How valuation and dilution works: a guide for the mathematically inclined
We tend to read articles on the internet that just offer formulas or little explanation of how your percentage in the company changes across the different funding rounds. In this post, we will explain how the mechanism works and give simple examples to help you understand how your percentage in the company changes as you go through the different funding rounds.
1. Intro
Imagine there are m outstanding shares in a company and this company is valued at $x. As outstanding shares, we refer to all shares held by institutional investors and restricted shares owned by the company’s officers and insiders. For simplicity, let’s assume that you are the only stakeholder, so you own 100% of the company. If we denote the price per share (pps) as $p, you can calculate it as:
In other words, it is the current valuation of the company divided by the total number of outstanding shares. For example, if there are 500,000 outstanding shares and your company is valued at $1,000,000, the price per share (pps) is p = 1000000 / 500000 = $2.
2. Pre-money vs post-money valuation
Pre-money valuation is the valuation of the startup before it begins to receive any investments. This valuation is subjective and is mainly determined based on the company's potential. Some factors are the company’s financials, whether it already has clients/revenue, comparable exits in the market, the makeup of the founders and team, how many investors want in on a deal, and so on. Pre-money valuation dictates the price per share (pps) $p of a startup and the ownership stake of an investor based on the capital they put in.
Post-money valuation is the valuation of the company after the investor has put in some capital (at the agreed-upon price per share (pps) $p). For example, if the pre-money valuation is $x and the investor puts in $y capital, the post-money valuation will be $x+y. The logic is pretty simple. The company was considered to be worth $x before the investment. Now, an investor has injected $y capital, so the company is worth $x+y.
2.1 How does your percentage change in post-money valuation?
The investor has injected $y capital and the company is worth $x+y. So, their percentage is y / (x+y). It follows that your percentage as a founder becomes 1 — y / (x+y) = x / (x+y). For example, assume your company is valued pre-money at x = $1,000,000. The investor puts in y = $200,000 capital. The post-money valuation of the company becomes $1,000,000 + $200,000 = $1,200,000. The investor would own $200,000 / $1,200,000 = 16.67% of the company, and you will own 83.33% of the company.
2.2 How do you calculate a company’s post-money valuation from the investor’s capital and percentage of ownership?
You can calculate a company’s post-money valuation by knowing the amount of capital an investor has put in and their percentage of ownership. If an investor puts in $y capital, which corresponds to a percentage of the business, the post-money valuation would be y / a. If we denote a company’s post-money valuation by $z and the investor owns a percentage of the business (post-money) by injecting $y capital, the post-money valuation $z is:
For example, if the investor contributes y = $200,000, and this $200,000 corresponds to 16.67% (=1/6) of the business (post-money), then what value corresponds to 100% of the business? We simply divide $200,000 by 1/6: $200,000 / 0.1667 = $1,200,000.
3. How does funding work
What is really happening when an investor wants to give you some capital for a percentage of your business? You first have to agree on the price per share (pps) $p. Then, based on the amount the investor wants to invest, you issue the corresponding number of new shares (at $p price per share). In general, if an investor invests $y and the price per share is $p, the investor will receive n = y / p shares.
Imagine, the company is worth $x pre-money and there are m outstanding shares. This means the price per share is p = x/m. Borrowing the example from the intro, if we have m=500,000 shares and the pre-money valuation is x=$1,000,000, the price per share (pps) will be p = $2. Now, if an investor wants to come in, the company needs to issue new shares for $2 per share. If the investor invests y = $200,000, this means that the investor will receive 200000 / 2 = 100,000 shares for their investment.
3.1 Why your ownership is reduced after funding and is this a good thing?
Your ownership in the company would be reduced because your number of shares didn’t change, but now the total number of shares has increased since you had to issue new shares for the investor. In the example, we just showed, you initially had m=500,000 shares and issued n=100,000 new shares for the investor. So, your percentage was reduced from 100% to 83.33% (=500,000/600,000). Why does it even make sense to issue new shares and have your ownership decrease? It has to do with the “pie analogy”. You basically go from owning all of a small pie to a smaller piece of a (hopefully) much larger pie. When the investor puts in some capital, this capital can go on to acquire new hardware, new equipment, help you achieve a more efficient sales campaign, and hire more engineers. You can really accelerate the growth (and potentially revenue) of your company, which would subsequently increase its valuation in the next financing rounds.
3.2 How many shares do you need to issue?
Most times, we don’t know the price per share $p. What we do know is the number of outstanding shares before investment and the percentage of the investor post-money. That’s enough to calculate the number of shares n that need to be issued for the investor. Continuing our example, assume an investor puts in $y and the post-money valuation is $x+y. This means the investor’s ownership post-money would be y / (x+y). For instance, if y = $200,000 and x+y=$1,200,000, the investor’s percentage in the business is 16.67%. This means that you as a founder now have 83.33% of the business (as a reminder before investment you owned 100%, but you now own 83.33%). Before investment, you had 500,000 shares and with these 500,000 shares, you now own 83.33% (=5/6) of the business. So, if 500,000 shares correspond to 5/6 of the business, how many shares correspond to 100% of the business? The answer is 500,000 / (5/6) = 600,000. Since you already own 500,000, the remaining 600,000–500,000 = 100,000 shares will go to the investor. You see that we arrived at the same conclusion as in the previous paragraph.
