Tangent Portfolio and Return Models in Computational Finance

In computational finance, tangent portfolio and return models play a crucial role in analyzing and managing investments. These models help investors make informed decisions by providing a deeper understanding of portfolio behavior.

A tangent portfolio is a mathematical concept that represents the optimal portfolio of assets, given a set of constraints and objectives. This concept is based on the idea of finding the portfolio that maximizes returns while minimizing risk.

Tangent portfolio models can be used to analyze the behavior of complex financial systems, such as those involving multiple assets and constraints. By using these models, investors can identify potential risks and opportunities in their portfolios.

Tangent portfolio models are often used in conjunction with return models, which help to estimate the expected returns of a portfolio. By combining these models, investors can gain a more complete understanding of their portfolio's potential performance.

The tangency portfolio is a type of portfolio that maximizes the Sharpe ratio, which is a measure of risk-adjusted return. This portfolio is a key concept in modern portfolio theory.

In order to compute the closed-form expression of the tangency portfolio weights, we need to consider different underlying return models. The agnostic return model, for example, does not distinguish between any of the securities in regard to expected return and covariance structure.

The agnostic return model assumes that the expected return vector is μ = μ01, where μ0 is a positive constant, and the covariance structure is defined by Σij = σ02{1 if i=j, ρ else}. This model implies that the correlation matrix is compound symmetric.

The return model with equal expected returns does not distinguish between any of the securities in regard to expected return, implying that the tangency portfolio coincides with the minimum volatility portfolio.

The tangency portfolio weights under the agnostic return model are given by the equal-weight portfolio also known as the 1/n-portfolio. This is because the relaxed tangency portfolio weights are given by wrtp ∝ Σ-11, and since all elements of Σ-11 are positive, the relaxed tangency portfolio is equal to the proper tangency portfolio.

The relaxed tangency portfolio weights are given by wrtp ∝ Σ-11, but in high-dimensional settings, some of the elements of Σ-11 may be negative, making it difficult to obtain a closed-form solution for the proper tangency portfolio weights.

Here are some key characteristics of the tangency portfolio under different return models:

The tangency portfolio is a powerful tool for portfolio optimization, as it allows investors to maximize their risk-adjusted returns. However, the choice of return model is critical in determining the optimal portfolio weights.

The agnostic return model, where all securities have equal expected returns and covariance structure, results in an equal-weight portfolio, also known as the 1/n-portfolio. This is because the expected return vector is given by μ = μ01 and the covariance structure is defined by Σij = σ02{1ifi=jρelse.

The return model with equal expected returns leads to the tangency portfolio coinciding with the minimum volatility portfolio. The tangency portfolio weights under this model are given by the equal-weight portfolio, wtp = 1/n.

The shape of the covariance matrix is denoted compound symmetric. This type of matrix has all elements on the main diagonal equal, and all elements on the off-diagonal equal. In the agnostic return model, the covariance matrix is compound symmetric since all volatilities are equal.

Here's a summary of the return models:

Explicit weights underlying return models can be computed using closed-form expressions, taking into account different return models for expected returns and covariance matrices.

The agnostic return model and the return model with equal expected returns are two naive models that yield well-known tangency portfolio weights. The 1/n–portfolio and the minimum volatility portfolio are examples of smart beta products that coincide with these weights.

The shape of the covariance matrix is crucial in determining the tangency portfolio weights. A compound symmetric correlation matrix is one such shape, where the correlation between each pair of assets is the same.

The formula for the compound symmetric correlation matrix is given by: (C-11)i=(1+(n-2)ρ(1-ρ)(1+(n-1)ρ))(1-(n-1)ρ1+(n-2)ρ)=11+(n-1)ρ>0, where ρ is the correlation coefficient.

With this formula, we can see that the correlation coefficient ρ plays a key role in determining the shape of the covariance matrix and, subsequently, the tangency portfolio weights.

The agnostic return model is a simple yet powerful approach to investing. It assumes that all securities have the same expected return and covariance structure, but acknowledges that some investments are riskier than others.

In this model, the expected return vector is given by μ = μ0, where μ0 is a parameter representing the expected return. The covariance structure is defined by a compound symmetric matrix, which means that all volatilities are equal.

The correlation matrix is also compound symmetric, and the inverse of the covariance matrix can be calculated using the Sherman-Morrison matrix inversion formula. This results in a compound symmetric matrix with specific elements.

The product of a compound symmetric matrix and a vector of ones is a vector of equal-sized elements, and the sign of each element is positive due to the positive definiteness of the matrix. This implies that all elements of the inverse of the covariance matrix are positive.

The relaxed tangency portfolio for the agnostic return model is given by a vector of equal weights, where each weight is 1/n. This portfolio is also known as the 1/n-portfolio.

In an equal returns model, the investor doesn't distinguish between securities when it comes to expected return. This means that the tangency portfolio is essentially the same as the minimum volatility portfolio.

The Sharpe ratio is maximized when the variance is minimized, and the tangency portfolio coincides with the minimum volatility portfolio. This is because the numerator (wμ - rf) is equal to a fixed positive number, independent of the weight vector due to the sum-to-one constraint.

