Markowitz uses personal probabilities in portfolio theory for two reasons: it is rational coherent behaviour, and personal probabilities in investment theory should be able to predict. But rational behaviour and predictability are two separate arguments: rational behaviour does not automatically lead to predictability. Moreover, Savage’s approach is about rational decision making, and not necessarily about prediction:
[…] that theory [the theory of personal probability] is a code of consistency for the person applying it, not a system of predictions about the world around him (Savage, 1954, p. 59, my insertion).
Let us now discuss how Markowitz (1959) relates the use of personable probabilities to predictability. Out of his discussion of Savage’s theory of
probability beliefs, he draws, in his own words, ‘two morals’, about the
predictability of the probability distributions of investment returns, which I will fully quote to support my analysis:
1. The existence of personal probabilities does not necessarily imply that, as of the moment, the individual is positive that his beliefs are ‘good beliefs.’ He may admit the possibility that he currently is either always overoptimistic or always overpessimistic, or in some other way subject to biased judgment. However, the idea of probability beliefs does imply a belief in an ability to learn with time and experience, to end a long life of predictions and constant education without substantial biases on the whole.
2. The connection between objective and subjective probabilities is quite close. We noted that they mixed on a par with each other in the calculation of expected utility. The discussion of this section indicates another connection. To assert that some physical experiment has a .6 probability of producing a result A is to assert that , if this physical experiment is carried out and if a large number of other
physically independent experiments (of the same or different kinds) all with a
probability of .6 of producing some particular (though perhaps different) result A are also carried out, then the relative frequency of A will almost certainly be .6. Similarly, to assert that a .6 personal probability is associated with an event is to assert the belief that it is virtually certain that the relative frequency of correct predictions among a set including this and a large number of other psychologically independent events is .6. Thus personal probabilities and subjective probabilities are connected via the notion of relative frequency in the long run (Markowitz, 1991/1959, pp. 272-273, his italics).
Concerning Markowitz’s first conclusion about subjective probability beliefs: he acknowledges that the exercising of personal probabilities can lead to a possibly biased judgment, but it also implies an ability to learn and become better at estimating. His argumentation suggests that he assumes that the probability distribution of investment returns can be known in principle, which means the probability distributions are not ontological, but merely epistemological uncertain. Yet, we have to ask ourselves under which conditions it is possible to learn about probability beliefs. We can learn about the ‘real’ probability distribution of investment returns if the distribution is stable over time, or if changes of the properties of the distribution can be predicted. I claim in the dissertation that the
probability distribution of investment returns is unstable. The current Chapter 3 reviews the arguments against stability in economic phenomena such as returns on financial markets and presents my own thought experiment about the possibility of predictability for investment theory.
Let us illustrate what happens when probability beliefs are updated. The probability distribution of investment returns applied could differ manifest from reality, if an event such as the credit crisis of 2008 with the accompanying negative results in the financial markets occurs. In the reasoning of Markowitz, the answer to a phenomenon like 2008, is the Bayesian approach of updating the probability estimates with the new information. But if the probability distribution of
investment returns is unstable, updating does not lead to a better level of predictability, because the ‘real’ distribution of investment returns cannot be learned. It is uncertain how often a crisis such as in 2008 will happen, and a next crisis could turn out even worse than 2008. If the probability distributions would be stable, objective probabilities will become available if the number of
observations becomes sufficiently high: both the frequentist and Bayesian method in statistics can be used to update and learn from new observations, both by a bigger sample. My conclusion is that personal probability beliefs about investment returns can only by coincidence be correct. So, though personal probabilities sharpen the intuition, they are not able to stochastically predict the future. The second conclusion of Markowitz claims that personal probabilities are like subjective probabilities by the notion of relative frequency in the long run. Meant here is seemingly that personal probabilities are individual probabilities, and that subjective probabilities are the accumulation of personal probabilities. My view is that he refers to Savage’s point that the personalistic view of probability can contain reasonable, objective views like the notion of relative frequency as well:
I would reply [to the critics] that the personalistic view incorporates all the universally acceptable criteria for reasonableness in judgment known to me and
that, when any criteria that may have been overlooked are brought forward, they will be welcomed into the personalistic view (Savage, 1954, p. 67, my insertion).
My view is that the second moral of Markowitz about similarity to the objective frequency approach is only valid if the probability distribution is, again, stable, or if the changes in the distribution are predictable. Just like his first moral, the second moral is founded on the assumption of stability. So, both his morals to justify the predictability of probability beliefs in investment theory depend on the assumption of stability, or the ability to predict changes in the stability.