In general, if the investor owns percentage a of the business, this means you own 1-a of the business (assuming you owned 100% of the business before the financing round). If you own m shares, this means that after the issuance of new shares for the investor, the total number of shares should be m / (1-a) and the investor should receive:
number of shares. For example, if there were m=500,000 outstanding shares before the financing round, and the investor put in $200,000 capital on a $1,200,000 post-money valuation, this means that their percentage in the business is a = 16.67% (=1/6). So, from the above formula, we conclude that we need to issue this number of shares to the investor.
Of course, we arrived at the same result as in our example at the beginning of Section 3, where we assumed that we knew the price per share (pps) $p.
4. Dilution
The dilution is the percentage that is left to you (or in general all existing founders/officers/investors) after the new investment. If you own 100% of the company pre-money, the pre-money valuation is $x, and the post-money is $x+y, dilution would be x / (x+y). This means that the investor would have put in $y capital and their percentage in the business would be a = y / (x+y). For example, if the investor puts in y = $200,000 and post-money valuation is $1,200,000, the investor’s percentage would be a = 16.67%. Similarly, if you own m shares and after the investment the total number of shares is m+n, dilution would be m / (m+n).
The dilution rate is defined as:
You should be getting the same result one or the other way because the valuation and number of shares are connected through the price per share $p as follows (remember $x and $x+y is the pre-money and post-money valuation respectively, m is the number of outstanding shares and n the number of newly issued shares for the investor):
In other words, the valuation of a company is the price per share times the total number of outstanding shares.
So:
4.1 How does your percentage change across funding rounds
If you start with a percentage r in the business, in the seed stage the dilution rate is 1-a1, in Series A the dilution rate is 1-a2, in Series B the dilution rate is 1-a3, and in Series C the dilution rate is 1-a4, your percentage in the company after Series C would be:
How do we come up with this? Assume you started with m shares and the total number of outstanding shares (before any financing round) is initially m0 shares. In other words, your initial percentage in the company is r = m / m0. I.e., if you own 400,000 shares and the total number of outstanding shares is initially 500,000, your percentage would be r = 0.8.
If by the end of series C, there are m4 outstanding shares, your percentage would be:
We can rewrite the above as a function of the previous rounds. Imagine after the end of the seed stage there are m1 outstanding shares (including the newly issued shares), after the end of Series A m2 outstanding shares (including the newly issued shares), after the end of Series B m3 outstanding shares (including the newly issued shares), and after the end of Series C m4 outstanding shares (including the newly issued shares). Then, you can express your percentage as follows:
In other words, 1-a1 is the dilution rate: the ratio of outstanding shares m0 before the seed stage divided by the outstanding shares m1 after the seed stage. Similarly, for the other rounds.
Let’s say you started by owning r = 80% of the business, in the seed stage the investor receives a1=15% of the business (post-money), in Series A the investor receives a2=20%, in Series B the investor receives a3=20%, and Series C the investor receives a4=30%, you will end up with:
A breakdown of typical ranges for equity allocation is as follows:
Seed Stage: investors receive 10–25% of total equity
Series A: investors receive 20–30% of total equity
Series B and later: investors receive 15–25% of total equity
You can read more about this here.
As a general rule of thumb, you should multiply your initial percentage with a factor between 0.3 and 0.5 to calculate your final percentage (after all the financing rounds). Of course, this can differ greatly based on the circumstances in your particular case, so the above range is common but not guaranteed.
Summary
Let’s do a recap of how the whole mechanism works: You agree on a pre-money valuation with a potential investor. This pre-money valuation determines the price per share (pps). You simply divide the pre-money valuation by the total number of outstanding shares to calculate the price per share. An investor puts in some capital and this capital gives them a number of shares (based on the agreed-upon price per share). The post-money valuation is simply the sum of the pre-money valuation and the capital the investor put in. The percentage of the investor in the company will be their injected capital divided by the post-money valuation. The percentage of all previous founders+officers+investors would be 100 minus the percentage of the new investor. That’s what is also called the dilution rate. To calculate your percentage after all financing rounds you simply multiply your initial percentage in the business with the dilution rates from all subsequent rounds. You can use this calculator to determine your equity dilution after a single round of fundraising.
Cheatsheet
- Price per share $p based on the number of outstanding shares m and valuation $x: p = x / m.
- Post-money valuation $z based on pre-money valuation $x and investor’s capital $y: z = x + y.
- Post-money valuation $z based on investor’s capital $y and their percentage of post-money ownership a: z = y / a.
- Investor’s ownership a based on their investment $y and company’s post-money valuation $x + y: a = y / (x+y).
- Number of new shares n issued for the investor who will own a percent of the business with m outstanding shares (before the financing round): n = m ∙ a / (1–a).
- Dilution rate if the investor puts in capital $y on a $x+y post-money valuation: 1–a = x / (x + y).
- Final percentage r4 of an officer/employee/investor after four financing rounds with dilution rates 1–a1, 1–a2, 1–a3, 1–a4 and with initial percentage (of this officer/employee/investor) of r: r4 = r ∙ (1–a1) ∙ (1–a2) ∙ (1–a3) ∙ (1–a4).
