A Price Paradox

This is a continuation of my previous post "Markets Don't Hate Uncertainty".

Don’t think about the stock market like you do for a specific good like, say, bananas. If a large group of the banana-consuming public suddenly decides they don’t like bananas very much anymore, the demand curve for banana shifts massively to the left meaning market prices will fall greatly and future quantities produced/consumed will too--remember the immediate supply curve is nearly vertical and the long run supply curve is much closer to horizontal.

However, that is not the case for stocks. The value of stocks is determined by the net present value of future cash flows. In simple terms: Buying a stock means being a part owner of a company. As an owner you are entitled to a share of future profits (eventually paid out as dividends--a reasonable, simplifying assumption). Those profits come in the future, so we have to value them today at a discount since a dollar today is worth more than a dollar tomorrow--the so called time value of money (TVM). Adding up all future cash flows individually discounted for how far out into the future they are gives us a figure for net present value (NPV). This is the "value" of the stock, which should equal its price. Let's assume for now there is no uncertainty about expected future profits.

An essential premise in the banana hypothetical is that the value of bananas has plummeted. Remember value is always a subjective concept. Suppose a large demographic group like the Baby Boomers in aggregate start reducing their desire to hold equity investments (stocks), it is not because they don’t believe in the value of stocks in general. Rather it is likely due to the fact they don’t want to have an asset exposure so much tied to the volatility associated with stocks. So they want to reduce their investment holdings in stocks. So in one sense there is no reason to believe that their selling activity should materially change the price of stocks because the investment value of stocks (NPV of future cash flows) is unchanged in the market overall.

But they are trying to sell, and their selling has to be met by buyers. In order to find a willing buyer they should have to offer a more attractive (lower) price than the current price. Hence, we have a paradox--prices shouldn't change but they have to change. The solution lies in a reframing of what investors are trying to achieve. They don't want stocks for the sake of stocks--this would be the banana model. They don't want the future cash flows per se--those are only attractive relative to the price paid for them given the risk associated with realizing them. They want the expected return--the future cash flows purchased at an appropriate price today adjusted for the risk.

So in order for Baby Boomers to sell their stocks they have to increase the expected return of stocks from the perspective of the buyer. Since they can't affect the future cash flows, they have to do one of two things: Either lower their prices to attract buyers or find buyers with different discount rates (buyers who don't value a dollar today as much as a dollar tomorrow). Realize, we are still assuming no disagreement about the expected future cash flows. That wrinkle is not needed to explain this hypothetical or the paradox. Everyone can agree on the expected future cash flows and still people want to exchange their positions (i.e., Baby Boomers want to sell and reduce market risk and buyers want to take on that risk).

Here is the graphical way to think about those two options available to the stock-selling Baby Boomers:

The colored lines B and C represent the change in price over time. The slope of these lines is the expected return. In the first case (top chart) prices today decline along path A and the lower price after Baby Boomers try to sell stocks implies a higher expected return for all stock investors. The market's expected return has increased from the slope of line B to the slope of line C.

In the second case (bottom chart) the difference in how each investor discounts the future price implies a different rate of return. Baby Boomers selling stocks have an expected return of B and buyers of their stock have an expected return of C. (Note: This is related to but not dependent upon Baby Boomers having a shorter life expectancy than buyers. The buyer can be the exact same demographically as long as they have a different discount rate. When compared to the stock-selling Baby Boomers, the buyers value a dollar tomorrow as being closer to the value of a dollar today. Boomers can be selling to other Boomers with lower discount rates.)

You should be feeling uneasy about this for two reasons:

1. Why don't prices today equal that far-off future price? Didn't we assume there is no disagreement about the amount of future cash flows?
2. In the second chart why wouldn't this have already come to fruition? Did stock-selling Baby Boomers suddenly just now increase their discount rates?

The answer to the first objection is this. Even if we assume there is no disagreement about expected future cash flows, there is a degree of uncertainty for them; hence the word "expected". The compensation a buyer/holder of stock receives for risk of investing is the expected return--technically speaking, the equity risk premium. 

Assume a stock will pay a dividend next year equal to all of its assets and profits and then cease to exist. All market participants agree that the dividend next year will be either $0 or $2 per share and that the probability of each outcome is 50%. In that case the expected future cash flow is $1 next year. This means the stock today is worth $1 before any TVM discounting. Once the year has passed and the dividend realized, investors will either have $0 or $2 to show for the $1 they put at risk by buying the stock. Therefore, the future price has to be something greater than the current price or no one would be interested in risking the investment.

The answer to the second objection is a little less elegant. Well, basically we have to be assuming Baby Boomers wanting to sell stock have increased their discount rate (i.e., become more risk averse). Otherwise, we can't have the hypothetical. But this leaves another problem: Why isn't there just one expected return in the market? Well, there is, but in the second chart we are breaking up the market into two segments. The blend of the two lines B and C would be the market expected return. That would be some new line D with a new destination in between the two lines B and C. 

So, what is the solution to the paradox? Will prices change? And if so, did the value of stocks change?

The answer is another paradox: Stock prices changed by going down because stock value went up so stock prices could come down because stock value had to go up.

Baby Boomers in this hypothetical all of a sudden wanted less stock because they wanted less risk. In order to reduce risk they have to sell that risk to someone else. That someone else either must want that same risk relatively more all of a sudden (the source of the second objection to the bottom chart mentioned above) or the Baby Boomers need to reduce the risk by offering a lower price. Some of both will happen meaning stocks get less risky simply because Baby Boomers want less risk--a rather surprising result. Here are the implications:

• The lower the price of stocks, the less risky they are holding expected cash flows and discount rates constant.
• The change of ownership from higher-discount rate investors (Baby Boomers all of a sudden in our hypothetical) to lower-discount rate investors (the buyers of the Baby Boomer's stock) means stocks are less risky to those who now hold stock without any needed change in price or expected future cash flows. 
• Stocks become more valuable when prices go down without any other change or expected future cash flows go up without any other change or discount rates go down without any other change. 
• If expected future cash flows go down and discount rates go up (as has been the case circa March 2020), then prices must go down. 

The moral to this story could be: DON'T REASON FROM A PRICE CHANGE! Prices reflect value. Value for consumption goods and services like bananas and hotel rooms are subjective. Value for financial investment assets like stocks, bonds, and real estate are subjective too. But they are subject to expectations about the future and the value we place on money today versus money tomorrow.

Prices don't simply change, and a change in price doesn't really tell us anything. The price of bananas declining could be because people stopped liking bananas as much or it could be because it just got a lot easier/cheaper to harvest bananas. The price decline tells us that the value of the next (marginal) banana is lower, but that fact by itself doesn't tell us why that is the case.

Stock prices declining could be because investors in aggregate think earnings from holding stocks will now be lower than estimated before or it could be because investors in aggregate value money today more than money tomorrow. The price decline in stocks tells us that on net one or both of these has happened, but the fact by itself doesn't tell us which.