Option amateurs underappreciate the role of funding in pricing derivatives. Professional options traders need to be obsessed with funding costs because they are trading for tiny, often sub-penny, margins.

Here’s a simple example to demonstrate the tyrannical effect of funding on pricing:

What is a 1-year American at-the-forward call option on a non-div paying, 20% implied vol, $100 stock worth?

You need to feed the model an interest rate to get an answer. You look at the yield curve and see a 5% rate (making this up) for 1 year. This yields a forward price of $105 (we can hand-wave simple vs compounded rates for this purpose).

Imagine the bid-ask for this call is 40 cents wide $7.80 – $8.20

If you buy on the bid and sell on the offer you make a .40 profit. Easy-peasy.

Now imagine you buy the bid and hedge the position until expiry. What implied vol did you buy?

The first thing to recognize is that you will be shorting the stock to hedge. Assuming it’s easy to borrow, you are still not going to receive a 5% rate on the cash proceeds. Your prime broker needs to earn its margin. If 5% is the risk-free rate, let’s assume they pay you 4.5% on cash balances. Conversely, the prime broker will lend at 5.5% (this is known as the “long rate” and it’s the rate you finance long positions at). If you sell the call on the offer you will need to pay that rate to finance the shares you buy.

Uh oh.

If you buy the call you need to use a 4.5% rate in the model to back out an implied vol and if you sell the call you need to use a 5.5% rate in the model. You can see where this is going.

• If you buy the call on the bid you are paying 20.06% implied vol.
• If you sell the call on the offer you are selling 19.95% implied vol.

(Check the math if you want)

You think you’re trading vol but because of the bid-ask spread on your funding rate, you are basically trading the same implied vol even if you buy the bid and sell the ask. Rho is the sensitivity of the option price for a 1% change in the interest rate. The vega of an option is the sensitivity of its price for a 1-point change in volatility.

The rho of this call option is 46 cents vs a vega of 40 cents.

A 1% difference in funding rate (ie 4.5% vs 5.5%) is an institutional level bid-ask. It can be much worse for retail.

If you are trying to make markets you think you’re trading vol but are you even?

Pricing and carrying longer-dated options is crucially dependent on funding costs and the bid-ask spreads might not even be wide enough to compensate a market maker for their funding spread. Another way of saying this: the market-maker with such a 1% wide funding rate is making a 20% “choice” market in the vol. If the bid-ask was tighter they would be bidding a higher vol than they were offering!

(Again this assumes they hold and manage the position as opposed to spreading the options off by say buying one call and selling another or having the privileged position of just getting ping-ponged on their posted bid-ask all day)

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

Assumptions:

• 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.]

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 Returns, we 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 betterexplained.com:

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:

e^x 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

• • e^x 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 e^x 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(e^x) 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
• e^x tells 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 e^x  growth
  • What rate we needed for a given time period to achieve that e^x  growth

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 e^x 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)
   and
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”) = Se^rt
• The standard deviation scaled to time = σ√t

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

• K_up = Se^{(rt + σ√t)}
• K_down = Se^{(rt – σ√t)}

[The rate in the e^x 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.

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:

$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. 

Review

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

where:

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:

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. 

Here’s a simple coin flip game. It costs $1 to play.

• Heads: you get paid an additional $1 (ie 100% return)
• Tails: you lose $.90

The expectancy of the game is $.05 or 5%.

We compute expectancy:

.5 * $1.00 + .5 * (-$.90)

It’s exactly the same calculation as a weighted average or arithmetic mean. This is a useful computation for many simple one-off decisions. Like should I buy an airline ticket for $1000 or the refundable fare for $1,100?

If there’s a 10% chance I need a refund then the extra $100 saves me $1,100.

10% * $1,100 =$110 which is greater than the $100 surcharge. 9% is my breakeven probability.

It’s tempting to use this logic in investing. Let’s say you expect the stock market to return 7% per year on average for 40 years. Start with $100 and plug in numbers:

$100 * 1.07⁴⁰ = $1497

Yay, you expect to have about 15x your starting capital after 40 years!

Eh. Sort of.

See the word “expect” in math terms and in colloquial terms is a bit different.

If I bet $1 on that coin game I theoretically expect to have $1.05 after 1 trial. In reality, I’m either going to end up with $2 when I double up or $.10 when I lose.

