In 1738 the Swiss mathematician and physicist Daniel Bernoulli came up with a simple thought experiment:
“A wealthy prisoner who owns two thousand ducats but needs two thousand ducats more to buy back his freedom, will value a gain of two thousand ducats more than another man with less money than himself.”
Let’s play this on and put Bernoulli’s prisoner in the context of modern markets and ask him to evaluate various investments. What immediately becomes clear is that his ducats are dedicated to one purpose: get out of prison to the devil!
Our prisoner has a purpose for his money, just like us.
Our prisoner can invest his ducats as he sees fit, and because he wants to maximize his chances of release, we can describe its use for various investments with the goal-based portfolio theory.
We don’t need to worry too much about the details right now, but clearly our prisoner will be evaluating both the expected returns and the expected volatility of a given security over time through the prism of realizing freedom. Its willingness to make trade-offs between return and volatility is shown in the following chart. The line is the minimum return it requires for a given level of volatility. As volatility, or the X-axis, increases, our prisoner demands ever-higher levels of return, as the Y-axis shows. This is no revelation: this is exactly what theory traditional would expect.
The Prisoner’s Dilemma: Return and Volatility
But what if we built a stock exchange in our prison and let our wealthy prisoners trade stocks with each other? This is where things get interesting.
In the second chart, we plot three different prisoners, A, B, and C, each with a different starting wealth, required final wealth, and time horizon. For the sake of simplicity, we’ll assume that everyone has exactly the same view of a security’s future volatility and return, which are labeled as s And m in the figure.
Three Prisoners Dilemma: Return and Volatility
Here’s the problem: every investor is willing to accept completely different returns for the same security!
Moreover, if the price of the security is simply the inverse of the yield — 1/Me, a simple but not unreasonable model — then each investor is willing to pay a completely different price for the exact same security!!
There is no difference of opinion on the characteristics of the security resulting in different acceptable prices, but rather a difference in the needs of investors.
When we place these three prisoners on the market, we would expect Prisoner A and Prisoner B to sell their shares to Prisoner C at a price of 1/vs until Prisoner C runs out of cash or Prisoner A and Prisoner B run out of inventory. Then the price drops to 1/band prisoner A continues to sell to prisoner B. From there, the price drops to 1/Aand Prisoner A would buy, but no one would be willing to sell.
Prisoner C is an enigma. Traditional utility models do not expect anyone to accept lower returns in response to upper volatility. But goal-oriented investors can seek variance when their initial wealth is low enough. Behavioral finance characterizes their goals as “aspirational”. This is why people buy lottery tickets and play: increasing the volatility of results is the only way to increase their chances of achieving life-changing wealth.
Of course, this is all more than just a thought experiment: it reveals some key lessons about markets.
First, when setting capital market expectations or target stock prices, analysts would do well to assess the market of buyers and sellers to determine how their needs and liquidity will influence the price going forward. This is of course more complicated than our example, because in addition to different needs, everyone also has a different vision for a given title.
This is no surprise to practitioners. Markets dominated by institutional buyers are very different from those dominated by ambitious investors and “YOLO” traders.
A very present example is our current regime of quantitative easing (QE) underway by central banks around the world. For investors baffled by exorbitant stock market valuations, the difference between Prisoner A and Prisoner B is illuminating. They are exactly the same except for one thing: Prisoner B is richer today.
In general, then, this means that adding liquidity to financial markets creates investors who are willing to pay more for the exact same security. Conversely, when excess liquidity is drained from the markets, prices should fall, all else equal, because investors with less cash today demand higher returns. Thus line B returns to line A.
Second, and most striking: there is no “correct” market price. No security has a “fair value” or “fundamental value”. Rather, price derives from the characteristics of a security that interact with the needs of investors in the market.
Another key element of the price is the relative liquidity of each investor in the market. If enough ambitious investors, or C prisoners, deploy their money in a securities market, prices can stay high or soar until their cash is exhausted. Sounds familiar, GameStop?
It may seem obvious, but it’s not the traditional perspective on the markets. The efficient market hypothesis affirms that securities are always trading at their fair value and that market timing cannot work. Of course, predicting the evolution of a stock’s fundamentals is a difficult task. But that’s only half of the equation. As our hypothetical prison stock market shows, understanding the market of investors and their behavior can yield equally valuable insights.
What’s even crazier: Every investor in the market acts rationally. Prisoner C offers a perfectly rational price for the title even though it is the highest bid on the market! Prisoner A acts just as sane despite the lower purchase price.
And here are some of the Goals-Based Portfolio Theory offerings. Behavioral finance would portray the price action of our prison market as irrational yet predictable investor behavior, and traditional theory would dismiss it as non-existent. But goal-oriented investors can see more clearly what is really going on.
Goals-based portfolio theory can, in fact, be a useful bridge between normative and descriptive theories.
Like the prisoner of Bernoulli’s thought experiment, we have specific goals to achieve with our money. And like the prisoner, we interact with public procurement with those goals in mind.
These goals influence prices in ways that traditional theory could not predict. And while behavioral finance offers some models for predicting irrationality, goal-based theory suggests that people can be more rational than initially thought.
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All posts are the opinion of the author. As such, they should not be construed as investment advice, and the opinions expressed do not necessarily reflect the views of the CFA Institute or the author’s employer.
Image credit: ©Getty Images / erlobrown