Cars and Physics

The Valuation Game

Read the valuation formula post for a more concise treatement using equations. But conceptually, the below should be the same. Without further ado,

A company’s share price appreciation comes from three sources:

(i) Lower expected rate of return for the same earnings. A 100$ recurring consistent quarterly earning is worth only $10,000 at 4% but is worth 40,000 at 1% returns. Of course real rates have not become this bad, but it has dropped preciptously in the last two decades to say 7% or so now, from above 12% in years past. But this earnings rate drop does not translate to a drop in market returns, as these income producting assets (stocks) have appreciated due to lower expected returns. So owners of stocks have bilssfully enjoyed price appreciation, but the future earning capacity of a set capital has now been greatly diminised. Theoretically, rising interest rates can turn the tide the other way. This does not seem to be in the cards due to reasons mentioned in the sixth and seventh blog posts in this series.

(ii) Earnings growth. A 100$ current quarterly earnings with 10$ yearly earnings growth upto 150$/quarter steady state is worth 10,000 currently but will be worth 15,000 in the future. So, the 5000$ gain would have a present value of say 3500$ if the rate of return is 4% again, for a total corporate value of $13,500. This only models a company which has 100$ earnings for a few years then suddenly jumps to 150$ quarterly earnings. To this amount, we need to add in the increased earnings of the ramp up for the couple years it takes to reach 150$. (the final answer will be less than 15000, since that’s the value of a company that makes 150$ in earnings today)

(iii) Earnings retention. If the company retains the earnings as cash (rather than paying out as dividends to the shareholders) then, the value of the company is not only its income generating value but also the amount of cash holdings, of course. For example, KO , the coca-cola company has a dividend payout of 75%, so it retains 25% of earnings which should increase the value of shares over time. (There are some other accounting rules w.r.t. depreciation and cash flow that could potentially make this case for appreciation even sweeter/richer, but this point is already valid as it stands)

After the passing of Jack Bogle in 2019, the legendary founder of Vanguard index funds, it only seems fitting to address this topic here. The dual prongs of compounded costs in investing and the low chance of beating the market are the true merits of the index fund. An index fund does not seek to beat the market, rather it seeks to track the performance of the market. It does this by owning a large representative list of (a certain class like large cap or even total market) stocks in proportion to the market cap of the companies. An index fund can be low cost since it does not take any effort (cost) to figure out which stocks are worth owning (or to rapidly trade to adjust the portfolio). Rather it owns stocks indiscriminately in accordance with the market value and prominence of companies.

I have one obvious lesson and one less so obvious lesson to share in this regard. The cost of a managed fund around 0.5% higher a year say, might seem trivial at first. But when you consider that the fee is assessed on the entire investment and not on the returns, it becomes substantial. In a decent year in a low inflation environment, the return might be 5%, which means you are giving away 10% of your gains! Case in point, your money doubles every 14 years at 5%, but takes almost 16 years to double if the return rate is 4.5%! And the expense ratios of funds can be even higher than this, particulary in times past. Jack Bogle truly helped a generation of small investors and institutional accounts in the form of pension funds, retirement savings etc., keep more of their money.

The second lesson is this – index fund returns are way above most active fund returns!! You might wonder surely merely tracking the market would yield only average returns? This is true. But this is where the difference between average (mean) and median come in. The average return of the stock market is way above the median return. Index funds outperform around 80% of active fund managers in a given year. The other 20% have such spectacular returns for the year or a period of few years, while some of the trailing managers don’t do too badly compared to the benchmark. What is better, historical back testing studies have found that there is no consistency in which 20% of the managers beat the market over longer time intervals, in other words even the minority success in beating the market is not consistently repeatable.

(Sometimes I get a bit theoretically idealistic – If it was 100% repeatable, then they would get all the investment money in the market, putting other funds out of business, which would mean the old 20% minority becomes the whole market, but that is another story. Let’s say they refuse to accept all the nation’s wealth under their management for argument sake). Practically, as investors, choosing the active funds based on past performance is an insufficient metric, rather than current holdings/strategy. But that exercise transforms the investor into a manager himself, making the point moot! In short, index funds are a timeless invention that cuts people a better than fair deal, ever more important at a time of declining interest rates and hence possibly declining income from other sources.

Furthermore, some back data gets skewed since poorly performing funds are merged or closed and transferred to other funds, so the case based on acutal data is even better for index funds.The surviving outperforming funds will naturally tend to gravitate toward the benchmark or below (in other words everybody cannot be above average!). This last bit of analysis is a bit loose, since new funds enter the market too. But all in all, there is a reason Warren Buffett is famous. No one else has ever done it to such scale. And even if you invest in his fund today, future returns above the benchmark are hardly a guarantee.

