Recent stock trading in an obscure company called Gamestop has brought into focus an age-old question in finance economics: what is the fundamental price of a stock? In the case of Gamestop, a concerted effort by a group of small traders engaged in a short squeeze has left large short sellers exposed. Stock prices rose to heights that clearly had nothing to do with the fundamental valuation of the company. If a stock price rises from US$18 to nearly US$400 in a few days, then the company has either invented a super-gizmo or miracle drug, or it is subject to massive speculation or market manipulation. If it looks too good to be true, it likely is.

Similarly, the stratospheric heights of the valuation of Tesla, a leading manufacturer of electric vehicles, has left many observers wondering: Is Tesla Stock Overvalued?. Look at the chart below that lists the market capitalization of the top 15 automakers in the world. Tesla's world market share is less than 1%, but it commands a market cap that is as large as the top eight or nine manufacturers combined. This market capitalization is beyond absurd even if Tesla is considered a multi-sector firm rather than just a carmaker. Defenders of Tesla's valuation make the claim that it is the leader in numerous sectors simultaneously, from making car batteries to making software for autonomous vehicles. Of course, on closer inspection, other carmakers are innovating in all these areas as well. At the end of the day, Tesla is just another carmaker—a very innovative carmaker to be sure, but not the wunderkind that has reinvented carmaking. Other carmakers have much better skills at design and manufacturing expertise, and enjoy much larger economies of scale.

What is the orthodox finance view of stock value?

The conventional economic theory behind stock valuation isn't overly complicated, but of course it depends all on expectations of future states of the world. This is domain of price-earnings (P/E) ratios and other metrics, which is all very well established.

The market value \(V=\tau\cdot\sigma\) is the product of stock price \(\tau\) and outstanding shares \(\sigma\) and in turn is the present-discounted stream of future profits \(\pi_t\), with discount rate \(\rho\). If profits are in steady-state, then \[V=\sum_{t=1}^\infty \frac{\pi}{(1+\rho)^t} = \frac{\pi}{\rho} \] If, on the other hand, our market would grow perpetually at rate \(\gamma < \rho\), then the market value grows to \[V=\frac{\pi}{\rho-\gamma}\] Profitability is reflected in the P/E ratio, the stock price relative to per-share earnings. In the above notation, the P/E ratio is \(\tau/(\pi/\sigma)\). Thus a typical P/E should be about \(1/(\rho-\gamma)\), so perhaps 20-30. The P/E ratio is the favourite yardstick for making comparisons across firms.

Some upstart firms don't have any earnings yet, so the P/E ratio is undefined. In those case we typically look at sales and sales growth and look at such metric as price to sales-per-share (P/S) as an alternative.

Ultimately, most heterogeneity in stock price valuations expresses differences in beliefs about where a company will be in a few years, or 10 years, or 20 years. Nevertheless, there are limits to valuation as markets don't grow infinitely. The assumption that a company's stock price can grow without bound is demonstrably false. All markets eventually reach saturation. The market for automobiles is not going to grow fundamentally larger in the decades to come as households will only own a fixed number of vehicles (and possibly less so in the future if autonomous vehicles really take off on a major scale). This implies that much depends on two types of expectation: how much market share a company can capture in the future if a market is near saturation, and how much a nascent market will grow before it approaches saturation. These are numbers that one can put rough ceilings on.

An unorthodox view of stock value over time: limits to growth

As pointed out earlier, companies can't grow forever in real terms because eventually markets saturate. So consider a more realistic scenario where profits grow logistically towards a long-term ceiling for profits \(\bar{\pi}\) so that profits rise over time according to \[ \pi(t) = \frac{\bar{\pi}}{1+(\lambda-1)^{1-t/T_{0.5}}} \quad\mathrm{with}\quad \lambda=\frac{\bar{\pi}}{\pi(0)}\] Here, \(T_{0.5}\) is the year in which the company is expected to reach 50% of the long-term profit ceiling ("the half-way point to its full size"), and \(\pi(0)\) is the size of today's (non-zero, pre-reinvestment) profits. The logistic growth function also implies that the most rapid profit growth occurs at \(T_{0.5}\). The ratio \(\lambda\) is the expected growth that we expect the firm to achieve in the long run. Now the market value will be \[V= \bar{\pi}\cdot\psi \quad\mathrm{with}\quad \psi(\rho,\lambda,T_{0.5})\equiv \int_0^\infty\frac{\exp(-\rho\cdot t)}{1+(\lambda-1)^{t/T_{0.5}-1}}\mathrm{d}t\] The \(\psi\) function delivers the multiple of the long-term profit maximum. The table below shows evaluations of this integral for different values of discount rate, profit ceiling, and half-way point into the future.

