Lessons From The .50 Delta Option

I was chatting with a quant friend who was bouncing an options idea off me. In the course of the conversation, he was surprised I did not assume the .50 delta option was the ATM (at-the-money) option. My friend is much smarter than me on finance stuff but options aren’t his native professional language. So if this idea had him tripped up I realized I had a reason to write a post.

If I do this correctly you will gain a better understanding of:

  1. Delta as a hedge ratio, not a probability
  2. How volatility affects the mean, median, and mode of these returns
  3. The relationship of arithmetic to geometric returns in option theory
  4. What these distributions mean for the value of popular option structures

Some housecleaning:

  • Option math is known for being calculus heavy. If you are a layperson, you are in luck, this tour guide likes to stick to the roads he knows. You won’t find complex equations here. If you are a quant, I suspect you can still benefit from an intuitive approach.
  • We are going to ignore the cost of carry (interests and dividends). While crucial to actual implementation it is just distracting to the intuition.

Delta Is A Hedge Ratio Not a Probability

Often delta and “probability of finishing ITM (in-the-money)” are indistinguishable. But they are not the same thing. The fact that they are not equivalent holds many insights.

Before we go there, let us revisit the most basic definition of delta.

Option delta is the change in option price per $1 change in the underlying

Consider the following example:

Stock is trading for $1. It’s a biotech and tomorrow there is a ruling:

    • 90% of the time the stock goes to zero
    • 10% of the time the stock goes to $10

First take note, the stock is correctly priced at $1 based on expected value (.90 x $0 + .10 x $10). So here are my questions.

  • What is the $5 call worth?

Back to the expected value:

    • 90% of the time the call expires worthless.
    • 10% of the time the call is worth $5

.9 x $0 + .10 x $5 = $.50

The call is worth $.50

  • Now, what is the delta of the $5 call?

$5 strike call =$.50

Delta = (change in option price) / (change in stock price)

    • In the down case, the call goes from $.50 to zero as the stock goes from $1 to zero.

Delta = $.50 / $1.00 = .50

    • In the up case, the call goes from $.50 to $5 while the stock goes from $1 to $10

Delta = $4.50 / $9.00 = .50

The call has a .50 delta

Using The Delta As a Hedge Ratio

Let’s suppose you sell the $5 call to a punter for $.50 and to hedge you buy 50 shares of stock. Each option contract corresponds to a 100 share deliverable.

  • Down scenario P/L:

Short Call P/L = $.50 x 100 = $50

Long Stock P/L = -$1.00 x 50 = -$50

Total P/L = $0

  • Up scenario P/L:

Short Call P/L = -$4.50 x 100 = -$450

Long Stock P/L = $9.00 x 50 = $450

Total P/L = $0

Eureka, it works! If you hedge your option position on a .50 delta your p/l in both cases is zero.

But if you recall, the probability of the $5 call finishing in the money was just 10%. It’s worth restating. In this binary example, the 400% OTM call has a 50% delta despite only having a 10% chance of finishing in the money.

I’ll leave it to you to repeat this example with a balanced distribution. Say a $5 stock that is equally likely to go to zero or $10. You will find the 50% delta call turns out to be ATM. Something you are used to seeing.

The key observation turns out to be:

The more positively skewed the distribution, the further OTM the 50% call will be. If a stock is able to go up 1000% and you sell a 400% OTM call on it you are going to need far more than a token amount of long stock to hedge.

The more positively skewed a distribution, the more the hedge ratio diverges from the “probability of finishing ITM”.


The Effect of Pure Volatility

Not to lead the witness too much, but an obvious feature of the binary example is the biotech stock is very volatile. That’s not a technical definition but a common-sense observation. “This thing is gonna move 100% or 900%!”.

Without math, consider how volatility alone affects a stock’s returns. If the stock price remains unchanged because we do not vary the expected value but instead inject more volatility what is happening?

  1. We are increasing the upside of possible payoffs.

In the biotech example, more volatility can mean the upside is not $10 but $20. 

  1. The counterbalance to the greater upside is a lower probability of rallying

If the stock is still worth $1 then the probability of the up scenario has just halved to 5% (95% x $0 + 5% * $20 = $1, the current price).

If we inject volatility into a price that is bounded by zero, the probability of the stock going down is necessarily increasing.

So volatility alone alters the shape of a stock’s distribution if you keep the stock price unchanged.

Let’s see how this works as we move from binary distributions to more common continuous scenarios.

How Volatility Affects Continuous Distributions

Let’s start with a simulation of a subjectively volatile stock.


  • The stock is $50
  • The annual standard deviation is 80%.

A basic presumption of option models is that returns are normally distributed but this leads to a lognormal distribution of stock prices 1.

Running the simulation:

  • I took a return chosen from a normal distribution with a mean of 0 1 and standard deviation of .80
  • I then ran that return through a simple log process to simulate continuous compounding. 2

S (T1) = S (T0) x e(random generated return)

  • I ran this 10,000 times.

Before we get to the chart note some key observations:

  • You get a positively skewed lognormal distribution bounded by zero. This is expected.
  • The median terminal stock price is $50 corresponding to a median return (aka the geometric mean) close to zero as expected.
  • The mean stock price is $68. corresponding to a mean return of 38%.
  • The modal stock price is $20 corresponding to a modal return of -60%.

Simulation Vs Theory

Let’s compare the simulation to what option theory predicts.

  • Median

As we stated earlier median expected return is 0 from theory and this lines up with the simulation.

  • Mode

The mode in the simulation lines up reasonably 3 with option theory which expects the mode to be:

S x e2 where σ is volatility

Note how volatility pulls your most likely outcomes lower. In this case, the most likely landing spot for the stock is $20 corresponding to a total return of -60%!

Average Arithmetic Returns

Look at the chart again. Note how the average arithmetic mean stock price is $68.89 in this sample. If the median return is 0, the positively skewed distribution has a mean arithmetic return of +37.8%! We don’t want to get excited about this since as investors we care about geometric returns which are zero here, but this 38% OTM strike is still very interesting.

It turns out it corresponds to the strike of the .50 delta option!

The equation for that strike:

S x e2/2)

That strike corresponds to 68.86 which is very close to the simulation result of 68.89.

This is the call that you must hedge with 50% of the underlyer.

The formula will look familiar if you remember that the geometric mean is pulled down from the arithmetic mean in proportion to the variance.

[This strike is special for option traders. This is the strike that has the maximum vega and gamma on the option surface. As implied vol changes the location of this strike can change, but it represents the maximum vega any strike can have for a given spot price. I’ll leave it to the reader to see how this relates to strategies that are convex in vol such as ratio’d vega neutral butterflies.]

Interesting Observations About Options

  • Even in a continuous distribution, the higher the volatility, the more positively skewed the distribution, the further OTM the 50d call strike lives.
  • The cheapest straddle will occur at the median outcome or the ATM4 strike. 
  • The most expensive butterfly will have its “body” near the theoretical mode. This makes sense since a butterfly which is just a spread of 2 vertical spreads is a pure bet on the distribution. If you chart the price of all the butterflies equidistantly across strikes you will have drawn the probability density function implied by the options market!

