we see that the CA home returns 1.44% more (close to 1.50%) over 30 years. If we use an inflation rate of 3% to keep consistent with what I chose as a mortgage rate, we find that the NJ home costs you $300,000 more over the 30 year holding period than the CA home, matching the result from the mortgage method.

Combining Effects

To compare the price of a home in CA to a home in NJ you need to account for both the difference in property taxes and how assessed values are treated. Let’s combine the results in one model with more realistic numbers:

  • A 4% annual home appreciation in both markets
  • A 2% inflation rate
  • CA tax rate is 1%
  • NJ tax rate is 2.5%
  • CA assessed values do not increase, NJ is re-assessed annually

CA, due to Prop 13 and a lower property tax rate, has an almost 2% edge in annual return (3.29% vs 1.34%). Since these are nominal returns and inflation is 2% per year we see that the NJ home end up actually losing value in real terms. The fact that the home is re-assessed every year means that even though the home’s value is growing faster than inflation the taxes are also growing very quickly.

I don’t want to have you miss the point — these CA and NJ homes were assumed to grow at the same rate of 4% per year and yet the CA home earned you an extra $900k in present value vs the NJ home. This is strictly due to lower property taxes and Prop 13.

We know that home appreciation in CA has been faster than NJ (my family considered moving to CA in late 70s, early 80s so we are very sensitive to the comparison). The difference in property tax policies has a staggering delta in terminal wealth when applied to CA real estate boom over the past 50 years.

Wrapping Up

Having grown up in NJ and now lived in CA, I have noticed a massive divide in how people have earned their money and wealth. You cannot live here and not notice the wealth built in real estate and not think about how policy has enabled it. When you start comparing apples-to-apples, the headline prices of CA homes are not as relatively expensive as they appear. Don’t hate on Californians though. Those SALT taxes are still burying all of us who still work for a living.

In sum:

  • Prop 13 allows homes to be a call option on home appreciation/inflation
  • High property taxes on homes that are re-assessed require rapid appreciation to not render the home ‘dead money’
  • Compare homes with different property taxes by amortizing the difference in monthly payments into a mortgage

Sending a thanks to @econompic who I discussed these topics with. As another NJ to East Bay transplant he has given these ideas plenty of thought as well. And on the math side, he gave me the idea to use IRRs instead of CAGRs. CAGRs are simpler because they are compounded returns which require no more than a start value, ending value, and time period. They are commonly used when calculating a return for a stock or fund that you buy and hold.

In this case, IRRs or NPVs are preferable since there are many cashflows.

Straddles, Volatility, and Win Rates

One of my favorite follows on #voltwit is @SqueezeMetrics. The account more colloquially known as “the Lemon” has a personal crusade against using implied vol to refer to option prices. Recall, volatility is just the asset’s standard deviation of returns. It’s usually an annualized number. So if the SPX has a 15% volatility that just means you expect the SPX to return +/- 15% about 68% of the time1

“Lemon” prefers using the average expected move, more commonly known as the straddle.

Thus tweeted the Lemon:

I think the convention of turning the straddle price into an annualized standard deviation is obfuscatory. Straddle gives you the average move that’s priced in. Why complicate that?

I can see how the distinction between average move (aka the “straddle”) and standard deviation (aka the “vol”) is “obfuscatory”.

So let’s clear it up.

Expect to learn:

  • The math relationship between the straddle and the volatility
  • How the distinction relates to win rates and expectancy
  • Why the spread between the straddle and volatility can vary in turn altering win rates
  • My own humble opinion on the matter

Turning Volatility Into A Straddle and Vice Versa

A handy formula every novice trader learns is the at-the-money straddle approximation2:

Straddle = .8Sσ√T

where S = stock price
σ = implied volatility
T = time to expiry (in years)

Ok, let’s pretend the SPX is $100, there’s 1 year to expiry, and implied volatility is 15%. Plug and chug and we get a straddle value of $12 or 12%. Pretty straightforward.

Straddle/S = .8σ√T

If we want to simply speak in annualized terms then we can assume T = 1 and can simplify:

Straddle as % of Spot = .8 x σ

Which of course means if you know the annualized straddle price as a percent of spot you can go in reverse to get the volatility:

σ = Straddle as % of Spot x 1.25

When is this useful?

Let’s say based on a stock’s past earnings move you see that it usually moves 5% per day. In other words, the earnings day straddle should be 5%. Then, you can find the standard deviation:

5% x 1.25 or 6.25%

The standard deviation is a volatility which you can annualize to plug into an options model which will spit out a 5% straddle price.

