# Beta Return

### Overview of Beta Returns Beta returns are derived from temporary fluctuations in the market.
There is an equation for calculating portfolio return.
$\text R={\text R}_\text f+{\beta}({\text R}_\text m-{\text R}_\text f)+{\alpha}$
Where:
\begin{aligned}\text R&=\text{Overall Portfolio Return}\\{\text R}_{\text f}&=\text{Risk-Free Return}\\\text{Beta Return}\;(\beta)&=\text{Market Volatility}\\{\text R}_{\text m}&=\text{Market Rate of Return}\\\alpha&=\text{Alpha Return}\end{aligned}
In the equation for portfolio return, $\alpha$ is the alpha return, or nonsystemic risk, and $\beta$ is the beta return. Alpha return is an excess return of an investment over a benchmark or index. Beta return is the measure of the volatility of the return of an investment, creating an opportunity for a quick return or loss. Beta is a coefficient of variation (CV) between the benchmark and the volatility. Coefficient of variation is a measure of the dispersion of the data used in statistics. (${\text R}_{\text m}-{\text R}_{\text f}$) is the market risk premium, which is the risk of the portfolio compared to risk-free rates. Because the beta is multiplied by the market risk premium, a beta of one would mean the security's price moves with the changes in the market; the portfolio moved as expected. Therefore, a beta of less than one indicates that the security's volatility is less than the volatility of the overall market. Furthermore, a beta greater than one indicates that the security has volatility greater than the overall market's volatility. Mathematically, the portfolio return equation can be depicted as a straight line graphed with the excess over the portfolio on the y-axis and the excess over the market on the x-axis. An excess, or excess return, is a return that surpasses the riskless rate on a security. The beta is the slope of the line as the asset's value increases compared to the overall market. The beta is a measure of volatility because calculating it gives the standard deviation, the statistical measurement of the distribution or spread of a data set, between the portfolio and the overall market. The standard deviation of the market returns would measure the overall spread of the samples, or the data used, in the market. If the overall market has an average return of 12 percent and the individual points that make up that average have a wide range above and below this percentage, the standard deviation would be high, and therefore the volatility would be high as well. If the data points are all close to the 12 percent return, then the standard deviation is low, and the volatility is low. As a ratio, a beta of one would mean the portfolio and the market benchmark for the market as a whole have the same standard deviation. A beta of two would mean the portfolio is twice as volatile as the market.

#### Regression Line for Alpha The rate over excess equation describes a line where the y-intercept is the alpha and the slope is the beta.

### Risk Profile for Beta Return Portfolios Beta returns give investors the opportunity to make large sums of money, but investors can also lose a great deal.

A risk profile is an analytical evaluation of an investor's acceptance of risk. It is used to develop a portfolio with the best balance between risk and return, as not all investors have the same reactions to risk. The risk profile will match the investing style with the level and type of risk the investor is willing to tolerate. Alpha and beta risk can be looked at differently if they are taken as historical measures that are related to each other. Alpha is the historical deviation of the return of an asset and the expected return, and beta is the historical move of the asset away from the benchmark point. If an investor has stock, the beta is the risk that the stock's value will change compared to the overall market. The alpha tells the investor how well the investment manager is doing compared to the market benchmark.

Cash, fixed interest, property and international shares are all different investment options that have varying levels of risk associated with them. Cash, fixed interest and property are less risky investments than international shares due to many risks such as geo-political and tax that may be specific to a country. Thus, when there is low risk, the return may not be as high. With greater risk comes greater return, but the risk must be a calculated risk to achieve expected returns.

#### Risk-Return Profile Investment managers use the risk-return profile to balance risk tolerance with the return. Cash, fixed interest, property and international shares have varying levels of risk with cash and fixed interest being the least risky investment options and international shares carrying the highest level of risk. Relative risk is also associated with the time that the investment is held.
Using these simplified definitions, risk is then a factor of the time that the asset will be held. However, while time will be a factor of risk there will always be other variables such as geo-political, tax, inflation, gross domestic product, and other economic or market and industry risks associated with any expected rate of return. Thus, if the investor must maintain relatively liquid assets, or assets that can be easily converted to cash, then the volatility associated with beta risk will be a problem.

For example, if Mary purchased shares in Big Systems Inc. at an above market price, and the shares then experience a rapid dip in price, Mary will not want to sell until the price increases so that there will not be any share, or profit loss. If she cannot tolerate loss, the stock will not be sold and will have a negative impact on Mary’s total portfolio meaning that any profit made with other equity or debt will be impacted due to the above price purchase of Big Systems Inc. stock. If she needs to make money, then alpha risk must account for volatility in order for the stock to make more money than can be gained by normal market movement. In this situation, the risk of low return is part of Mary’s risk profile.

Retirement investments are usually not within the same category due to age and need. This means that retirees are less inclined to take risks and are in greater need of liquid assets to assure they have enough cash, or cash equivalents, on hand to meet their needs. Thus, Mary’s needs will be different and when considering the concepts of alpha and beta risk age, need, and overall cash and investment needs must be considered to meet specific return criteria while always keeping in mind the risk factors and how varied risks can negatively impact returns.

### Beta Return Calculations Beta returns can be calculated using a set of data points.
Volatility can be used to give the formula for beta.
$\text {Beta}=\frac{\text{Covariance of the Return of the Asset Compared to the Return of the Market}}{\text{Variance of the Return of the Market}}$
Variance is a measurement of the spread of data points. Variance shows the volatility of an investment, or how far an investment moves compared to its mean. The mean is the average of all of the assets or the average of a single asset over time. Mary has a portfolio that is composed of stocks. One of the stocks is in Big Systems Inc., which has a mean price of $30. If the price rises to$50, the difference between the mean and the new higher price is its variance, \$20. This example only refers to one point in time; a good analysis will refer to many points, such that the square root of the standard deviations is the variance, typically written as ${\sigma}$.

Covariance is the measure of how two assets move together. Correlation is the correspondence of movement between one variable and another, showing that there is a link between the two. A positive correlation is a relationship between two variables in which one increases when the other increases. A negative correlation is a relationship between two variables in which one increases when the other decreases.

Like the variance, the covariance for multiple comparisons becomes a standard deviation. Using standard deviations, the beta formula changes.
$\text {Beta}=\text{Correlation Between Returns}\times\left(\frac{\sigma\;\text {of Individual Return}}{\sigma\;\text {of Market Return})}\right)$
As an example, Mary wants to calculate the beta for investment in Big Systems Inc. stock compared to the S&P 500 index. The first thing Mary would do is look at the last five years of the stock and calculate the correlation between the Big Systems Inc. stock and the S&P 500 index. Assume that she calculated the correlation to be 0.79. Next, Mary calculates the standard deviation of return for the stock to be 21 percent, with the standard deviation of the index as 31 percent. Individual return is the average return for an individual stock for a particular company. Market return is the average return for an index such as the S&P 500, for example. The difference between beta and standard deviation is that a beta measures the risk of the market as a whole and standard deviation measures the risk of individual stocks. As a result, there is a correlation between beta and standard deviation because to arrive at the overall risk of any stock, or company, also referred to as standard deviation, we first must understand overall market risk, beta, and how that risk affects the individual company. By providing a risk correlation on the macro, or market level, to the micro, or individual company, we arrive at proper risk calculations. Using this, the beta is $0.79\times(0.21/0.31)=0.54$. This means that this stock is less volatile than the index. It has approximately one-half the volatility of the index.