Enter Black Scholes

In a positively skewed distribution, the probability of finishing in the money for a call was lower than the delta. In the binary example where the stock had only a 10% chance of being worth $10, the probability of the $5 call was much lower than the delta of the $5 call.

What does this have to do with Black Scholes?

In Black Scholes:

  • The term for delta is N(d1).
  • The term for the probability of finishing in the money is N(d2).

What’s the relationship between d2 and d1?

  • d2 = d1 – σ√t

The math defines the relationship we figured out intuitively:

The higher the volatility 5 the more delta and probability will diverge!

Delta and probability are only similar when an option is near expiration or when it’s vol is “low”.

From Theory to the Real World

Markets compensate for Black Schole’s lognormal assumptions by implying a volatility skew. While a biotech stock might have a positive skew on steroids, a typical stock’s distribution looks more normal than positive. By pumping up the implied volatility of the downside puts and lowering the implied vols on the upside calls, the market:

  • Increases the value of all the call spreads.
  • Shifts the implied mode rightward.
  • Shifts the 50d call closer to ATM. Actually, it lowers all call deltas and raises all put deltas. This is important since deltas are the hedge ratios.
  • Fattens the left tail relative to a positive distribution and at least in index options even more than a normal distribution.

These adjustments reconcile the desirability of a simple, easy to compute model like Black Scholes which uses lognormal distributions with empirically consistent asset distributions that we observe in markets.



The next time you hear delta used as probability, remember this is only really useful when options are near-dated. Since most option activity occurs in the front end of the term structure the assumption is typically harmless.

Taking the time to understand why they differ turns out to be a great exercise in building an intuition of investment returns and their distributions. 



Shorting Bimodal Stocks


My friend and former colleague Jason took exception to the viral tweet I referenced last week about how shorting a bi-modal company is like an option. Not because it isn’t but because all equity is an option.

In short, the entire viral tweet is a tautology.

Jason joined Twitter to respond to it. Jason’s gripe was that all equity is effectively a call option struck at zero (you can argue that a positive book value sets the call strike higher but it doesn’t materially change the point).

Jason argues that if the viral tweet pretends it is saying anything beyond “being short stock is like being short an option”, you are mistaken. There are several reasons why, and I’ll use my intro to twitter to go thru them…(Jason’s reply)

Where do I stand?

I liked the original @HedgeDirty tweet because I am just a fan of presenting ideas in different ways. The flaws in the tweet are real and technical but it conveyed a correct impression even if it got there incorrectly. To Jason and @HedgeDirty the riskiness of shorting a bimodal company whose equity is probably worthless is obvious. I appreciated the narrative style which reinforced that point. Even if it’s self-evident to anyone who shorts stocks.

But I also see learning opportunities in deconstructing the flaws in the tweet.

You can read Jason’s reply to see the flaws he found. Especially resonant was the observation that the 0 strike call is a 100 delta call. It has no gamma or theta. The original tweet claimed otherwise.

What would I focus on?

1) Let’s do a little math to find the annualized vol. The first thing to note is how the bi-modality creates a very volatile stock. 125% per year standard deviation.

(Warning: it doesn’t make much sense to use standard deviation based on a normal looking return when we already stated it was bi-modal but the obvious takeaway is “damn, this thing is going to make large moves in return space.” If you know nothing else, you know going short this thing is like barebacking a bucking bull. Size appropriately.)

2) My biggest gripe with the example. The stock doesn’t trade for $185 because it’s hard to be short stocks.

If the stock is trading for $185 the market is implying different distribution. Either the 80% and 20% are not true, the stock’s upside is more than $250 (1.25B EV), or the recovery value is much higher.

It’s not trading for $185 with a theta that pushes towards zero or anything like that. We don’t need to invoke Greeks for an alleged $50 fair stock trading for $185. If it’s trading $185 the market doesn’t agree with the assumptions that compute a $50 fair value. Full stop.


Despite the flaws, I enjoyed the original tweet. You can read plenty of @HedgeDirty threads and see it’s a good account to follow. For example, the Why You Should Never Hold Levered ETFthread is great. I’ve written about the brutal math of levered, especially inverse levered funds before.

Shorting bi-modally distributed stocks is hairy. If it feels like it’s being short an option it’s only because the stock is volatile, equity is an option, and being short options is volatile. Nobody shorting stocks should have needed an education from that tweet. For everyone else, they should be aware of errors in mapping it to option theory even if I think overall the thread was net positively educational.

As for Jason who has been trading options since the late 90s, I imagine he felt that @HedgeDirty borrowed his bass and played it with a pick. Only Paul McCartney gets away with that.

Why You Don’t Get Paid For Diversifiable Risks?

Finance theory dictates that an investor does not get paid for “diversifiable” risk. You do not get paid for idiosyncratic risk, only systematic risk. I have not formally studied the CAPM pricing model and prefer more intuitive explanations anyway. So I went to #fintwit:

Twitter Roundup

You can click on the Tweet to see the responses. Here’s a roundup of the Tweets that most informed my understanding.

Here’s @Value_Quant:

Here’s @spreekaway:

My Take

The responses helped me consolidate my own non-technical understanding. I’ll walk through my own take and how I’ve seen the idea that you do not get paid for diversifiable risks in practice.

A Bidding Game

First, a quick game derived from this Bogleheads thread:

Imagine there are 2 boxes. One of them holds 100k and one is empty.

You have a net worth of $50,000. Multiple people are allowed to bid and none of them has more than $50,000.

How much would you bid for each box?

This is an extremely risky bet. Perhaps you bid $40,000 for one of the boxes, for an implied 25% return (you are buying an asset worth $50k for $40k)?

In this narrow example, a box will trade for a healthy discount to its fair value but at the price that the least risk-averse investor is willing to pay.

What happens if everyone in the world is invited to bid?

Jeff Bezos will bid $99,999.99 to buy both boxes and have a guaranteed profit. Even if he were only allowed to bid on a single box, he would bid $49,999.99 since he still has positive expected value and the potential loss is an invisibly small percentage of his net worth.

This Is A Good Model For How Markets Actually Work

Think of how market-making firms profit.

  • They are willing to trade for mere basis points of edge. Often it’s simply rebates.
  • They make thousands of trades a day that get thrown into a giant pool of positions.
  • The systematic risk or beta is hedged out so that they are left with a diversified, offsetting portfolio of idiosyncratic risks. Over time the law of large numbers crystallizes the expected edge into hard p/l dollars.
  • This model validates the idea that providing liquidity is a long-term positive expectancy.

The business model rests on two concepts.

  1. Bet an appropriate fraction of the bankroll for a given amount of edge.
  2. Diversification

It’s just like the casino business. The house doesn’t care what happens on any individual bet as long as the bet is a small fraction of its bankroll.