Another example:

I roll a die. If it comes up “1”, I win $600. Otherwise, nothing happens. Theoretically, I expect to win $100:

1/6 * $600 + 5/6 * $0 = $100

But if I asked you what you “expect” to happen if you play this game…you “expect” to win nothing. You only win 1/6 of the time after all.

Back to the investing example.

Investing is not a one-off game. It’s a compounding game where you plow your total capital back into the sausage machine to get that 7%

That’s why we use 1.07⁴⁰.

You are counting on your $100 growing by 1.07 * 1.07 * 1.07…

So that 15x number…that’s mathematical expectancy the same way the dice game is worth $100 or the coin game is worth $1.05 even though those outcomes are never actually experienced.

What you expect to happen in the colloquial sense of the term is the geometric mean. The arithmetic average is a measure of centrality when you sum the results and divide by the number of results. (In our examples you are summing results weighted by their probabilities, but you are still summing). The geometric mean corresponds to the median result of a compounding process. Compounding means “multiplying not summing”. The median is the measure that maps to our colloquial use of “expected” because it’s the 50/50 point of the distribution. That’s the number you plan life around.

The theoretical arithmetic mean result of playing the lotto might be losing 50% of your $2 Powerball ticket (which is another way of saying you are paying 2x what the ticket is mathematically worth). The median result is you lit your cash on fire. You plan your life around the median, especially when it’s far away from the mean. We’ll come back to that.

With investing we are multiplying our results from one year to the next together. The geometric mean is what you actually “expect” in the colloquial sense of the term. The geometric mean is more familiarly known as the CAGR or ‘compound annual growth rate’.

What is the relationship between the arithmetic mean to the geometric mean? This is the same exact question as “what is the relationship of mathematical expectancy and the CAGR?”

It’s an important question since that theoretical arithmetic mean is only expected if we live thousands of lives (actually there are ways to experience the arithmetic mean without relying on reincarnation. This is pleasant news because what good is being rich if you come back a pony.) We want to focus on the CAGR, which is much closer to what we might experience.

It turns out that number is lower.

How much lower? It depends on how volatile the investment is. The formula that relates the arithmetic mean and CAGR:

CAGR = Arithmetic Mean – .5 * σ²

where:

σ = annualized volatility

If an investment earned 7% per year with a standard deviation(ie volatility) of 20% you can estimate the CAGR as follows:

CAGR = .07% – .5 * .20² = .05

In arithmetic expectancy, over 40 years you expect to earn 1.07⁴⁰ = 15. You expect to have 15x’d your money.

But the median outcome, which corresponds to the geometric mean is 1.05⁴⁰ = 7.

7x is much closer to what you “expect” in the colloquial sense of the term. Less than 1/2 the arithmetic expectation!

The formula tells us that the arithmetic and geometric mean (“CAGR”) will diverge by the volatility. And that volatility term is squared…which means the divergence is extremely sensitive to the volatility.

This is a table of CAGRs where you can see the destructive power of volatility:

Why is volatility so impactful on a compounded return?

An easy way to see the impact of high volatility is to imagine making 50% and losing 50%. The order doesn’t matter. You have net lost 25% of your initial capital.

We can compute the geometric mean by weighting each possibility by its frequency in the exponent (in this case the exponents must sum to 2 because that’s the sample space — up and down):

.5¹ x 1.5¹ = .75

Go back to the first game in the post. You invest $1 in a coin game. Heads to make 100%, tails you lose 90%. This game had a positive arithmetic expectancy of 5%.

What is our arithmetic expectancy if you compound (ie re-invest) by playing 2x then the total possibilities are:

HT: 2 x .1 = .2

HH: 2 x 2 = 4

TH: .1 x 2 = .2

TT: .1 x .1 = .01

Since each scenario is equally likely (25% each) the arithmetic expectancy is simply the average = 1.1025

This jives with 1.05² = 1.1025

The average arithmetic return compounds as expected.

But our lived (median) experience is much worse. The median result is .20, a loss of 80%!

We could have seen that by computing the geometric mean:

2¹ x .1¹ = .20

Driving the point home with an extreme example

Consider a super favorable bet.

You roll a die:

• Any number except a ‘6’: 10x your bet
• Roll a ‘6’: Lose your entire bet

The arithmetic expectancy is ridiculous.

