In my last article1, I explored what the GameStop (GME) situation could tell us about markets and capitalism. My major conclusions were that (1) the run-up in GameStop’s share price was a normal market reaction to increased demand (capitalism was not “broken” by GameStop), and (2) the sale price of any asset/commodity is determined solely by the parties involved in the transaction - no one else’s opinion matters. Supply and demand dynamics (where demand is subject to the whims of individuals) drive prices, nothing else. It is the second of my points that we will be dealing with for the most part in this article, but only in the very specific setting of the stock market. In a later article (or many articles), I will return to the most general setting of what I called capitalist transactions2 in my last article and their implications.
It was during the pandemic lock down summer of 2020 that I began to think deeply about the impact of point (2) above. How could I make money in the stock market when prices are, as Ludwig von Mises said in Human Action, “the subjective value judgments of individuals?” I recognized, of course, that stock prices were just valuations of businesses, so I began to research and think about how to value businesses. To illustrate what I learned, let’s take an example.
Imagine a company started by five individuals. Each individual puts $20,000 into this company. In exchange, each individual gets 20,000 shares in the company. This would imply that the total amount of shares outstanding is 100,000, and the company has total net assets (or book value in the common jargon) of $100,000. Let’s further imagine that all this company does is invest the $100,000 that it has. It is an investment fund. Further, let’s restrict it to take no more money in: if someone wants to buy into the fund, they need to buy a share from one of the founders. The founders paid $1/share each, so if they sell their shares for more than that, they have made money.
What happens as this business operates? Over time, the fund will buy assets and sell them, making a profit or a loss on each investment. Let’s say the fund is rather successful - after one quarter of operation, the fund has sold all of its investments3 and now has $104,000. It made $4,000 this quarter. Let’s also say that the fund has paid all of its taxes and fees too, so that this $4,000 is its total earnings for the quarter, or its net income. What happens with this money? The fund has two options: either (1) the fund can keep the money and reinvest it in more assets, thus growing its book value, or (2) the fund can pay its shareholders this money directly in the form of a dividend. The fund can do a mix of each - say it keeps $3,000 to increase its book value to $103,000 and pays $1,000 to its shareholders. Since there are 100,000 shares outstanding, each shareholder gets $0.01 per share that they own. It should be stressed that either way, the shareholders own the full value of this $4,000 profit, since the book value of the company is theirs.
The obvious question now is this: if I wanted to buy a share of this company, how much should I pay for it? To price it, I need to think about what I am buying. As we noted above, the company has some book value that represents its net assets. If I bought a share, I would be buying a share of these assets. That is not all, though. That share would also entitle me to a share of all of the future profits, in the form of increases in book value or dividends as above. We now come to the key point of this entire article- make sure you understand this next sentence and have convinced yourself that it is true in the example above. When I buy a share, I am buying the current book of the company plus any future increases in the book value and any future dividends. That is the tangible thing I am purchasing when I buy a share in this fund.
If I now wanted to find the monetary value of what I am getting for the share I buy, it seems like the calculation is straightforward. I should just use a model to project the amount of future book value increases and dividends and add them to the current book value. There is a problem with this method, though. If I think the business will never fail, then I will get infinity for this sum (in math terms, the sum will diverge - I will just add an infinite amount of big numbers to get infinity). Luckily, we get saved by another general consideration. If I am trying to buy a share of this fund, it is because I have cash I want to invest. I can invest this cash anywhere. Let’s say that I can either invest this cash in the fund we are looking at or I can put it in a bank account that would yield me 10% interest per quarter4. When I count the future book value increases and dividends, I have to account for the fact that I could be getting 10% elsewhere.
There is an opportunity cost I endure when I put my money in the share as opposed to elsewhere. Let’s see what effect this has on the calculation of the value (the math in the next part can be confusing, but don’t fret - I will present easier formulas to understand at the end). Say the dividend paid by the fund next quarter is $1,000, as above. The effect of this opportunity cost is that $1,000 next quarter is not equal to $1,000 today. If I want the present value of that $1,000, I need to discount it by the 10% rate I could get elsewhere. The present value of the $1,000 next quarter is the amount of money I would need to invest5 at the discount rate of 10% to have $1,000 next quarter, namely $1,000/1.10 = $909.09. It can be a little hard to conceptually understand why it is necessary to discount money delivered to you in the future. The reason is that money delivered today can be invested at the discount rate, so that in one quarter it will be worth more than it is today. Discounting future returns accounts for this lost investment potential.