Competition Drives Efficiency

Competitive equilibrium will mean that the casinos who can bid the highest for the “customer” is the house that can:

a) source the most uncorrelated offsets to the wager


b) has the biggest bankroll

In the trading business, condition A is satisfied by the market makers with the best data/analytics and “see the most flow”. A firm entrenched in both equity markets and futures markets with licenses from both the SEC and CFTC is going to be more efficient at laying off the risks it acquires from serving tourists regardless of the venue they choose to play in.

A and B will create a virtuous loop. The best players will build larger bankrolls which allow them to outbid competitors further which earns them first look at the flow which improves their models and so forth.

Note the role of bankroll and diversification in the following examples:


Banks are able to trade options outside the bid/ask when they do bi-lateral deals with energy companies. The resources they pour into doing custom deals and loans earns them the extra spread. They create a high touch VIP area with fatter margins. They do this by being horizontally integrated across credit and derivatives markets 1


Bookies are able to take the other side of Mayweather’s sports bets because they have offsetting flow (phrased differently: they can make 100% negative correlated bets). They can take their vig between many offsetting bets providing an ample cushion to justify sweating the residual position they cannot expect to hedge…the actual outcome of the game.

Life Insurance

By pooling risk, insurance companies can underwrite policies at an affordable premium that still leave room for an actuarial profit. If you, as an individual, wanted to write an insurance policy for your neighbor you would not be able to offer a price that was simultaneously affordable to your neighbor and compensated you enough to tolerate the risk of their premature death. And even if you had enough money to insure a hundred people, you’d need the infrastructure to source them from multiple geographies so as to minimize the correlation between the deaths in the event of a natural disaster, terrorist attack, localized pandemic, or asteroid.

Stock Investing

You are faced with 2 stocks with the following attributes.

The portfolio math to construct the chart is described in my earlier post.

Interesting things to note:

  • Stock B is more volatile and a much worse risk/reward as proxied by its Sharpe ratio.
  • At positive correlations, the optimal weighting is to load up on Stock A. Not surprising. In fact, the more of A you have, the better the portfolio Sharpe ratio is.
  • But when correlations flip negative, optimal weights are now recommending significant weightings of Stock B.
  • As I tinkered with combinations of assets I found that assets with low or even negative Sharpe ratios improve a portfolio if they are both negatively correlated and highly volatile.

The insight for me is that negative correlations make assets outstanding diversifiers even if they have negative expected returns. And if the asset has a negative correlation, high volatility can even be attractive.

This is one more reason to suspect that volatile assets can be justifiably overpriced and not a source of excess return premium. In fact, they are valuable despite their unattractive stand-alone attributes.


In an efficient market, prices setters:

  • Are maximally-funded or have relatively low costs of capital
  • Have broadest perspective/market access

The emergent properties of markets will lead to idiosyncratic risks being held by the player most optimized to hold it. If the risk is borne by the most efficient holder, it is by definition priced so that there is no portfolio to which the risk is more valuable. What remains are systematic risks that no entity has discovered a hedge for. Those are the premiums you are allowed to pick up.

When there is no diversification left to hide behind, the systematic risk can only be compensated by a pure risk premium. At the end of the day, there may be no hedge for getting 2 to 1 on a coin flip. The long-run is going to have to do.

Further reading:

Quant and author Aaron Brown explains on Quora (Link)

My brief post on market efficiency: Dinosaur Markets (Link)

Negative Prices Make Sense

Negative Prices

Oil prices for the prompt future traded negative this week. Matt Levine covered it well all week, I won’t re-hash. I just wanted to explain negative prices a bit more since it sounds crazy. It’s not. Trader lingo is oil was “trading for a credit”. In other words, you get paid to own it. You get paid to own it because the inconvenience of owning it exceeds its value of using it. If I brought some radioactive uranium to your house you might pay me to take it away. Even if it has lots of value in the right hands.

What are other examples?

  • Asbestos. If you google “price of asbestos” you only find prices to “remove asbestos”. This was not always true.
  • Old beater cars. My Highlander costs $500 to register every year (thanks California). At some point, the cost to carry a car can be greater than its value. If you couldn’t give it away you are happy to pay someone to take it.

The Wider Lesson of Questioning Assumptions

And if you have failed to examine assumptions in these crazy times you are wasting an opportunity to think outside the box about risk. Interactive Brokers failed to realize futures could trade negative (this oversight cost them $88mm). Well in your model of what a rental property is worth have you considered what happens if a law circumvented limited liability? What if a law changed that made insurers raise prices dramatically or even withdraw from markets? What would that do to affected properties? Modernity if nothing else appears to be hyper-optimization on the stability of assumptions which may turn out to have an expiration. Assumptions that outlive their convenience.

The Curious Case of USO

When it comes to abstractions built on faulty assumptions take a look at the U.S. oil fund more recognizable by its ticker, USO.

Until this week, it held near-dated futures but when those traded as low as $7 a barrel, USO was a few circuit-breakers away from liquidation. Since oil futures we learned can trade below zero, you could argue that USO had about a 25% chance of going bankrupt if you looked at the delta of the 0 strike put.

USO decided to halt trading to announce it would roll into longer-dated futures that were trading near $20. In other words, they changed their mandate (which was legally within their right according to the prospectus) to give themselves more distance from zero. An act of self-preservation that more than halved their chance of going bankrupt.

So the fund managers extended USO’s life. They have a management fee to clip so their motivation is clear. But should USO exist?

USO is a public fund that cannot trade below zero but holds an asset that can. It seems like a mismatch of legal structures. USO posts margin for the oil futures it holds with a portion of the proceeds from what investors have paid for the fund. At negative futures prices USO has no way to collateralize the futures losses since it can not go into shareholders’ pockets for more money.

(Side note: Assuming it doesn’t gap down, my guess is the CME or their clearing broker forces them to delist for cash first. After all, who is on the hook for the future’s losses if the prices go negative if not the shareholders).

Hockey Stick Diagrams

At risk of giving CFA and Series 7 vets PTSD let me use USO to demonstrate optionality. Oil futures can trade negative. USO cannot. If you buy shares of USO and oil futures go negative nobody can ask you for more money. Your shares just go to zero.
So USO can only go to zero but futures can go negative. For the visual, let’s look at the payoff diagrams.

  • In the first pair we look at the payoff of buying oil for $5/barrel using futures OR USO.
  • In the second pair we look at the payoff of shorting oil at $5/barrel in both cases.
  • In the 3rd chart, we take advantage of the fact that USO can only go to zero. Read ahead to see what happens.

A savvy reader will see that the last diagram is the payoff diagram of a put option struck at the zero strike price. And you’ll note that it’s a free option. There’s no oil price at which you lose money.

Well, markets aren’t stupid. USO has been trading at a premium to NAV. Options have premiums.

Questions for advanced readers

1. USO’s premium to NAV has varied greatly this week from 10% to almost 40%. What do you think it’s worth? What drives the fair value of the premium?

