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I studied Kelly portfolio and tangent portfolio and found that they have the same weights. But the empirical studies that I have seen so far show that Kelly portfolio has a smaller number of stocks with higher returns and higher variances. I do not understand why this happens, because the empirical study has to be based on theoretical results. Is there any one who understands this phenomenon?

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So to begin with the terms mentioned ,The Kelly portfolio is based on optimizing the growth rate of wealth over the long term, considering the probabilities of different outcomes. On the other hand, the tangent portfolio is a part of the Capital Market Line (CML) that represents the best combination of the risk-free asset and risky assets, aiming to maximize the risk-adjusted return.

I think the Kelly portfolio and the tangent portfolio can have the same weights but only under certain conditions. This could occur when the marke's risk-free rate aligns with the expected return of the risky assets, making the tangent portfolio's weight distribution similar to that of the Kelly portfolio.

However, your confusion arises from the empirical studies that show a different scenario. It's important to note that empirical studies are based on real-world data, which may not always perfectly align with theoretical assumptions. Factors such as market dynamics, investor behavior, and unexpected events can influence portfolio performance in ways that theoretical models might not capture accurately.

The discrepancy between theoretical expectations and empirical observations could be due to various reasons:

Market Dynamics: Real-world market conditions can vary significantly from theoretical models. Factors like market volatility, liquidity constraints, and changing economic conditions can impact portfolio behavior.

Data Limitations: Empirical studies are based on historical data, which might not perfectly represent future market behavior. The dataset's scope, quality, and the period considered can affect results.

Behavioral Biases: Investor behavior and sentiment can lead to deviations from theoretical expectations. Investors may not always act rationally or in accordance with theoretical models.

Model Simplifications: Theoretical models often make assumptions to simplify complex realities. These assumptions might not hold true in all circumstances.

Market Anomalies: Empirical studies might uncover anomalies or patterns that challenge theoretical predictions, leading to further research and exploration.

I hope this helps :)

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