Startup options v. cash | I'm trying some experimental tiers on Patreon to see if I can get to substack-like levels of financial support for this blog without moving to substack!

I often talk to startups that claim that their compensation package has a higher expected value than the equivalent package at a place like Facebook, Google, Twitter, or Snapchat. One thing I don’t understand about this claim is, if the claim is true, why shouldn’t the startup go to an investor, sell their options for what they claim their options to be worth, and then pay me in cash? The non-obvious value of options combined with their volatility is a barrier for recruiting.

Additionally, given my risk function and the risk function of VCs, this appears to be a better deal for everyone. Like most people, extra income gives me diminishing utility, but VCs have an arguably nearly linear utility in income. Moreover, even if VCs shared my risk function, because VCs hold a diversified portfolio of investments, the same options would be worth more to them than they are to me because they can diversify away downside risk much more effectively than I can. If these startups are making a true claim about the value of their options, there should be a trade here that makes all parties better off.

In a classic series of essays written a decade ago, seemingly aimed at convincing people to either found or join startups, Paul Graham stated "If you wanted to get rich, how would you do it? I think your best bet would be to start or join a startup. That's been a reliable way to get rich for hundreds of years" and "Risk and reward are always proportionate." This risk-reward assertion is used to back the claim that people can make more money, in expectation, by joining startups and taking risky equity packages than they can by taking jobs that pay cash or cash plus public equity. However, the premise — that risk and reward are always proportionate — isn’t true in the general case. It's basic finance 101 that only assets whose risk cannot be diversified away carry a risk premium (on average). Since VCs can and do diversify risk away, there’s no reason to believe that an individual employee who “invests” in startup options by working at a startup is getting a deal because of the risk involved. And by the way, when you look at historical returns, VC funds don’t appear to outperform other investment classes even though they get to buy a kind of startup equity that has less downside risk than the options you get as a normal employee.

So how come startups can’t or won’t take on more investment and pay their employees in cash? Let’s start by looking at some cynical reasons, followed by some less cynical reasons.

Cynical reasons

One possible answer, perhaps the simplest possible answer, is that options aren’t worth what startups claim they’re worth and startups prefer options because their lack of value is less obvious than it would be with cash. A simplistic argument that this might be the case is, if you look at the amount investors pay for a fraction of an early-stage or mid-stage startup and look at the extra cash the company would have been able to raise if they gave their employee option pool to investors, it usually isn’t enough to pay employees competitive compensation packages. Given that VCs don’t, on average, have outsized returns, this seems to imply that employee options aren’t worth as much as startups often claim. Compensation is much cheaper if you can convince people to take an arbirary number of lottery tickets in a lottery of unknown value instead of cash.

Some common ways that employee options are misrepresented are:

Strike price as value

A company that gives you 1M options with a strike price of $10 might claim that those are “worth” $10M. However, if the share price stays at $10 for the lifetime of the option, the options will end up being worth $0 because an option with a $10 strike price is an option to buy the stock at $10, which is not the same as a grant of actual shares worth $10 a piece.

Public valuation as value

Let’s say a company raised $300M by selling 30% of the company, giving the company an implied valuation of $1B. The most common misrepresentation I see is that the company will claim that because they’re giving an option for, say, 0.1% of the company, your option is worth $1B * 0.001 = $1M. A related, common, misrepresentation is that the company raised money last year and has increased in value since then, e.g., the company has since doubled in value, so your option is worth $2M. Even if you assume the strike price was $0 and and go with the last valuation at which the company raised money, the implied value of your option isn’t $1M because investors buy a different class of stock than you get as an employee.

There are a lot of differences between the preferred stock that VCs get and the common stock that employees get; let’s look at a couple of concrete scenarios.

Let’s say those investors that paid $300M for 30% of the company have a straight (1x) liquidation preference, and the company sells for $500M. The 1x liquidation preference means that the investors will get 1x of their investment back before lowly common stock holders get anything, so the investors will get $300M for their 30% of the company. The other 70% of equity will split $200M: your 0.1% common stock option with a $0 strike price is worth $285k (instead of the $500k you might expect it to be worth if you multiply $500M by 0.001).

The preferred stock VCs get usually has at least a 1x liquidation preference. Let’s say the investors had a 2x liquidation preference in the above scenario. They would get 2x their investment back before the common stockholders split the rest of the company. Since 2 * $300M is greater than $500M, the investors would get everything and the remaining equity holders would get $0.

