What role does probability theory play in the design of a retirement fund? This article touches (very briefly) on this topic. For concreteness, the reader should read this article from the lens of the working-class individual seeking financial security for retirement.
1. Background
The overarching research effort is to investigate the random dynamics of a retirement fund. Retirement savings play an important role in every society. Failure to adequately save for retirement increases reliance on social security and the risk of financial hardship in old age.
This article highlights the role of probability in the design of retirement funds. Additionally, we provide a precise mathematical mapping of the time evolution of a single random variable introduced in the first article.
2. Framing Investment Problem
Random multiplicative processes are ubiquitous real-world phenomena. In retirement savings, a random multiplicative process emerges from how a retirement portfolio compounds through time. The process happens sequentially through reinvesting of wealth over multiple periods.
The random multiplicative process is written in mathematical form as follows:
Equation 1: Wealth compounding through time
Further specifications.
An elementary version of these processes is called a binomial multiplicative process. We denote returns as independent identically distributed random variables1. The return distribution is specified as Bernoulli which means that each year we have a binary payoff: did the investor’s wealth go up or down? It’s possible to go beyond the Bernoulli distribution and apply a distribution that captures a return continuum. However, for our purpose, the binary return function suffices.
Computer simulation.
A well-known feature of the random multiplicative process is an asymmetry between ups and downs. If you lose 50% of your retirement savings, you need 100% investment returns to break even.
So what happens if an individual has a 50% probability of losing 55% of their savings and a 50% probability of increasing their savings by 100%? It seems to be a good investment since each year you either double your savings or lose only 55% of your capital.
Well, that’s not the case. Figure 1 below shows sample paths generated from a computer simulation and it’s clear that the retirement pot shrinks to zero over time. The simulations can be independently verified by accessing the link to the source code. The pattern persists as we increase the number of sample paths. The reason for this is the asymmetry mentioned above: you need more than 100% gains in the next period to break even after losing 55% of your capital.
Figure 1: ten sample paths, with Bernoulli shocks.
Free source code: https://embracingx.gumroad.com/l/donlb
Additional points.
By its very nature, retirement savings is a multi-period investment process. Individuals and pension funds have the demanding task of safeguarding the retirement pot over many years. Part of that demanding task deals with the fact that it is hard to predict random events. Even if you have superior skills and a good sense of probabilities and distributions of events, you still don’t know what’s going to happen.
The challenge is that individuals have only one life. In this lifetime, we will only experience events, nothing more. No one will experience probabilities, averages, distributions, etc. After all, when was the last time you saw a retiree using probabilities to pay for an electricity bill?
This point focuses our attention on what counts the most for a retirement portfolio: how random events impact the portfolio. Most investment strategies are different ways to deal with this issue. The aim is to ensure that a retiree experiences a better path as compared to the paths in Figure 1. “Ensuring” is a strong word when it comes to random outcomes, but we won’t get into additional layers of analysis at this point.
3. A dynamic random variable
Few technical ideas are required to explain the concept of time evolution of a single random variable. Firstly, we need the traditional probability space defined as follows:
The last step is to combine this probability space and the transformation acting on the probability space as follows:
This dynamical probability space can be wrapped around thousands of real-world problems. It’s a mathematical representation of what I call a human struggle. A struggle to figure out how to move “stuff” through time. I wrote this section while opening a book, Entropy and Information Theory, by information theorist of the highest rigour, Professor Robert Gray. The stuff Professor Gray has devoted his life to is information moving through a noisy channel.
Based on this formalism, it’s clear that Equation 1 above can be seen as taking one random variable and repeatedly transforming it while mapping the outcomes into the same random variable. Hence time evolution of a single random variable. It is important to note that the transformation does not necessarily have to be multiplicative.
4. Conclusion
The main goal was to present a few deep concepts in probability with maximum simplicity. These concepts provided a different angle of viewing investing and the challenge of saving for retirement. The next step is to progressively expand on these ideas.
Let’s conclude on a lighter note with a message in the movie Miami Vice by undercover detective Sonny Crockett to his lover, Isabella, who works for a drug lord:
“Probability is like gravity, you cannot negotiate with gravity.”
Nothing is certain in life and there is only one logical conclusion…
EMBRACE RANDOMNESS…
hapsburg
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