\(\rho\)	\(\lambda\)	\(T_{0.5}\)	\(\psi(\rho,\lambda,T_{0.5})\)
0.05	5	5	15.45
0.05	5	10	12.59
0.05	5	15	10.70
0.05	10	5	15.64
0.05	10	10	12.60
0.05	10	15	10.43
0.05	20	5	15.67
0.05	20	10	12.52
0.05	20	15	10.18

A rapid-growing firm that is expected to quintuple its profits in the long run and take 10 years to get their half-way can be expected to have a firm valuation about 12.6 times its expected profitability ceiling, or 63 times it current profits. Without growth and at the same discount rate, he same firm would only be worth 20 times its profits.

The \(\psi\)-function can be compared with \(1/(\rho-\gamma)\) in the simple valuation model above. Essentially, it implies much smaller P/E ratios than the conventional model. Indeed, you can read the \(\psi\) values as if there were P/E ratios. To justify a high P/E, a firm must grow rapidly (short \(T_{0.5}\)) and grow far (high \(\lambda\)). These are lofty goals for even the most innovative companies.

In my admittedly unorthodox view of the the financial world, many firms have P/E ratios that are unjustifiably high based on their growth potential. Simply put, markets don't grow forever. Even the most innovative companies eventually mature and reach the point of market saturation.

What does microeconomic theory tell us about the comparative value of stocks within a mature industry?

The discussion above focused on rapidly-growing firms. What about firms in an oligopolistic market where firms compete for market share? In mature markets, growth is only possible when production costs fall significantly through innovation. However, companies can and do switch positions through competition, by way of product and process innovation, or through (mild) economies of scale.

Intuitively, market share should be reflected in corporate value. Firms with a larger market share should also have a larger market capitalization, unless there are significant differences in expectations about the path of a company. Innovation and product quality can certainly influence the valuation, but overall market capitalization should at least be roughly in line with the ranking of market shares. However, how strong is the effect of market shares? Should we expect a simple linear relationship or something else? Let us look at a simple Cournot model, where I employ negative-exponential demand to get slightly more elegant solutions than with linear demand.

Negative exponential demand has the form \(Q=a\cdot\exp(-p/b)\) with demand elasticity \(|\eta|=p/b\), where \(a\) captures the size of the market and \(b\) is the price at which the demand elasticity is exactly equal to one. The \(n\) firms in the market have marginal cost \(c_i\) and fixed costs \(f_i\) per unit of capacity. Then the solution to the Cournot equilibrium yields market shares \[s_i\equiv\frac{q_i}{Q}=\frac{1}{n}+\frac{\bar{c}-c_i}{b} \quad\mathrm{with}\quad \bar{c}\equiv\frac{1}{n}\sum_{i=1}^n c_i\] and equilibrium price \(p=\bar{c}+b/n\). Each firm's profits are \[\pi_i=(p-c_i)q_i-f_i z_i=(b s_i^2-f_is_i)Q\] where capacity \(z_i=s_iQ\) follows firm size ex tempore and \(f_i\) is the fixed cost per unit of capacity. If we ignored fixed costs, then observing the market size \(M=pQ\) and the demand elasticity \(|\eta|\) would allows us to infer firm \(i\)'s profits (and thus its valuation and stock price) simply through the relationship \[\pi_i\approx s_i^2\frac{M}{|\eta|}\] The firms with the larger market share are disproportionately profitable. If we further assume similar unit fixed costs \(f\) across the industry, then the profit ratio of two firms \(i\) and \(j\) is then given by \[\frac{\pi_i}{\pi_j}=\frac{s_i(s_i-f/b)}{s_j(s_j-f/b)} \approx\left(\frac{s_i}{s_j}\right)^2\] where the approximation holds if fixed costs \(f\) are small. The market values of the two firms should follow the same proportion as the profit ratio, assuming we use identical discount rates. We can therefore conclude:

• The market capitalization of oligopolistic firms is roughly proportional to the square of their market shares, especially in industries where variable costs are a large part of total costs.
• In industries with high fixed costs relative to total costs, the market capitalization of oligopolistic firms will be less than proportional to the square of the market share, and instead will be closer to linear-proportional to the market share.

Economic theory tells us that market share should give us a rough indication of the relative value of companies. The larger companies are large because they have a cost advantage and higher profitability than their smaller cousins. When looking at movement within the market, today's market capitalization may be indicative of tomorrow's market share. We can reverse the profit and valuation equations to predict the implied future market share of firm \(i\) as approximately \[ \widehat{s}_i = \sqrt{\frac{\rho\cdot\tau_i\cdot\sigma_i}{M\cdot|\eta|}} \]

The market share story explained above has numerous caveats. Some firms compete in multiple product markets and span more than one industry. Others sell differentiated products and it therefore sell in separate markets with imperfect substitution between them. Some firms face barriers to international trade, such as tariffs, which limit their ability to compete globally. Nevertheless, most automakers are really just automakers, and they compete globally. So at least for the automotive industry the above story should provide a reasonable approximation.

