Investor Paradox. How can you calculate how much is a company worth when buying part of it changes the total value?

You are confusing investment in the company itself with purchasing shares of the company from another person.

In your example, the company has $1 worth of assets, and you have 100% of the company.

When your friend wants to invest his own $1, there are two ways he can do this. He can purchase the entire company from you outright, in which case you now have 0% and he has 100% of the $1 company, but you personally now have his $1 in cash. Or, you can form a partnership, where his $1 of assets are merged with the $1 of assets in your company, and you now each have 50% equity in the company that now has $2 in assets.

Your company wasn't worth $2 until after his contributed assets were included.

The concept is know as "enterprise value," "firm value" or "pre-money" in different contexts. It is the market value of the company minus the company's cash plus the company's debt.

In your example, the enterprise value of the company is $1 before the deal is made and is also $1 after the deal is made. $2 market value minus $1 company cash equals $1 enterprise value.

tl;dr: You asked: Can someone help me understand this? No, because you understand it perfectly already! From a purely mathematical point of view on its own, the sharks are indeed ripping them off. (But it's probably still worth it!)

Details: Wow. After reading your question, I just watched 2 episodes of Shark Tank to be sure, and you're absolutely correct. The Sharks got the valuation wrong every time a number was spoken. The company owner makes a request, such as:

I'm seeking $300K in return for a 30% stake in my company.

Then the sharks will respond with:

"You're valuing your company at one million? Are you out of your mind?"

But the reality is in this case the owner is actually only valuing their business at $700K.

Similarly, I just watched one person who was seeking $250K for 10% equity. So that means he feels his company is worth $2.25M before the investment. He ended up accepting an offer of $300K for 20% equity. Right before he accepted it, he thought out load and said "$1.5M...you've got a deal." But the valuation they gave him was actually only $1.2M.

It appears the Sharks are "cheating" and aren't trying to hide it, and everyone seems to just go along with it. (Surely the Sharks know this and are taking advantage of it.)

So, you're correct that from a valuation standpoint the business owners on Shark Tank are getting "ripped off". However, in the case of Shark Tank specifically, they are not silent investors, and in fact are likely worth far more to the business than the money they provide. It wouldn't surprise me if some businesses could simply offer 10% equity for free to a specific Shark, and come out way ahead as a result.

Update: in case I'm explaining this poorly, here is some further reading that might make it clearer:

Why Does Shark Tank Lie About Valuation?

Too Many Entreprenuers Don't Understand Valuation on this Show

Too long for a comment, so here:

The main source of confusion here is arguably the undefined expression "join forces". Under one interpretation, you sell a share of your firm to a friend at a value you accept uncoerced: there is no effect on firm value; only an effect on your portfolio: more cash, fewer shares. Under another interpretation, you convince a friend to bring capital into the firm in exchange for shares: now the firm's value changes and, when and if this change in value is reflected in share prices, so will your portfolio (same amount of cash, fewer but more valuable shares).

Bear in mind that the company's value and your portfolio's value cannot be confounded. Also, not all share prices are determined by market forces: if you sell your shares at a price you consider too low (a friend's price), that's a redistribution of assets from you to your friend.

Bottom line: do not confuse business assets and personal assets and do not confuse business partner and friend.

The important thing to always remember is that you must maintain a separation in your mind between you as individual(s) and the company. Think of the company as a box. Inside that box are assets (e.g. cash, real estate, income-generating contracts) and maybe liabilities (e.g. unpaid bills, loans). The value of the box is what someone else (e.g. someone participating in the stock market) is willing to pay for the combination of everything that is inside the box. If the assets are worth $2 and the liabilities are $1 then the box is worth $1.

A share represents ownership of a part of the box. In practice there are different types of shares with different types of rights but let's assume we are talking "ordinary shares" which give full and proportionate rights to the portion of the company that they represent.

Let's say your company starts with 100 shares, assets of $2, and liabilities of $1 (so that it is worth $1 altogether):

                                  ---------------------
Daniel --- 100 shares (100%) ---> | Assets      = $2  |
                                  | Liabilities = $1  |
                                  | Value       = $1  |
                                  ---------------------

In this case, your 100% ownership of the company is worth $1.