Another difference between your common stock and preferred stock is that preferred stock sometimes comes with an anti-dilution clause, which you have no chance of getting as a normal engineering hire. Let’s look at an actual example of dilution at a real company. Mayhar got 0.4% of a company when it was valued at $5M. By the time the company was worth $1B, Mayhar’s share of the company was diluted by 8x, which made his share of the company worth less than $500k (minus the cost of exercising his options) instead of $4M (minus the cost of exercising his options).

This story has a few additional complications which illustrate other reasons options are often worth less than they seem. Mayhar couldn’t afford to exercise his options (by paying the strike price times the number of shares he had an option for) when he joined, which is common for people who take startup jobs out of college who don’t come from wealthy families. When he left four years later, he could afford to pay the cost of exercising the options, but due to a quirk of U.S. tax law, he either couldn’t afford the tax bill or didn’t want to pay that cost for what was still a lottery ticket — when you exercise your options, you’re effectively taxed on the difference between the current valuation and the strike price. Even if the company has a successful IPO for 10x as much in a few years, you’re still liable for the tax bill the year you exercise (and if the company stays private indefinitely or fails, you get nothing but a future tax deduction). Because, like most options, Mayhar’s option has a 90-day exercise window, he didn’t get anything from his options.

While that’s more than the average amount of dilution, there are much worse cases, for example, cases where investors and senior management basically get to keep their equity and everyone else gets diluted to the point where their equity is worthless.

Those are just a few of the many ways in which the differences between preferred and common stock can cause the value of options to be wildly different from a value naively calculated from a public valuation. I often see both companies and employees use public preferred stock valuations as a benchmark in order to precisely value common stock options, but this isn’t possible, even in principle, without access to a company’s cap table (which shows how much of the company different investors own) as well as access to the specific details of each investment. Even if you can get that (which you usually can’t), determining the appropriate numbers to plug into a model that will give you the expected value is non-trivial because it requires answering questions like “what’s the probability that, in an acquisition, upper management will collude with investors to keep everything and leave the employees with nothing?”

Black-Scholes valuation as value

Because of the issues listed above, people will sometimes try to use a model to estimate the value of options. Black-Scholes is commonly used because well known and has an easy to use closed form solution, it’s the most commonly used model. Unfortunately, most of the major assumptions for Black-Scholes are false for startup options, making the relationship between the output between Black-Scholes and the actual value of your options non-obvious.

Options are often free to the company

A large fraction of options get returned to the employee option pool when employees leave, either voluntarily or involuntarily. I haven’t been able to find comprehensive numbers on this, but anecdotally, I hear that more than 50% of options end up getting taken back from employees and returned to the general pool. Dan McKinley points out an (unvetted) analysis that shows that only 5% of employee grants are exercised. Even with a conservative estimate, a 50% discount on options granted sounds pretty good. A 20x discount sounds amazing, and would explain why companies like options so much.

Present value of a future sum of money

When someone says that a startup’s compensation package is worth as much as Facebook’s, they often mean that the total value paid out over N years is similar. But a fixed nominal amount of money is worth more the sooner you get it because you can (at a minimum) invest it in a low-risk asset, like Treasury bonds, and get some return on the money.

That’s an abstract argument you’ll hear in an econ 101 class, but in practice, if you live somewhere with a relatively high cost of living, like SF or NYC, there’s an even greater value to getting paid sooner rather than later because it lets you live in a relatively nice place (however you define nice) without having to cram into a space with more roommates than would be considered reasonable elsewhere in the U.S. Many startups from the last two generations seem to be putting off their IPOs; for folks in those companies with contracts that prevent them from selling options on a secondary market, that could easily mean that the majority of their potential wealth is locked up for the first decade of their working life. Even if the startup’s compensation package is worth more when adjusting for inflation and interest, it’s not clear if that’s a great choice for most people who aren’t already moderately well off.

Non-cynical reasons

We’ve looked at some cynical reasons companies might want to offer options instead of cash, namely that they can claim that their options are worth more than they’re actually worth. Now, let’s look at some non-cynical reasons companies might want to give out stock options.

