Beating The SP500 > Winning The NBA Title

76ers GM Daryl Morey is one of the pioneers who brought Moneyball-type thinking to basketball during his tenure with the Rockets.

His interview with Patrick on Invest Like The Best is insightful and entertaining. I want to zoom in something Morey says:

You are weighing championship odds. And generally, we look over a three year time horizon with that. You could really pick any time horizon, but three years seems to work best with the data. And we basically do a sharp ratio like you would in investing, which is like here’s how championship odds increase, here’s the variance of that move.

Is it on the efficient frontier of return to risk basically and Shane [Battier], obviously, fit that for us.

None of our information is anywhere as good as the financial models. Actually, our underlying data is more predictive, quite a bit predictive. I talk to a lot of quants on Wall Street, and I tell them our signal to noise ratio using whatever measure you want….And they go like — yes, they go like, whoa, you guys are — that’s incredible. And I’m like, yes, but you remember, we have to be best of 30. You guys just have to beat the S&P by 2% and you are geniuses. So each industry has its own challenges.

We’re like a pure play. It’s the lifeblood of our business, whereas in other businesses, I’d say execution probably matters a lot more. In all aspects, including coaching, a well-executed, slightly suboptimal strategy generally will be the best strategy poorly executed. I mean you know that.

That’s generally true in basketball as well. But I would say in our realm of decision-making, it’s really almost a pure decision-making thing. This draft pick beats that draft pick. This free agent for $5 million beats that free agent for $5 million. It’s more of a pure play.

Sports is actually way simpler than most of the people you talk to, way simpler. Our sport, it changes, but not much. Our data is pretty good. Our competitors aren’t coming out with new products. Our competitive dynamics are known.

They’re hard, but they’re — no, we don’t have the Rumsfeld problem of unknown unknowns, like some start-up in stealth mode that might emerge, like, that’s why academics have done more and more papers about sports.

Because if you’re trying to isolate how to make good decisions, sports is really the right area to do that in

This is a great section because it highlights how different domains just have different size error bars. Sports signals are stronger than investment signals. The counterbalance to that fact is when Morey says:

I talk to a lot of quants on Wall Street, and I tell them our signal to noise ratio using whatever measure you want….And they go like — yes, they go like, whoa, you guys are — that’s incredible. And I’m like, yes, but you remember, we have to be best of 30. You guys just have to beat the S&P by 2% and you are geniuses. So each industry has its own challenges.

Umm, beating the SP500 by 2% consistently is rarified air even if that number sounds small. Morey admits that only a handful of teams have the requisite talent to even compete for the title. So your probability of winning the championship is either 0 or likely much better than a professional fund manager beating the SP500 by 2%.

Asset managers win by being good salespeople (a friend called this the Matt Levine model — being a good hedge fund is about gathering assets when you get hot and keeping them when you get cold. It’s a scheme for getting rich that has a lot less to do with returns than the industry will admit. Come to think of it, being a valuable sports franchise probably has more to do with the logo and stadium than actually winning…it’s not that winning and returns don’t matter, it’s the gap between how much they matter and how much we think they matter).

I’m guessing Morey threw the 2% number out there without much thought. He was actually making a deep point that if an adversarial game is technically easier (say checkers vs chess) the competition enjoys the same low-difficulty advantage and you are in the same place of having a low chance of winning. But I was curious…how hard is it to beat the SP500 by 2%?

I’ll admit a question like this is in my friend Nick Maggiulli’s wheelhouse so when he reads this he’ll almost certainly have a more complete answer. But I decided to take a quick stab at it.

I pulled up the fund screener and filtered for US equity large-cap funds with at least a 5-year history benchmarked to the SP500 total return (this is an appropriate benchmark for a large-cap US equity fund.)

My criteria for beating the SP500 without getting lucky was the fund needed an information ratio (IR) of .50 or greater. An information ratio is outperformance normalized by tracking error. Tracking error is the standard deviation of the difference in returns between the fund and the SP500. If a fund outperforms by 2% per year but the tracking error is 10% (ie an IR = .2) that feels like noise vs a fund that outperforms by 2% with only 4% tracking error [I realize I’m using a simple, satisficey method for separating signal from noise, so if you are an allocator who just threw up in their mouth, brush your teeth then email me with an education so I can learn too!].

What did the screen turn up?

  • 45 out of 677 funds had IR of .5 or greater (caveat: the IRs use a 3-year lookback)
  • 8 funds out of 677 had at least .5 IR AND outperformed the SP500 total 3-year returns by 200 bps
  • Only 3 funds outperformed by 200 bps for 5 years (the IR ratio is still a 3- year lookback)

Daryl your point is well-taken but beating the SP500 by 2% with skill is 90s Bulls-level for public fund managers.

If You Wait For All The Info You’ll Be Too Late

In the past few days, I’ve been getting around to the feedback and follow-ups from last week’s StockSlam sessions. Here’s a reaction and my response worth sharing widely.


Just wanted to shoot you a quick note – loved the game last week, thanks for putting it on!

I had a hard time playing the game because I didn’t have intuition for the odds of the game… I’m way more of a Quant- the only thing I could think of was trying to execute the optimal strategy.

To figure out the optimal strategy I’d run a Monte Carlo simulation – play the game 100,000 times (programmatically using python or something) and see the distribution of outcomes as well as figure out some conditional probabilities (like what are the odds of last place winning given current relative location). Getting a sense of this would help price different bets – not a sure thing all the time, but better odds!

Generally, I ended up playing the game buying out-of-the-money “horses” (i.e. last place)… I figured with the mean reversion built into the game combined with behavioral biases to dump losers would be a winning strategy… and I ended up with a positive PnL so maybe I was into something!

I don’t know how you did that for a career for so long… so stressful and I was wound up all night from it, haha…

My reply:

An anecdotal observation — I’ve noticed that quants and accountants actually get a bit paralyzed sometimes and it highlights the fact that crunching the numbers to perfection isn’t the core skill of trading.

It really is handicapping how wrong you could be and then acting with a margin of safety commensurate with the possible reward. Basically, if you wait to have the best info you’ll be too late. So the constraint is “how do I act optimally subject to being fast?” Everyone is in the same boat. That’s a key point. The game would be different if everyone had infinite time to crunch the numbers. Trading is playing the game at hand — and that has a speed component. This is inescapable. It’s also true in reality even if the form varies. Buffet might wait for a fat pitch, but when it comes the bat speed still needs to be fast.

Whatever your game, you ultimately get a feel for it by being able to hold your attention on what matters and tuning out the rest. There’s some visualization…being ready to pounce on an incorrect market that you’ve been studying. In StockSlam, you really get a sense of what consensus is for a color in a certain relative position and then your antennae is up for aberrations. You are gathering and measuring data via listening and memory while in real-life the same functions are performed in code. But they are the same functions. And both are downstream from “what do I need to be paying attention to?” That will vary by the time horizon of your strategy.

[The attendee also mentioned that the penalty for not executing the game’s “broker cards” was too low.

My response:

As far as the penalty we are actually thinking to ditch it anyway and use carrots for doing things on your card rather than punishments. But I hear you on the $5 not mattering much but it remains a useful part of the game by letting us examine if players can find the least expensive way to execute the card. You are effectively benchmarking a trade not to “does this have edge” but “is this better than negative $5”.

This is a critical concept in real life. Broadly, satisficing is often better than making perfect the enemy of the good. Also, there are some strategies that are not profitable if you have to cross a spread but are profitable if the benchmark is “it saved me from crossing a spread” (very relevant for an org that has to make many hedging trades per day). Academic papers are notorious for finding strategies that underappreciate indirect transaction costs. But you may be able to repurpose such strategies to warehouse risks instead of crossing bid-asks to shed them. That’s a lower bar than a strategy that needs to cross a spread. In a world of rebate liquidity this is especially true. The cost/rebate structures for taking/ supplying liquidity is like a 4-point swing in a basketball game.]

Related reading (as an exercise you can think of why these posts are so related to what I described above):

  • If You Make Money Every Day, You’re Not Maximizing (28 min read)
  • The Paradox Of Provable Alpha (1 min read)

Mock Trading

We played StockSlam after dinner. The kids (well not the 1st grader) and adults were all into it.