6.25% x 252 = 99.2% vol

Knowing the 1-day implied volatility is useful when you are trying to estimate a term volatility for a longer period that includes the earnings day (topic for another time).

What’s the practical difference between straddles and volatility?

Volatility is a number you stick into a model to generate a price for an instrument you actually trade. In this case, a straddle. If you input 15% vol into our above example, you will find that a 1-year straddle will cost you 12% of spot.

If you buy this straddle your return is equal to:

Absolute value of SPX return – 12%

Your worst case scenario is the SPX is unchanged and you lose your entire 12% premium. You are “long volatility” in that you want the SPX to move big one way or another.

So let’s talk about what we really care about — expectancy and win rates.


The point of the model is to generate a price that is fair for a given volatility. 12% was the fair theoretical value for a 15% vol asset.

If you pay 12% for the straddle on a 15% vol asset you have zero expectancy.

But that’s not the whole story.

Win Rates

Expectancy and win rate are not the same. Remember that the most you can lose is 12% but since there is no upper bound on the stock, your win is theoretically infinite. So the expectancy of the straddle is balanced by the odds of it paying off. You should expect to lose more often than you win for your expectancy to be zero since your wins are larger than your losses.

So how often do you theoretically win?

A fairly priced straddle quoted as percent of spot costs 80% of the volatility. We know that a 1- standard deviation range encompasses about 68% of a distribution. How about a .8 standard deviation range?

Fire up excel. NORMDIST(.8,0,1,True) for a cumulative distribution function. You get 78.8% which means 21.2% of the time the SPX goes up more than .8 standard deviations. Double that because there are 2 tails and voila…you win about 42% of the time.

So in Black-Scholes world, if you buy a straddle for correctly priced vol your expectancy is zero, but you expect to lose 58% of the time!

Outside Of Black-Scholes World

The Black Scholes model assumes asset prices follow a lognormal distribution. This leads to compounded or logreturns that are normally distributed. This is the world in which the straddle as percentage of spot is 80% of the annualized volatility.

In that world, you lose when you buy a fairly priced straddle 58% of the time. Of course fairly priced means your expectancy is zero. What happens if we change the distribution?

I’m going to borrow an example of a binary distribution from my election straddle post:

  • 90% of the time the SPX goes up 5.55%
  • 10% of the time the SPX goes down 50%

    Expected move size = 90% x 5.55% + 10% x 50% = 10%

Expected move is the same as a straddle. The straddle is worth 10% of spot. Your expectancy from owning it is 0.

If this was Black-Scholes world, we would say the volatility is 1.25 x 10% = 12.5% (not annualized). But this is not Black Scholes world. This is a binary distribution not a lognormal one. What is the standard deviation of this binary asset?

We can compute the standard deviation just as we do it for coin tosses or dice throwing.

σ= √(.9 x .05552 + .1 x .502)

σ = 16.7% (again, not annualized so we can compare)

Note that your straddle is 10% but your volatility is 16.7%. That ratio is not the 80% we saw in the lognormal world, but instead it is 60%.

Note you cannot repeat the earlier process to find the win rate. You can’t just NORMDIST(.6,0,1,True) because the distribution of returns is not normal. Luckily, with a binary distribution our win rate is easy to see. In this example, if you pay 10% for the straddle you lose 90% of the time.

Even if you paid 6% for the straddle you still lose 90% of the time. However if you bought the straddle that ‘cheap’, your expectancy will be massively positive!

My Own Humble Opinion

When there is a short time to expiration, arbitrarily let’s say a few weeks, my mind’s intuition might latch on to a straddle price. I might think in terms of expected move as one does for earnings in getting a feel for what is the right price. But on longer time frames I prefer to think of implied vol because I am going to be dynamically hedging. Measures of realized vol can be readily compared with implied vol.

If I look at a straddle price for a long period of time, say 1 year, I might fall into a trap thinking “20%? That just sounds high.” I’d rather just compare the implied vol which would be 25% (remember 1.25 x straddle), to realized vol since I am interested in the expectancy of the trades, not the win-rate.

There are all kinds of house of mirrors when looking at vols and straddles and thinking about winning percentages. As Lemon says, it’s “obfuscatory”. Everyone should do what works for them.

If you tend to be long vol, be aware having more losing months than winning months might be completely normal. It’s baked into the math. And the more skewed the distribution, the worse your batting average will be.

But in the long run it’s your slugging percentage that matters.