Now your friend approaches you and says "I want to put $1 into the company and I want us to own 50% of the company each". What happens is that the company issues an additional 100 shares to your friend, and he pays $1 in to the company in return for those shares. That $1 goes into the box:

                                  ---------------------
Daniel --- 100 shares (50%) ----> | Assets      = $3  |
                                  | Liabilities = $1  |
Friend --- 100 shares (50%) ----> | Value       = $2  |
                                  ---------------------

You now each own 50% of a company which is worth $2, so your shares are worth $1 just as before.

"But this can be seen as me selling 50% of my company for 1 USD (which will mean my company was worth 2 USD, instead of 1)"

The reason that this is wrong is because at no point did you "sell" 50% of your company to your friend. You still have the 100 shares that you owned before. Nothing was sold. Instead, the company was increased in size and new shares were created for your friend which represent the increased "part" of the company. You can think of this new part as being an extra box with $1 inside it which your friend "glued" on to the original box to make a bigger box:

                                  -----------------------------
Daniel --- 100 shares (50%) ----> | Original company = $1     |
                                  -----------------------------
Friend --- 100 shares (50%) ----> | New part of company = $1  |
                                  -----------------------------

The value of your friend's shares represents the new part that they added by investing the $1 into the company, and your shares represent the $1 that was already in the company.

"Owner: My company is worth 1 USD. Rich Guy: Fair, I´ll buy 50% for 0.5 USD. We can use these 50 cents to buy inventory. The company is now worth 1.5 USD, but each didn´t contribute equally."

Normally in this type of scenario the Rich Guy really means he will invest $0.50 into the company (in which case the scenario above applies), so the word "buy" in your sentence is incorrect. However, let's pretend that instead of the original scenario, your friend will in fact buy 50% of the company for $0.50. "Buy" means that he will buy it from you instead of investing into the company. Now the new diagram looks like this:

                                  ---------------------
Daniel --- 50 shares (50%) -----> | Assets      = $2  |
                                  | Liabilities = $1  |
Friend --- 50 shares (50%) -----> | Value       = $1  |
                                  ---------------------

Here, instead of your friend getting 100 new shares (and you keeping your original 100 shares) as happened in the first scenario, you've given 50 of your shares to him. Notice that in this new scenario the company is worth exactly what it was when you started ($1). That's because your friend did not put any money into the company. Instead, he gave you $0.50. That money is in your pocket, not inside the box. He gave you $0.50 and now you own shares worth $0.50 and he owns shares worth $0.50. The part where you say "The company is now worth 1.5 USD, but each didn´t contribute equally." is wrong because the company is not worth 1.5 USD.

If you want to take Rich Guy's suggestion and use the $0.50 to buy new inventory, then because that $0.50 is in your pocket, you will need to lend it to the company so that it can buy the new inventory. In that case the company's value is exactly the same as before. That's because its assets (in the form of cash) have gone up by $0.50, but its liabilities (in the form of money owed to you) have also gone up by $0.50, resulting in no net change:

                                  -----------------------
Daniel --- 50 shares (50%) -----> | Assets      = $2.50 |
                                  | Liabilities = $1.50 |
Friend --- 50 shares (50%) -----> | Value       = $1    |
                                  -----------------------

Scenarios of the first type (where money is invested into a company) is known as equity financing and usually takes the form of shares. Scenarios of the second type (where money is lent to the company) is known as debt financing and usually takes the form of loans or bonds. They are both common methods but are different to each other, and you need to be clear which you are using if you want to understand the resulting valuations of your stake and the company.

The other comments have explained the accounting valuations of investments in large, steady businesses, but I want to a perspective that's more specific to Shark Tank.

Let's say a shark accepts 30% of a business for $300k. The business is worth $1M after, so it was worth $700k before, right? Not so fast.

The sharks are not only trying to value their share, they are trying to value the potential of the business. They are thinking: if everything goes well, this business will be worth more than $1M plus interest, and I'll own 30% of it. They are not thinking: if I give this business $300k, then since it's already worth $700k, it will be worth $1M.

That line of thinking works for established businesses. If you invest $100k into a restaurant that's worth $100k already, maybe it can afford to open another location, and then make twice as much money, so you'd own 50% of a twice-as-big business. But we're not talking about established businesses, we're talking about startups. That investment may be the difference between the business being worth $1M, and being worth zero.