2. USO is a basket of futures plus a premium. What does that mean for the volatility of USO? (Hint: there’s some Inception type stuff going on here)

3. The poker question. Can you form a view on USO volatility based on the incentives of the fund? How about the incentives of the risk holders?

Have fun thinking about it. You can share your answers with me but full disclosure, I will not confirm or disconfirm. Your only upside in telling me is I might be impressed.

On Trading and Aptitude

I know there are many younger readers of this letter. I’m not a quant. I took Calculus BC in HS and one stats class in college (although I do want to take more math online — I’m not advocating ignorance). Many of you are very strong in math. There’s a perception that options are about graduate degree stochastic calculus and differential equations. There are research-oriented jobs for which this is true. These jobs require raw mental horsepower and lots of training to tackle technical problems.

On the trading side, don’t get discouraged by academic notation in option papers. Here reasoning and numeracy are the pen and hammer. The tools of the trade. I should add for college students looking to get into trading coding is now table stakes. You need to have something to give in exchange for learning. The business is harder than ever, fetching lunch is not enough. (I know what you are thinking. Every generation in trading always thinks “if I was just born 10 years earlier it would have been so much easier to rake in the bucks”. It’s as stupid as a 300 lb lineman who wishes he could have come up in a time when linemen only weighed 250. He’s committing a time travel fallacy where he gets to go to the past with knowledge of recent innovations in diet, drugs, and exercise). Continuing on. The ability to code is also self-reliance. My own ability is very limited and I’m sure a junior will look at me the way I used to look at older traders who struggled with Excel. Circle of life.

Perhaps more so than the pure quant roles, in trading there’s a lot of room for grit. The analogy is as simple as the fact that most poker players are not quants, but there’s no doubting their discipline, endurance, ability to focus, number-sense, and logic skills. Your liberal arts (and no economics and business degrees are not science) degree is not a life sentence in ops.

(To be clear, this is not an affront to ops…my wife went that way. In fact, there is a whole conversation to be had about why a career in ops can be the more lucrative route. But it’s a parallel route and if a person wants to trade and take risk, anything else will feel like they failed even if objective standards might say otherwise).

On the other hand, tying this all back to the Parable of the Talents essay above — trading is not for everyone. It’s not even for many. You can do anything, except for what you can’t do.

The Moontower Retirement Model

With all the market chaos, a bunch of friends, most of them outside the field of finance, have dove into a Whatsapp chat devoted to money strategies. The chat is aptly named “Early Retirement Inc”. This is a smart group of people with good careers. The nature of public money discourse seems to revolve around what stocks are going to do or even worse what single stocks are going to do. This means everyone, even smart people, are breathing polluted air.

I’m gonna take a stab over the next couple weeks at a healthier approach. A lot of financial advisors read this so I’m sure I’ll be corrected. But if you want to get the right answer to something, you publically say something obviously wrong. Let’s see what happens.

The Retirement Problem

You work for 40-50 years but you live for 20 to 30 years more. The problem you must solve: don’t run out of money. Retirement finance is a vast field filled with overly precise mathematical treatments of “safe withdrawal rates” and investment allocation “glide paths” (think a target-date fund that is aggressive when you are young and conservative as you age). William Sharpe who won the Nobel Prize for the CAPM model called it the hardest problem in finance.

I’m not going to bore you with all that. I’m also not going to bore you with my soapbox rant about how destructive I think the whole vision of retirement is as portrayed by Charles Schwab commercials. The fact that it takes a moment to figure out if it’s a financial commercial or a Viagra ad is enough of hint that drugs are being sold in both cases.

Instead, I will encourage you to walk through an exercise that identifies the most important levers of the problem. I’ve put together a Moontower Retirement Model that you can make a copy of and play with your own numbers. I built it years ago and because of back and forth in my Whatsapp chat had a chance to dust it off and improve it. The model’s value is not in its output. “Hardest problem in finance”, remember?. It’s laden with assumptions and simplifications. It also spits out a path without confidence intervals. You can’t get away from the shortcomings. But if you have never worked through something like this, you are going to think about money in a much more enlightened way afterward.

Here’s the Moontower Retirement Model

(Please don’t hesitate to ask questions, point out errors and so forth. I’d like to enhance it over time.)

Here’s a screenshot of the input screen:

You will have fun trying to estimate numbers for your household income and expenses and if not it may alert you to get a grasp on these things.

Once you have some reasonable assumptions start tinkering with:
1) Inflation and investment return rates

Inflation is applied to pay and expenses evenly. Investment returns are applied to your entire net worth. Since net worth includes cash, homes, and stocks try to estimate a decent weighted-average return.

2) Retirement Age

Every year you delay retirement has a double impact. You increase savings that can grow AND you don’t withdraw return-generating assets.

3) Savings

A dollar saved is another source of big swings. You not only reduce the expense but you reduce the compounding of that expense. Or you can say that a dollar saved is also extra dollars in the future due to compounding. This is the most interesting lever because it’s high impact and insofar as you can control your expenses, it’s the lever you can steer the best. In contrast to, say, inflation. Good luck understanding inflation nevermind controlling it.

Don’t focus on the specific numbers as if you saw your future. You didn’t. This exercise has immense value despite that.

Do Professional Investors Understand Fees?

Fees Are In Focus


Giant fund manager/brokerages like Vanguard and Fidelity have made fees front and center. Like Walmart, if you are the lowest cost provider and wield blue whale scale, you are going to compete on price. Competition has spurred a race to the bottom on fees. With many investment choices commoditized, the focus on fees has served customers well. 

If I wanted to nit-pick, I might say investors don’t fully account for more opaque fees when choosing funds. These can swamp the management fees. Turnover, slippage costs, borrowing costs and abysmal sweep account rates all have significant impacts on net performance. These hidden costs are not easily reduced to a number that can be compared to a management fee. Hint: it’s a good place to search for how managers are able to drive fees to zero. But that’s a digression. I’m not especially interested in retail. Their financial advisors are doing a good job using steak and wine to box out the fund managers. There’s only so much fee to go around.


Allocators have a more difficult job. They devote teams to parsing alternative investments. A sea of private investments and complex hedge fund strategies. Within that context the allocators must construct portfolios that trade-off between tolerable risks and the probability of meeting their mandates. 

The allocators rummage through a diverse mix of strategies each with their own mandates. Growth, wealth preservation, defensive, hedged alpha. A fund can be thought of as a payoff profile with an associated risk profile. A thoughtful allocator is crafting a portfolio like a builder. They want to know how the pieces interlock so the final product is useful and can withstand the eventual earthquake. 

A builder cannot think of materials without considering cost. Wood might make for a better floor than vinyl but at what price would you accept the inferior material? When builders estimate their costs they must consider not only the materials, but transportation costs and how the cost of labor may vary with the time required to install the material. 