5/6 x 10 + 1/6 x -1 = 8.167 or ~700% return

But if you keep reinvesting your proceeds in this bet, you will go bust as soon as the 6 comes up. The median experience is a total loss, even though the arithmetic expectancy compounded is wildly positive. If you played this game 20 times in a row you’d [arithmetically] expect to make ~ 700%²⁰.

But you have a 97.4% chance of going broke because you need “not a 6” to come up 20 times in a row = 1 – (5/6)²⁰

That arithmetic expectancy of ~ 700%²⁰ is being driven by the single scenario where the 6 never comes up (that occurs 2.6% of the time). In that case, your p/l is $10²⁰ or between a quintillion and sextillion dollars.

But the geometric mean is 0 because multiplying over the 6 sample spaces:

10⁵ x 0¹ = 0

I chose such extreme examples because nothing illustrates volatility like all-or-nothing bets. The intuition you need to keep is that high volatility means you should expect to lose your money even if the arithmetic expectancy is high.

As soon as you start re-investing (ie compounding) your results are going to be governed by that geometric mean which hates volatility.

For the people who tout lotto ticket investments like crypto or transformative technologies with talks of “asymmetrical upside” or “super positive expectancy” remember even if they might be right, the most likely scenario is they lose all their money on that investment. Even literal lotto tickets can tip into positive expectancy. When that happens how much do you put into it?

Exactly. Not much. Because you know what to expect.

The role of rebalancing and diversification

Investing is not a one-off game. You always re-invest. By re-balancing, you “create” more lives by not concentrating your wealth in a single bucket which swamps the rest of your portfolio as it grows. If you never rebalanced BTC on the way up it would have eventually become nearly 100% of your portfolio and then 2022 happened.

If you don’t ever rebalance you are effectively praying that “not a 6” comes up for the 40 years you are compounding wealth. It’s not as extreme as that because market volatility isn’t as extreme as dice or coins. But the principle holds.

You only get one life so you care about the median. Diversification plus rebalancing gives you the god-perspective of getting to invest a fraction of your wealth into many lives.

Keep in mind — rebalancing is not changing your overall expectancy; it’s changing the distribution of returns by pushing the median return (geometric mean or CAGR) up to your theoretical arithmetic return. This trade-off is not free. If you rebalance you don’t get the 1000x payoff that occurs when a single concentrated position hits 50 heads in a row.

Money Angle For Masochists

Imagine a $100 stock that can either go up or down 25% every year.

It’s 50/50 to be up or down.

Let’s look at the distribution of the stock after 4 years (with the probabilities of each price below it)

Look at the extremes after 4 years:

• $31.64

  A -25% CAGR over 4 years = cumulative loss of 68%

• $244.14

  A +25% CAGR over 4 years = cumulative gain of 144%

If you sumproduct every terminal probability by terminal price you get $100. And yet, while the stock is fairly valued at $100, after 4 years, you have lost money in 11/16th of scenarios (~69%). The right tail is driving the fair value of $100 while most paths take the stock lower.

This is the mathematical nature of compounding. The most likely outcomes are lower even if the stock is fairly priced.

In the real world, stocks don’t just flip up and down like coins. The probabilities are not 50/50 and there aren’t just 2 buckets they can rest in from one year to the next. The beauty of option surfaces is they allow us to separate the probabilities from the distance of the buckets (and the number of buckets is continuous…there’s no price the stock is not allowed to go to).

Here’s some homework you can do with the above data:

1. What’s the value of the 4-year $146.68 strike call worth?¹
2. What’s the value of the 4-year $75 strike put? ²
3. How about the 4-year $125 call? ³

Bonus Questions

Imagine this stock is an ETF and there’s a 2x levered version (which means it’s 2x as volatile) of it.

• What strike call on the levered ETF is equivalent to the $146.48 strike on the unlevered ETF?⁴(Hint: It’s further than $46.48 OTM)
• What’s the value of the call at that strike? ⁵
• If I was a market-maker and I got lifted at fair value on the 2x levered ETF 4-year 200 strike call and I go buy the regular ETF 150 4-year 150 calls to cover my risk how many do I need to buy to be perfectly hedged? (Assume you can buy them for what they’re worth…you have enough information to compute their fair value). ⁶

If you got through this then you have a new appreciation for how far certain prices are from a spot price and how it depends on time and volatility!

