ESOP and dilution maths

Published on Dec 23rd, 2022

Recently we were evaluating an investment into an early stage company. One of our requirements is that the company creates a VSOP, with enough shares that it represents 10% of the cap table after dilution.

This is a surprisingly interesting mathematical problem, but before getting into it I want to go other the maths of cap tables in general.

Let’s imagine a company, Acme Inc. whose cap table is as follows:

Acme Inc. decide to create a VSOP for their employees, by issuing 50 new shares, and diluting everyone else equally.

The cap table now looks like this:

So far so good.

Now a new investor comes along, and decides to invest £100,000 at a £1m pre-money valuation.

The current value of the shares are given by £1m/550 = £1,818 per share.

Therefore the investor’s £100,000 leads to 55 new shares.

We’re at 550 outstanding shares + 55 new shares, so 605 shares in total. The cap table is now:

We can see that all existing parties have been diluted, but the dilution isn’t even. If we take a slight detour to understand the dilution better:

So we see that the absolute change is largest for the founders, but proportionally its the same for each person. With a few definitions and some basic algebra we can come to formula for the proportional change:

We start with N shares, and there is a $\gamma$ percent increase in outstanding shares. $\gamma$ obviously scales with the amount of money being invested. After investment we have N’ shares.

So the original percentage and the percentage after investment for a particular party can be found quite easily:

Starting with those equations and doing some basic rearranging, we find that the proportional change can be found by:


So now we can quite easily work out how much each person will be diluted, based on the percentage of new shares being issued. This is a handy tool.

Now let’s go back to the original example. Acme Inc. is looking for investment, and their cap table looks like this:

I want to invest £300,000 at a £1m pre-money valuation. However, before I invest they need to create a VSOP, and the VSOP must be large enough that after the dilution from my investment it accounts for 10% of the table. So how big must the VSOP be?

Can we use our new formula to help us?

With an investment of £300k at a £1m pre, $\gamma =30\%$. The VSOP is 10% post dilution, and we know ${\%}_{pre}=(1+\gamma )\times {\%}_{post}$. So the VSOP pre-dilution represented 13% of the company.

We also know that before dilution:

where N is the total number of shares, and $N=n_{old}+n_v$.

We want to find $n_v$ and we know the other variables. Some algebra later and we get to:

Now let’s test it. We know $(1+\gamma )=1.3$, ${\%}_o=0.1$, and $n_o=500$. This gives us 74.7 shares. We’ll round up to 75 for completeness. This represents 10% of the company, so we should have 747 shares after the financing round. There were 500 existing, plus the 75 for the VSOP, so the financing round should create 172 shares. Does this work?

Trying our calculated values, our shares look like this:

And that leads to percentages like this:

That looks pretty good. Everyone is diluting, the VSOP is at 10% post investment, and the points are adding up to 100%. Let’s just make sure the proportional and actual changes are still working.

And then proportionally:

Perfect! Now these are calculations which VCs do very frequently. However, this is often done with Excel and tools like Goal Seek, or just playing around till the numbers balance. I’ve not seen an algebraically driven approach yet, and this seems like a more elegant way of doing it (to me at least).

I’m going to leave this here for tonight, but in future I’d like to revisit this and in particular look at the cumulative proportional change for the founders, and see if we can predict that based on the amount being invested and the VSOP requirements.

Update: I’ve since built a liquidation preference calculator that models startup liquidation waterfalls with pre-seed and seed investor tiers. It calculates whether preferred shareholders should convert to common stock or take their liquidation preference at various exit values — implementing the standard “convert vs. take LP” decision logic. The code is a single Python class and could be a useful companion to the maths above.