We finally have all of the pieces we need to calculate the monetary value of what I am purchasing when I buy a share of the fund above. This number, which is called the intrinsic value6 of the fund, can be found by adding together the present book value of the fund plus the discounted future dividends and increases in book value (I’ll give you a formula in a minute). That’s it - that’s all the information I need to value this fund. What do I do when I have the intrinsic value? I divide it by the number of shares outstanding to get an implied price for the shares. This price of course depends on the model I have chosen for the fund’s growth - if the model is wrong, then I have the wrong implied price. Fear not - most people use some kind of margin of safety/conservatism to protect themselves from modeling inaccuracies; we will not concern ourselves with them here (these are mostly a personal choice anyway).
If the stock price is below my implied price, then I should buy shares. Why? I calculated that the cash value of the share is higher than the price. From my perspective, the situation here is the same as someone trying to sell me a $100 bill for $70. We know, from our calculation and projection model, that the present value of the amount of cash we will have in the future from this business (the book value plus the discounted dividends and book value increases) is greater than the amount we would have to pay for the share. In the case of this fund, we are literally buying money (as all of the assets are in cash) for less than it is worth. The practice of looking through the market for opportunities like this is called value investing.
Think about what I just said. If this is true, and value investors can buy $100 bills for $70, then why would stock prices ever deviate from the intrinsic value? Wouldn’t everyone take the deal to buy the $100 for $70? We already know the answer to this question from the GameStop article. Price is subjective - just because I think the fund should have some value does not mean that anyone is willing to pay that amount. Moreover, it does not mean that anyone is willing to sell at that amount. If the original founders of the fund did not want to sell for anything less than $100/share, they could do that. The fact that stock prices do not match consensus valuations does not mean that the stock market or capitalism are broken - it is a natural feature (some would call the fact that value investing opportunities exist market inefficiency, but I do not like this term because it is what the market is actually supposed to do - it is not inefficient in any way). Thus, value investing opportunities occur naturally in the market.
We should now return from the case of our fictional fund to businesses in general. Real businesses are not just all cash, they have assets. These assets all have other markets, however. The general business is an amalgamation of many different assets (real estate, machinery, cars, etc.), all of which have a price unrelated to the fact that they are part of this business. It is this fact that there is demand for the individual pieces of the business outside of its operations that gives value investing its teeth. A business is, of course, more than the sum of its parts. These other markets for goods help us find a “theoretical floor” for the stock price by giving us a valuation. The whole point of value investing is that, even if a stock’s price is not tied down to a valuation, the actual business that the investors are buying is. To the value investor, it does not make sense that a (hypothetical) business which owns $1 million in real estate should be trading with a valuation of $800,000. The value investor views buying the company’s stock as buying part of that $1 million in real estate at lower than market value. The thesis here is that, eventually, other people will realize that the company is undervalued and buy, raising the stock price and delivering a profit.
I should emphasize again that stocks can trade at whatever price market forces dictate. Supply and demand are all that matters. My intrinsic value calculation is my opinion, which we already know does not impact the stock price at all. The most important part about this realization is that stocks can trade below their intrinsic value for transient reasons, giving me buying opportunities that will turn into profitable trades. Of course, this all sounds good in theory, but how do I know it is true? I have literally made these trades and profited from them.
Before we get to those formulas that I promised you earlier, I should note one thing. Regular businesses have one slightly different thing from our all-cash fund from earlier, and this difference actually presents a point of difference between my method for calculating intrinsic value and those of other value investors. For this reason, I will highlight it (if you are new to value investing, then, the rest of this paragraph might not mean much to you because of how I presented the fund valuation above). Non-cash assets can depreciate and/or need to be replaced. Thus, the book of a business will have some maintenance costs. Some of the net income will go to this cost; however, the intrinsic value is still the sum of the current book value plus the discounted dividends and future book value increases. These future increases and dividends will just not add up to the net income (they will be equal to the net income plus the maintenance costs for the book); for those of you who may already know a thing or two about value investing, know that this is the reason I have used the dividend and book value increases instead of cash flows or net income for the projection part of my intrinsic value calculations. Book value increases and dividends are the real things that matter.