From an employee standpoint, one non-cynical reason might have been stock option backdating, at least until that loophole was mostly closed. Up until late early 2000s, many companies backdated the date of options grants. Let’s look at this example, explained by Jessie M. Fried

Options covering 1.2 million shares were given to Reyes. The reported grant date was October 1, 2001, when the firm's stock was trading at around $13 per share, the lowest closing price for the year. A week later, the stock was trading at $20 per share, and a month later the stock closed at almost $26 per share.

Brocade disclosed this grant to investors in its 2002 proxy statement in a table titled "Option Grants in the Last Fiscal Year, prepared in the format specified by SEC rules. Among other things, the table describes the details of this and other grants to executives, including the number of shares covered by the option grants, the exercise price, and the options' expiration date. The information in this table is used by analysts, including those assembling Standard & Poor's well-known ExecuComp database, to calculate the Black Scholes value for each option grant on the date of grant. In calculating the value, the analysts assumed, based on the firm's representations about its procedure for setting exercise prices, that the options were granted at-the-money. The calculated value was then widely used by shareholders, researchers, and the media to estimate the CEO's total pay. The Black Scholes value calculated for Reyes' 1.2 million stock option grant, which analysts assumed was at-the-money, was $13.2 million.

However, the SEC has concluded that the option grant to Reyes was backdated, and the market price on the actual date of grant may have been around $26 per share. Let us assume that the stock was in fact trading at $26 per share when the options were actually granted. Thus, if Brocade had adhered to its policy of giving only at-the-money options, it should have given Reyes options with a strike price of $26 per share. Instead, it gave Reyes options with a strike price of $13 per share, so that the options were $13 in the money. And it reported the grant as if it had given Reyes at-the-money options when the stock price was $13 per share.

Had Brocade given Reyes at-the-money options at a strike price of $26 per share, the Black Scholes value of the option grant would have been approximately $26 million. But because the options were $13 million in the money, they were even more valuable. According to one estimate, they were worth $28 million. Thus, if analysts had been told that Reyes received options with a strike price of $13 when the stock was trading for $26, they would have reported their value as $28 million rather than $13.2 million. In short, backdating this particular option grant, in the scenario just described, would have enabled Brocade to give Reyes $2 million more in options (Black Scholes value) while reporting an amount that was $15 million less.

While stock options backdating isn’t (easily) possible anymore, there might be other loopholes or consequences of tax law that make options a better deal than cash. I could only think of one reason off the top of my head, so I spent a couple weeks asking folks (including multiple founders) for their non-cynical reasons why startups might prefer options to an equivalent amount of cash.

Tax benefit of ISOs

In the U.S., Incentive stock options (ISOs) have the property that, if held for one year after the exercise date and two years after the grant date, the owner of the option pays long-term capital gains tax instead of ordinary income tax on the difference between the exercise price and the strike price. In general, the capital gains has a lower tax rate than ordinary income.

This isn’t quite as good as it sounds because the difference between the exercise price and the strike price is subject to the Alternative Minimum Tax (AMT). I don’t find this personally relevant since I prefer to sell employer stock as quickly as possible in order to be as diversified as possible, but if you’re interested in figuring out how the AMT affects your tax bill when you exercise ISOs, see this explanation for more details. For people in California, California also has a relatively poor treatment of capital gains at the state level, which also makes this difference smaller than you might expect from looking at capital gains vs. ordinary income tax rates.

Tax benefit of QSBS

There’s a certain class of stock that is exempt from federal capital gains tax and state tax in many states (though not in CA). This is interesting, but it seems like people rarely take advantage of this when eligible, and many startups aren’t eligible.

Tax benefit of other options

The IRS says:

Most nonstatutory options don't have a readily determinable fair market value. For nonstatutory options without a readily determinable fair market value, there's no taxable event when the option is granted but you must include in income the fair market value of the stock received on exercise, less the amount paid, when you exercise the option. You have taxable income or deductible loss when you sell the stock you received by exercising the option. You generally treat this amount as a capital gain or loss.

Valuations are bogus

One quirk of stock options is that, to qualify as ISOs, the strike price must be at least the fair market value. That’s easy to determine for public companies, but the fair market value of a share in a private company is somewhat arbitrary. For ISOs, my reading of the requirement is that companies must make “an attempt, made in good faith” to determine the fair market value. For other types of options, there’s other regulation which which determines the definition of fair market value. Either way, startups usually go to an outside firm between 1 and N times a year to get an estimate of the fair market value for their common stock. This results in at least two possible gaps between a hypothetical “real” valuation and the fair market value for options purposes.