So let’s go back to the allocators. If the menu they were choosing from wasn’t complicated enough, they must also evaluate the costs. This is a daunting topic. They face all the opaque costs the retail investors face. But since they are often investing in niche or custom strategies that are not necessarily under a public spotlight they have additional concerns. A basic due diligence process would review:

  • Which costs are allocated to the GPs vs the LPs
  • Liquidity schedules
  • Fund bylaws
  • Specific clauses like “most-favored-nations”
  • Netting risks1

Unlike their retail counterparts, the professional investor’s day job is devoted to more than just investments but terms. Like our builder, this cannot be done faithfully without understanding the costs. Mutual funds sport fixed fees but complex investments often have incentive fees (a fee that is charged as a percentage of performance, sometimes with a hurdle) making them harder to evaluate. Regretfully, I suspect a meaningful segment of pros do not have a strong grasp on how fees affect their investments. 

Understanding Fees

While it is challenging to price many of the features embedded in funds’ offering documents, there is little excuse for not understanding fees whether they are fixed or performance-based.  After all, if you are an investor this is one of the most basic levers that affect your net performance and does not rely on having skills. It’s a classic high impact, easy to achieve objective. It’s the best box in that prioritization matrix that floats around consulting circles. 

Let’s take a quick test. 

You have a choice to invest in 2 funds that have identical strategies.

They have the same Sharpe ratio of .5

There are 2 differences between the funds. The fee structure and volatility.

  Fund A Fund B
Expected Return 5% 15%
Annual Volatility 10% 30%
Annual Fee 1% 2%

Let’s assume the excess volatility is simply a result of leverage and that the leverage is free.

Which fund do you choose?

Normalizing Fees By Volatility

The correct way to think about this is to adjust the fee for volatility.

  • Fund A’s fee is 10% of its volatility (1% / 10%).
  • Fund B’s fee is 6.7% of it volatility (2% / 30%)

If you doubt that Fund B is cheaper from this reasoning you could simply sell Fund A and buy 1/3 as much of Fund B.

Let’s use real numbers. Suppose to had a $300,000 investment in Fund A. You would be paying 1% or $3,000 in fees. 

Instead, invest $100,000 in fund B. Your expected annual return and volatility would remain the same, but you would only pay 2% of $100k in fees or $2,000. Same risk/reward for 2/3rd the price. Compound that.

I am not alone in this observation. From his book Leveraged Returns, Rob Carver echoes that a fund’s fees can only be discussed in context with its volatility:

I calculate all costs in risk-adjusted terms: as an annual proportion of target risk. For target risk of 15%/year and costs of 1.5%/year, your risk-adjusted costs are 1.5%/15% = 0.10. “This is how much of your gross Sharpe ratio will get eaten up by costs.


A Clue That Some Allocators Get This Wrong

Allocators will often target lower vol products for the same fee when a higher vol fund would do. To be fee-efficient they should prefer that managers ran their strategies at a prudent maximum volatility. Optimally some point before they were overlevered or introduced possible path problems. There are many funds and CTAs that would just as easily target higher volatility for the same fee. Investors would be better off for 2 reasons:

  • Allocators could reduce their allocations

As we saw in the Fund B example, it is more fee-efficient for vol targeting to be done at the allocation level not the fund level.

  • Limit cash drag.

They would stop paying excess fees for a fund that had been forced to maintain large cash reserves since it was targeting a sub-optimal volatility. Why would an allocator be ok with paying fees for funds that are holding excessive t-bills?

If you are not convinced that investors’ preference for lower vol versions of strategies demonstrates a lack of fee numeracy then check out this podcast with allocator Chris Schindler.  As an investor at the highly sophisticated Ontario Teachers Pension he witnessed firsthand the folly of his contemporaries’ thinking around fees. While mingling at conferences he would hear other investors bragging that they never pay fees above a certain threshold.

As we saw from our example, these brags are self-skewers, revealing how poorly these managers understood the relationships between fees and volatility. Not surprisingly, these very same managers would be invested in bond funds and paying optically low nominal fees. Sadly, once normalized for volatility, these fees proved to be punitively high. 

This brings us to our next section. How would you like to pay for low volatility or defensive investments?

Tests to Compare Fixed Fee Funds with Incentive Fee Funds

A Low Volatility Example

Let’s choose between 2 identical funds which only vary by the fee structure.

Both funds expect to return 5% and have a 5% volatility. Yes, a Sharpe ratio of 1.

  • Fund A charges a fixed .75%
  • Fund B charges 10% of performance from when you invest. Fund B has a high watermark that crystallizes 2 annually.

Which fund do you choose?

A Large Cap Equity Example

This time let’s choose between funds that have SPX-like features

Both funds expect to return 7% and have a 16% volatility.

  • Fund A again charges a fixed .75%
  • Fund B again charges 10% of performance from when you invest. Fund B has a high watermark that crystallizes < annually.

Which fund do you choose?

Studying The Impact Of Fee Structure

I wrote simulations to study the impact of fees on the test examples.

The universal setup:

  • Each fund holds the exact same reference portfolio
  • 10 years simulation using monthly returns
  • Random monthly returns drawn from normal distribution 
  • 1000 trials
  • Fixed Fee Fund charges .75% per year deducted quarterly
  • Incentive Fee Fund charges 10% of profits crystallized annually

Case 1: Low-volatility 

Simulation parameters:

  • Monthly mean return of .42% (5% annual)
  • Monthly standard deviation of 1.44% (5% annually)3

This chart plots the outperformance of the fixed fee return vs incentive fee return fund annually vs the return of the portfolio which they both own. The relative performance of the 2 funds is due to fees alone. 


  • It takes a return of about 7% or higher for the fixed fee fund to outperform.
  • This makes sense. A 75 bp fee is difficult to overcome for a 5% vol asset.
  • If the asset returns 5% the performance fee would only be 50bps and we can see how the difference in fees approximates the underperformance of the fixed fee fund for 5% level of returns.

Case 2: Large Cap Equity Example

The universal setup remains the same. 

We modify the simulation parameters:

  • Monthly mean return of .58% (7% annual)
  • Monthly standard deviation of 4.62% (16% annually)


  • Most of the time the fixed fee fund outperforms. So long as the return is north of about 4% this is true.
  • The most the fixed fee fund can underperform is by the amount of the fixed fee. Consider the case in which both portfolios lose value every year. The incentive fee fund will never charge a fee, while you will get hit by the 75bps charge in the fixed fee fund. You can see these cases in the negative points on the left of the chart where the portfolio realizes an annual CAGR of -5%.
  • Conversely, the incentive fee can be very expensive since it captures a percentage of the upside. In cases where the underlying portfolio enjoys +20% CAGRs, the simple fixed fee fund is outperforming by about 150 bps per year. 

Bonus Case: The High Volatility Fund

Finally I will show the output for a low Sharpe, high volatility fund.

The universal setup remains the same. 

