How to Value a Lemonade Stand

The other day I realized I have made a very serious mistake with these posts: not starting at the beginning.

All my posts assume a certain level of financial and mathematical knowledge. Often more than the average retail investor has. Today I come seeking redemption. To invest, you need to model. To model, you need to understand. Every good investment, and every good strategy, starts with knowing the basics well.

You need to be clear on one simple but fundamental idea: what makes something a good investment. Which is not the same thing as a good company.

How do we know what a good investment is?

You may not believe this, but this is one of those problems that has already been solved on paper. Solved. In a stroke of celestial irony, the gods of finance (people older than you and me) solved these problems many years ago. And the formula is so simple that a primary school kid could calculate it.

Imagine we have the simplest business in the world: a lemonade stand 🍋

You buy the ingredients. You make the lemonade. You sell it from a small stand on the street. How much would you pay for a business like that?

The intuitive answer is that it depends on how much money the stand generates and, above all, how much money I can take out of it.

If it generates $10,000 of profit every year, it is obvious that $10,000 would be a cheap price. After all, in one year I would have already recovered my investment. In year 2, I would also have doubled my initial investment. Everything after year 1 would be "free" money. A wonderful business.

For now I am using "profit" intuitively. We will refine it below, because when valuing companies we are not interested in just any kind of profit: we are interested in free cash.

$100,000 would be much more demanding, because it would take me 10 years to recover my money, even before adjusting for risk, inflation, or opportunity cost. And besides, the further money is in the future, the less it is worth. I prefer $10,000 today over $10,000 in 100 years. That is time preference.

On top of that, we live in a changing world. The business may change over the years. The price of lemons may go up, taxes may go up, or people may stop wanting lemonade and start drinking other things. Fewer people may walk past my stand. Or a thousand other risks may appear. Investing involves risk. And the further into the future we look, the harder those risks are to account for.

That is why profits from future years are worth less the further away they are from the present. It may be because of my own time preference - I prefer the money now - or because of the risk that those profits never actually materialize.

We also need to account for a little phrase I use to torture everyone I love: opportunity cost. If we invest that money in buying the lemonade stand, that is money we no longer have available for other things.

Now suppose we live in a strange world. A world where time preference does not exist and where I know with certainty that the lemonade stand will deliver real profits of $10,000 for 10 years. After that it will close. And the bargain is over.

To keep things simple for now, we will assume those $10,000 are free cash flow: money the owner can take out without damaging the business.

If I buy the stand for $50,000 and receive $10,000 per year, the stand gives me a simple return of 20% on the price paid. Or, said differently, I recover my initial investment in 5 years.

Careful: this does not mean I earn a 20% annual compounded return. If the business disappears after 10 years and has no terminal value, part of those cash receipts are simply recovering my initial investment. The real annualized return would be lower.

Even in this world, the price paid may still be too high depending on the other opportunities available to me. Imagine I can also buy an ice cream stand for the same price, and that stand will generate $20k per year for 10 years. That represents a simple return of 40% on the price paid. Better option. It would be stupid to buy the lemonade stand if I could buy the ice cream stand.

I would be "losing" money. Or, more accurately: I would be failing to earn it.

The foundations of company valuation live inside this very simple example.

We need to understand how much money a business will earn in the future. Then we need to adjust those future earnings based on the risk we are taking, the opportunity cost, and our time preference.

Here comes the first important realization:

The business does not generate different cash flow for you and me. But the maximum price each of us should pay can be different, because our alternatives, needs, and tolerance for risk are not the same.

• If I can invest in my brother's business, my opportunity cost is different from yours.
• If I have no savings and urgently need the money, my time preference also changes.
• If I am very risk averse and afraid of losing my money, I will demand a higher return than you for taking the same risk.

Cash flow

So nobody gets lost: when I say cash flows, I mean the money that enters and leaves a business.

Cash enters the lemonade stand when we sell cups. Cash leaves when we buy lemons, sugar, cups, or ice. It also leaves when we pay taxes. Or when we repair the stand. Cash flow is the difference between those inflows and outflows.