First, the valuation is updated relatively infrequently. A common pitch I’ve heard is that the company hasn’t had its valuation updated for ages, and the company is worth twice as much now, so you’re basically getting a 2x discount.

Second, the firms doing the valuations are poorly incentivized to produce “correct” valuations. The firms are paid by startups, which gain something when the legal valuation is as low as possible.

I don’t really believe that these things make options amazing, because I hear these exact things from startups and founders, which means that their offers take these into account and are priced accordingly. However, if there’s a large gap between the legal valuation and the “true” valuation and this allows companies to effectively give out higher compensation, the way stock option backdating did, I could see how this would tilt companies towards favoring options.


Even if employees got the same class of stock that VCs get, founders would retain less control if they transferred the equity from employees to VCs because employee-owned equity is spread between a relatively large number of people.


This answer was commonly given to me as a non-cynical reason. The idea is that, if you offer employees options and have a clause that prevents them from selling options on a secondary market, many employees won’t be able to leave without walking away from the majority of their compensation. Personally, this strikes me as a cynical reason, but that’s not how everyone sees it. For example, Andreessen Horowitz managing partner Scott Kupor recently proposed a scheme under which employees would lose their options under all circumstances if they leave before a liquidity event, supposedly in order to help employees.

Whether or not you view employers being able to lock in employees for indeterminate lengths of time as good or bad, options lock-in appears to be a poor retention mechanism — companies that pay cash seem to have better retention. Just for example, Netflix pays salaries that are comparable to the total compensation in the senior band at places like Google and, anecdotally, they seem to have less attrition than trendy Bay Area startups. In fact, even though Netflix makes a lot of noise about showing people the door if they’re not a good fit, they don’t appear to have a higher involuntary attrition rate than trendy Bay Area startups — they just seem more honest about it, something which they can do because their recruiting pitch doesn’t involve you walking away with below-market compensation if you leave. If you think this comparison is unfair because Netflix hasn’t been a startup in recent memory, you can compare to finance startups, e.g. Headlands, which was founded in the same era as Uber, Airbnb, and Stripe. They (and some other finance startups) pay out hefty sums of cash and this does not appear to result in higher attrition than similarly aged startups which give out illiquid option grants.

In the cases where this results in the employee staying longer than they otherwise would, options lock-in is often a bad deal for all parties involved. The situation is obviously bad for employees and, on average, companies don’t want unhappy people who are just waiting for a vesting cliff or liquidity event.

Incentive alignment

Another commonly stated reason is that, if you give people options, they’ll work harder because they’ll do well when the company does well. This was the reason that was given most vehemently (“you shouldn’t trust someone who’s only interested in a paycheck”, etc.)

However, as far as I can tell, paying people in options almost totally decouples job performance and compensation. If you look at companies that have made a lot of people rich, like Microsoft, Google, Apple, and Facebook, almost none of the employees who became rich had an instrumental role in the company’s success. Google and Microsoft each made thousands of people rich, but the vast majority of those folks just happened to be in the right place at the right time and could have just as easily taken a different job where they didn't get rich. Conversely, the vast majority of startup option packages end up being worth little to nothing, but nearly none of the employees whose options end up being worthless were instrumental in causing their options to become worthless.

If options are a large fraction of compensation, choosing a company that’s going to be successful is much more important than working hard. For reference, Microsoft is estimated to have created roughly 10^3 millionaires by 1992 (adjusted for inflation, that's $1.75M). The stock then went up by more than 20x. Microsoft was legendary for making people who didn't particularly do much rich; all told, it's been estimated that they made 10^4 people rich by the late 90s. The vast majority of those people were no different from people in similar roles at Microsoft's competitors. They just happened to pick a winning lottery ticket. This is the opposite of what founders claim they get out of giving options. As above, companies that pay cash, like Netflix, don’t seem to have a problem with employee productivity.