There are 8 colors or shares that take a random walk over 10 rounds. The shares of the color that climb the highest are worth $100 at expiration. The rest of the shares are worthless. So you are trading a derivative contract (a future) not the share prices directly.

Purple is in the lead:

The rules are simple. You are mock trading in an open outcry environment. You start with 4 shares of each color and cash. It’s a free-for-all where you can trade with anyone at any price. You can bid, offer, or make 2-sided markets. It’s exactly what we did when we trained although simpler since we aren’t using options (although depending on the audience we will also trade options as side bets…”what’s your offer on the blue 150 call?”— if you get lifted, you can buy blue shares to delta hedge and isolate the vol).

The game is a deeply layered experience. You can just play for fun. It’s wildly energetic — we make sure everyone gets involved and there are gentle ways to do that, different personalities manifest in so many ways…some sling from the hip, some are shy or don’t want to open their mouths until they think they know the value of everything but then it all changes and you realize that approach won’t work.

But what attracted me to the game, beyond the fun, was how it bursts with trading lessons. Based on the audience we modulate the experience up and down. We give homework leading up to the event and bridge the rationale of the questions to insights embedded in the game. We connect real-life investing and trading concepts to the game (and honestly we don’t even get to them in these 2+ hour events…everyone wants to play not listen to lecture).

The single most powerful lesson though is one I harp on all the time — trading is about measurement not prediction. In the game, prediction is not even possible. The walk is random. But skill expresses itself strongly! Your ears pipe in pricing data so you can triangulate fair value and find aberrations. The visceral feel of playing skillfully is well-matched to the feeling of trading effectively in real life. When I pull you aside and ask why you did X or Y, a good answer will take the same form of sound trading rationale — “well, I bid 17 for green because red which is in the same position just traded 20 and I know Sam bought a bunch of green last round for 12 and is looking to flip a quick profit”. Your transacting like crazy but you can kind of tell without stopping to count if you are making or losing money when you get into the flow.

Getting In The Trading Headspace

Let’s pose some questions and entertain some scenarios.

At the start of the game, all the colors start at 100. I might start by just throwing out a 14 bid for red or a 9 offer in yellow just to see or a 16 offer in green, etc to get a read of the thought processes when the game is a blank slate.

Let’s look at a scene futherer along:

Suppose the following montage represents the situation in the pit:

Purple: 28-32

Green: 20-24

Blue: 20 bid

Gray: 10 bid

Jane yells “Pink/orange 1×3…even bid for the pink. I’ll buy pink, sell 3 orange for even”

What do you do?

If you sell the 1×3 you will get long 3 orange and short 1 pink. You can then turn around and lift the 1 green at 24 while hitting the 10 bid in the gray 3x.

What’s your net position:

+3 orange

-1 pink

+ 1 green

-3 gray

Chunking the risk:

  • You’re long 3 orange and short 3 grays (they are worth about the same, as they are 96 and 95 respectively in the race).
  • You are long 1 green and short 1 pink (again worth similar amounts based on their race position)

The risk on these positions is basically a wash…but you collected $6!

[You sold 3 grays at $10 each and bought 1 green for $24. The pink/orange 1×3 traded premium neutral]

If you keep doing positive expectancy trades and manage to not get too unbalanced in your positions you will have a high Sharpe and be profitable by expiration. If you just try to load up on the color you think will win, that’s a zero expectancy strategy that’s high risk/high reward and will have a garbage Sharpe over many games.

As we play the game I might come over and nudge you:

  • “Hey, do you think the gray bid had any room? If you can squeeze an 11 bid out of them then you would have collected $9 instead of $6.”
  • “What if the gray bid was thin and you could only sell 2 on the 10 bid? Do you see how liquidity and gauging the size on the bid/offer is important? You are now ‘hung’ on 2 grays that you couldn’t offload. Is the trade still worth doing if you have to hit a 9 bid on the remaining 2 lot for an average price of 9.33?”
  • “The green bid was only 20, you could have bid 22 and maybe the 24 offer would have stepped down and offered 23s or better yet just hit your mid-market bid.”
  • “Blue is 20 bid…maybe those oranges and grays were kinda cheap relatively and the good side of the trade was just buying the orange 3x via the first ratio trade but not locking it in by selling the grays. Don’t do a trade good by $5 and then do a trade bad by $2 to lock it in if you don’t have to…you have to maximize when you have the best of it because you may find yourself needing to give up edge sometimes to manage risk”.
  • “With the green offered at 24, maybe you can dangle a 22 offer in the blue…if you get lifted turn around and take the greens. You’ll have legged the spread for 2…maybe you try to offer out the pink/blue spread at 7 fishing for a 5 bid. Paste those and your net position is long green/short pink for a $3 credit!”

This is trading.

Replace colors with option strikes/maturities and all the many combinations of vertical spreads, synthetics, straddles, and underlying… churn all day, and let the chips fall where they may.[see Mock Trading Options With Market Makers]

If you trade enough with a positive edge the expiration results are just noise — you win some, you lose some. The p/l over time converges to your edge.

Knowing the arbitrage relationships in options is the same as knowing that the field of colors can’t be worth less than or more than $100. Today we measure fair value from liquid consensus using machines — in the game we gather consensus by listening. In the pit, it’s loud and busy and orders are flying around everywhere. You learn to focus attention on what matters. And that changes depending on the context. The same is true in modern trading.

Today we enter trades with code or mouse clicks not vocal cords but the concepts are the same. That’s why prop firms still use mock trading to train. The arena is a Socratic forum that opens up conversations about practical scenarios. It’s like having a poker coach press you on “Why did you call that bet? What did you think they were holding? With what odds? If you think they just caught a 2 pair with that Jack of clubs on the river, do you think they really would have called the big bet on the turn with a low pair and no draw?”

Mock trading in the presence of an experienced trader is an opportunity to debug your thinking.

This was Friday night:

And then Saturday night with the family:

rea events ranged from 10 to 25 people. I’m still in awe of a 6th grader who could just see the Matrix. The kid was fast and a total shark, preying on people that were still getting their bearings. After the game, he had opinions about shifting some of the probabilities in the algo and adding skew. I asked his mom if he was coding or using Excel and she said “no, not yet”.

”Umm, give him to me”. With some tools for expression he’ll be off to the races!

Otherwise, with respect to the game, I will share more as appropriate. We did have a videographer at all the sessions so at some point there will be more to see. In the meantime, if you are interested in having us do a team-building or educational seminar at your office, conference, or school hit me up and we’ll figure something out. By the way, the game shown above is just one of several games we actually trade on. The attendees will remember their favorite “bunny” I’m sure.]

Financial Literacy

In a recent interview with Tim Ferriss, VC Bill Gurley admits:

If there was a scale of financial sophistication between one and 10, and you would say a really smart person in New York is an 8.5, the average Silicon Valley person on financial literacy is a two.

And it’s funny because they make fun of Wall Street, but it’s just out of ignorance, they don’t know anything.

Bill said it, not me (the transcript is worth reading for the full context but I’m not twisting him… those words are the spirit). I don’t know enough to have an opinion on this but I do find it surprising. Financial literacy starts at home and VCs don’t strike me as a cohort that rose from the gutter so either I’m wrong about the source of financial literacy or maybe poor kids play lacrosse after all.

Either way, the workings of money are abstractions like code. It touches almost every decision since it prices time (interest rates function as an exchange rate between time and money). It’s a basic life skill in an increasingly abstract, financialized world. Teaching our kids about it is basic hygiene.

Last night, we had our regular family dinner with my wife’s sister’s fam (4 adults, 4 kids — grades 1, 4, 5 and 7 plus another 4th grader who was spending the night). We usually go around the table asking each person about something they were grateful for that week or what’s something they tried at and failed (I know, I know it’s a bit cliche. These prompts do lead to provocative discussions and serve to put kids and adults on the same level).

But this time we did something different.