If $20,000 comes in during a year and $10,000 goes out, the business has generated $10,000 of cash.

It looks like profit. But it is not always the same thing. A company can show accounting profits and still not be generating real cash. For example, because its customers have not paid yet, because it needs to spend a lot of money on inventory, or because it needs to reinvest constantly just to keep operating.

That is why, when valuing a company, we are usually more interested in free cash flow, or FCF.

Free cash flow is the money left after paying what is necessary to keep the business alive.

In simple form:

FCF=cash generated by the business-investment needed to keep it running

In real life, those $10,000 are often not completely free.

Back to the lemonade stand. The stand generates $10,000 of cash. But every year I have to spend $2,000 repairing the structure, renewing the license, or buying a new fridge. So I cannot really take out $10,000. I can take out $8,000.

Those $8,000 are free cash flow.

And that is the number that matters to the owner. Because it is the money they can use to pay themselves, repay debt, save, distribute dividends, buy another stand, or reinvest to grow.

Put even more simply:

• Cash flow: how much real money the business generates from operating.
• Free cash flow: how much money is left for the owner after keeping the business running.

That is why the valuation formula uses FCF, not an abstract idea of "profit". In the end, the value of a business depends on the cash it can deliver to its owners over many years.

In economic theory, all of this is expressed in a more complicated way:

"The value of an asset is the sum of its future cash flows brought back to the present, adjusted by an appropriate discount rate."

• "Future cash flows" refers to the real money the business generates. Not accounting profits I cannot use.

• "Brought back to the present" means those real profits are not worth the same today as they will be in 10 years.

• "An appropriate discount rate" is the little number we use to adjust the value of future money back to the present.

  Since this little number reduces the value of future cash flows, it will be larger if money in the future is worth much less than money in the present. For example, if there is a lot of risk, a lot of inflation, or much juicier alternatives available right now. Also if I urgently need the money today.

In mathematical terms:

V=n∑

t = 1

FCF_t

(1 + r)^t

Where

V = business value

The maximum it would make sense to pay today for the lemonade stand, based on the cash flows we expect to receive.

FCF_t = free cash flow in year t

The real money the business can deliver to its owner each year, after paying what is necessary to keep operating.

r = discount rate

The minimum return we require for waiting, taking risk, and giving up other opportunities.

t = specific year

The further away the cash flow is, the more it is divided and the less it is worth today.

n = estimated life of the business

The number of years during which we expect the stand to generate cash.

Before putting numbers into it, let's translate each part of the formula into the stand example:

• V is what we want to calculate: how much the stand is worth today.
• FCF_t is the money it generates each year, in this case $10,000 every year.
• r is the return we require, so if we require 10%, we use 0.10. That is why the denominator is 1 + 0.10 = 1.10.
• t is the specific year. Here it goes from year 1 to year 10.
• n is the life of the business we are valuing. Here it is 10 years.

So each annual cash flow is divided by 1.10 raised to the year in which it arrives. The $10,000 from year 1 is divided by 1.10¹, the $10,000 from year 2 by 1.10², and so on until year 10.

The first years would look like this:

Year	FCF	Discount factor at 10%	Present value
1	$10,000	1 / 1.10	$9,091
2	$10,000	1 / 1.102	$8,264
3	$10,000	1 / 1.103	$7,513

If the stand generates $10,000 per year for 10 years and we require a 10% return, the applied version would be:

V=$10,000 / 1.10¹$10,000 / 1.10²$10,000 / 1.10³...$10,000 / 1.10¹⁰=$61,446

And here you can see why the discount rate matters so much:

Required return	Value of the stand
5%	$77,217
10%	$61,446
15%	$50,188
20%	$41,925

And that's it. With this magic formula, it looks like we can know exactly how much the stand is worth.

Very easy.

Too easy...

The first thing you are probably wondering is: why a r of 10%?