The very initial phase

I like to start a blog post with the very fundamentals because the simpler your ideas, the easier it is to get started. Feel free to skip down lower if this is too obvious. So what are shares and what do companies do? Companies make money by engaging in business activities and they are owned by people. In the case of public companies, anybody can own a share or piece of the company by buying a unit stock or more.

The captial to conduct business and inventory (if needed) comes from either:

(a) Investment from people, who fund the business in exhange for new or increased ownership

(b) Debt, borrowed against the company’s assets/reputation

In this blog we’ll be mostly discussing public companies. Stock market trading does not yield the company any capital, as simply the old owners and new owners of the company are trading ownership stakes in the company. The firm only raises money in a new equity offering or on the initial sale of stock.

The Modern Economy

Once we have estabilshed that companies exist to make money for owners by serving clients, we come to the earnings history and cycle of the public companies traded on the modern stock exchange. This is usually done on a quarterly cycle that lines up with the calendar year. Modern trading is much better due to being electronic and algorthmic, with liquidity and price determination driven both by volume of stock of the big companies and by instant matching of bid-ask between buyers and sellers by the computers.

The value of a company stems from its ability to make money in the long term. The recurring nature of consistent or increasing earnings is key, not just profitability itself. So, the value of a company based on a set earning is determined by (dividing by) the total return rate. The total return rate is the sum of the inflation rate plus the market rate of return plus the risk-reward rate of return for that company. For the purposes of this post, let’s consider stable medium cap and large cap established companies that have relativiely low risk of bankruptcy or heavy capital loss, so the primary driver is the (inflation adjusted) market rate of return.

Let’s say a company makes 100$ a quarter in profit. Then the question one needs to aks oneself is what amount of money can generate this return. In modern economies, growth is releatively low and ordinately inflation tends to be low. This also drives interest rates to be low as well.

So continuing with our numbers finally, the 1% annual inflation rate and a 100$ quarterly earnings would imply the company is worth $40,000, if the competing market had zero productivity and only held on to inventory or commodities that kept pace with inflation. Of course this is with zero risk and zero actual market return, so we might see trading prices in the market closer to $10,000 for this company, with a 4% total return. This is known as a discounted cash flow model and is not the subject of this blog post. Rather the details of value fluctuations under macro-conditions are of interest here.

The role of debt and leverage in valuation

The ratio of the debt to the equity investment a company has raised by selling ownership interests is called the leverage ratio and is an improtant facor in risk multiplication, as a rule of thumb. (Risk being defined as a chance of company bankruptcy under adverse performance, with higher leverage being more high risk/high reward)

Quaternions are generalized complex numbers that enable rotations in 3D rather than in just a complex plane. The complex number i is replaced by a general vector v which can be expressed in a 3D orthogonal basis i,j,k.

The rules are defined as ijk=-1 and any basis squared is also minus one, like ordinary complex numbers.

ij=k, jk=i, ki = j are the desired cyclic relations that motivate the above definition and follow from the definition, as can be seen by multiplying both sides by any single basis vector and applying the square rule.

So essentially, it combines the complex number rules with the rules of a cross product of vectors to come up with an elegant and useful mathematical structure.

Legend has it when Hamilton discovered them, he carved the rules into the side of the bridge he was walking on.

How is it useful for rotations you ask? Well, pick any quaternion of the form a+bv.

Here a and b are real numbers and v is a vector. We normalize the quaternion to the form cos(x)+sin(x)v.

This quaternion rotates vectors perpendicular to v by an angle x, about the rotation axis v. This is easily seen by just multiplying it out and observing the sine is perpendicular and cosine retains the input direction. But what happens to parallel components?

It does not work! So what do we do? Well parallel components should remain unaltered after rotation around the axis. So the formulation that Hamilton came up with to split up the rotation into two pieces:

(cos(x/2) + sin(x/2) v) (u) (cos(x/2) – sin (x/2) v)

The post multiplication by the conjugate ensures that when the parallel component commutes out.. the conjugate cancels the first factor. Hence the parallel vector remains unchanged after rotation.

But does this disrupt the original perpendicular vector? No, it still works. The perpendicular component is agnostic to the conjugation process and the two half angles add up to yield the full angle rotation. This can be proved by using the commutation rule ab = -ba, that applies to perpendicular vectors only ( derived from the basis rules). In this case, u and v are perpendicular.

So, given any input vector, it can be resolved to the sum of parallel and perpendicular components to the rotation axis and we can see the quaternion formulation works correctly!

Now we can conclude that this approach will work to formulate an axis-angle representation. Let me also mention Euler’s rotation theorem which states that any combination of 3D rotations can be reduced to rotation about an axis drawn from the origin. The particular axis might change based on the rotation sequence but the resulting transformation (final effect) is always so expressible.