### A Share Is Meaningless

March 10th, 2014
I was recently talking to a friend who was trying to figure out how much their stock options were worth. When joining the company they got options for `X`, and looking up similar companies that were publicly traded they saw they had prices averaging \$`Y`/share. They thought their company might have a one in four chance of growing to the size of these competitors, so they estimated they had options for stock that might be worth \$`X*Y/4`. Unfortunately this approach doesn't work at all, because "a share" doesn't tell you anything unless you know the total number of shares. One company may have fifty thousand shares while another has ten million, and if they're otherwise identical companies you'd much rather have a share of the former.

Thinking in percentages can help make this clear: if there are fifty thousand shares each one is 0.002% of the company while if there are ten million each one is only 0.00001%. If a company is publicly traded you can just look at the market price of a share, but to estimate the price of a share from what similar companies are worth you have to know how much of the company you're talking about. This means that the first thing you need to do when someone offers you stock options is ask how many shares there are total.

Once we know how much of the company we're talking about we can redo their calculation. Say the shares represent `Z`%of the company and competitors are worth various amounts averaging \$`W`. They still guess a 1:4 chance of success, so now we have \$`Z*W/4`.

(This still isn't quite right, even assuming you're risk neutral, because of strike prices. Imagine I'm trying to evaluate options for 0.1% of two companies A and B that I think both have an expected value of \$10M. My options are for stock worth 0.1% of \$10M in both cases, or \$10k. But the way standard employee stock options work is you have to pay a certain price to turn them from options into shares of stock. Let's say for my options on both A and B that's \$4k. You might think then that my options are worth \$6k in both cases, but it actually depends on the distribution behind my expected value estimate. Let's say A is certain to succeed and will be worth exactly \$10M when it does, while B has a 1% chance of being worth \$1B and a 99% chance of being worth nothing. The options for A are still worth \$6k, but the options for B are worthless 99% of the time and worth \$1B less the \$4k strike price 1% of the time for an expected value of \$9.96k. The key is that you only pay the strike price if your options become worth something, so you need to subtract strike prices before averaging over the distribution. Which means the more uneven the distribution the lower the effective strike price. [1] Most people are risk averse, though, which is a force even stronger the other way.)

[1] Unless you expect to leave the company before you learn whether the options are worth it, in which case you'll have to decide then, losing this benefit of having an option.

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