This is a complicated question. We have already said that r is the discount rate we require for taking risk, uncertainty, and everything else. But it is worth making something clear from the beginning: we should not see it as "extra profit", but as the minimum compensation we require for waiting, taking risk, and giving up other opportunities.

How do we choose the discount rate?

This could be an entire blog post. For this one, one rule is enough: the discount rate is the minimum return I require for the investment to be worth it.

First I look at a reasonably safe alternative. The typical example is the US Treasury bond. It is not perfect. But for most people it is safe enough. If that bond pays 5%, it makes no sense to accept less for buying a lemonade stand. For that, I buy the bond and avoid the hassle.

Then I add uncertainty. The more fragile the business is, the more return I should demand. If it can lose customers, take on too much debt, or suffer changes that are hard to foresee, I need a higher return. Otherwise, taking that risk is not worth it.

And then opportunity cost comes in. If I have another similar investment that can give me 20% per year, the lemonade stand competes against that 20%. If it cannot beat that alternative, I would be "losing" money. Or, more accurately: I would be failing to earn it.

There is no perfect rate. There are more academic formulas to try to refine it, but we will leave those for another day. The important thing is to understand the intuition: small differences in this little number can change the final value a lot.

The discounted cash flow formula is simple. The problem is getting the little numbers right. Small differences can have huge impacts on the final result. That is where the difficulty lies.

The two big problems are:

1. Correctly estimating future cash flows, meaning the "free" money the company will earn in the future.
2. Choosing the right discount rate.

And that is how simple, and how complex, investing in the stock market is.

Closing the loop

Shares are pieces of ownership in a real business. Although we have already seen that having control over the company is not the same thing as being a small shareholder. Those shares give you the right to participate in part of the company's cash flows. That is exactly what we want: money. That is why we invest. To make money.

And this point matters. In the end, what we want is to make money for ourselves. Not for the company. For ourselves. If you buy 100% of a company, this distinction may not seem especially important; after all, the company is yours. But it does matter when you buy a small piece. If you do not have decision-making power, the cash flow the company has is not entirely yours.

Imagine you own a small piece of the lemonade stand business. Even if the stand generates $10k per year of free cash flow, you do not necessarily see a single dollar. If the people running the stand spend that free cash flow on unproductive things, does it really help you?

Here we enter another swampy area.

The company can decide to return that free cash flow to its shareholders. In that case, yes, we could say that the company's free cash flows are approximately equal to the cash flows we receive as individuals, proportional to our ownership of the company. But what if the company never returns that money to shareholders? In that case, it would not be unreasonable to think that we do not care about the cash flows the company generates, because we receive 0.

In practice, not everything is black and white. Companies often retain those free cash flows inside the company to keep a financial cushion, acquire other companies, grow the business, and so on. If done intelligently, all those things are positive. They increase the value of the company and, therefore, the value of your position.

There is an implicit promise: in the future, the company will return money to shareholders when the moment is right. You do not want to distribute money too early if the company can earn a higher return on it than you can. In the meantime, you can "create" that cash flow by selling the stake to someone else, using the asset as collateral, or in other ways.

Normally, it is reasonable to assume that free cash flow more or less corresponds to the cash flow the shareholder will receive, except for taxes and similar issues. But, from time to time, there are companies that do not distribute that money. They burn it. They waste it. Or they simply leave it on the balance sheet. For practical purposes, from the shareholder's point of view, that company does not generate free cash flow.

A classic example is companies with a huge amount of accumulated cash. Imagine the lemonade business is selling for $50k, but it has $100k of cash sitting there. What a bargain! You are buying dollars for 50 cents. And, if you have decision-making power, that is true. But if you are a tiny shareholder, with no voice and no vote, be careful.

These opportunities appear because the company has a history of not returning money to shareholders. In other words, that money will never reach your pocket unless something changes.

This is why many people declare themselves "dividend investors". They only invest in companies that pay a dividend. That way they make sure the company returns at least part of those free cash flows to them.

