Investing is often described in terms of returns: which stock will rise the most, which portfolio has the highest expected return, or which strategy produces the best average outcome. But when wealth is compounded over time, a different kind of mathematics becomes important.
The central idea of growth-optimal investing is simple:
The objective is not merely to maximize average return, but to maximize the long-term rate at which wealth grows.
This seemingly small distinction leads to a remarkable connection between investment theory, logarithms, probability, information theory, data compression and even option pricing.
Compounding Changes the Mathematics
Suppose an investment earns a return of 20% one period and loses 20% the next.
An arithmetic average would suggest a return of zero:
(20% − 20%) / 2 = 0%
But an investor who starts with $100 ends with:
$100 × 1.20 × 0.80 = $96
The investor has actually lost money.
The reason is that wealth compounds multiplicatively, not additively.
For this reason, the natural mathematical measure of long-term investment growth is the geometric mean, or equivalently the average of the logarithms of returns.
This leads to the growth-optimal strategy: choose the portfolio that maximizes expected logarithmic growth.
Why the Logarithm Appears
The logarithm converts multiplication into addition.
If wealth evolves as:
Wₙ = W₀ × R₁ × R₂ × ... × Rₙ
then:
log(Wₙ/W₀) = log R₁ + log R₂ + ... + log Rₙ
A complicated compounding process therefore becomes a sum of individual outcomes.
This is more than a mathematical convenience. It creates a bridge between investment growth and information theory, where logarithms are fundamental to the measurement of information and entropy.
Growth Optimality and Information Theory
Information theory studies uncertainty, information and the efficiency with which information can be represented or transmitted.
Investment markets can also be viewed as systems in which uncertain outcomes occur repeatedly.
The lecture develops an intriguing parallel:
- Investment growth is governed by logarithmic returns.
- Information is measured using logarithms.
- Entropy measures uncertainty.
- Data compression exploits the statistical structure of uncertain outcomes.
- Portfolio selection can similarly exploit the statistical structure of market outcomes.
This produces a deep connection between two subjects that at first appear unrelated: investing and information theory.
The Universal Portfolio
One of the most interesting ideas is the universal portfolio.
Imagine that, with hindsight, we knew which constant-rebalanced portfolio would have performed best over a particular period. That would be useful—but impossible to know in advance.
A universal portfolio attempts to overcome this problem.
Rather than trying to predict exactly which portfolio will win, it distributes its investment across many possible constant-rebalanced portfolios. Over time, its performance approaches that of the best portfolio that could have been selected with hindsight.
The important concept is universality.
You do not necessarily need to know the best strategy beforehand. A sufficiently well-designed strategy can asymptotically approach the performance of the best strategy in a specified class.
This idea has a striking parallel with universal data compression, where a compression method can approach the performance of the best coding strategy without knowing the underlying probability distribution in advance.
Side Information Can Have Economic Value
Information can also directly affect investment growth.
Suppose an investor receives useful information about future market outcomes before making a decision. That information can allow the investor to allocate capital more effectively.
The lecture connects the resulting improvement in growth with mutual information, a central concept in information theory.
In the special case of a horse-race market, the relationship becomes particularly clear: information about the outcome can be translated directly into additional investment growth.
This provides an elegant interpretation of the value of information:
Information is valuable when it allows capital to be allocated more efficiently.
From Portfolio Selection to Option Pricing
The same mathematical framework also reaches into financial derivatives.
The lecture discusses the relationship between growth-optimal investing and the Black-Scholes theory of option pricing.
Black-Scholes is normally introduced through concepts such as replication and no-arbitrage. But the growth-optimal perspective provides another way of looking at the same mathematical structure.
The broader lesson is that seemingly different areas of financial mathematics can emerge from common principles involving probability, compounding, optimization and information.
More Than One Definition of “Optimal”
Another important point is that the growth-optimal portfolio has several related properties.
It can be viewed through different notions of optimality, including:
- maximizing long-term growth,
- expected-ratio optimality,
- competitive optimality,
- and certain martingale or supermartingale relationships.
The convergence of these different mathematical viewpoints is presented as evidence that growth optimality is not simply one arbitrary investment criterion, but a particularly natural structure arising from compounded wealth.
Amplifying Growth
A long-only growth-optimal portfolio may sometimes appear conservative because it is constrained by the available assets and the requirement not to take short positions.
The framework can be extended using long/short strategies, allowing the exposure to be amplified.
This does not change the underlying principle. It changes the degree of exposure to the growth-optimal opportunity.
The Bigger Picture
The most interesting part of this theory is perhaps not any individual formula. It is the chain of ideas connecting apparently separate subjects:
Compounding → logarithms → entropy → information → compression → universal portfolios → option pricing → competitive investment.
What begins as a simple question—How should wealth be invested so that it grows as rapidly as possible over the long run?—eventually leads into one of the central ideas of information theory.
The mathematics suggests that investment and information are not as separate as they first appear.
A market generates uncertain outcomes. An investor allocates capital among those outcomes. The resulting wealth compounds. Logarithms provide the natural language for that compounding, while information theory provides a language for uncertainty and knowledge.
The result is a fascinating perspective on investing:
To understand long-term investment growth, we may need to think not only about returns, but also about probability, information and the mathematics of uncertainty.
https://www.youtube.com/watch?v=wsd8nsJg6D8