Yinh wanted to use the Silicon Valley Bank run as a learning moment. She started by explaining how banks invest deposits in longer-dated loans to earn a yield. To nudge the kids towards understanding the risk, she said the bank invests in loans that only pay back once a year. While not mechanically true, the point was to have them recognize the liquidity mismatch between the long-dated loan and the deposits that can be withdrawn anytime.

I taught them how rising interest rates cause the value of the loans to fall. But I also dispensed with mechanical accuracy in favor of intuition. I told Zak he plays the role of the bank. He loans me $100 and I promise to repay him $110 in one year. But then, immediately, mom asks to borrow that $100 from me but she’ll pay me back $120 in a year.

How should Zak feel? Well, sad. He’s going to get $110 in one year but since his mom is willing to pay $20 for a loan he could have lent her only $90 and still known he’s going to get back $110 in a year. Of course, this isn’t accurate interest rate math, but save that for a 7th grader. For a 4th grader, this delivers the point intuitively. [And for adults who think buying individual bonds instead of a bond fund somehow is less risky because they know how much nominal money they’ll get back, think through Zak’s position here — he is still getting $110 back but he’s definitely sad even though the counterfactual universe where he invests in a bond fund that gets marked down to $90 is optically worse.]

I was fortunate that my mother taught me about money. I can still remember my brain hurting when she explained a mortgage to me. It took a while to get my head around it. Remembering that keeps me patient — I’m grateful she persisted until it got through my dense skull. She didn’t push, she just repeated herself calmly every time I was frustrated “how does this work again?”. It sinks in eventually. If anything, the exposure will prime them to learn faster when they do encounter it down the line when the stakes are higher in school or real life.

[If you think my difficulty in understanding a mortgage was stupid, I got a better one for you. When I was about 12 or 13 an older kid told me a prostitute is “someone that gets paid to have sex with you”.

Sit down for this.

My mental model for “someone gets paid to X with or at you” was…a hitman.

I now believed that there was a person whose career was to have someone pay them to have sex with a 3rd party. Until then I had the impression that sex was a desirable activity but then hearing it connoted as something that is delivered as revenge or assault made me wonder if sex might actually be a gross punishment.

Dazed and confused is a fitting description of my existence so I’ve got that going for me and this blog.]

Reasoning Through A Housing Trade Out Loud

Today I’ll share a personal investing story. It’s in the thinking-out-loud category. I can see the spots where someone could say “that’s stupid” (don’t let that deter you from pointing them out). And that’s why I want to share it — this is the messy process of making a decision. It’s imprecise. It has more “vibes” than I’m supposed to admit. But at the end of the day, there’s an irreducible amount of “putting your finger in the air” with most investing decisions.

The Housing Trade

At the start of 2022, I felt housing might be screwed. Home prices and inflation were red-hot and the risk of the Fed’s hand being forced to raise interest rates was beginning to materialize. Mortgage payments were extra sensitive to bond duration math if rates were to start lifting from such a low base. This would slow housing demand. On the supply side, there were still materials and labor supply shortages. Superficially this is bullish housing but that was already in the price. Looking ahead, this combination felt (notice the vibes…I’m not looking any data up. It’s pure staring out the window) like it could destroy demand. The idea of demand destruction reverberates from my oil trading past. OPEC doesn’t optimize for the maximum price the way you might expect from a cartel. They can be quick to supply the market because they don’t want to kill their customers. Sure a high price means the inventory in the ground is worth more but the business of producing oil, the business that enjoys a multiple, is burnt toast.

The most vulnerable part of the stack felt like the homebuilders because, like an oil refiner, they sit in between the raw materials and the finished goods. They would be squeezed on both sides. Cancellations + high costs.

I pulled up a chart in March of 2022 (this is what it looks like through this weekend of course).

Since the beginning of the year, in less than 90 days, XHB underperformed SPY by nearly 20%.

The market was well ahead of me. Dammit. It appears there’s nothing to do. In the liquid market at least.

I had 2 ideas that could be applied to stale markets.

  1. Decline to invest in the next batch of Austin flips. We had been bankrolling a friend’s short-term flips in Austin since the pandemic. We were just receiving our return from the most recent one and while we’d normally just re-invest, we took a break.
  2. Sell the house we bought in Texas the prior summer. We had a renter in place and we still hadn’t owned the house for a year (meh, short-term gains). We asked our realtor what he thinks the house could fetch and he indicated the market was still hot. He thought we could get 35-40% more than we paid the prior July (which is really nuts since the house had already appreciated since the pandemic and our purchase price was a 12% overbid to the listing price). The realtor’s number sounded optimistic but looking at comps I thought there was maybe a 15-20% chance of catching his number and in most other cases get some kind of quick profit. But I wasn’t really pricing it off profit. I was worried about risk. The cap rate would be terrible if rates went up even 1% and since we were committed to CA we didn’t want the property anymore anyway. The liquid markets were a sell signal. The illiquid market was lagging.

A family with small kids and another on the way was renting the house so our ability to move quickly was a bit hampered with respect to showing but we did get the house on the market by April. We immediately caught a bid above our ridiculous asking price! 2 days later, the stock market dove. Yinh and I were convinced they would back out.

We were right. A day later we got the call. They’re out. Apparently, their financial advisor told them to cancel. I feigned annoyance while secretly thinking “smart advisor”.

Skipping ahead, we cut the price and caught one single bid. But we needed to agree to a long closing period. We’d wake up every day “please no whammy”.

It finally closed in October. We made a touch over 20% before commission which felt so lucky. By now it was also a long-term capital gain.

But what do we do with the cash?


You sell the thing up 20%, what’s on sale to buy?

We would reallocate the cash to stocks on a relatively vol-neutral basis (if we sold $1 of house, maybe buy about $.50 of stocks if we think stocks are twice as volatile as residential RE).

But there was another risk on my mind.

Being renters ourselves we were effectively “short” or underweight housing after selling the property. From a liability-matching investing lens, this was unsettling. Conveniently, the homebuilders were now down about 40% compared to SPY — the thing I wanted to short a few months earlier I now wanted to buy because it filled a risk hole AND was pricing in pain. So we put 1/4 of the proceeds from the house sale into IWM and 1/4 into XHB.

(I just cut half the position a couple weeks ago as we reduced our net equity exposure and rolled into T-bills. I keep our equity exposure in a band and I chose to sell XHB based on its outperformance.)

Things I believe

  1. Markets are smart. Liquid markets adjust quickly.
  2. My life’s work is not figuring out what prices are right, so my allocations are driven by desired exposures or non-exposures to risk. That’s the best I can do given how much time I am willing to spend thinking about things I have no control over.
  3. Within that framework, my choice of diversified exposure is relative value voodoo and vibes. But you know what…even in my professional trading that was true. In that case, my life’s work was to measure option prices at much closer resolutions than anything I’m doing here, but pulling the trigger felt pretty much the same. What’s the liquid market telling me about about fair value and what do I do with that info? Any individual trade is noise, but if I’m disciplined about risk then no decision carries the risk of the whole portfolio and the framework is left to converge to its logic over time — capture a risk premium without mortally wounding yourself along the way.
  4. Luck will betray you one day so enjoy it when she smiles upon you. We felt like we caught the last bid in America on that house. If we listed the house a few months earlier (which we might have except for the complications with the tenant — no fault of anyone was just a matter of details) we would have been extra lucky, but we would have gotten a worse price on our stock buys.And if we don’t make the sale? Pain parade. We miss the profits, don’t get to rebalance, and I curse myself for getting into an illiquid asset. I hate illiquidity already. As I get more experience, I want to rule out illiquidity more and more. Ruling-in needs to be for a justifiably unique exposure. The option to rebalance has a value — whether you choose to ignore it or not is up to you. See How Much Extra Return Should You Demand For Illiquidity?

A note on taxes

We will pay LT gains taxes of about 30% between Fed and CA. Why not 1031 exchange? Well, I thought real estate prices would be too sticky (ie they won’t come down enough) before our 6-month window to close on a new property. I expected wide bid-asks as sellers locked into low fixed rates try to wait out market weakness. I didn’t want to sell something up 20% to buy something down 5 or 10% when I could buy something down 40% (which is more standard deviations — again, think in vol-adjusted terms. This is also why buying high growth wasn’t attractive even if they were down more than housing…they are higher vol plus the skew in their distributions means volatility is understating the risks — that’s a post for another time).