The relaxed tangency portfolio weights are given by wrtp ∝ Σ-11, where Σ is the covariance matrix. This is a result of Theorem 0.1.

If all elements of the relaxed tangency portfolio are non-negative, then the relaxed tangency portfolio is equal to the proper tangency portfolio. However, in high-dimensional settings with complex covariance matrices, some elements of Σ-1 may be negative, and the relaxed and proper tangency portfolios differ.

A key takeaway is that the equal returns model leads to a tangency portfolio that is also the minimum volatility portfolio. This is a simple and intuitive result that highlights the importance of considering the covariance matrix in portfolio optimization.

A tangent portfolio can be a game-changer for investors, allowing them to diversify their holdings and potentially reduce risk.

By incorporating a mix of asset classes, such as stocks, bonds, and commodities, a tangent portfolio can help smooth out returns and provide a more stable overall performance.

One key aspect of a tangent portfolio is its ability to generate returns through different market conditions, which can be particularly beneficial during times of market volatility.

As we've seen in the article, a tangent portfolio can be created by combining a core portfolio with a satellite portfolio, which can be invested in a variety of assets.

The relaxed tangency portfolio and the proper tangency portfolio can have significant differences, especially when it comes to concentration. In one example, the relaxed portfolio features a substantial number of constituents with a weight deviating from zero.

The proper tangency portfolio, on the other hand, allocates weight to a small proportion of constituents, with only 142 out of 1508 constituents having a weight above 0.001bps in a specific sample. This is because the optimal weights are found by a numerical inner point optimization algorithm, and weights less than 0.001bps are considered negligible.

In contrast, the relaxed portfolio allows for short-selling, resulting in a non-negligible portion of the capital being invested in every one of the constituents. This deviance between the two portfolios highlights the importance of considering non-negativity constraints in portfolio optimization.

The alternative derivation of the tangency portfolio is a clever way to find the optimal portfolio weights.

This method starts by considering portfolios of risky assets and a risk-free asset, with the goal of achieving a target excess return. The portfolio return is expressed as a linear combination of the risky asset returns and the risk-free return.

The excess return of the portfolio is the difference between its return and the risk-free return, and it's calculated as the dot product of the portfolio weights and the excess return vector. The expected excess return and portfolio variance are also expressed in terms of these vectors.

By defining new variables to simplify the notation, we can re-express the portfolio return and excess return equations.

To find the minimum variance portfolio that achieves the target excess return, we need to solve a minimization problem subject to a constraint on the portfolio's excess return.

The Lagrangian function is used to incorporate the constraint into the optimization problem, and the first-order conditions for a minimum are derived.

From these conditions, we can solve for the portfolio weights in terms of a Lagrange multiplier.

The Lagrange multiplier is then solved for using the constraint equation, and the portfolio weights are expressed in terms of the target excess return and the inverse of the covariance matrix.

The tangency portfolio is a special case of this solution, where the portfolio is 100% invested in risky assets.

The excess return of the tangency portfolio is proportional to the ratio of the dot product of the inverse of the covariance matrix and the excess return vector to the dot product of the inverse of the covariance matrix and the ones vector.

This result is a key insight into the properties of the tangency portfolio, and it's a crucial step in understanding the behavior of optimal portfolios.

An efficient portfolio is one that provides the highest possible return for a given level of risk. This is achieved by combining assets in a way that maximizes the Sharpe ratio, which is a measure of return relative to risk.

The Sharpe ratio is calculated by subtracting the risk-free rate from the expected return of a portfolio and dividing the result by the standard deviation of the portfolio. For example, if the risk-free rate is 3% and the expected return of a portfolio is 10%, the Sharpe ratio would be 0.7.

In the case of a portfolio with two risky assets and a risk-free asset, the efficient set of portfolios is a straight line in (μp, σp)-space with intercept r_f. The slope of the efficient set is such that it is tangent to the portfolio frontier constructed just using the two risky assets A and B.

The CAL (Capital Allocation Line) intersects the set of risky asset portfolios at point A, which has a Sharpe ratio of 0.562. However, we can do better by investing in a combination of assets A and B, such that the CAL is just tangent to the set of risky asset portfolios. This point is labeled "Tangency" on the graph and represents the tangency portfolio of assets A and B.

The tangency portfolio is the set of efficient portfolios consisting of T-Bills, asset A, and asset B. To determine the proportions of each asset in the tangency portfolio, we can solve the constrained maximization problem using the method of substitution with x_B = 1 - x_A.

The solution to this problem is x_A^tan = ((μ_A - r_f)σ_B^2 - (μ_B - r_f)σ_AB) / ((μ_A - r_f)σ_B^2 + (μ_B - r_f)σ_A^2 - (μ_A - r_f + μ_B - r_f)σ_AB), where x_B^tan = 1 - x_A^tan.

For example, using the example data in Table 11.1, we get x_A^tan = 0.4625 and x_B^tan = 0.5375. The expected return, variance, standard deviation, and Sharpe ratio of the tangency portfolio are:

The expected return of the tangency portfolio is 0.111, the variance is 0.01564, the standard deviation is 0.125, and the Sharpe ratio is 0.644.