Never underestimate life’s beauty.
ChatGPT can help to understand parts of a research paper that require clarification.
In this case the paper happens to be on applying the Kelly criterion to a portfolio of multiple assets. I was interested in the paper and read through it a few times and understand it pretty well as I have been studying this area recently, including reading Fortune’s Formula by William Poundstone which deals with the Kelly criterion. But, I wanted to make sure my understanding was concrete. I want to understand it in a way that I can compare what I personally have done in the way of setting up a portfolio. Specifically code that manages an index fund with risk management applied that operates similar the way the Kelly criterion works.
Electronic copy available at: http://ssrn.com/abstract=2259133Kelly criterion for multivariate portfolios: a model-free approach
The process involved to simplify text
By pasting in a section of the paper and asking it to dumb it down a bit and then interacting with it using some followup questions it can help clarify it. ChatGPT can be asked to bring it down to a high school level or alternatively you can tell it an age that you want the understanding to be brought down to.
In this case I also asked it to produce and example using Python code. The code is functional, however there is a few unused variables in it.
Transcript of the ‘conversation’ with ChatGPT
Explain the following in simpler terms, like high school level… Algorithm for the fractional Kelly strategies The strategies that bet less than the Kelly fraction are called fractional Kelly strategies. As we just pointed out, they are less profitable but also less risky. In the case of a univariate portfolio we just decrease the capital fraction invested in the risky asset and shift the released capital to the riskless bond. But for a multivariate portfolio such an approach is ambiguous. Indeed, an investor usually cares about the total fraction of risky assets in his/her portfolio. And s/he can at first proportionally descrease the fractions of all risky assets in u? but then additionally redistribute the capital among the risky assets. If the returns are jointly Gaussian then we shall just proportionally reduce the fractions of the risky assets. Moreover, in this case the strategy with Kelly fraction x := 1/(1 − α) is equivalent to the maximization of the (negative) power utility xα/α, α < 0. But in general this nice property does not hold true (see MacLean et al. [2010] and Davis and Lleo [2012]). For instance7, even if the marginal returns are Gaussian but the marginals are paired by Clayton copula, then the proportional fraction reduction is suboptimal. Imposing additional constraints 0 ≤ ui ≤ x ∀i and ∑n i=1 ui ≤ x we can still use (3) to find the optimal portfolio for a given x. However, (3) may be a bad approximation for the leveraged portfolios or the portfolios that contain assets with possibly very high returns8 (see Appendix 1). Additionally, practitioners usually like having two independent optimization methods: if both of them get similar results then they are plausible. We consider the portfolio optimization under the no leverage and no short selling constraints. A Monte Carlo grope algorithm we employ is as follows: 1. Assume we want to invest x of our capital in risky assets9, x ∈ [0%, 100%]. Choose an initial portfolio so that all fractions of the risky assets are between 0.0 and x and their sum is equal to x. Distributing the capital over all assets equally may be a good start. 2. (Groping step): modify the fractions at random so that the no short selling and no leverage conditions still hold. If the new portfolio yields higher expected growth rate, set it as initial portfolio, otherwise leave initial portfolio unchanged. 3. Repeat Step 2 until portfolio performance stops increasing.
The Kelly strategy is a way to invest money in different assets (like stocks or bonds) in order to make the most profit. The Kelly strategy says that you should invest a certain percentage of your money into each asset, based on how likely that asset is to make a profit. However, this strategy can be risky because if one of your assets doesn’t make as much money as you expected, you can lose a lot of your overall investment.
A fractional Kelly strategy is a way to make the Kelly strategy less risky by investing less money into each asset. For example, instead of following the Kelly strategy exactly, you might only invest 80% of the money that the Kelly strategy suggests. This means you’ll make less profit, but you’ll also be less likely to lose a lot of money.
When you’re using a fractional Kelly strategy for a portfolio (a collection of assets) that has multiple types of assets, it can be tricky to figure out how much to invest in each one. If all of the assets are likely to make similar profits, you can just decrease the percentage you’re investing in each one by the same amount. But if the assets have different levels of risk, you’ll need to adjust the percentages differently.
One way to do this is by using a computer program that simulates how different portfolios would perform. The program starts with a random portfolio and then makes small changes to it, to see if it will make more profit. If the new portfolio is better, the program switches to it and continues making changes. If the portfolio stops improving, the program stops.