More generally, let’s examine the math of 1031 tax savings. Imagine the house I sold went from $800k to $1mm. My tax liability is about 30% of 200k or $60k. But the brokerage cost of what I buy on the backend is pretty close to that (5% of $1mm when I eventually sell the 1031 property). It’s true that the cost is deferred but the cost is also inflation indexed since it’s a percent of the home value. You are not saving nearly as much as you think because you are forced into a high transaction cost asset and the cost is a percentage of the entire asset value, not just the profit.

[Note 1: If you don’t have to pay that fat state tax and your LT gains rate is closer to 20% than this argument is even stronger.]

[Note 2: This argument is much less compelling if you plan to never sell and get stepped-up basis for your heirs. But you get stepped-up basis on stocks when you die too. But anyway, I’m not in the never-sell camp because the tax tail isn’t going to wag my risk dog. There’s always a price that warrants saying “sold” to. If a HODLer wins they get concentrated. That might be ok for your human capital but that’s not a strategy for a random number generator. And from my unenlightened seat, the market’s job is to set prices for great assets so that they are effectively random. If you disagree, you should invest for a living. I heard you can get rich doing that. Actually, you have a better chance of getting rich by convincing people you could do that.]

It’s Not The Merit It’s The Price

My past self makes me cringe.1

I remember a weekend Yinh and I spent in Big Sur before having kids. We stayed at a resort/hotel place for free in exchange for listening to the timeshare spiel. I’m just pushing back on every point, complaining about the math this poor lady on the bottom-of-the-realtor-totem-pole is conveniently ignoring. Looking back, I’m genuinely sorry to have been acting myself in that moment.

When you feel your blood pressure rising you can channel some grace by just thinking of someone you know who would be smooth in that situation. The aspirational move here is just smile and nod. I had the situation exactly backward — it was me who was embarrassing himself, not her with the canned pitch as pushy and nonsensical as it was.

Luckily I have this moon letter thing as an outlet for my teeth-grinding financial complaints. I’m over the timeshare sales thing (well, actually I just pay for a room and save myself the grief. I admit this feels more like a hair dryer solution 2 than addressing the root of my anger) and onto another — I can’t stand when a life insurance salesperson pretends they are doing god’s work by telling me about their widow client’s big settlement. I’m not against buying insurance — I have car insurance and life insurance. But I’m against motte-and-bailey persuasion techniques. If a widow getting paid is deemed a self-congratulatory act of corporate benevolence then Warren Buffet is the priest of puts, a hokey paragon of virtue, backstopping markets with the heart of a patriot. Ok.

Defending life insurance by focusing on the settlements that get paid out is as silly as branding calls sold as income. And for the same reason — there is no consideration of price. Let’s compare:

Defense of insurance: “Look at the settlement the policyholder received. It has so many zeros in it.”

Rebuttal: That would be true even if the insurance cost twice as much. So the issue isn’t whether there would be a settlement it’s the proposition on the whole.

Defense of covered calls: “The premium you collect is extra income, and if the calls go in-the-money you’ll be happy anyway”

Rebuttal: This would be true if I sold the calls for 1/2 the price that I actually sold them for.

In other words, both of these defenses are empty words because they skirt the defining point:

It’s not the merit of the idea — it’s the price.

The wrong price will ruin any proposition. Ideas without prices are worthless. “It’s a good idea to brush your teeth.” But if brushing your teeth took 8 hours a day, you’re better off pulling them all and getting implants.

“It’s a good idea to get insurance” has the invisible qualifier “assuming the price is reasonable”. From there we can debate “reasonable” and we should. But I assure you the percentage of time spent in a life insurance consultation that’s devoted to decomposing its cost is not commensurate to how important it is in the decision.

Money Angle

Let’s harp on this “merit cannot exist independent of price” idea. We’ll return to insurance for a moment.

The griftiness of insurance sales as a function of complexity is an inverted U curve. Term insurance is not complex, it’s highly competitive and low margin. Private placements, which I’ve written about, are sold to very wealthy people who likely have a CFO-type managing their money. It’s the midwit crowd from all ends of the income spectrum that express their snowflake exceptionalism in exactly the wrong place and end up paying for their agents’ kids’ private school tuition.

Many insurance products are complex and seriously difficult to understand — every now and then I’ll take a hard look at one and just think, “they expect the average person to comprehend what’s actually going on inside this black box?!” And of course, the answer is “no”. That’s actually the point.

Here’s a tip — run away if you can’t understand the insurance product better than the salesperson. This is not as high a bar as you think. Salespeople are experts at sales not financial engineering. If they weren’t selling annuities they’d be selling cars or homes. (It’s a blanket statement so there are exceptions — but you know who will agree with me the most? Nerdy advisors who don’t have perfect teeth. This is the old Taleb bit “surgeons shouldn’t look like surgeons”.)

When I look at insurance products, especially structured products, I look for the options embedded in them. The costs for these options is opaque. Many of them have analogs in the listed options markets, but ultimately the ones buried in insurance policies resemble illiquid flex options with long-dated maturities and substantial padding added to their prices. If you wanted to be rigorous about valuing an insurance policy you’d need to know everything from the value of these hidden options to how much credit risk to discount the various issuer’s policies by. Apples-to-apples comparisons are impossible. This de-commoditizes the products giving unscrupulous salepeople ample room to practice their dark art.

An aside about options thinking

I know someone who negotiates and prices leases for commercial office space. They work on huge leases with clients like FAANG. One of the things they mentioned was how they would try to embed provisions in leases which were basically hard-to-price options. The person also spent a couple years with an options market-making group and is generally very quantitative — I would use the person for math help regularly.

I also know of a few wildly successful option traders who did quite well in personal RE investing by structuring options with potential sellers (one of these stories was focused on an ex-colleague of mine which was discussed in a certain big city’s media post-GFC).

And one more related bit — an option manager I know is friends with a fund manager who deals exclusively in the pre-IPO share market. This is a class of funds that provide liquidity to late-stage VC portfolio company employees. The manager was able to help the fund manager by showing them how a particular option embedded in their structures was deeply mispriced.

A final aside on the usefulness of option thinking…in Option Theory As A Pillar Of Decision-Making, I include this:

Getting to The Price

A current example of the need to assess a proposition by understanding its price comes from the boom in covered-call ETFs. Jason Zweig of the WSJ recently published:

Why Investors Are Piling Into Funds That Promise Not to Beat the Stock Market (paywalled)

After great returns last year, covered-call funds are all the rage among income-oriented investors. But their high yields aren’t a free lunch.

The article covers the explosion in AUM in covered-call funds like the JPMorgan Equity Premium Income ETF (JEPI) or Global X Nasdaq 100 Covered Call ETF (QYLD).

These ETFs manage roughly $20B and $6B aum respectively.

We’ll talk about QYLD because its holdings are published while JEPI is a discretionary, actively managed ETF. (But I still want to know who gets to hungry-hungry hippo those option orders!).

QYLD sells covered calls on the Nasdaq 100. That means it sells a call option while owning the underlying index. If you buy 100 shares of QQQ and sell a call option you could do the same thing. That’s not an argument against this product though. Ease is a valid use case for a product.

More background: it sells the 1-month at-the-money call as opposed to out-of-the-money calls which is what people generally think of with covered-call strategies (when I was just a boy they called these “buy-writes” but I haven’t heard that term since Arrested Development was on the air).

I’ve addressed “selling options for income” as euphemistic, sales-led framing. I’m not necessarily opposed to selling options but when you brand it as “income” you are blatantly misrepresenting reality. You are pretending the option premium is income when the bulk of it is just the fair discounted weighted average of a set of possible futures. My bone with the marketing pitch is that there’s no discussion of price. Again, whether this is a good strategy depends on price and the price isn’t static. (I feel like like I’ve force-fed you like foie gras on this topic. If I have to hear about this “strategy” from one more medical professional I hope I better be sedated on an operating table so I can finally drown it out)

When the marketers show me the level of implied correlations they are selling in the calls then we can have a good-faith conversation. Or how about when they tell me who the buyer for those calls is? Because I can assure you there’s no natural buyer — the boys and girls buying those calls are only doing so because they are too cheap. They didn’t wake up in the morning and think “I’m not going to look at prices, I just think owning call options that go to zero is a reasonable way to invest my money.” You know what traders are thinking when they see the marketers pitch: “Thank you for stocking the pond, we’ll be waiting”.