Starting from basics like the volatility tax, progressing to how path influences the volatility tax (trends are more like a volatility rebate and choppiness is a tax….the ratio of trend to chop will determine the ultimate cost of the volatility), and finally bridging these concepts to Black Scholes this series will take your understanding of compounding and how returns work to a deeper level.

1. The Volatility Drain
2. Path: How Compounding Alters Return Distributions
   [Between this post and the bonus questions you can start to see why pricing OTM options on levered ETFs given a liquid options market on the unlevered version is an application of these concepts]
3. Solving A Compounding Riddle With Black-Scholes

Shout Out To Matt Hollerbach

Despite trading options for nearly 20 years at the time, it wasn’t until 2019 that I thought really hard about compounding. I knew how to manipulate formulas and how it related to options but it wasn’t until I discovered Matt’s work that I started to see it from a new angle. Matt makes it approachable and builds up insights in small steps. His blog inspired mine, especially many of my earlier posts. The entire blog is worth spending time working through. It’s similar to what I’ve said about gambling — it’s a place where you will learn how to think about risk and return far better than what finance texts will teach.

These are all-time great ones:

Trend Following is Hot Air

Investing Games

Solving the Equity Premium Puzzle, and Uncovering a Huge Flaw in Investment Theory

It’s painful to watch the median (or should I say average) “investor” reason about how markets work because without these intuitions (you don’t need to know formulas necessarily) you are innumerate. That’s like being illiterate but for like numbers and stuff. And the deficiency is as obvious as illiteracy is to a literate person.

The good news is we can all get better.

Simple Interest

If you pay someone $90 today and they promise to give you $100 in 12 months, you are making a loan. This is the same idea as buying a bond. To back out the simple interest rate denoted (ie assuming no compounding) we solve for r:

90 * (1+r) = 100

r = 100/90 – 1

r = 11.11%

Annualizing

If the loan was only for 6 months, then we’d annualize the interest rate by multiplying by 2 (12 months / 6 months) for a rate of 22.22%

Compound Interest

Let’s return to the 12-month loan and say that the rate is compounded semi-annually. Then the computation is:

90 * (1+r/2)² = 100

r /2 = (100/90)^.5 – 1

r = 10.82%

If you compound more frequently than annually, it makes sense that the implied interest rate is lower. Consider the path of the principal + accrued interest:

Compounding semi-annually means interest gets credited at the 6-month mark. So the rate for the next 6 months is being applied to the higher accrued value amount which means the implied rate to end up at $100 (the same way the simple interest case ends up $100) must be lower than the simple interest case.

Continuous Interest

We can compound interest more frequently. Quarterly, monthly, daily. Since the number we are backing out, namely the implied rate, is being applied to a growing basket of principal + accrued interest at each checkpoint (I think of the compounding interval as a checkpoint where the accrued interest is rolled into the remaining loan balance), the implied rate to end up at $100 must be smaller. If we take this logic to the extreme and keep cutting the time interval into smaller increments we eventually hit the limit of Δt → 0. The derivatives world models everything in continuous time finance so interest rates get the same treatment.

Mechanically, the math is no harder.

To compute the continuously compounded interest rate we still just solve for r:

90 * e^rt = 100

t is a fraction of a year. So for the 12-month case:

90 * e^r*1 = 100

e^r = 100/90

r ln(e) = ln(100/90)

r = 10.54%

As expected, this is a lower implied rate than the 11.11% simple rate and the 10.82% semi-annual rate. Again, because we are compounding continuously.

Annualizing remains easy. If $90 grows to $100 in just 6 months, we compute the continuously compounded rate as follow:

90 * e^r*1/2 = 100

e^r*1/2 = 100/90

r *1/2 = ln(100/90)

r = 21.07%

This can be contrasted with the 22.22% 6-month loan using simple interest we computed earlier.

Application To Real Life

Note in all these cases, $90 is growing to $100. We are just seeing that the implied rate depends on the compounding assumption. In real life, when you see “compounded daily” or “compounded monthly” and so on, you are now equipped with the tools to compare rates on an apples-to-apples basis. If a rate is lower but compounds more frequently than another rate the relative value between both loans is ambiguous.

APYs disclosed on financial products make yields comparable. But now you understand how APYs convert different rate schedules into a single measure.