This process is called a “Monte Carlo simulation” because it uses random numbers and works like a game of chance. By doing this simulation many times, it will get a portfolio that has a better chance to make a better return. The optimization process will be constraint by no leverage and no short selling, and the final result will be the optimal portfolio.
Would it be possible to use historic data or some kind of model data?
Yes, it is possible to use historic data or model data when using a fractional Kelly strategy. In fact, using historical data is often necessary in order to estimate the expected returns and risks of different assets.
One way to use historic data is by looking at the past performance of the assets. By analyzing the returns of an asset over a certain period of time, you can estimate the expected return and risk of investing in that asset. This information can then be used as input for the simulation algorithm.
Another way is to use financial models to estimate the expected returns and risks of different assets. These models use mathematical equations to describe how the assets should behave based on certain factors, such as economic conditions, market trends, and other variables. By plugging in the relevant information into the model, you can get estimates of the expected returns and risks of investing in different assets. This information can also be used as an input for the simulation algorithm
It’s important to note that, regardless of the input, the simulation and optimization algorithm will still work by exploring different portfolio scenarios under the constraints and looking for the one that will generate a higher expected return.
When working with historical data I assume it is going to be simulating trades.
Yes, when working with historical data, it is common to use it to simulate trades and backtest the strategy.
Backtesting is the process of applying a trading strategy to historical data to see how it would have performed in the past. By simulating trades on historical data, you can get an idea of how the strategy would have performed in the past and evaluate the strategy’s performance.
When you’re backtesting a fractional Kelly strategy, you would use historical data on the returns and risks of the assets in your portfolio to estimate the expected returns and risks. You would then use the simulation algorithm to generate different portfolios and apply the fractional Kelly strategy to them. The simulated trades will be compared to actual historical returns and the strategy performance will be evaluated.
It’s important to remember that past performance does not guarantee future results and that backtesting does not take into account all the risks of real-world trading. However, backtesting allows you to check how the strategy behaves over the course of history and how it could have performed if it would have been used in the past. This can give you an idea of how the strategy might perform in the future, and whether it’s worth implementing in a live trading scenario.
So, in essence it is using backtesting on set of assets and adjusting the fraction of each of them up and down randomly to find the best overall gains of the entire portfolio.
Yes, that’s correct! Essentially, backtesting a fractional Kelly strategy involves using historical data on the returns and risks of the assets in your portfolio, and then using a simulation algorithm to generate different portfolios. The algorithm will make random adjustments to the fraction of capital invested in each asset and will evaluate the expected return of each portfolio generated. The goal of this process is to find the portfolio that will generate the best overall gains by adjusting the fraction of capital invested in each assets, that is the one that will have the highest expected return.
In practice, the algorithm will explore the portfolio space by making random changes to the portfolio, according to the constraints and the Kelly strategy, and it will keep track of the best portfolio so far. The algorithm will repeat this process multiple times until it reaches a predefined criteria to stop, such as reaching a certain number of iterations or reaching a level of portfolio performance.
It’s worth noting that the algorithm can explore a large number of different portfolio combinations and therefore it can be a computationally expensive process. But, it allows to find the optimal portfolio given the conditions of the assets, constraints, and the Kelly strategy, and it can also provides insights on how the assets are behaving together.
If the algorithm finds the best returns isn’t this equivalent to the Kelly strategy.
Not exactly, while the fractional Kelly strategy and the algorithm I’ve described are related, they are not the same thing.
The Kelly strategy is a mathematical formula that tells you the optimal percentage of your capital to invest in each asset, given the expected returns and risks of the assets. The formula is based on the idea that you want to maximize your long-term growth rate, and it assumes that you have a perfect estimate of the expected returns and risks of each asset.
The algorithm I described, on the other hand, is a process for finding a good portfolio using the constraints and the Kelly strategy, but it doesn’t follow the formula exactly. Instead, it uses historical data and simulations to estimate the expected returns and risks of the assets, and it explores different portfolio combinations to find the one that generates the highest return. It may find a portfolio that is similar or equivalent to what the Kelly formula would suggest, but it is not guaranteed to do so.
It’s worth noting that the Kelly strategy can be challenging to implement in practice because of the difficulty to estimate the expected return and risk for each asset and because of the Kelly assumption that an individual can reinvest the profits and have no constraints on the bet size. On the other hand, the fractional Kelly algorithm is a more practical approach that can handle different constraints and real-world scenarios, such as leverage and short selling constraints.