We modify the simulation parameters:

  • Monthly mean return of .42% (5% annual)
  • Monthly standard deviation of 10.10% (35% annually)


  • This case demonstrates how complicated the interactions of fees and volatility are. The fixed fee fund will massively outperform by even as much as 200bps per year when the portfolio compounds at 20% annually.
  • The fixed fee fund even outperforms at low to mid single-digit returns albeit modestly. 
  • The high volatility nature of the strategy means lots of negative simulations, thanks to geometric compounding (for further explanation I discuss it here). When a fund performs poorly you pay less incentive fees so it’s not surprising that in many of these case the fixed fee fund underperforms by nearly the entire amount of the management fee. 


Fixed Fees

  • Best when the volatility of the strategy is high and the returns are strong (again you are warned: most high volatility strategies don’t have strong returns because of geometric compounding).
  • The most a fixed fee investor can underperform an incentive fee investor is by the amount of the fixed fee.

Incentive Fees

  • Best when the strategy is low volatility or returns are negative. Or the asset is defensive in nature. For hedges or insurance like funds, you may prefer to pay a performance fee to minimize bleed.
  • The amount an incentive fee investor can underperform is technically unbounded since it’s a straight percent of profits.


  • Fee structures must be considered relative to the volatility and goals of the strategy. There are no absolutes. 
  • By dividing fixed fees by the fund’s volatility you can normalize and therefore compare fund fees on an apples-to-apples basis. Even seemingly low fixed fees can be very expensive when charged on low volatility funds. 
  • Incentive fees look like long options to the manager (which implies the investor is short this option). The investor has unbounded potential to underperform a fixed fee solution and can only outperform by the amount of the fixed fee (the left hand side of those charts). To further study the embedded optionality of incentive fees see Citigroup’s presentation.
  • Incentive fees are meant to align investors and management. Who can argue with “eat what you kill”? But they can also create bad incentives. If trapped below the high watermark, the manager has nothing to lose and may swing for the fences irresponsibly. In addition, a staff working at a fund that is underwater might be dusting off their resumes instead of focusing on getting back on track knowing that they need to work through uncompensated p/l before they see another bonus. 
  • Fixed fees can encourage management to diversify or hold more cash to lower the fund volatility. These maneuvers can be combined with heavy marketing in a strategy more colloquially known as “asset-gathering”.


Fees need to be considered in light of the strategy. This requires being thoughtful to understand the levers. Unless you are comparing 2 SP500 index funds, it’s rarely as simple as comparing the headline fees. If we all agree that fees are not only critical components of long-term performance, while being one of the few things an allocator can control, then misunderstanding them is just negligent. A one size fee doesn’t fit all  alternative investments so a one size rule for judging fees cannot also make sense. Compared to the difficulty of sourcing investments and crafting portfolios getting smart about fees is low-hanging fruit. 

How Tails Constrain Investment Allocations

You would need to be living under a rock to not know about the importance of small probabilities on asset distributions. By 2020, every investor has been Talebed to death by his golden hammer. But knowing and understanding are not the same. I know it’s painful to give birth. But if I claimed more than that I’d end up only understanding what it felt like to be slapped in the face.

I’m hoping the above discussion of the devilish nature of small probabilities makes the seemingly academic topic of fat-tails more visceral. But if it didn’t I’m going to try to drive it home in the context of a real-life investing decision.

Step 1: Understand the impact of fat tails

I ran a simple monte carlo assuming the SPX has a 7% annual return (or “drift” if you prefer to sound annoying). I assume a 16% annual vol or standard deviation and ran a lognormal process since we care about geometric returns. We’ll call this model the “naive simulation”. It does not have fat tails.

Based on these parameters, if you invest on January 1st:

  • You have a 5% chance of being down 23% at some point during the year.
  • You have a 50% chance of being down 7% at some point during the year.

Now be careful. These are not peak-to-trough drawdowns. They are actually a subset of drawdown since they are measured only with respect to your Jan 1st allocation. The chance of experiencing peak-to-trough drawdown of those sizes is actually higher, but these are the chances of your account being X% in the red.

That’s the naive simulation. To estimate the odds in a fat-tailed distribution we can turn to the options market which implies negative skewness and excess kurtosis (ie fat tails). I used 1-year option prices on SPY. Option prices answer the question, “what are the chances of expiring at different prices?” not “what are the chance of returning X at any point in the next year?”. To estimate what we want we will need to use the pricing from strikes that correspond to the equivalent one-touch option. Walking through that is overkill for this purpose but hit me offline if you want to see how I kluged it.

Let’s cut to the market-implied odds.

  • You have a 5% chance of being down 39% at some point during the year.
  • You have a 50% chance of being down 11% during the year.

Now you can see the impact of fat-tails: the gap between 23% and 39%. This is the impact of kurtosis in the options. Meanwhile, in the heart of the distribution, the downside moves from 7% to 11%. Not as dramatic and attributable to market skew.

When we shift probabilities in the tails of distribution vs the meat the impact on the payoffs is significant.

Repeating this insight in a different way may help your understanding. Consider tossing a pair of dice. Imagine playing a game that pays the fair odds for a roll (i.e. craps).

Now let’s chip the dice to change the probability of how they land.

  • In scenario 1, add 1% to the “7” and shave .5% from each tail.
  • In scenario 2, add 1% to the “7” and shave .5% from the meat, the “6” and “8”

By shaving from the tails we take a fair game and turn it into a negative 30% expected value per toss. This is far worse than almost any casino game you might play. By changing the tail probabilities the effect on the game is magnified because the odds are multiplied across an inversely proportional payoff!

Step 2: How should tail sensitivity affect allocations?

By now, the danger of poorly estimating should be a bit more clear. How do we use this when making allocation decisions? After all, most of the time whether they are 1% or 2% events, huge moves are usually not in play. But we must care because when these events hit the impact is huge.

Tail outcomes should dictate constraints based on what you can tolerate. I’ll work through a conservative framework so you can see the impact of naive tail probabilities versus market-implied tail probabilities. The exact answers don’t matter but I’m hopefully offering a way to make tail-thinking relevant to your allocation decisions.

Reasoning through sizing decisions

Suppose things are going well and you are able to save $50,000 per year after paying expenses. You decide that losing $50,000 in the stock market is the largest loss you can accept, reasoning that it’s a year’s worth of savings and that you could make up the lost sum next year. If you impose a restraint like that, well, the most you can allocate to stocks is $50,000. That’s too conservative especially if you have accumulated several hundred thousand dollars in savings.

So you must relax your tolerance. You decide you are willing to accept a $50,000 loss 5% of the time or 1 in 20 years. Roughly a generation. If we use the naive model’s output that we lose 23% of our investment with 5% likelihood then the maximum we can allocate to stocks is $50,000/.23 = $217,000.

The naive model says we can allocate $217k to stocks and satisfy our tolerance of losing $50k with 5% probability. But if the market’s fat-tails are implied more accurately by the option skew, then our max allocation can only be $128k ($50,000/.39).

