# The Empirical Differences Between Kelly and Tangent Portfolios

When it comes to portfolio management, two popular strategies that often come up in discussions are the Kelly criterion and the tangent portfolio approach. Both aim to optimize investment returns, but they differ in their underlying principles and empirical outcomes.

The Kelly criterion, developed by John Larry Kelly Jr. in the 1950s, is based on the concept of maximizing the geometric mean of portfolio returns over the long term. It takes into account both the expected return and the risk associated with each investment. The formula for calculating the optimal fraction of wealth to allocate to each investment is:

*f _{opt} = (bp – q)/b*

Where:

*f*is the optimal fraction of wealth to allocate to the investment_{opt}*b*is the net odds offered by the investment*p*is the probability of success*q*is the probability of failure

The Kelly criterion suggests that investors should allocate a fraction of their wealth to each investment based on the ratio of the investment’s expected return to its risk. This means that investments with higher expected returns and lower risks should receive a larger allocation.

On the other hand, the tangent portfolio approach, also known as mean-variance optimization, was developed by Harry Markowitz in the 1950s. It is based on the principle of maximizing the portfolio’s expected return for a given level of risk, or minimizing the portfolio’s risk for a given level of expected return. The optimal portfolio is found at the point where the tangent line to the efficient frontier is tangent to the investor’s indifference curve.

While both the Kelly criterion and the tangent portfolio approach aim to optimize investment returns, they differ in their assumptions and empirical outcomes. The Kelly criterion assumes that the probabilities of success and failure are known with certainty, which is often not the case in practice. It also assumes that investors have a logarithmic utility function, meaning that they are risk-neutral and only care about maximizing long-term wealth accumulation.

The tangent portfolio approach, on the other hand, does not make any specific assumptions about the probabilities of success and failure. It takes into account the expected returns and variances of the individual assets in the portfolio to find the optimal allocation. This approach is more flexible and can be applied to a wide range of investment scenarios.

Empirically, the Kelly criterion has been shown to outperform the tangent portfolio approach in certain situations, especially when the probabilities of success and failure are known with certainty and the investor has a logarithmic utility function. However, in real-world scenarios where these assumptions do not hold, the tangent portfolio approach may perform better.

In conclusion, while Kelly and tangent portfolios may have the same weights under certain assumptions, they differ empirically in terms of their underlying principles and outcomes. The Kelly criterion is based on maximizing the geometric mean of portfolio returns, while the tangent portfolio approach aims to maximize expected return for a given level of risk. Understanding the differences between these two strategies can help investors make informed decisions about portfolio management.