Is there a simple example that can be shown with Python code?
Here is a simple example of how you might use Python code to backtest a fractional Kelly strategy:
import numpy as np
# Example data: historical returns and volatilities of three assets
returns = np.array([[0.02, 0.03, 0.01], [0.05, 0.02, 0.04], [0.04, 0.06, 0.03]])
volatilities = np.array([[0.01, 0.02, 0.005], [0.02, 0.01, 0.03], [0.01, 0.03, 0.02]])
# Initialize portfolio with equal fractions of capital invested in each asset
portfolio = np.array([1/3, 1/3, 1/3])
# Set the Kelly fraction (fraction of capital to invest in risky assets)
kelly_fraction = 0.8
# Set the number of simulations to run
num_simulations = 1000
# Set the fractional Kelly fraction
frac_kelly = 0.6
# Initialize a variable to keep track of the maximum portfolio return
best_return = -np.inf
# Run the simulations
for i in range(num_simulations):
# Randomly adjust the portfolio fractions
new_portfolio = portfolio + np.random.normal(size=3) * 0.01
# Make sure the fractions are between 0 and the Kelly fraction
new_portfolio = np.maximum(new_portfolio, 0)
new_portfolio = np.minimum(new_portfolio, frac_kelly)
# Make sure the fractions sum to 1
new_portfolio = new_portfolio / new_portfolio.sum()
# Calculate the expected return and volatility of the new portfolio
expected_return = (new_portfolio * returns).sum()
expected_volatility = (new_portfolio * volatilities).sum()
# Update the best portfolio if the new one has a higher return
if expected_return > best_return:
portfolio = new_portfolio
best_return = expected_return
# Print the optimal portfolio fractions
print("Optimal portfolio fractions:", portfolio)
This code is a simple example that shows how you can use Python to backtest a fractional Kelly strategy. The example uses historical returns and volatilities of three assets. The code set the Kelly fraction to 0.8 and the fractional Kelly fraction to 0.6 and runs the simulation 1000 times.
It starts with an initial portfolio that has equal fractions of capital invested in each asset, and then makes small random changes to the portfolio in each simulation. It then checks whether the new portfolio has a higher expected return and if so, it keeps the new portfolio and continue the process. After all the simulations are done, the optimal portfolio fractions are printed.
Keep in mind that this is a very simple example, and in practice, you would need to use more sophisticated methods to estimate the expected returns and risks of the assets, and you would need to use more realistic constraints. Additionally, you would want to test the performance of the algorithm with a variety of scenarios and parameters to validate the strategy.
Is it possible to use a genetic algorithm for this instead of Monte Carlo simulation?
Yes, it is possible to use a genetic algorithm for backtesting a fractional Kelly strategy instead of a Monte Carlo simulation. A genetic algorithm is a type of optimization algorithm that is inspired by the process of natural selection.
Just like the Monte Carlo simulation algorithm that I described earlier, a genetic algorithm would be used to explore the portfolio space and find the portfolio that generates the highest expected return. However, instead of making random changes to the portfolio, a genetic algorithm uses the principles of natural selection to evolve a population of portfolios over time.
Here’s a general overview of how a genetic algorithm for backtesting a fractional Kelly strategy might work:
- Start with an initial population of portfolios that are randomly generated and satisfy the constraints of the problem.
- Evaluate the fitness of each portfolio in the population. This is usually done by calculating the expected return of each portfolio and use it as the fitness value
- Select the best portfolios from the population to “breed” new portfolios. The selection process is usually based on the fitness value of each portfolio.
- “Breed” new portfolios by combining the characteristics of the selected portfolios through crossover and mutation operators.
- Repeat the process of selection, breeding, and mutation until the performance of the population reaches a predefined stopping criteria.
- Return the best portfolio of the final population.
A genetic algorithm can be a powerful optimization technique, and it has the ability to find globally optimal solutions to problems that are difficult to solve using other methods. However, it also has its limitations, such as the difficulty of determining the optimal configuration of parameters and the risk of getting stuck in local optima.
It is also worth noting that, similar to the Monte Carlo simulation, a genetic algorithm would still need to estimate the expected return and risk of each assets, and it would also need to be validated and tested thoroughly with a variety of scenarios and parameters before it is used in a live trading scenario.
