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Steve Paulone Facilitator

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Standard Deviation in Risk Measurement Expected returns on investments are derived from various numerical results from a business which are collected over time.The distribution in any large sample or population becomes a normal distribution (bell curve).

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Bell Curve (Normal Distribution) All data points fall under the curve. The average response is also the most numerous and it falls in the center of the distribution (mean=median=mode). Since the majority of responses are expected at the median (middle) this becomes the point from which we measure deviation from and the spread of that deviation is called the standard deviation (sigma). As we deviate from the median (50% mark) we move farther away from the average or expected outcome.

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Standard Deviation The mean (expected value) is the sum of the expected values times the probability of occurrence. The standard deviation is the square root of the sum of the possible values for a variable minus the expected value times the probability of occurrence of the variable squared.

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Standard Deviation (Beta) One way to measure risk is to compute the standard deviation of a variables distribution of possible values. The standard deviation is a numerical indicator of how widely dispersed the possible values are around a mean. The more widely dispersed a distribution is, the larger the standard deviation, and the greater the probability that the value of a variable will be greatly different than the expected value. The standard deviation, then, indicates the likelihood that an outcome different from what is expected will occur.

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Beta & CAPM things to consider Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index unique risk Total risk = Systematic + Unsystematic (Unique) Greater levels of risk aversion lead to larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

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Beta Beta is nondiversifiable risk from the marketplace. It is understood that the marketplace has a beta of 1.0 as the ultimate group of diversified assets. Risk-free portfolios have a beta of 0 since the return does not deviate from expectation. The more return of the portfolio fluctuates relative to the market the higher the beta. Companies with risk equal to the marketplace will have a beta of 1 while those that are less risky than the market will be lower than 1 and those with more risk will be higher than 1.

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CAPM The Capital Asset Pricing Model (CAPM) is used to calculate the appropriate required rate of return for an investment project given a degree of risk as measured by beta.

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Assumptions of the CAPM Investors evaluate portfolios by looking at the expected returns and standard deviations of the portfolio over a one period horizon. Investors are never satisfied. When given portfolios with the same standard deviation they will choose the one with higher return. Investors are risk averse When given portfolios with the same return they will choose the one with lower standard deviation. There is a risk-free rate at which an investor may either lend (that is invest) or borrow money. The risk free rate is the same for all investors. Information is freely and instantly available to all investors. Investors have homogeneous expectations about returns, standard deviation and covariance of securities.

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CAPM The three factors to consider in this equation are: The risk-free rate of return The required rate of return on the overall market the investments beta (risk) You remember from previous discussions that the risk-free rate is the rate of return investors demand for a no risk investment usually represented by US long term treasury Notes and inflation. The required rate of return of the overall market minus the risk free rate represents the additional return required by investors for investing in the market. Called the market risk premium.

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CAPM Beta = nondiversifiable risk or systematic risk So to compute the CAPM we: Risk free rate + [(market risk - risk free rate) x beta] The CAPM helps measure portfolio risk and the return an investor can expect for taking that risk.

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CAPM formula

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CAPM