At the end of 2020, Tesla had a global market share of 0.8%. Perhaps the most optimistic scenario for the company is that it reaches a market share of 8%, similar to the current automotive giants Toyota and Volkswagen. The valuation difference is thus \((0.08/0.008)^2=100\) — a huge growth potential that gives plenty of scope for Tesla's market capitalization to be larger than its market share implies. However, in a mature market for automobiles there is simply no scope for growing profitability beyond the existing market. Manufacturing cars still involves tons of hardware, and increasingly costly batteries. There is no magic innovation that could reduce the cost of producing cars. Logically, even if Tesla's market share grows by an order of 10, Tesla's valuation can't be larger than the largest car makers today. In a world where product differentiation and variety matters, Teslas will never be the only cars that people will want to buy. As I said, even rising to the top still means capturing only a fraction of the world's automobile market. The conclusion is clear: Tesla is fundamentally overvalued, and something has got to give.

If short-selling overvalued stocks is risky, how can markets correct the price of overvalued stocks?

In an ideal world, short-selling pushes excessively optimistic stock valuations toward more rational levels. Short-selling has a moderating influence on market exuberance and can help burst valuation bubbles.

Short sellers can get caught in a short squeeze. Short-selling is inherently risky and it is entirely possible to get caught on the wrong side of the market movement. The risk of long and short positions is asymmetric: the risk of the short seller is unlimited, whereas the risk of the long buyer is limited by the purchase price. Short-selling based on fundamental valuations can help with price finding in markets. However, the problem with short-selling is timing: even if one knows that a stock is overvalued, it is hard or impossible to tell when market prices will decline. When short positions are accumulating, this actually gives markets a signal that a correction may be immanent. However, it also increases the incentive for market manipulators to bid up the price to put the short sellers into a short squeeze when they have to clear out their positions. Short-selling is therefore not for the faint-hearted, and is the domain of hedge funds rather than retail investors. So take my word of advice: stay away from short trading unless you can well afford incurring large losses at times.

If short-selling alone can't fix the problem of overvalued stocks, what can? The answer is remarkably simple: portfolio exclusion. Overvalued stocks don't belong in a value-based portfolio. This is the paradigm of value investing — buying under-priced stocks and selling over-priced stocks based on fundamental valuation techniques. This paradigm was pioneered by Columbia Business School professors Benjamin Graham and David Dodd, and implemented successfully by Warren Buffet's Berkshire Hathaway. Value investing is generally a wise paradigm for mature markets. It is tweaked to fully realize the potential of growth in novel markets with completely new value propositions — the Amazon and Netflix type. Remarkably,despite its rise in value, Tesla was excluded from the S&P500 in September 2020, but as prices rose it was eventually included in the S&P 500 with 1.69% weighting, fifth largest in December 2020. Anyone who has an S&P500 index fund will now be exposed to Tesla's valuation.

Tesla is not the next Apple. I've bought plenty of Apple products over the years (I am writing these lines using one), but I don't think I'll ever buy a Tesla because there are plenty of great new electric cars coming on the market from other carmakers. As much as I admire Tesla's innovative spirit (and the sheer pizzazz of Elon Musk's other ventures such as privately-held SpaceX), Tesla is simply not the wunderkind of the electric vehicle age. The future of electric mobility is mostly in the hands of traditional automakers with their deep expertise in design and manufacturing. Tesla beat them to the start line, but I very much doubt that Tesla will beat them to the finish line.

Has there been a fundamental shift in stock valuations?

A recent working paper by finance researchers Dmitry Kuvshinov and Kaspar Zimmermann illustrates a very prominent structural shift in the 1990s. They calculated the ratio of stock market capitalization to Gross Domestic Product (GDP) in 17 countries a and found that this ratio (about one third) was quite stable between 1870 and about 1980-1990. There was a doubling of this ratio in the following decade and stock market capitalizations have been both higher and more volatile. They called it the "big bang" in market capitalizations. How does this observation relate back to my comments above? Kuvshinov and Zimmemann attribute their findings to a significant increase in the profits of publicly-traded firms as a share of GDP and capital income, which in turn is related to increasing market power of large firms. In other words, rather than a secular shift in high future growth potential, the observed rise in market capitalizations seems to be a shift in market power that is becoming more worrisome. A new paper by de Loecker et al. (QJE 2020) documents the evolution of market power based on firm-level data for the U.S. economy since 1955. The three authors measure both markups and profitability and find that since 1980, aggregate markups have started rising from 21% above marginal cost to 61% today. Expect future finance research to focus much more on market structure.