If we constrain our allocation by our sensitivity to extreme losses, the max allocation is extremely sensitive to tail probabilities. In this example, we simply varied the tail probability between a naive model using a mean and variance to a market-implied model which adjusted for skew and kurtosis. The recommended allocation based on our tolerance dropped a whopping 42% from $217k to $128k.

Many will point out that this approach is extremely conservative. Constraining your max loss tolerance to the amount of money you can save in a year seems timid. But the probabilities we used here did understate the risk. Again these were not peak-to-trough drawdown probabilities but the narrower chance of incurring losses on your start of year allocation. If we are thinking about the true experience of investing and how you actually feel it, you probably want to consider the higher drawdown probabilities which are out of scope for a piece like this. I know many financial advisors read this letter, I’m curious how allocation models reason through risk tolerance.

Current examples to consider in context of small probabilities

1) Bernie

There are market watchers who believe that electing Bernie Sanders would send us back to living in caves. Democrats are trading for about 40% to win the election. Bernie is trading at about 45% to win the nomination, implying an 18% chance to win the election. Market watchers who fear a Bernie presidency are either totally overstating his alleged market impact or the market is already discounting his odds. If the latter is true and the market is efficient, math dictates that it should shoot much higher in the event he loses.

At 18%, Bernie is no longer in the tail of the distribution. So you could argue that as he went from single-digit probability to his current chances, the market strongly re-calibrated either his impact or the sustained rally in the meantime would have been much larger. One of these things must have happened by the necessity of math as odds shifting from a few percents to 18%.

Or there is a third option. The market never really believed that Bernie’s impact would be as deep as his detractors contend.

2) Tesla

We have all seen this stock double in the past month. There has been a lot of talk about far out-of-the-money call options trading on the stock. These are bets on the upside tails of the stock over relatively short time frames. I won’t comment too much on that other than to point out a different tail in the matter. All the credit for this observation goes to a friend who keenly remembered that a year ago the Saudi’s collared their position in TSLA. That means they bought puts and financed by calls sold on the stock. Given the size of the move, the calls they sold are definitely deep in the money. This hedge likely cost them over 3 billion dollars. Billion with a “b”. That’s 6% of there projected government deficit. Their investment in TSLA stock was supposed to be a tail hedge against electric cars destroying demand for oil permanently. In the meantime, they got smoked hedging the hedge. The other tail in this story is going to be that of the official who recommended the hedge. This is a government that nearly executed a 13-year old for protesting. Fair warning to anyone looking to be an execution trader for the kingdom. You are probably short the mother of all puts. Make sure you are getting paid at least as much as a logger.

And one last TSLA note. This keen observation by Professor Bakshi.

Sometimes Keynes’ beauty contest doesn’t just judge beauty. It can create it.

Market Mutations

recently described markets as biology not physics in recognition of how players adapt. Let’s discover 2 more opaque examples and their causes.

1) Structured products

Historically your bank would happily sell you an investment note which guarantees your principle (insofar as you are ok with your bank’s credit risk) and earns you a return which is linked to return of an equity index. To manufacture this investment product the bank would invest in bonds and a portion of the interest income would be directed to buy call options on the index. There are more shortcuts they use to create the product (for example, the investor typically doesn’t capture the dividends which are a significant portion of the expected return), but the important thing to understand is these notes require enough interest income to finance the call options. With interest rates near zero in most of the world, banks have had to get more…creative.

To keep these notes promising attractive rates of return, the issuers buy insurance against a sell-off from the investors. Not explicitly of course. Instead they embed a feature that “knocks” your note out and exposes you to the losses if the reference index falls far enough. Yes, the prospectus spells this out. But for whatever reason, retail investors fail to wonder why an investment product can offer seemingly attractive returns in a low risk-free rate environment. They continue to gobble them up, not realizing they are self-financing these returns by underwriting catastrophic risk.

Here’s where it gets interesting. Since interest rates have never been this low and the aging developed nations have never been this large, there is unprecedented demand for these notes. These products are intensely popular in Asia and Europe (a friend once quipped you could buy them at a 7-11 in Italy. I want to believe this because it sounds so ridiculous so I refuse to fact-check it). The issuing banks, who are not in the business of taking directional or outright volatility risk, must recycle the optionality that these notes spit off. The associated option flows from these popular products are correspondingly massive.

From a “market is biology” perspective, it’s useful to remember that anybody using historical data to make their case may not be fully appreciating that our current landscape includes a bunch of dormant, non-linear payoffs that kick in only when the market has already made a large down move. An extreme analogy would be like comparing NFL wide receivers through time without noticing that they got rid of pass interference rules.

Although the bulk of these notes have historically been tied to Asian indices like Korea, they are becoming increasingly linked to the SP500. Will the tail wag the dog? Let options fund manager Benn Eifert explain on his latest appearance on the Bloomberg Odd Lots episode titled How To Create Havoc In The U.S. Options Market. (Link)

2) How corporate governance responds to the age of passive indexing
Consider these points taken from Farnum Street Investment’s latest letter. (Link)

  • In 1965, the CEO-to-worker pay ratio was 20-to-1. By 2018, it had jumped to 278-to-1. How did pay structures get so lopsided? Shouldn’t someone have stepped in? Yes, someone should have stepped in: the owners of the companies. But if you’re a passive index holder, you abdicated that responsibility to Vanguard, Blackrock, State Street or Fidelity. It wasn’t a custodian like Vanguard’s job to mind the henhouse. It was the job of the owners of the company.

Hard Truth: If you own an index fund, you waive your right to complain about CEO compensation.

  • In 2019, Lyft went public. With the increased transparency of SEC filing, it was discovered the company had 46 million restricted stock units (RSU) outstanding. RSUs are a way to incentivize employees, but they can become a big bill for owners. In the case of Lyft, the RSUs would cost owners $2-4 billion, depending on the IPO price. This represented a 20-25% ownership stake of the company being granted to employees. Corporations who grant extravagant stock options do so at the expense of the owners. There are no free lunches.

Hard Truth: If you own an index fund, you waive your right to complain about option dilution.

  • From 2008-2017, the pharmaceutical giant Merck distributed 133% of profits back to shareholders via dividends and share buybacks. Yes, they paid out more than they took in. Those resources could have gone toward research, saving lives, and the next blockbuster drug. The strategy seems obviously shortsighted. How come no one stepped up to tell them to think long term? Analysis initiated by SEC Commissioner Robert Jackson Jr. revealed that in the eight days following a buyback announcement, executives on average sold five times as much stock as they had on an ordinary day. Management is effectively cashing out at the owners’ expense when they know the price will be supported by internal buybacks. How come no one is stopping them?

Hard Truth: If you own an index fund, you waive the right to complain about myopic corporate strategy and share buybacks.

  • Sir Winston Churchill once said, “Capitalism is the worst economic system, except for all the others.” That remains true, but proper capitalism requires thoughtful stewards, meritocratic outcomes, and engaged owners. If we all abdicate our responsibilities, we risk perversion of the system that’s created more positive effects for humanity than arguably any other single phenomenon. Hope is not lost as history tends to move in cycles. We’re in need of the pendulum to change direction.

