Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

Measure of Risk of any Security in Equilibrium


The standard Capital Asset Pricing Model (CAPM) is an equilibrium model which construction allows us to determine the relevant measure of risk for any asset and the relationship between expected return and risk for asset when market are in equilibrium. Within CAPM the expected return on security $i$ is given by:
$$
\bar{R}_i = R_F + \beta_i(\bar{R}_M-R_F)
$$ where $\beta_i$ is argued to be the correct measure of a security’s risk in a very well-diversified portfolios. For such portfolios the non-systematic risk tends to go to zero and the reminder – the systematic risk – is measured by $\beta_i$. For the Market Portfolio its Beta is equal one. Therefore, the aforementioned equation defines the security market line.

It is possible to go one step further and write the same CAPM formula as follows:
$$
\bar{R}_i = R_F + \left( \frac{\bar{R}_M-R_F}{\sigma_M} \right) \frac{\sigma_{iM}}{\sigma_M}
$$ what keeps its linear relationship between the expected return but in $\sigma_{iM}/\sigma_M$ space. Recall that the term in braces is the market price of the risk at riskless rate of interest given by $R_F$. $\sigma_{iM}$ denotes the measure of the risk of security $i$ but if left defined as $\sigma_{iM}/\sigma_M$ is provide us with more intuitive understanding itself as the measure of how the risk on a security affects the risk of the market portfolio.

In other words, $\sigma_{iM}/\sigma_M$ is the measure of risk of any security in equilibrium and, as we will show further below, it is equal:
$$
\frac{\sigma_{iM}}{\sigma_M} = \frac{X_i^2\sigma_i^2 + \sum_{j=1,\\ j\ne 1}^{N} X_j \sigma_{ij}} {\sigma_M}
$$ We may get that performing a calculation of the first derivative of the standard deviation of the market portfolio $\sigma_M$, i.e. for
$$
\sigma_M^2 = \sum_{i=1}^{N} X_i^2\sigma_i^2 + \sum_{i=1}^N \sum_{j=1, j\ne i}^{N} X_iX_j \sigma_{ij}
$$ the risk of a security is the change in the risk of the market portfolio. as the holdings of that security are varied:
$$
\frac{d\sigma_M}{dX_i} = \frac { \left[ \sum_{i=1}^{N} X_i^2\sigma_i^2 + \sum_{i=1}^N \sum_{j=1, j\ne i}^{N} X_iX_j\sigma_{ij} \right]^{1/2} }
{ dX_i }
$$
$$
= \frac{1}{2} \left[ 2X_i\sigma_i^2 + 2 \sum_{j=1, j\ne 1}^{N} X_j\sigma_{ij} \right] \times \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left[ \sum_{i=1}^{N} X_i^2\sigma_i^2 + \sum_{i=1}^N \sum_{j=1, j\ne i}^{N} X_iX_j\sigma_{ij} \right]^{-1/2}
$$
$$
= \frac{X_i^2\sigma_i^2 + \sum_{j=1, j\ne 1}^{N} X_j \sigma_{ij} } {\sigma_M}
= \frac{\sigma_{iM}}{\sigma_M}
$$ Above, $\sigma_{ij}$ denotes a covariance between any $i$ and $j$ security in the market portfolio and $X_i$ is the share of capital invested in $i$’s security.

Interestingly, the only and remaining question is: how do we know what is best choice of the market portfolio? I will try to address a separate post around the research one may conduct covering that topic. As for now, one may assume that S&P500 or S&P900 universe of stocks would serve here as a quite good example. If so, that would also mean that in order to find $\sigma_{iM}/\sigma_M$ for any security we invested our money in, we need to find $\sigma_M$. Is it hard? Well, if we assume the market to be S&P500 then for any $i$ its $X_i=1/500$. Now, all we need to care is to have an idea about the security’s variance $\sigma_i^2$ and the measure of covariance, i.e. how two securities like each other in out market portfolio. A clever reader would ask immediately: But what about a time-window? And that’s where the devil conceals itself.

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