Businesses work by spending money on things, and earning money from other things, and for our purposes, those two numbers have no relation whatsoever. Spending money doesn't mean you'll make money. You spend money on employees, premises, equipment, electricity, inventory, and so on. You earn money by selling, let's say, robot vacuum cleaners. You hope you can make $1M of robot vacuum cleaners, but for that you need all of the above. If you can't get all of the above, you can't even make two (you made one already in your garage as a prototype).

Sure, if your business is worth $10k as a concept (because you have a prototype and a few pre-orders) and then a shark gives you $300k of cash, it's now technically worth $310k. But then you go out and spend $300k hiring employees and setting up a factory and buying parts. Now your business is worth $20k as a concept (because people are more willing to invest now that you have a factory), plus the theoretical value of the factory and parts. Probably still less than $310k because you lost money on all those things. But then you turn it on, and start churning out robot vacuum cleaners. And here's the magical part - the robot vacuum cleaner is worth more than the sum of its parts. Now your business is worth $100k as a concept (it's a lot more attractive now that you've actually made a bunch), plus the theoretical value of the factory, plus the theoretical value of all the robot vacuum cleaners sitting in boxes ready to sell. And then you sell them, and now you have a bunch of money - more than the cost of the parts and employees in that time. And then you repeat.

The key point here, is that without that original $300k, you wouldn't have a factory at all. You'd have no employees. You'd just be a guy with a prototype robot vacuum cleaner, which is worth something, but not nearly as much as a guy with a robot vacuum cleaner factory that's actively running.

The sharks are trying to gauge the value of your business after you build the factory. If paying $300k enables you to create a $1M business, that's worth 30%, even if the current value is nearly zero. Of course, they'll try to get 40% or 50% or 60% out of you if they can, using arguments like "I'm the one contributing all the money here" and "your business is currently valued at nearly zero".

I don't see an answer that gives a clear explanation of the difference between "pre-money" and "post-money" valuations, which is critical to understanding this kind of scenario.

Say I have a company, and I am told that it is worth 1 USD (100% of the company).

So far, so good.

My friend, who has 1 USD, asks me if we can join forces. He puts in 1 USD, I put in my company (1 USD), and we now each own 50% of the company.

Here we have an investment transaction -- let's talk about valuation. Your friend put in 1 USD, and he got 50% of the company, which implies that the company is now worth 2 USD in total. That is the "post-money valuation", which is how much the company is worth after the investment. From this we can also compute a "pre-money valuation": if we subtract the invested 1 USD, we can see that the company was worth 1 USD before the investment. Everything seems good.

We both put in the same amount, we now each own the same amount. But this can be seen as me selling 50% of my company for 1 USD (which will mean my company was worth 2 USD, instead of 1).

Yep, that's correct. Selling 50% for 1 USD implies that the whole company is worth 2 USD. This in turn implies that the company BEFORE the investment was worth 1 USD (the "pre-money valuation", as discussed above.) That happens to be the same amount that you thought your company was worth before the investment happened, but that DOES NOT have to be true. An investment is a negotiation, and "how much the company is worth" is part of what is being negotiated.

Let's say you instead agreed to sell your friend 50% of the company for 2 USD. Then the post-money valuation is 4 USD, and the pre-money valuation is 2 USD. That means you effectively agreed, as part of the investment, to consider "your end" of the deal -- that is, the existing value of the company -- as being worth 2 USD. You say "I am told that is is worth 1 USD", but that's assuming that the "value" of a company is some fixed well-known fact. In reality, you might know that the company was worth 1 USD at some point in the past, but you are hoping to make that value grow over time, so it's TOTALLY NORMAL that the "pre-money valuation" of an investment in a company will be MORE than what the company was worth the last time you checked.

Does this make sense? My reasoning is that in e.g. Shark Tank, the rich guys value the company in 1 USD, they "buy" a percentage of the company at that valuation, but that money goes into the company instead of the owner. Doesn't this mean that the owner is getting ripped off?

You can look at the transaction several different ways, but in the end it should always be the case that the owner is getting the same amount of value as they are trading away. Let's look at your example.

Owner: My company is worth 1 USD.

Rich Guy: Fair, I'll buy 50% for 0.5 USD. We can use these 50 cents to buy inventory.

The company is now worth 1.5 USD, but each didn't contribute equally.