  • Straddles as a percent of spot are 80% of the volatility (all annualized)
  • Straddles tell you the average move.
  • Fair straddles have zero expectancy.
  • You lose more often when you win when you are long a straddle.
  • Your win sizes are larger than your losses.
  • Skewed distributions change the relationship between win rates and expectancy. They also change the relationship between straddle prices and standard deviations.

The Curse of the Reserve Currency

I’m familiar with the US dollar as the world’s reserve currency through a conventional lens. As a deliberate bargain between the US and the rest of the world.

It goes something like this:

The US enjoys a stable currency effectively lowering her cost of capital. In exchange, US Naval might enforces order on maritime trade routes. The safety of shipping lanes is a global good lifting all economies through the efficiencies of comparative advantage and arbitrage. This global good would be difficult to coordinate without a single cop like the US so the world accepts this bargain as reasonably fair even if it might nitpick aspects of it.

If you are a just being introduced to this idea you can see my notes on:

I recently read a different perspective on this global arrangement. In this alternative view, the status quo was not an explicit or even implicit deal between the US and the rest of the world but an emergent phenomenon. The distinction is important because the force that maintains it is not international diplomacy shaped by national interests. Instead, it is simply the position at which the equilibrium is at rest according to economic gravity. The invisible hand working bottom-up not authority working top-down.

Yakov Feygin and Dominik Leusder explain:

The dollar system evolved not as a tool of imperial statecraft, but as the project of a transnational elite that has effectively usurped control of an international public good.

Frameworks for understanding the persistence of the dollar system tend to vary from from reductionist to outdated, often examining international politics with discrete nation states as the main unit of analysis. In this view, the dollar is a product of hegemonic US interests, wielded as a tool of statecraft. But global financialization has upended this framework: elite interests are not aggregated domestically but internationally, and are transmitted via the balance-of-payments mechanism and the financial system…Herman Mark Schwartz, one of the foremost experts on the dollar and American hegemony, offers a better way to think about the dollar—namely, as the state money of a quasi-imperial global system, in which the different economic regions are tied together by a shared reserve currency. This ‘imperial currency’ is more of a by-product, and less of an enabler of (or even an enabling constraint on) American expansionism and military adventurism, both of which preceded the reserve currency status of the dollar.

In this version of world order, the status quo is not actually to any nation’s benefit but to a political and economic class whose interests transcend sovereign borders. This leads to a counterintuitive conclusion:

to the extent that the world has prospered since Bretton Woods, it is in spite of, not due to, the USD being the reserve currency.

The full case is laid out in The Class Politics Of The Dollar System (Link)

My Selected Excerpts And Notes

The Soft Power Of Issuing The Reserve Currency

Two clear geopolitical advantages accrue to the US because of its reserve currency status:

  • Sanctions

  • Dollar liquidity swap lines

The source of the Federal Reserve’s power over the eurodollar system—and the vulnerability of emerging markets within it—is the global reliance on central bank backstopping. In the 2008-9 crisis, the Fed deployed so-called central bank liquidity swap lines to backstop the global system. These took the form of reciprocal currency arrangements between central banks: The Fed replenished the dollar reserves of other central banks in exchange for local currency. The real power of the swap lines is not who gets them but rather who doesn’t. In a recent piece for the Nation, Andres Arauz and David Adler highlight how these swap lines can be used for a form of monetary triage, in which the United States decides which countries have better prospects for weathering economic storms.

Questioning the Narrative

Despite the advantages, dollar eminence should not be a goal. The long-run cost outweighs the near-term benefit.

Dollar primacy feeds a growing American trade deficit that shifts the country’s economy toward the accumulation of rents rather than the growth of productivity. This has contributed to a falling labor and capital share of income, and to the ballooning cost of services such as education, medical care, and rental housing. With sicknesses like these, can we say for certain that the reserve currency confers substantial benefits to the country that provides liquidity and benchmark assets denominated in that currency?

How The Plumbing Works

Offshore dollar pools depend on the liquidity of treasuries and near substitutes as collateral to raise cash in the event of a margin call.

The reason for these dollar pools is twofold. First is the need to fund trade. The Eurodollar system facilitates trading relationships between countries with different currencies by giving them access to a common stable currency in which to denominate trade—the dollar. Dollar credit allows the execution of contracts without actual, US-issued currency being exchanged. Instead, the system functions as an exchange of IOUs to deliver receipts at various periods of time.