By the way, a large fraction of the people who were made rich by working at Microsoft joined after their IPO, which was in 1986. The same is true of Google, and while Facebook is too young for us to have a good idea what the long-term post-IPO story is, the folks who joined a year or two after the IPO (5 years ago, in 2012) have done quite well for themselves. People who joined pre-IPO have done better, but as mentioned above, most people have diminishing returns to individual wealth. The same power-law-like distribution that makes VC work also means that it's entirely plausible that Microsoft alone made more post-IPO people rich from 1986-1999 than all pre-IPO tech companies combined during that period. Something similar is plausibly true for Google from 2004 until FB's IPO in 2012, even including the people who got rich from FB's IPO as people who were made rich by a pre-IPO company, and you can do a similar calculation for Apple.

VC firms vs. the market

There are several potential counter-arguments to the statement that VC returns (and therefore startup equity) don’t beat the market.

One argument is, when people say that, they typically mean that after VCs take their fees, returns to VC funds don’t beat the market. As an employee who gets startup options, you don’t (directly) pay VC fees, which means you can beat the market by keeping the VC fees for yourself.

Another argument is that, some investors (like YC) seem to consistently do pretty well. If you join a startup that’s funded by a savvy investors, you too can do pretty well. For this to make sense, you have to realize that the company is worth more than “expected” while the company doesn’t have the same realization because you need the company to give you an option package without properly accounting for its value. For you to have that expectation and get a good deal, this requires the founders to not only not be overconfident in the company’s probability of success, but actually requires that the founders are underconfident. While this isn’t impossible, the majority of startup offers I hear about have the opposite problem.


This section is an update written in 2020. This post was originally written when I didn't realize that it was possible for people who aren't extremely wealthy to invest in startups. But once I moved to SF, I found that it's actually very easy to invest in startups and that you don't have to be particularly wealthy (for a programmer) to do so — people will often take small checks (as small as $5k or sometimes even less) in seed rounds. If you can invest directly in a seed round, this is a strictly better deal than joining as an early employee.

As of this writing, it's quite common for companies to raise a seed round at a $10M valuation. This meeans you'd have to invest $100k to get 1%, or about as much equity as you'd expect to get as a very early employee. However, if you were to join the company, your equity would vest over four years, you'd get a worse class of equity, and you'd (typically) get much less information about the share structure of the company. As an investor, you only need to invest $25k to get 1 year's worth of early employee equity. Morever, you can invest in multiple companies, which gives you better risk adjusted return. At rates big companies are paying today (mid-band of perhaps $380k/yr for senior engineer, $600k/yr for staff engineer), working at a big company and spending $25k/yr investing in startups is strictly superior to working at a startup from the standpoint of financial return.


There are a number of factors that can make options more or less valuable than they seem. From an employee standpoint, the factors that make options more valuable than they seem can cause equity to be worth tens of percent more than a naive calculation. The factors that make options less valuable than they seem do so in ways that mostly aren’t easy to quantify.

Whether or not the factors that make options relatively more valuable dominate or the factors that make options relatively less valuable dominate is an empirical question. My intuition is that the factors that make options relatively less valuable are stronger, but that’s just a guess. A way to get an idea about this from public data would be to go through through successful startup S-1 filing. Since this post is already ~5k words, I’ll leave that for another post, but I’ll note that in my preliminary skim of a handful of 99%-ile exits (> $1B), the median employee seems to do worse than someone who’s on the standard Facebook/Google/Amazon career trajectory.

From a company standpoint, there are a couple factors that allow companies to retain more leverage/control by giving relatively more options to employees and relatively less equity to investors.

All of this sounds fine for founders and investors, but I don’t see what’s in it for employees. If you have additional reasons that I’m missing, I’d love to hear them.

_If you liked this post, you may also like this other post on the tradeoff between working at a big company and working at a startup.

Appendix: caveats

Many startups don’t claim that their offers are financially competitive. As time goes on, I hear less “If you wanted to get rich, how would you do it? I think your best bet would be to start or join a startup. That's been a reliable way to get rich for hundreds of years.” and more “we’re not financially competitive with Facebook, but ... ”. I’ve heard from multiple founders that joining as an early employee is an incredibly bad deal when you compare early-employee equity and workload vs. founder equity and workload.

Some startups are giving out offers that are actually competitive with large company offers. Something I’ve seen from startups that are trying to give out compelling offers is that, for “senior” folks, they’re willing to pay substantially higher salaries than public companies because it’s understood that options aren’t great for employees because of their timeline, risk profile, and expected value.