If the rich guy buys 50% of the company for 0.5 USD, that implies that the whole company is worth 1 USD ("post-money valuation"), and therefore the company before his investment must have been worth 0.5 USD ("pre-money valuation"). If the owner agrees to this, that means accepting that the company was only worth 0.5 USD before the investment -- that's part of the deal. If the company was previously worth 1 USD, this would be what's called a "down round" in startup investing -- the company has lost value over time, which you really don't want your company to do!

I don't know whether your specific example from Shark Tank was indeed a 'down round', or whether the entrepreneur misstated what the company was worth, or whether the sharks are saying false/misleading things about valuation (as another answer claims), or whether they merely explained it poorly for television.

"The simplest possible answer."

I have thought a LOT about the pedagogy here and I am now attempting to put in the simplest possible answer.

Say Mr. Wonderful does a transaction, he writes a cheque for $1m and he now owns 20% of the company.

Is the million bucks in the company?

Friends! For now, let us say that the $1m in fact goes in to the company bank account.

(That is nonsensical, but for now assume that.)

Let me repeat, I am theoretically adopting this outcome so we can see what happens.

Friends! For now, let us say that the $1m in fact goes in to the company bank account.

So now Biff owns 80%, Mr.W owns 20%, and indeed - the million bucks is in the company bank account.

So! What is the company "valued" at?

It is valued at ...................

and this is so easy ....................

Five million bucks.

It's just that easy.

A company having a certain amount of cash in the bank account adds, or subtracts, value to the company. If credit is trivial and cheap for the company, it's a bad thing. If it's a startup owned by someone with a known record of spending frugally, it's a positive. If it's a startup owned by a wastrel, it's a negative. The current company with the most money in their bank account is Apple, and analysts disagree on whether it's a good or a bad thing.

None of that matters.

The company now has a reasomable valuation, since:

Mr. Wonderful just paid $1m for 1/5th.

Note that Mr. Wonderful >> of course knew << that would be the state of the company when he bought his 1/5th.

Let's say Mr.W thought like this: "Hmm. I think this company will be worth $N million once this deal is put thtough ..." Then the deal goes through. Then Mr.W suddenly says ....

.... "Oh no! With my experience buying and selling companies I completely forgot that that money would be in the company! I didn't figure that in my estimates! Wow! What a bonus!!! It's worth more than I thought! I'm, such a dummy! Who would have thought of that?!"

Of course ... that is not the case.

Regardless of the cash sitting in the bank account, the company is (by definition of the most recent meaningful purchase) worth $5m. Again, in this post we are going along with the idea that the $1m in fact goes in to the company bank account. (Somehow or other .. whatever.) Obviously (if you subscribe to that) he and everyone knew that was the mechanism. So the value is $5m.

It's really that simple.

I think you are confusing a couple of things. How do you do the valuation of your company? If you sold 50% of your company for $1, then the buyer thought it was worth $2. You can sell part of your company for any amount that the buyer and seller agree on.

If you are doing a valuation of your company and it's only asset is $1 in petty cash, and your friend buys a 50% share for $1, then you have $1 and a 50% interest in the $1 in petty cash. You just screwed your friend out of $0.50.

If the company has $1 in petty cash to create the $1 valuation and your buddy invests $1 in the company for a 50% share, presumably the company would now have $2 in petty cash and you would each be 50% owners in the company.

In the real world you would hopefully come together and use the $2 to create more money. Invest in some Kool Aid and sell it on the street to thirsty people. Say you spend $1.50 and make $10, now your each have a 50% share of $10.50. You have increased the value of the company by $8.50. The $1 investment by your friend enabled you to buy the Kool Aid which your company couldn't afford before. Your company might have been Kool Aid based before, but maybe your friend saw an opportunity for selling your Kool Aid along the route for a walkathon that you didn't know about, so his expertise helped grow the company.

You could have invested another $0.50 yourself to afford the Kool Aid, but maybe you didn't have it or didn't want to take the risk yourself. Also you might not have had the knowledge to plan for the event.

The "worth" of something in econ 101 terms is some price between the price someone would sell it for and the price someone would buy it for.

For non-liquid investments (and non-commodities really), that gap tends to be large, and there are transaction costs -- learning enough about the product to determine if it is what you think it is -- that are significant.

And an entire startup business is a very non-commodity item.

