If someone buys 25% of your business for $2 million, it is easy to determine the value. If someone invests $10 million in your business for 50% it is a little bit tougher. How do you determine value if someone says, “I’ll pay you $ 2 million and I’ll invest another $10 million in the company and then we’ll both own 50%”?

### Equity Purchase

Let’s start with the easy example. Say you sell 25% of your company for $2 million. That means your company is worth $8 million, correct? Not much algebra there.

### Capital Investment: Classic Pre-money / Post-Money Valuation

Now let’s say that someone offers to invest $10 million in your company for 50% of the company. What is it worth then? Well, after the investment the company will be worth $20 million, but really the enterprise value of the company itself will be $10 million (the same value as before the deal), with $10 million of invested cash on the balance sheet. In venture-capital-speak, the pre-money valuation will be $10 million while the post-money valuation will be $20 million. Another way to look at it is that the owner did own 100% of a $10 million dollar company, and now owns half of $20 million company but he still owns $10 million of value.

### A Mix of Equity Purchase and Capital Investment

Now here is the tough one. We recently had a buyer make a compelling offer to our fast-growing client: “I’ll pay you $2 million and I’ll invest another $10 million in the company to fund growth and then we’ll both own 50%” (By the way, I changed the numbers from the actual deal). It wasn’t until later that the question came up: “Um, what does that make the company worth?”. It was an interesting deal because no one actually cared but the attorneys and accountants – the seller and buyer were happy with the statement above.

The first cut at this may be, “Well, the buyer just paid $12 million total for half the company, so it’s obviously worth $24 million”. In fact, that is exactly what the attorneys said. Those guys are really bright on legal matters.

However, the day after the close the buyer would own half the company and therefore half the balance sheet, which now has another $10 million in cash on it. In other words, he would have 50% of the company and rights to $5 million in excess cash, which theoretically he could take home. So really, he paid only $7 million for half because he still “owns” $5 million. That would indicate it is valued at $14 million. In venture-capital-speak the pre-money would be $14 million, and the post-money would be $24 million. Or another way of saying it, the enterprise value of the core business is $14 million, with $10 million of excess cash on the balance sheet. This method, however, is really a short cut and doesn’t use much math.

Taking it even further, what is really going on with the ownership of the company? The owner is selling some of the company to the buyer for $2 million, however the buyer is also being issued new shares for the $10 million investment they are making. At the end of it all, including dilution, they both own 50%. How much of the company is the seller selling, and how many new shares should be issued? Now this is deal algebra.

To work this out I set up some equations with variables like shares sold, new shares issued, 50% ownership, enterprise value, etc. and solve them simultaneously. To make it easy I assumed 1,000 shares existed in the company although that wasn’t actually the case. One key constraint is that the share price for the shares sold and the shares issued is the same. In other words, the buyer is buying some shares from the owner, and some from the company treasury but is paying the same per share price for both.

For this example the owner would sell (143 shares) 14.3% of his company to the buyer for $2 million and $14,000 per share. 714 new shares would be issued, also at $14,000 per share. This would mean both the buyer and seller own the same number of shares at 857 shares.

### The Actual Algebra – for Those that Care

For those that actually want to look at the math, here it is:

First, let’s assume 1,000 shares are originally outstanding. Then let’s say:

X = Shares sold by seller to buyer

Y = New shares issued to buyer

Z = Total amount of shares outstanding post-merger

P = Price per share

We know the buyer will own 50%, so I set up the equation:

*X+Y = Z/2*

We also know that the price per share (the same for both shares sold and newly issued shares) is:

*P = 2,000,000/X (or X=2,000,000/P)*

*P = 10,000,000/Y (or Y=10,000,000/P)*

Substituting these into the X+Y=Z/2 equation:

*2,000,000/P + 10,000,000/P = Z/2 * Or

*Z * P = $24,000,000*

Z is the total shares after the merger, so Z * P is the “post-money” valuation of $24 million. The enterprise value, or pre-money valuation, is always the post-money less the investment cash, so the pre-money would be $24 – $10 million or $14 million.

We also know that the pre-money valuation is the share price (which doesn’t change pre to post-money) times 1,000 shares:

*$14,000,000 = P * 1,000*

So* *

*P = $14,000 / share*

Using the equations above for X and Y in terms of P, we can easily calculate:

*X = 2,000,000 / P = 143 shares* are sold to the buyer (you can also think of this as 14.3% of the company is sold to the buyer for $2 million)

And

*Y = 10,000,000 / P = 714* new shares issued to the buyer for his investment of $10 million

A quick check is now the seller has 1,000 – 143 = 857 shares. The buyer now has 714 new shares + 143 shares he bought from the seller = 857 shares. Dang, it worked.

There are obviously numerous ways of solving this, but that is how I did it. I have some cap table spreadsheets I dug out, but none of them could handle both an equity purchase and investment at the same time, so it really did come down to algebra.

If you got this far, do me a favor. I would be curious if anyone actually got any use from this algebra exercise. If you did, drop me a note at ney [at] compasspointcapital.com and let me know.

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