Now, finally, some formulas. We will do only the most basic model and example here7. First, let’s define our symbols. The book value will be B(n), where n is the quarter we are looking at. B(0) is the book value for the previous quarter (we know this number from the company’s quarterly reports), B(4) is the book value one year from now (we project this), and so on. D(n) is the dividend paid each quarter. D(0) is the dividend paid last quarter, D(1) is the dividend paid next quarter, etc. The discount rate will be denoted DR, written as a fraction (a discount rate of 5% will be DR=0.05, for example). Now we need to model the company’s growth. The simple model we will assume is that the book value and dividend are growing at some fixed rate G every quarter, with one rate for the book value and one for the dividend (these will again be written as a fraction: 5% growth will be G=0.05, etc). As functions, B(n) and D(n) become
To find the intrinsic value, we need to find the book value increase for each quarter in the future. Since we stipulated growth at a constant rate, the increase in book value is just that rate times last quarter’s book value (see term two below). We can finally put everything together and calculate the intrinsic value. It is a sum of three terms: (1) the current book value, (2) the discounted sum of all future book value increases, and (3) the discounted sum of all future dividends. In symbols, it looks like this:
For those of you who have never dealt with infinite sums before, it may seem like we cannot actually calculate anything here. Fortunately, there are formulas we can use for the sums in this case8 if each growth rate G is less than the discount rate DR. I will spare you the details9, but the final result is that the intrinsic value for this growth model can be calculated as follows:
This formula for the intrinsic value is the culmination of this article. We now know how to judge the value of a business growing at a fixed rate just from its fundamentals. The share price implied from this valuation is just the intrinsic value divided by the number of shares outstanding (this number can be found on the company’s quarterly report). I feel obligated to again mention that the intrinsic value (and thus the implied price) of a business is simply the opinion of the person who did the modeling; like all other opinions, it has no bearing on the price action of a market.
That is all of the material I wanted to cover in this article. Let’s recap what we did: (1) we reviewed some basic principles and facts about pricing in markets (namely, that the market will do what it is supposed to do given supply and demand regardless of anyone’s opinion), (2) we motivated the definition of intrinsic value as the sum of the current book value plus the discounted future dividends and increases in book value by analyzing the situation of a hypothetical investing fund, (3) we argued that this definition applies to all businesses in general, (4) we discussed how value investors use the intrinsic value as a theoretical floor for the price of a stock, (5) we argued that value investing opportunities (stock price < intrinsic value’s implied price) occur naturally in markets, and (6) we calculated the intrinsic value for the simple (but not so realistic) situation of a company growing its book and dividend by a fixed percentage each year.
I hope you enjoyed this article. If you found it useful or enlightening, I would appreciate it if you joined my mailing list (for free) or subscribed (first 30 days free). If you know people who would enjoy this article, I would appreciate you sharing it with them. As always, you can send me comments here or on Twitter (@ReportIv). See you next time!
It really is great; you should read and share it. Also, have you read the disclaimer? This is a good opportunity for me to remind you that I am not a financial advisor, this is not financial advice, and I have no registrations. Now that that’s out of the way, we’ll get back to the good stuff.
The term “capitalist transaction” is probably non-standard, but I think it is the best way to refer to a positive-sum market transaction (the type which form the basis of free-market capitalism). Throughout the rest of this piece, I will be using bold and italics to introduce various terms/jargon.
Of course, we don’t have to assume it sold everything. Each investment will have some kind of monetary value at all times based on the current market rate for it, so the assets do not all have to be in cash. I am just moving everything to cash to keep this example simple and convince you of the basic premise that the value added during any given quarter is the dividend plus the book value increase. We have not mentioned any business costs from, say renting an office or paying salaries. All of this would be subtracted from the total profit of the company to give the net income.
There is obviously no bank account that yields 10% (the reason why will be the subject of another article). This rate is just pulled out of thin air. Constraints on the discount rate do not really concern us for now, but I will be writing an article on them at some point. In some models of growth, the discount rate must be higher than the growth rate, otherwise the sum will diverge. This issue can be fixed in a variety of ways, which I will explore in the future article on discount rates that I am planning.
The easy way to remember this formula is (present value)*(1 + discount rate as a percent / 100)^n = future value. Thus, if I was given $1,000 in five quarters’ time, and the discount rate was 10%, the following equation would hold: (present value of $1,000 delivered to me 5 quarters from now) * (1.10)^5 = 1000. You can then solve for the present value by dividing. Interestingly, the discount rate I need to use can be affected by my own time-preference for money - if I really need money now, then money five quarters from now is not as valuable to me. Thus, my discount rate would be higher. Choosing a discount rate is more art than science.
The name of my newsletter makes its appearance! I really hope that reading this line is like those moments in movies or songs when they say the title. Maybe it could even be like that scene in Breaking Bad where they play the opening theme during the show for the only time. That would be cool.
I rarely use this model myself, since it does not actually apply to many stocks. I will be reviewing my other modeling methods in subscriber only articles - subscribe so you don’t miss them!
Because of the simplicity of our model, the sums actually are just geometric series. Most students of calculus will be familiar with them - if you are, it is a good exercise to prove the formula I give for the intrinsic value. Of course, if we change our book value/dividend growth model (to, say, linear growth), then we will no longer have a geometric series and will most likely have to compute the sums numerically.
If you do not trust my analytical math skills (I don’t blame you), fear not. I verified the final formula for the intrinsic value numerically in a few different cases.