Because 80% of trade in emerging market economies is denominated in dollars, firms with receipts in a domestic currency acquire unsustainable debt in dollars if the domestic currency falls. For this reason, central banks attempt to stockpile dollar assets, most commonly US debt. To acquire them, they usually run a persistent trade surplus by repressing the real wages of their workers. (I need more clarification on this point)

This might be sustainable in the short run, but in the long run, it leads to periods of economic stagnation, or international trade and currency wars.

The second driver of these offshore dollar pools is wealth inequality and outsized corporate returns. Large corporations, pension funds, and extremely wealthy individuals cannot bank their money in the retail banking system. Instead, they hold them in pools of dollar liquid denominated assets that can be converted into dollars quickly. While this ‘shadow banking’ system has legitimate uses, it also facilitates tax evasion and kleptocratic corruption.

The dollar system thus facilitates and fuels the power of elites who have an interest in maintaining the status quo. A globalized system with a dominant key currency aids the accumulation of rents at the expense of higher consumption from workers in exporter countries and the hoarding of those rents in the legal black hole of offshore finance.

Zooming In: How It Hurts The US

  • Financial “Dutch Disease”

    Talent or ample resources have a downside. Some might even say a curse. It can make you lazy or overly reliant on your intrinsic advantage. Here’s the idea applied to USD dominance.

    Demand for high quality dollar-denominated assets saddles the United States with a financial ‘Dutch Disease’; a situation in which the reliance on exporting a single commodity raises the exchange rate and thus squeezes out the production of tradeable, value-added goods in favor of services and financial rents….Dutch diseased economies often result in a shrinking, narrow elite whose power rests on income from sales of the single commodity, or the services and management that bloom around the cash flows generated by this commodity. For the United States, this single commodity just happens to be the dollar.

  • The evidence

    The most visible cost of the disease is the steady appreciation of the dollar since the 1980s, despite a falling US share of global gross domestic product. The main domestic symptom has been the rising costs of non-tradable goods—such as medicine, real estate rents, and education—over tradable goods. This disconnect is at least in part responsible for the country’s low rate of inflation, falling wage share, and increased economic insecurity despite access to a wider range of consumer goods. While the American consumer can now purchase an ever-expanding set of appliances, electronics, and small luxuries, services that are necessary for economic mobility and household sustainability are increasingly out of reach.

  • MMT as full blown financial Dutch disease

    Justin Czyzsczewski writes:

    In the MMT view, there is no recourse against a government going off the rails. Some developing countries are said to suffer from a “resource curse“, when an abundance of natural resources means the government doesn’t rely on taxes, and so becomes unresponsive to the wants and needs of the populace. In the past, kings with powerful armies ruled in the same way. There is a very real risk of the same phenomenon when government spending becomes untethered from taxation.

Zooming In: How It Hurts Developing Nations

  • The need to hoard dollars crowds out productive domestic investment

    For the rest of the world, the ills are clear enough. In developing countries, the need to insure their economies against currency crises and debt deflation has meant the accumulation of dollars at the expense of necessary domestic investment. These policies are usually accompanied by a suppression of consumption and incomes to establish a permanent trade surplus vis-à-vis the dollar system.

  • Dollar liquidity lowers the cost of corruption

    The dollar system allows corrupt elites to safely transport their ill-gotten earnings to global banking centers located in jurisdictions with opaque ownership laws.

While the dollar system has undoubtedly had a disproportionately negative effect on developing countries, the main fault lines that emerge from the dollar system are along class, rather than national lines.

In other words, a rich Chinese national has more in common with the US elite than their fellow citizens.

Obstacles and Remedy

  • Elites’ preference for status quo

    Developed world exporters like Japan and Germany also maintain a growth model based on cost competitiveness and wage suppression. An increased role for the Euro or the Yen would undercut these models. For resource exporters, it facilitates corruption and tax evasion through simple capital flows. In the United States, it benefits financial industry elites, who can reap the rewards from intermediating capital inflows into US markets, while the cost of non-tradable services like tuition, healthcare and real estate rises for everyone else. Across all countries, elites win.
  • Reducing Inequality

    Too great a share of the national income is in the hands of high-saving entities with dollar liquidity preferences, such as high net worth individuals and large corporations. To reverse this imbalance, income would have to be transferred from these powerful interests to China’s workers—a dynamic described by Albert Hirschman as early as 1958.

The fact that the dollar system is primarily based on social, rather than geopolitical conflict means that the best solutions suggest at a reform of the system in a manner that empowers people at the bottom of the global social hierarchy.