And they will be waiting. Market-makers are lions in the bush who know the dinner’s migration patterns. Unlike lions, they need to be discreet. You can’t just pounce and scare everyone off. You don’t want to make a scene. So they pre-position.

The market-makers’ pre-positioning serves a dual purpose.

  1. It spreads the market impact over a longer window of liquidity. This is actually pro-social — it’s “markets properly working”. The telegraphed order is not as scary even though it’s a large size because the end of it is known and there’s no adverse selection risk. It’s what’s known as a “dumb” or uninformed order. It’s not reasonable to expect zero market impact because unless there’s someone who wants to buy all these options, the pool of greeks need to be absorbed by a get-paid-to-warehouse-risk-in-exhange-for-profit entity. The market is just an auction for that clearing price and the greeks dropped on the market will be recycled in adjacent markets emanating from the original disturbance. (I.e. the market makers will buy vega from you and sell it in some other correlated market where the entire proposition presents an attractive relative value play — it’s just a big web. Market-makers are the silk between the nodes.)
  2. You want the option seller to get filled near the offer so they feel good about the fill. That’s what it means to “not leave a scene”. So now that you are short vol 3 days ahead of the anticipated arrival of the order, knowing that the current vol level incorporates the impact of your own selling, you are ready to buy the new supply “in line”. Remember this is not frontrunning. It’s a probabilistic bet. The market-makers have no fiduciary duty to the fund (as opposed to actual frontrunning where the broker trades ahead of an order they control). Market-makers want the brokers to “feel” like they got a good fill. There are no fingerprints. A TCA that looks at execution price vs arrival price is already benchmarked to a mid-market price that has been faded to absorb the flow.

What does this mean for the cost of something like QYLD?

A napkin math approach


  • At the current AUM, they sell about 5,000 NDX at-the-money call options (equivalent to 200,000 QQQ options) every month.
  • Implied volatility is about 25% so the fund collects 2.89% of the index level 3 in premium monthly. (Can you see how ridiculous it is to call this income? Would you call it income regardless of how little premium it collected? What if the option was in-the-money and they collected the same amount of premium? Conflating premium with income is a timeshare tactic except it’s pushed by corporations who know better not Jane “it’s this job or dogfood for dinner” Doe.
  • The ATM call is pure extrinsic value.

The question is how much vol slippage can we expect on that order. I asked around and a full vol point seems like a reasonable estimate. Because of the “setting the table” pre-positioning effect it’s hard to get a perfect answer. So we’ll use 1 vol point and you can adjust the final analysis by changing it.

If there is 1 full vol click of slippage and the option you sell is pure extrinsic, than you are losing:

1 vol point / 25 vol points x 2.89% of AUM x 12 months in annual slippage.

That’s 139 bps in annual slippage. That needs to added to the 60 bp expense ratio for the fund.

So you are paying 1.99% per year for a beta-like exposure created with vanilla products. And the alleged income is not income. It’s a correctly priced option premium in one of the most liquid equity index markets in the world.

Even if I grant you a 10% VRP (variance-risk-premium is an idea that options are bid beyond their fair value for any number of reasons like convexity-preference, hedging demand, or the possibility that markets allocate prices according to efficient portfolios and single assets being mispriced might not be from a portfolio point-of-view) that means the alleged income is 10% of what the marketers claim.

This whole trend in covered-call ETFs feels more like an innovation for getting paid for commoditized exposures in a fee-compressed landscape than an innovation that actually improves investing outcomes.

An (Overly) Candid Opinion

I’m not some socialist arguing against giving people an abundance of choice. I just want to remind you that no smart-sounding idea gets a free pass without consideration of its cost. And my own wholly personal opinion is you are paying a lot for convenience here. Plus the more AUM these things get the worse the slippage.

A saying I repeat too much: Asset management is the vitamin industry. It sells placebos. It sells noise as signal.

The proliferation of option products seems like something devised by products people not alpha people, a complaint I’d charge against most of the asset management world (which probably means I’m being too harsh but also I’m not criticizing any single firm — I don’t even know anything about these large fund companies because they were not part of my career genealogy. To me, they were always just the names of customers). Another reason I should be softer on all this is that, in aggregate, active management is critical. But there’s a paradox of thrift thing where we should (and this is dark) encourage it for others but not subscribe ourselves.

If you are truly obsessed and love investing then you can figure out your own way and maybe I’m just a faint admonishing voice in the background that you mostly ignore (I do hope I help you think better around the edges at least). But for the casual investor whose targeted by pitches and thinks they are missing out, you are given permission to live FOMO-free. There’s nothing to see except a midwit trap.

[And definitely don’t look at these. Gag me.

Actually, any TSLA options mm wants to gag me for raining on their parade. That should tell you something.]

Spending As Self-Discovery

There are a couple of ideas that tug on me with a force I haven’t felt before my experience in the last 2 years.

Regarding my career

It’s become more of a priority to sustain a living in a way that feels deeply tied to others. There are a lot of ways to make money, but finding a way that doesn’t feel alienating, finding a way that feels like I’m lifting others is harder to pull off. To be clear, any constraint limits your options. Expectancy-wise I’m paying for this preference the same way a remote worker might be taking a pay cut compared to the counterfactual. But this is only a concern if I measure expectancy only in dollars. That’s falling into a trap of letting legible accounting dictate all of what matters. You know what’s more baller — giving your soul a bank account and finding a way to stack it. That means more energy for everyone else around you.

Look, if you have the luxury of reading a Substack at your desk, then simply making money is easy. You just have to be good at something. By definition, mediocrity is everywhere. It’s a low bar for ambition. Making enough money on your terms takes either top 1% talent or being good at something plus courage. If you stop at just being above average at something, some overlord will always stand ready to make you part of their portfolio. But you’ll just be another asset to be rebalanced on their schedule.

You don’t become irreplaceable until you display how you are different (I’m obviously talking about the differences that are helpful while remaining aware that many strengths are weaknesses in other contexts). You cannot do this from a place of fear. You can’t maximize opportunities if you can’t afford to say no. So it stands to reason that you don’t want to build your life in a way that makes you unable to say no. That’s the real trap of conformity. Conformity isn’t clothing. It’s not even what you say necessarily. It’s relegating your agency for comfort to someone with a different mix of values because you were too lazy to identify your own. And the irony is they probably did the same.

[It’s out of scope for this post, but this idea is deeply intertwined with learning which I believe is really about agency and freedom from conformity. And I don’t mean that in some truther way which is just conformity sold as alternative. Like what “grunge” became. And I say this as someone with all the albums.]

Spending Money Strategically

I mentioned that emotionally moving was expensive. It’s expensive to transact real estate and travel. But the costs were worth the information. That’s a concept that exists in poker or even trading. For example, you can “fish” mid-market by dangling a one-lot to see where the bots live (and to defend yourself against such tactics, in a dark pool for example, you can define a minimum trade size).

In The Art and Science Of Spending Money, Morgan Housel offers a menu of ways to spend money that you may not have considered. #10 explains why:

Not knowing what kind of spending will make you happy because you haven’t tried enough new and strange forms of spending.

Evolution is the most powerful force in the world, capable of transforming single-cell organisms into modern humans.

But evolution has no idea what it’s doing. There’s no guide, no manual, no rulebook. It’s not even necessarily good at selecting traits that work.

Its power is that it “tries” trillions upon trillions of different mutations and is ruthless about killing off the ones that don’t work. What’s left – the winners – stick around.