As I already said: not too much, not too little. There is no need to demand a dividend if the company can make better use of that money. And we should remember that there are other ways to return money to shareholders, such as share buybacks. But you also should not always trust that the money the company generates will end up in your pocket. All of this, in theory, should be considered in the valuation.

So how much is a share worth? Well, it depends on the cash flows shareholders will receive in the future, of course, adjusted for risk, opportunity cost, and all the things we have already seen. We ourselves have calculated how much our lemonade stand should be worth. Although we only calculated the value of the whole business. Roughly $60k, to round a little.

How much should one share of the lemonade stand be worth? $10, $50, $10,000?

It depends on the number of shares. More precisely, it depends on what percentage of the business each share gives you. If there is only 1 share and it represents 100% of the company, it is logical that the share is worth the full $60k. If there are 2 shares and each represents 50% of the company, it is fair that each one is worth $30k. If there were 10 shares and each represented 10%, its price would be $6k. And so on.

Once we have valued the business, we need to divide its value by the number of shares to know the fair price of each share. I say this to avoid a common beginner mistake. A $5 share of company A is not necessarily a better investment than a $500 share of company B. It does not work like that. No. You need to divide the total value of the company by its number of shares. Stocks trading at $2 do not necessarily have to rise more than stocks trading at $2,000.

Fine. So far, that makes sense. But why do share prices move so much? What sense does it make for their value to change day by day?

There are several reasons. We already know some of them. For example, we know that the maximum price each investor should pay can be different, remember? Because of time preference, risk aversion, opportunity cost, and so on. Our circumstances change. That simple fact already justifies share values changing day by day. Even more so if we remember that small changes in the formula can have large impacts on valuation. The stock market is not a casino just because prices move. It is perfectly rational and natural.

Even so, there are more reasons. Another major one is that new information is created every day. This information can change expectations about the company's future cash flows or its risks. As our information changes, valuation should change. In theory, day by day. Second by second. But the observant eye will notice that valuations of the largest companies in the world vary enormously even within the same year. Trillion-dollar companies that become worth half as much or twice as much. Is that rational? The answer is that often it is not.

Price changes themselves are rational. Huge price changes, not always. Sometimes they are. And this is the good thing about the stock market. Sometimes it is investors' emotions that move the market: the fear of losing money or the greed for more. People are not always rational, and that creates price moves that can be irrational.

This is the advantage for the rational investor: taking advantage of these market moves to buy businesses at ridiculously cheap prices. We said the lemonade business should be worth close to $60k, but one day you can wake up and see that some investor is offering it to you for only $30k. If we remember that the business generated $10k per year, we quickly realize that this is a fantastic deal.

Sometimes the opposite will happen. The market will offer you the same business for $120k. Your job as an investor is simple: buy when the market offers you the business cheaply and sell it when the market wants it expensive. That simple.

Once again, it sounds easier than it really is. But that is all.

And this is the central idea: we do not invest in "good companies". We invest in future cash flows bought at an attractive price. Price rules. An excellent company can be a bad investment if you pay too much. And a mediocre business can be a great investment if you buy it absurdly cheap.

This is the foundation of stock investing and company valuation. If by some divine miracle you managed to value every business perfectly, it would take you a few months to become a millionaire. Days, if you are smart. The problem is that valuing businesses is anything but simple, and controlling your emotions is even harder.

The discounted cash flow formula, the one we have shown here, has several problems. Among them, the difficulty of estimating the discount rate or the fact that businesses do not "die" after 10 years, as our lemonade stand did. To solve these problems, clever and lazy investors have invented other ways to value stocks. Mutations of the original formula. Methods that, under certain assumptions, simplify this very difficult task.

Do valuation multiples, the P/E ratio, or terminal value ring a bell? All of those are extensions of the discounted cash flow formula. They are shortcuts. Each with its own advantages and problems. Simplifications to work better in this very difficult world. And I said simplifications. That is what they are. And a common mistake, in investing and in life, is simplifying when you should not.

In another post we will see how to derive any comparison-by-multiples method from the discounted cash flow formula. Because yes, they all come from there. And if they do not, they probably make no economic sense.