There’s a huge amount of variation in offers, much of which is effectively random. I know of cases where an individual got a more lucrative offer from a startup (that doesn’t tend to give particular strong offers) than from Google, and if you ask around you’ll hear about a lot of cases like that. It’s not always true that startup offers are lower than Google/Facebook/Amazon offers, even at startups that don’t pay competitively (on average).

Anything in this post that’s related to taxes is U.S. specific. For example, I’m told that in Canada, “you can defer the payment of taxes when exercising options whose strike price is way below fair market valuation until disposition, as long as the company is Canadian-controlled and operated in Canada”.

You might object that the same line of reasoning we looked at for options can be applied to RSUs, even RSUs for public companies. That’s true, although the largest downsides of startup options are mitigated or non-existent, cash still has significant advantages to employees over RSUs. Unfortunately, the only non-finance company I know of that uses this to their advantage in recruiting is Netflix; please let me know if you can think of other tech companies that use the same compensation model.

Some startups have a sliding scale that lets you choose different amounts of option/salary compensation. I haven't seen an offer that will let you put the slider to 100% cash and 0% options (or 100% options and 0% cash), but someone out there will probably be willing to give you an all-cash offer.

In the current environment, looking at public exits may bias the data towards less sucessful companies. The most sucessful startups from the last couple generations of startups that haven't exited by acquisition have so far chosen not to IPO. It's possible that, once all the data are in, the average returns to joining a startup will look quite different (although I doubt the median return will change much).

BTW, I don't have anything against taking a startup offer, even if it's low. When I graduated from college, I took the lowest offer I had, and my partner recently took the lowest offer she got (nearly a 2x difference over the highest offer). There are plenty of reasons you might want to take an offer that isn't the best possible financial offer. However, I think you should know what you're getting into and not take an offer that you think is financially great when it's merely mediocre or even bad.

Appendix: non-counterarguments

The most common objection I’ve heard to this is that most startups don’t have enough money to pay equivalent cash and couldn’t raise that much money by selling off what would “normally” be their employee option pool. Maybe so, but that’s not a counter-argument — it’s an argument that the most startups don’t have options that are valuable enough to be exchanged for the equivalent sum of money, i.e., that the options simply aren’t as valuable as claimed. This argument can be phrased in a variety of ways (e.g., paying salary instead of options increases burn rate, reduces runway, makes the startup default dead, etc.), but arguments of this form are fundamentally equivalent to admitting that startup options aren’t worth much because they wouldn't hold up if the options were worth enough that a typical compensation package was worth as much as a typical "senior" offer at Google or Facebook.

If you don't buy this, imagine a startup with a typical valuation that's at a stage where they're giving out 0.1% equity in options to new hires. Now imagine that some irrational bystander is willing to make a deal where they take 0.1% of the company for $1B. Is it worth it to take the money and pay people out of the $1B cash pool instead of paying people with 0.1% slices of the option pool? Your answer should be yes, unless you believe that the ratio between the value of cash on hand and equity is nearly infinite. Absolute statements like "options are preferred to cash because paying cash increases burn rate, making the startup default dead" at any valuation are equivalent to stating that the correct ratio is infinity. That's clearly nonsensical; there's some correct ratio, and we might disagree over what the correct ratio is, but for typical startups it should not be the case that the correct ratio is infinite. Since this was such a common objection, if you have this objection, my question to you is, why don't you argue that startups should pay even less cash and even more options? Is the argument that the current ratio is exactly optimal, and if so, why? Also, why does the ratio vary so much between different companies at the same stage which have raised roughly the same amount of money? Are all of those companies giving out optimal deals?

The second most common objection is that startup options are actually worth a lot, if you pick the right startup and use a proper model to value the options. Perhaps, but if that’s true, why couldn’t they have raised a bit more money by giving away more equity to VCs at its true value, and then pay cash?

Another common objection is something like "I know lots of people who've made $1m from startups". Me too, but I also know lots of people who've made much more than that working at public companies. This post is about the relative value of compensation packages, not the absolute value.


Thanks to Leah Hanson, Ben Kuhn, Tim Abbott, David Turner, Nick Bergson-Shilcock, Peter Fraenkel, Joe Ardent, Chris Ball, Anton Dubrau, Sean Talts, Danielle Sucher, Dan McKinley, Bert Muthalaly, Dan Puttick, Indradhanush Gupta, and Gaxun for comments and corrections.