There’s a theory in evolutionary biology called Fisher’s Fundamental Theorem of Natural Selection. It’s the idea that variance equals strength, because the more diverse a population is the more chances it has to come up with new traits that can be selected for. No one can know what traits will be useful; that’s not how evolution works. But if you create a lot of traits, the useful one – whatever it is – will be in there somewhere.

There’s an important analogy here about spending money.

A lot of people have no idea what kind of spending will make them happy. What should you buy? Where should you travel? How much should you save? There is no single answer to these questions because everyone’s different. People default to what society tells them – whatever is most expensive will bring the most joy.

But that’s not how it works. You have to try spending money on tons of different oddball things before you find what works for you. For some people it’s travel; others can’t stand being away from home. For others it’s nice restaurants; others don’t get the hype and prefer cheap pizza. I know people who think spending money on first-class plane tickets is a borderline scam. Others would not dare sit behind row four. To each their own.

The more different kinds of spending you test out, the closer you’ll likely get to a system that works for you. The trials don’t have to be big: a $10 new food here, a $75 treat there, slightly nice shoes, etc.

Here’s Ramit Sethi again: “Frugality, quite simply, is about choosing the things you love enough to spend extravagantly on—and then cutting costs mercilessly on the things you don’t love.”

There is no guide on what will make you happy – you have to try a million different things and figure out what fits your personality.

Recently, I’ve seen a wild example of this.

My wife has a close friend that was in the rat race grind. After joining Yinh on the board of OrFA, the friend has been regularly visiting the orphanage in Vietnam. She’s a tall blond, as American as apple pie. A stranger in a strange land in the Vietnamese countryside. But she found herself deeply moved by not only the children but the local culture and all its people. She has dramatically re-arranged her entire life to prioritize her involvement and presence in Vietnam. Witnessing the impact on an otherwise familiar, professional life that started with a donation and some curiosity has been a powerful frame shake. (My family, 13 of us in total, are going to Vietnam for a few weeks this year and have already planned a soccer match between all the kids at the orphanage. If my kids moan about what’s for dinner after that trip, they’re getting lit the f up).

This is a reminder. You can just do things. You can’t introspect your way to knowing what you want. It’s too much projection of your current self into a different reality. It doesn’t recognize that the feedback changes you.

Spending money in a new way is just another method to try on different versions of yourself. To explore personal frontiers on the not-so-crazy lark that there are deeply rewarding modes to explore the world that you are completely ignoring. Your fixation on the crowded paths your surroundings have directed you towards might frustrating, not because of any personal failings, but because those paths are overbid (look no further than the college admission Hunger Games).

If you can’t be happy unless you get that house or that wedding or that title, it’s not because they are your destiny — it’s because you haven’t taken your imagination off-leash.

Using Log Returns And Volatility To Normalize Strike Distances

Basic Review

Consider a $100 stock. In a simple return world, $150 and $50 are each 50% away. They are equidistant. But in compounded return world they are not. $150 is closer. This blog post will progress from an understanding of natural logs to normalizing the distance of asset strikes.

The use of log returns in financial and derivatives modeling is useful because investing contexts usually involve re-investing your capital. In other words, the growth process is multiplicative, not additive. But if it’s multiplicative we find ourselves needing to specify a compounding interval. This is an invitation to attach a cumbersome asterisk to every model.

Logarithms offer an elegant solution — they allow us to standardize an assumption:  returns are continuously compounded.

If you are uncomfortable already, these short primer posts will help you catch up. And don’t worry, we will revisit HS math intuitively in this post before getting to the main course.

  • In Examples Of Comparing Interest Rates With Different Compounding Intervals, we saw how to convert back and forth between simple returns and compounded returns by dividing a holding period into different intervals.
  • In Understanding Log Returnswe showed how log returns are an extreme case of compounded returns — it assumes that compounding occurs continuously. In other words as you divide the holding period into smaller and smaller intervals, you find a rate that is smaller than the growth rate for the entire holding period. If the growth from $1 to $2 is fixed than the more compounding periods there are, the lower the rate must be in order for $1 to end up being $2.

Math Class Made Intuitive

You probably remember hearing about the constant e and the natural log from math class. You also repressed it. Because it was taught poorly.

Understanding e

We’ll turn to

e is NOT just a number!

Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point. Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on.

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit. 

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more.

Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate.

Let’s say our basic unit of time is a year.

e is the constant that says “if I start with $1 and continuously compound at a rate of 100%, how much do I end up with…$2.71828”

Understanding the natural logarithm (ln)

It’s true that the natural log is the inverse of an exponential of base e just as logs answer the question “what power do I raise 10 to in order to get to X?”. But defining the natural log as an inverse is circular not intuitive. Again, we turn to BetterExplained. From Demystifying the Natural Logarithm (ln):

The natural log gives you the time needed to reach a certain level of growth.

e and the Natural Log are twins:

ex is the amount we have after starting at 1.0 and growing continuously for x units of time

ln⁡(x) is the time to reach amount x, assuming we grew continuously from 1.0

If e is about growth, the natural log (ln) is about how much time it takes to achieve that growth.

The Natural Log is About Time

    • ex lets us plug in time and get growth.
    • ln(x) lets us plug in growth and get the time it would take.

For example:

    • e3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with.
    • ln⁡(20.08) is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate).

Let’s apply e and natural logs to asset returns to understand how to normalize distances.

Normalizing Distance

Let’s return to the $100 stock. We said $150 is closer than $50 in the world of compounding. Let’s assume our growth occurs over 3 years. Here’s a summary of simple returns vs annually compounded returns (or CAGR):

So far so good. The compounded returns are lower than the simple average return. Since log returns are just compounded returns sampled continuously we’d expect them to be even lower.

The total log return is indeed lower than the total simple return.

We can also see that in logspace -50% total return is “further” away than up 50%. This is the first encounter we get with the concept of distance where we see that 50% in either direction is not the same. But by the end of this post, you will learn how to normalize even 2 log returns that look the same, but don’t mean the same thing.

But before that, we will need to complete our understanding of log returns. We saw that the 3-year total log returns are lower than the 3-year total returns. To do that I pose the question:

Can you compute the annualized log returns?

Pattern-matching the computations for average simple returns and CAGR, it appears we have 2 choices respectively:

  1. Total log return / 3or
  2. (1 + Total log return) 1/3 – 1

Remember what e and ln mean in the first place:

The expression ex is a total quantity of growth. It’s actually assumed to be e 1 * x where the 1 represents 100% continuously compounded growth and X represents a unit of time. The natural log or ln(ex) then solves for how much time (ie x) did it take to arrive at the total quantity of growth assuming 100% continuous compounding. 

A key insight is that we don’t need to assume a 100% rate and x to be time. We can simply think of x as the product of “rate multiplied by time”. This allows us to substitute any rate for the assumed rate of 100% to find the time. Once again we turn to BetterExplained:

We can use their logic to return to our question: Can you compute the annualized log returns from these total 3-year  log returns?

Down Case:

log return = -69%

rate x time = -69%

rate x 3 = -69%

The annualized rate must be -23.1%

To annualize log returns, we simply take the total log return and divide by the number of years!

The complete summary table:

All is right in the world…the more compounding intervals we divide the total period into the lower the return must be. Continuous compounding represents the most intervals we can slice the period into and therefore it is the smallest rate.

Recapping so far:

  • Compounded rates are lower than simple rates for the same total return
  • Log returns are convenient measuring sticks because we just assume continuous compounding
  • etells us how much continuously compounded growth we get if we know the time period and rate
  • The natural log can tell us:
    • How much time we needed at a given rate to achieve that egrowth
    • What rate we needed for a given time period to achieve that egrowth

Normalizing Distances For Volatility

Let’s return to the $100 stock and assume continuous compounding. What price on the downside is the equivalent of the stock moving up $20? By now, we understand, the equivalent downside move is less $20 away. Let’s compute the equivalent distances in log space.

ln(120/100) = 18.23%

We solve for a negative 18.23% log return:

ln(x/100) = -18.23%

x/100 = e-18.23%

x = .8333 * 100 = $83.33

If the stock starts at $100 then $120 and $83.33 are equidistant in log space.

We want to take this further. To compare distances, especially in different assets, we want to normalize for volatility.

