Financial Market Analytics

Financial Market Analytics

Fabrizio Cominetti

In this group work, we want to better understand the structural characteristics that risk brings to real investment portfolios.

In order to understand this empirically, we need to build real portfolios that are concentrated/tilted with respect to a specific level and kind of risk.

Our reference will be the definition of "Total," "Systematic," and "Specific" risk as defined and introduced through the CAPM we studied. We will then rely on the SML to investigate the risk level of individual securities and then proceed to the subsequent groupings.

$E(r_i) = r_i + \beta_i (E(R_M) - r_f)$ [1]

In pursuit of the analysis the above SML equation, must be reconsidered in the form of a regression equation as in the Market Model, also known as single index model.

We can interpret such equation as the “ex-post” version of the SML.

$r_i = \alpha + \beta_i(R_M) + e_i$ → we excluded the expectations

it is usually applied, for equivalence, in excess returns form:

$r_i − r_f = \alpha_i + \beta_i(R_M − r_f) + e_i$ [2]

In this ex-post framework, we see two new parameters $\alpha$ and $e_i$

Given that even in the "ex-ante" version of the SML, the Beta is derived from time series, then the $\beta_{SML(i)} = \frac{COV(R_M, R_i)}{\sigma^{2}{M}}$ and the $\beta{MM}$ as the regression slope will match, but only if the time series are of same length.

In the setting of the Market Model we get two other coefficients, the $\alpha_i$ that should be zero if the CAPM hold and $e_i$ proxying the specific risk.

In addition, the equation [2], since it is a regression, yields an $R^2$ informing on the goodness of fit.

This way there are many possible profiles on which to do stock groupings. Firstly, the profile of fitting based on $R^2$. Then going in more detail the relations between total risk decomposed in specific risk and systematic risk:

$Total risk = Systematic Risk + Specific Risk$

$\sigma^{2}{i} = \beta^{2}{i} \sigma^{2}{M} + \sigma^{2}{ei}$ [3]

The ratio of (Systematic Risk)/(Specific Risk) is the $R^2$, but such a separation could lead to different characteristics in case of stocks with high or low level of total risk. The last profile interesting to investigate it is the one of excess returns represented by the $\alpha_i$. Such a return could be positive, null, or negative, but also significant or not as expressed by the specific t-test. Also, the ratio between this potential excess return and the total stock return could be of interest.

Finally, the return/risk profile, varying the adopted measure of return and risk should be of interest.

About

In this group work, we want to better understand the structural characteristics that risk brings to real investment portfolios. In order to understand this empirically, we need to build real portfolios that are concentrated/tilted with respect to a specific level and kind of risk.

The results showed that the portfolios created over-performed compared to the index during the period under consideration. However, the same results also showed higher volatility than that of the index.

Repository Overview

├── README.md
├── data
├── figures
├── notebooks
└── report