Fundamentally, there is no one true "worth" or value for a business. You can use various schemes to approximate the value of a business; you can look at its liquid, and less liquid, assets, you can approximate its future revenue stream, you can examine how those values change with an injection or withdrawal of capital, etc etc.

In this case, the "worth" being referred to is the implied worth of the company after the capital injection (and any other support) by the Sharks.

A 300k investment for 30% means that the Shark valued the company with 300k of extra liquid assets, with the current people running it with certain contract terms, at 1 million dollars. It doesn't actually mean they'd buy it for 700k from the current owners.

Part of that businesses "worth" is going to be that the current people running it have a lot of skin in the game (their own capital etc), and another part is the belief that they can generate more value with this extra capital.

So saying it was "worth" 700k before is wrong, because nobody is offering 700k for the company. Saying it is "worth" 1 million afterwards the investment is fundamentally more accurate, because someone did offer 300k for 30% equity stake.

It is possible, and even plausible, that the people involved are making the math mistake you are describing. And the wording definitely leads to that implication.

But, "properly capitalized and with my advice, this business is worth X$" is a reasonable statement to make.

Amusingly, this means that the shark food could say "instead of 300k for a 30% stake, how about 200k for a 20% stake" and come out ahead, in theory. The resulting company is still "worth" 1 million dollars under the above math, except now the owner is an 80% instead of 30% owner (and under the "more correct" math, the investors would own 20% of a 900k business, or 180k worth, and the original owner would own 720k of a 900k business, 20k more than taking the 300k investment. But that is actually nonsense if they need that other 100k.)

However, they are probably capital hungry, and going from 80% of a smaller business to 70% of a larger one is a win in their minds. And they may be right; I mean, in the limit, 0.1% share for a 1k stake looks better mathematically, but they wouldn't be playing Shark Tank if they didn't think they need the cash to make their business grow.

You did not "sell" 50%…

You "sold" the whole thing when it was worth what the two of you agreed it was worth at the time, then bought back half of it at the different value the two of you agreed at a (very slightly) different time.

There was never any change in the overall value of the business… only in what the two of you agreed should be contributed for a half share… and please note that "a half share" at that stage is not the same as "half the shares" later.

Ask yourself whether the cost or price, worth or value would be the same if this had been a real commercial transaction with millions of shares in play among thousands of investors, as compared to your agreement between two people, whether friends or colleagues or what?

Just a partial answer here to add to what has already been said.

Many of the other answers are talking about the company having assets, and about your investor adding his $1 to your company's assets, etc. However, a complicating factor is that many companies, especially fast-growing startups, have intangible assets whose values can change rapidly based on what people think they are worth.

For instance, it is quite possible that you have a company that you think is worth $1, but that as events unfold the value of your company's assets increases, seemingly "magically". Suppose for simplicity that you actually have 100 shares of stock and you own them all and you believe they are worth $1 in total. Now someone comes along and offers to buy 50 of your shares for $1. What does this mean? It means that you were wrong in thinking your 100 shares were worth $1; in fact, they were worth $2, because you just found someone willing to pay you $1 to get only half of them.

This way of thinking about things does not correspond precisely to the logic you indicated in your question, but it is closely related. You discuss a situation where the company's value "is" $1 before the investment and "became" $2 after. In the example above, the company's value actually "was" $2 just before the investment --- you just didn't know it yet.

This may seem strange, but it happens all the time. Many startups have few tangible assets; rather, people are investing in them because they believe the company's intangible assets are worth a lot. These intangible assets can include things like "future revenue potential" or an "innovative business plan". If one day someone thinks your social media company can grow to 10 million users over the next year, but then the next day people start thinking it can grow to 20 million users over the same time period, the value of your company will increase even though you didn't gain any tangible assets.

The thing about these intangible assets is that the only way you know how much they are worth is by seeing what someone else will offer to pay for them. They are thus similar to "sui generis" assets like paintings. If you own a painting by some artist, you (or an art appraiser) can come up with an estimate of its value, but you won't actually know what it could sell for unless you actually try to sell it. If for instance you put it up for auction, it may sell for far more than you predicted. Similarly, if you own a company, you (or a corporate valuation expert) can come up with an estimate of its value, but that value can change dramatically if it turns out that investors are willing to pay more than you thought for a certain stake.