Chapter 9 Log-optimal strategy

The “biased coin” game (see for example: https://explore.paulbutler.org/bet/) provides an excellent illustration of risk management and optimal betting strategy in scenarios where recurring investments are involved.

To succeed, it is crucial to identify favorable bets (favorable bets are those with a positive expected value). Only such bets should be taken, as they ensure increase of wealth over time. However, success does not depend solely on choosing favorable bets; one needs to manage the amount of wealth risked in each bet.

An overly cautious approach, where too little is risked, results in minimal returns on investment. On the other hand, a recklessly “bold” strategy, when too much is risked, leads not only to the failure to grow wealth but also to significant losses, potentially wiping out what you already have.

The so-called Kelly strategy shows that — in the context of recurring investments — the guiding principle for should be the expected log return on assets. Maximizing the expected log rate (not the simple net rate!) for the total wealth (across all assets, and not just on individual investments!) ensures the maximization of wealth in the long run.

In the following exercises, your task is to determine the answers by maximizing the expected log rate of return on assets.

Please note that the average rate of capital growth per individual investment period can be determined using the expected logarithmic rate of return on total wealth for a single investment period.

\[R_G = e^{\bar{r}}-1\]

9.1 Exercises

Exercise 9.1 Suppose we have the opportunity to make regular (repeated) investments in a specific financial market structure. For each investment, the capital can either double in the short term or be lost entirely. The probability of doubling the capital is 0.6, while the probability of losing it is 0.4.

Given an initial capital of $100, what strategy should we adopt to optimize our returns? Additionally, what average rate of capital growth can be achieved with this strategy?

Exercise 9.2 Suppose we have the opportunity to make regular (repeated) investments in a specific financial market structure. For each investment, the capital can either double in the short term or be lost entirely. The probability of doubling the capital is \(p\), while the probability of losing it is \(1−p\).

Given an initial capital of $100, what strategy should we adopt to maximize long-term capital growth?

Exercise 9.3 Suppose we have the opportunity to make regular investments in a venture where the invested capital can either be tripled or lost entirely. The probability of tripling the capital is 0.5, and the probability of losing it is 0.5.

What strategy should we adopt to maximize long-term capital growth? Additionally, what average rate of capital growth can be achieved with this strategy?

Exercise 9.4 Suppose we have the opportunity to regularly invest in a venture where the invested capital can either be tripled or lost entirely. The probability of tripling the capital is \(p\).

What strategy should we adopt to maximize long-term capital growth as a function of \(p\)?

Exercise 9.5 Suppose that by counting cards in every game of blackjack, we have a 50.75% probability of doubling our invested capital. Otherwise, we lose the entire invested amount.

  • How much of the total pot should we invest in each game to maximize long-term capital growth while minimizing risk?

  • If one game lasts 2 minutes and we play for 8 hours a day, how long will it take (under optimal risk management) to double our capital?

Exercise 9.6 Suppose we have the opportunity to make recurring investments in two types of assets:

  1. A stock, whose price is equally likely to double or halve in each period.

  2. A risk-free asset (cash), which does not change in value over time.

Questions:

  • What strategy should be adopted to maximize long-term capital growth while managing risk?

  • What average rate of capital growth can be achieved under this strategy?

  • What will be the long-term average rate of capital growth if all funds are invested in stocks?

  • What will be the long-term average rate of capital growth if all funds are kept in cash?

Exercise 9.7 Suppose we have the opportunity to make recurring investments in three types of assets:

  1. Stock A, which has a 50% probability of yielding a simple rate of return of +100% and a 50% probability of yielding a simple rate of return of −50%.

  2. Stock B, an uncorrelated stock with the same parameters as Stock A.

  3. Cash, a risk-free asset that does not change in value over time.

Questions:

  1. What strategy should be adopted to maximize long-term capital growth while managing risk?

  2. What average rate of capital growth can be achieved under this strategy?