In 1738 Swiss mathematician and physicist Daniel Bernoulli proposed an easy thought experiment:
“A rich prisoner who has two thousand ducats, but needs two thousand ducats more to buy his freedom, will place a higher value on a gain of two thousand ducats than another man who has less money than he.”
Let’s play this out further and put Bernoulli’s prisoner within the context of recent markets and ask him to judge various investments. What becomes immediately clear is that his ducats are dedicated to 1 goal:
Our prisoner has a goal for his money, similar to we do.
Our prisoner can invest his ducats as he sees fit, and since he wants to maximise his possibilities of being released, We can describe its use for various investments using the goal-based portfolio theory.
We don’t must worry an excessive amount of about details in the mean time, but our prisoner will evaluate each the expected returns and the expected volatility of a given security over time from the angle of gaining his freedom. His willingness to trade off return and volatility is shown within the graph below. The line is the minimum return he demands for a given level of volatility. As volatility, or the X-axis, increases, our prisoner demands ever higher returns, as represented by the Y-axis. This is hardly a revelation: it is precisely what traditional theory would expect.
The prisoner’s dilemma: return and volatility
But what if we arrange a stock exchange in our prison and let our wealthy prisoners trade stocks amongst themselves? This is where it gets interesting.
In the second graph, we depict three different prisoners, A, B, and C, each with a distinct starting wealth, a distinct required ending wealth, and a distinct time horizon. For simplicity, we assume that every has the exact same view of a security’s future volatility and return, that are marked and within the figure.
The triple prisoner’s dilemma: return and volatility
That’s the purpose: every investor is willing to just accept completely different returns for a similar security!
If the value of the safety is just the alternative of the return – 1/ an easy but not unreasonable model – then every investor is willing to pay a very different price for the exact same security!!
It is just not a difference in opinion concerning the characteristics of the safety that results in different acceptable prices, but reasonably a difference within the needs of investors.
If we put these three prisoners available on the market, we might expect Prisoner A and Prisoner B to sell their shares to Prisoner C at 1/ until Prisoner C exhausts his liquidity or Prisoner A and Prisoner B exhaust their supplies. Then the value falls to 1/ and Prisoner A continues to sell to Prisoner B. From then on, the value falls to 1/ and Prisoner A would buy, but nobody could be willing to sell.
Prisoner C is a puzzle. Traditional utility models wouldn’t expect anyone to just accept lower returns in response to volatility. However, goal-oriented investors may seek variance if their initial wealth is low enough. Behavioral economics characterizes their goals as “aspirational.” This is why people buy lottery tickets and play: just by increasing the volatility of outcomes can they increase their probability of achieving life-changing wealth.
Of course, that is greater than just an easy thought experiment: it gives us some necessary insights concerning the markets.
First, when setting capital market expectations or price targets for stocks, analysts should evaluate the market of buyers and sellers to find out how their needs and liquidity affect the longer term price. This is in fact more complicated than our example because, along with different needs, everyone also has a distinct attitude toward a selected security.
This is not any surprise to practitioners. Markets dominated by institutional buyers look very different from markets dominated by ambitious investors and “YOLO” traders.
A really recent example is our current regime of sustained quantitative easing (QE) by central banks world wide. For investors confused by sky-high stock valuations, the difference between Prisoner A and Prisoner B is instructive. They are the exact same apart from one thing: Prisoner B is richer today.
In general, which means that injecting money into financial markets creates investors willing to pay more for the very same security. Conversely, as excess liquidity flows out of the markets, prices should fall because investors who’ve less money today will demand higher returns. Thus, line B shifts back to line A.
Second, and most strikingly, there isn’t a “correct” market price. No security has a “fair value” or “fundamental value.” Rather, price is decided by the characteristics of a security combined with the needs of investors out there.
Another necessary consider price is the relative liquidity of every investor out there. If enough aspiring investors or Prisoner C put their money right into a securities market, prices can stay high or skyrocket until their liquidity is exhausted. Sound familiar? GameStop?
This could seem obvious, nevertheless it is just not the normal view of markets. The efficient markets hypothesis claims that securities all the time trade at their fair value and that market timing cannot work. Of course, it’s difficult to predict the event of a security’s fundamentals. But that is barely half the battle. As our hypothetical prison stock market shows, understanding the market of investors and their behavior can provide equally beneficial insights.
What’s even crazier is that each investor out there is behaving rationally. Prisoner C offers a superbly reasonable price for the safety, even whether it is the best bid out there! Prisoner A is behaving just as rationally, though he has the bottom purchase price.
And that is a part of the promise that goal-based portfolio theory offers. Behavioral economics would describe the value motion of our prison market as irrational, albeit predictable, investor behavior, and traditional theory would dismiss it as nonexistent. Goal-based investors, nevertheless, can see more clearly what’s really happening.
Goal-based portfolio theory can actually be a helpful bridge between normative and descriptive theories.
Like the prisoner in Bernoulli’s thought experiment, we have now certain goals we would like to realize with our money. And just like the prisoner, we interact with public markets with those goals in mind.
These goals influence prices in ways in which traditional theory wouldn’t expect. And while behavioral economics offers some models for predicting irrationality, goal-based theory suggests that individuals could also be more rational than initially thought.
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Photo credit: ©Getty Images / erlobrown