Hard Truth: This too shall pass.

Investing Is Biology Not Physics

Since the 1980s, there has been a tradition of Wall Street luring physicists from academia. Option math has more in common with the laws of thermodynamics than it does with accounting. But if the nature of markets themselves resembled any science it would be biology. Markets are governed by predator-prey dynamics. Models are adaptive. The actors learn. Doublethink and tradecraft.

In physics, the rules are fixed. No matter how many of us use the laws of gravity to keep firmly planted on planet Earth, gravity doesn’t get crowded. It keeps me just as bound to its surface as it did the Neanderthals. In markets, if I raise a bunch of money by showing people that selling volatility “harvests” a risk premium and the strategy continues to work then people will give me money to do it even more. So the strategy’s assets will grow both via inflows and via returns. The only problem is that to continue delivering the same performance on the larger asset base the strategy needs to sell ever more options. Assumptions of market liquidity when a strategy manages X will not hold when the strategy manages 10x or 100x. That’s about as close as we get to a physical law in finance.

The nature of liquidity is biological. It is subject to the whims of masses. It is the physical point where the backtest meets reality. Reality is a recursive, perma-learning system, with constraints and desires whose steers are pulled by investors, politicians, and corporations.

One of the best discussions I’ve ever listened to about what this looks like in practice is investor Andy Redleaf on Ted Seides’ Capital Allocators podcast. Redleaf has been in the game for over 40 years and was an early options market maker when they were listed in the 1970s. Since then he has followed opportunities that present themselves as markets change. A true agnostic on the hunt for profitable niches. Especially niches with structural reasons for being extra profitable. The advantage of this approach is that when the reasons go away, you know it is safe to cut and run. The disadvantage is that you cannot be a one-trick pony. You need to keep finding easy games.

For the full discussion of market history, where sources of edge often lurk, investing challenges today, and why he bought a bank check out the episode including my notes. (Link)

Susquehanna took their understanding of markets as biological to a logical recruiting conclusion — hire game players. Poker, Magic, chess, sports bettors. All games that require multi-order thinking and adapting to your environment. If you know anyone with a strong game background (and ideally some programming chops) check out Moontower reader Metaling Mage’s call for an intern. He’s a former Susquehanna PM.

You can reach out to him for details but it’s safe to say based on where he is now that this is could be one of the most selective Wall Street internships on the markets side of the business.

Is There Actually An Equity Premium Puzzle?

The equity risk premium, or ERP, is defined as the excess return you get for investing in stocks over the risk-free rate. Simply, it’s the premium return you earn in exchange for dealing with path. The fact that you might experience a 20% drawdown every few years (with U.S. equity markets currently sitting on all-time highs it’s hard to believe that just 1 year ago the SP500 had a 20% drawdown). I admit this “no pain, no gain” explanation sounds a bit weird.

Student: Hey prof, why do I get paid extra for buying stocks instead of t-bills?

Master: Because if you weren’t offered a discounted price to buy stocks you wouldn’t. Duh.

Proof by induction can be unsatisfying. To be fair, my use of the word proof is straining its English definition. Instead, it’s typical to hear ERP referred to in the context of a puzzle since some economists with calculators decided that this roughly 6% historical premium has been excessive compared to what they would expect even risk-averse investors to demand.

Enter the Witch

But what if I told you that there is actually no ERP and therefore no puzzle. Well, you’d accuse me of heresy since I’m directly contradicting widely accepted financial orthodoxy. After all, I’m ignoring the fact that equities have in fact outperformed t-bills by a wide margin.

Let’s look at that assertion again — equities have outperformed t-bills by a wide margin.

Well, what do we mean by equities? Single stocks or indexes? This is where I let the witch take over. The heretic, BreakingTheMarket who states:

The Equity Premium Puzzle has lasted for 37 years without anyone recognizing the market index doesn’t represent stocks.

Mistaken Equivalency

Turns out the existence of an ERP depends on your definition of equities and an index of equities is just not just equities. It’s a strategy. An index is a rule-based weighting that rebalances intermittently. The difference cannot be overstated. Why?

“Stocks” and the “Stock Market Index” are not the same thing and never have been. One is an asset class, the other is a trading strategy of that asset class. They don’t behave the same and don’t have the same properties, return, or standard deviation. You can’t use one to replace the other.

The math makes it clear.

When you compare the geometric return of stocks not a stock index you do not find an ERP!

The key here is that the historical volatility or standard deviation of single stocks is .33 which is about twice what it has been for U.S. stock indices. He makes the case that a .55% premium is much more in line with what economists would predict or just dismiss it as noise.

Enjoy the full post Solving the Equity Premium Puzzle, and Uncovering a Huge Flaw in Investment Theory. (Link)

How This Ties Together With What We Have Learned In The Past

As you digest this, there should hopefully be a comforting reinforcement of past ideas, namely:

  • When we deal with multiplicative processes, like returns that compound wealth, we care about geometric or logreturns not arithmetic returns because of the “volatility drain”. (Link)
  • Portfolio components are not perfectly correlated so when we rebalance, we capture a premium geometric return. (Link)
  • The imperfectly correlated aspect of a portfolio contributes to what Fernholz called the excess growth component that diversification earns when you are in logreturn space. (Link).

If we presume stock index volatility is only 17% (as opposed to the 33% for single stocks), we can use napkin math to make additional observations.

  • Index ERP is closer to 6% – .5 * (.17^2) = 4.56%…the extra 4% represents Fernholz’s “excess growth rate”. This is why some pros refer to diversification as the only “free lunch” in investing.
  • The average cross-correlation of stocks in an index can be approximated by the ratio of index variance to average weighted stock variance. Using our estimates (.17^2) / (.33^2) = .27 which is in the ballpark of where long term average SP500 index correlations have realized (although option folks know how spikey that number can be, especially on short measures).

Summing Up

ERP doesn’t exist if you look at stock; only stock indexes!

  • Researchers commonly mistake equivalency between a single asset and a portfolio:
    • Treasury bills (and bonds) are a single investment item. An equity market index (SP500 for the original study and many others) is a portfolio of many investments, who’s composition changes all the time. They are not the same thing and shouldn’t be compared as if they are!

A Final Note

I chat with BreakingtheMarket on Twitter and follow his discussions with quants. So much of the merit of Twitter, and the internet in general, is the beauty of being able to learn and engage in conversations with talented, curious people whom you may not have found otherwise. Breaking the Market is not in finance. He’s an engineer with a strong math background who approached markets with a “beginner’s mind”. I don’t think it’s an accident that two of my favorite finance writers on the internet are from scientifically minded people from a different field. I think the best finance blog is PhilosophicalEconomics.com which is penned by another finance outsider, the pseudonymous Jesse Livermore. Jesse did his first interview this year and it’s worth checking out, along with his widely influential writing. (Link to interview with my notes)