Volatility is just another word for standard deviation. A 10% log return in BTC means a lot less than a 10% log return in 5-year Treasury notes. We should measure log returns in terms of how many standard deviations away a specified amount of growth is. Note, this is exactly what the concept of a z-score is in statistics. It tells us how far away from the mean a particular observation is.

Let’s stick with our $100 stock and give it a volatility of 18.23%.

  • A 1 standard deviation move to the upside in 1 year is $120
  • A 1 standard deviation move to the downside in 1 year is $83.33

If we define K as a strike price, we can back into a general formula for how far K is from the spot price in terms of standard deviations. Let’s define all our variables first:

K = strike price

S = Spot price

σ = volatility

t = time (in years)

We start with an intuitive expression for a Z-score using our variables:

We can confirm this makes sense with numbers from the previous example. We’ll set t to 1 (ie 1 year) and the Z-score is 1 corresponding to 1 standard deviation:

The formula makes sense. In English, it says “divide the distance in logspace by the annualized volatility scaled to 1 year”.

This simply validated the expression for Z-score. We still want to define any strike price, K, as a function of its volatility and time.

Algebra ensues:

  • If you input a positive volatility number, the formula spits out what a 1 standard deviation up move is.
  • If you input a negative volatility number, the formula spits out what a 1 standard deviation down move is.

If you recall, the big insight from earlier:

The expression ex is a total quantity of growth…we don’t need to assume a 100% rate and x to be time. We can simply think of x as the product of “rate multiplied by time”.

This fact can allow us to decompose the Z-score expression to account for the fact that our underlying stock process has both:

  1. a drift component (option theory uses the risk-free rate for reasons that are beyond this post)
  2. a random component drawn from a distribution defined by a mean (spot + drift) and volatility.

Defining the expressions:

  • Risk-free rate or drift = r
  • The mean of the distribution (aka the “forward”) = Sert
  • The standard deviation scaled to time = σ√t

The Z-score formulas that incorporate drift for 1 standard deviation up and down respectively:

  • Kup = Se(rt + σ√t)
  • Kdown = Se(rt – σ√t)

[The rate in the ex portion is part drift and part random. Why do we combine them with addition instead of multiplication? Because the time portion affects each component differently. We can’t double the variance and halve the time because time also factors into the drift (ie the interest rate)]

Let’s wrap with an example, this time including the drift.

Set r = 5% and t = 1

Fwd = 100e.05 = $105.13

If we are just considering the one standard deviation around the mean (as opposed to a full standard deviation up or down) this is the theoretical stock distribution:

What’s the point of all this?

For anyone within sneezing distance of a derivatives desk, these are rudiments. These computations are the meaning behind the Black Scholes’s z-scores (d1 and d2) and probabilities. These standardizations are critical for comparing vol surfaces. If you can’t contextualize how far a price is you cannot make meaningful comparisons between option volatilities and therefore prices.

If you only trade linear instruments because you are a well-adjusted human then hopefully you still found this lesson helpful. Seeing math from different angles is like filling in the grout in the tiles of your mental processing. You can measure the distance (or accumulated growth, positive or negative) in log space to account for compounding. You can standardize comparisons by using the asset’s vol as a measuring stick. And after all that, if you still don’t enjoy this, you can feel better about your life choices to do work that doesn’t rely on it.

If you do rely on understanding this stuff, hopefully you got e.00995-1 better today.

Understanding Log Returns

If you draw a return a simple return at random from a normal (ie bell curve) distribution and compound it over time, the resultant wealth distribution will be lognormally distributed with the center of mass corresponding to the CAGR return.

Imagine your total 1-year return is 10%. So your terminal wealth is 1.10.

If you compounded monthly to end up at a terminal wealth of 1.10 we can compute the monthly compounding rate as:

1.10 ^ (1/12) = .797% per month or annualized (ie x12) =  9.57% 

Let’s instead compound daily to end up with a terminal wealth of 1.10.

1.10 ^ (1/365) – 1 = .026% or annualized (x365) = 9.53%

The more frequently we compound while keeping the total return the same the lower the compounded rate or average rate that prevails to get us from initial to terminal wealth.

Log returns are returns compounded continuously (as if you were going to compound even more frequently than every single second but at a tiny rate). When we annualize that rate as we did in the prior examples we end up with a log return.

Or simply:

Ln(1.10) = 9.53%

Similar after rounding to just compounding daily.

Let’s say your $1 grows to $1.50 after 1 year, then

  • your simple return is 50%
  • your log return is ln(1.5) = 40.5%

This chart reveals 2 facts:

  1. Log returns are always smaller than simple returns just as compounded returns are lower than simple returns. This makes sense because log returns are just compounding where the interval between compounding is reduced to zero so it takes a lower rate applied more frequently to get to the same total return.
  2. Higher volatility (ie the larger changes) means a wider gap between the simple and log return. Again, reminiscent of the formula relating geometric and arithmetic returns.

The chart raises a question. We know that volatility increases the gap between simple and compounded returns but why is this exacerbated on the downside? There was nothing in the formula (CAGR = Arithmetic Mean – .5 * σ²) that points to any such asymmetry.

The answer lies in an illusion.

In the chart, 1.5 and .5 appear to be equidistant away. They are both 50% away, right?

That’s true…but only in simple terms!

In compounded terms, .50 is “further away” than 1.5.

A thought exercise will make this clear:

If I start at 100 and can only move in increments of 10%, I can get to 150 in 5 moves.

100 * 1.10 * 1.10 * 1.10 *1.10 * 1.10 = 1.61

But on the downside, compounding by a fixed amount means more moves to cover the same absolute distance.

100 * .9⁵ = 59

In fact, I need 2 more moves to “cross” 50. With 7 moves I finally get to 47.8

The chart masks the fact that in logspace .5 is much further than 1.5 and therefore to have moved 50% from the start the volatility (ie the move size) must have been higher. And that’s exactly what the log returns show:

Price Simple Return Logreturn
50 -50% -69%
150 50% 41%

$50 is further away in logspace corresponding to a higher compounded volatility. If the volatility is higher, the gap between the simple and log-returns is wider.

Application to options

The analogy to options is the x-axis in this chart is strike prices because they are absolute distances apart. They are not equidistant apart in logspace!

We make the x-axis equidistant in logspace by making the log returns 10% apart.

Now we can chart the log returns on the x-axis. The distance of each total return from the diagonal shows the divergence between the log returns and simple return. It widens as you expect as we get to larger move sizes, but the chart is more symmetrical because the distance between the “strikes” is now normalized to compounded returns. 

Geometric vs Arithmetic Mean In The Wild


In ‘Well What Did You Expect’? we learned:

  • Mathematical “expectation” is a simple average or arithmetic mean of various outcomes weighted by their probability
  • Arithmetic means are familiar. Your average score in a class is the sum of your test scores divided by the number of tests. If you score 85, 90, 98  your average for the class is:  (85+90+98)/3 = 91

    Note the scores are weighted equally. Here’s what the number sentence looks like without factoring out the 1/3:

    .33 * 85+ .33 * 90 + .33 * 98 = 91

    If the final test is worth 50% of the total grade the weighted average is computed: .25 * 85 + .25 * 90 + .50 * 98  = 92.75

    Whether we are weighting the results equally or not, we are still computing the average by summing, then dividing.

  • Geometric means are like arithmetic means except quantities are multiplied instead of summed. Since investing is the process of earning a return and reinvesting the total proceeds we are multiplying, not summing results. If you invest $100 at 10% for 5 years your final wealth is given by:

    $100 * (1.10) * (1.10) * (1.10) * (1.10) * (1.10)  or simply $100 * (1.10)⁵ = $161.05

    In life, we often know the ending amount and the initial investment but want to know “what was my average growth rate per year?”

    The answer to that question is not the simple arithmetic average but the geometric average because we were re-investing or multiplying our capital each year by some rate. That rate is known as the CAGR or “compound annual growth rate”

    If we start with $100 and have $161.05 after 5 years we compute the geometric average in an analogous way to arithmetic averages, but instead of dividing by the number of years, we take Nth root of our total growth where N is the number of years we compounded for.

    CAGR for 5 years = ($161.05/$100) ^ (1/5) -1 = 10% 

    [we subtract that 1 at the end to remove our starting capital and just have the rate]

  • CAGR vs Simple Average Returns

With investing we are almost always re-investing our capital. That means our capital is being multiplied by a rate from one period to the next. When we want to know the average rate, we really want to pick the geometric average not the arithmetic one (there are other types of averages too like the harmonic average!). We want to compute the CAGR.

As a last proof that the CAGR and simple arithmetic average are different we can revisit the example above. If we compound an initial capital of $100 at 10% per year for 5 years we end up with $161.05 for a total return of 61.05%.

If we compute the simple average:

61.05% / 5 = 12.2%

This is higher than the CAGR of 10%

This is a consistent result. The geometric mean is always lower than the arithmetic mean!

How much lower?

It depends on how volatile the investment is. The reason is intuitive.

Imagine making 50% and losing 50%. The order doesn’t matter. You have net lost 25% of your initial capital.

The formula that relates the arithmetic mean and CAGR:

CAGR = Arithmetic Mean – .5 * σ²


σ = annualized volatility


This Is Not Just Theoretical

I grabbed SP500 total returns by year going from 1926-2023. Here’s what you find:

Simple arithmetic mean of the list: 12.01%

Standard deviation of returns: 19.8%

These are actual sample stats.

What did an investor experience?

If you start with $100 and let it compound over those 97 years, you end up with $1,151,937. 

What’s the CAGR?

CAGR = ($1,151,937 / $100)^(1/97) – 1 

CAGR = 10.12%

These are the actual historical results. An average annual return of 12.01% translated to an investor’s lived experience of compounding their wealth at 10.12% per year. 

Comparing the sample to theory

If you knew in advance that the stock market would increase 12.01% per year and you used the CAGR formula with our sample arithmetic mean return and standard deviation, what compound annual growth rate would you predict?

CAGR = Arithmetic Mean – .5 * σ²

CAGR = 12.01% – .5 * 19.8%²

CAGR = 10.06%

An average arithmetic return of 12.01% at 19.8% vol predicted a CAGR of 10.06% vs an actual result of 10.12%

Not too shabby. 

I used the same parameters to run a simulation where every year you draw a return from a normal distribution with mean 12% and standard deviation of 19.8% and compounded for 97 years.  

I ran it 10,000 times. (Github code — it works but you’ll go blind)

Theoretical expectations

CAGR = median return = mean return .5 * σ²

CAGR = .12 – .5 * .198² = 10.04% 

Median terminal wealth = 100 * (1+ CAGR)^ (N years)

Median terminal wealth = $100 * (1+ .104)^ (97) = $1,072,333

Arithmetic mean wealth = 100 * (1+ mean return)^ (N years)

Arithmetic mean wealth = $100 * (1+ .12)^ (97) = $5,944,950

The sample results from 10,000 sims

The median sample CAGR: 10.19%

The median sample terminal wealth = $1,2255,90

The mean terminal wealth: $5,952,373

Summary Table 

The most salient observation:

The median terminal wealth, the result of compounding, is much less than what simple returns suggest. When you are presented with an opportunity to invest in something with an IRR or expected return of X, your actual return if you keep re-investing will be lower than if you take the simple average of the annual returns.

If the investment is highly volatile…it will be much lower. 

The distribution of terminal wealth

The nice thing about simulating this process 10,000x is we can see the wealth distribution not just the mean and median outcomes.

Remember the assumptions:

  • Drawing a random sample from a normal distribution with a mean of 12% and standard deviation of 19.8%
  • Assume we fully re-invest our returns for 97 years

And our results:

  • The median sample CAGR: 10.19%

  • The median sample terminal wealth = $1,2255,90

  • The mean terminal wealth: $5,952,373

This was the percentile distribution of terminal wealth:

The mean wealth outcome is 5x the median wealth outcome due to a 2% gap between the arithmetic and geometric returns. The geometric return compounded corresponds exactly to the median terminal wealth which is why we use CAGR, a measure that includes the punishing effect of volatility. 

In terms of mathematical expectation, if you lived 10,000 lives, on average your terminal wealth would be nearly $6mm but in the one life you live, the odds of that happening are less than 20%.

The chart was calculated from this table:

Percentile Wealth 97-year CAGR
0.95 $22,323,532 13.5%
0.9 $12,048,311 12.8%
0.85 $7,955,791 12.3%
0.8 $5,601,855 11.9%
0.75 $4,098,451 11.6%
0.7 $3,210,573 11.3%
0.65 $2,480,813 11.0%
0.6 $1,981,453 10.7%
0.55 $1,604,153 10.5%
0.5 $1,275,987 10.2%
0.45 $1,009,583 10.0%
0.4 $804,035 9.7%
0.35 $627,807 9.4%
0.3 $476,756 9.1%
0.25 $357,112 8.8%
0.2 $257,498 8.4%
0.15 $186,552 8.1%
0.1 $115,257 7.5%
0.05 $58,646 6.8%

Note that, also 20% of the time, your $100 compounded for 97 years turns into $257,498 or a CAGR of 8.4%. A result that is 1/5 of the median and 1/20 of the mean. Ouch. 

So when someone says the stock market returns 10% per year because they looked at the average return in the past, realize that after adjusting for volatility and the fact that you will be re-investing your proceeds (a multiplicative process), you should expect something closer to 8% per year. 

And one last thing…you should be able to see how rates of return, when compounded for long periods of time, lead to dramatic differences in wealth. Taxes and fees are percentages of returns or invested assets. Make sure you are spending them on things you can’t get for free (like beta).

A Question I Wonder About

If you draw a return a simple return at random from a normal (ie bell curve) distribution and compound it over time, the resultant wealth distribution will be lognormally distributed with the center of mass corresponding to the CAGR return.

We saw that theory, simulation and reality all agreed. 

Or did they?

The simulation and theory were mechanically tied. I drew a random return from N [μ=12%, σ = 19.8%] and compounded it. But reality also agreed.

It may have been a coincidence. Let me explain. 

Stock market returns are not normally distributed. They are well-understood to differ from normal because they have a heavy fat-left tail and negative skew.

  1. The fat-left tail describes the tendency for returns to exhibit extreme (ie multi-standard deviation) moves more frequently than the volatility would suggest.
  2. Negative skew means that large moves are biased toward the downside.

These scary qualities are counterbalanced by the fact that the stock market goes up more often than it goes down. In the 97-year history I used to compute the stats, positive years outnumbered negative years 71-26 or nearly 3-1. 

The average returns, whichever average you care to look at, is the result of this tug-of-war between scary qualities and a bias toward heads. With the distribution not being a normal bell curve it feels suspicious that the relationship between CAGR and arithmetic mean returns conformed so closely to theory.

I have some intuitions about negative skew (that’s a long overdue post sitting in my drafts that I need to get to) that tell me that in the presence of lots of negative skew, volatility understates risk in a way that would artificially and optically narrow the gap between CAGR and mean return. By extension, I would expect that the measured CAGR of the last 97 years would have been lower relative to the theory’s prediction. 

But we did not see that.

I have 2 ideas why the CAGR was held up as expected, despite non-normal features that should penalize CAGR relative to mean return. 

  1. Path

    In Path: How Compounding Alters Return Distributions, we saw that trending markets actually reduce the volatility tax that causes CAGRs to lag arithmetic returns. It’s the “choppy” market that goes up and down by the same percent that leaves you worse off for letting your capital compound instead of rebalancing back to your original position size. The volatility tax or “variance drain” occurs when the chop happens more than trends (holding volatility constant of course). But since the stock market has gone up nearly 3x as often as it went down perhaps this trend compounding “bonus” offset the punitive negative skew effect on CAGR. 

  2. What negative skew?
      Qty Avg Return St Dev of Returns
    Up years 71 21.3% 12.7%
    Dn years 26 -13.4% 11.4%

    Using annual point-to-point returns, I’m not seeing negative skew. 

I’ve exhausted my bandwidth for this topic so I’ll leave it to the hive. Hit me up with your guesses.