Learning Objectives
1.Define volatility measurements
2. Differentiate between volatility and downside risk
3. Quantify tail risk both statistically and historically
4. Analyze systematic risk (beta) and non- systematic risk (idiosyncratic)
volatility measurements
a) Standard deviation
b) Variance
c) Covariance
Standard Deviation
Standard deviation (SD) is represented by Greek letter sigma (?)
� Measure of volatility (i.e., risk)
� Measures amount of variation or dispersion from an average
� SD is considered a measure of "total risk"
� The SD of a random variable is the square roo
? is used in
SD is used in the
*Capital Allocation Line (CAL)
Sharpe Ratio, M2, information ratio*
Calculating Standard Deviation
arithmetic mean rate of returns
Subtract the mean rate of return from each year's returns
Square the differences
Take the square root of variance = standard deviation
Expected Return
E(r) = E p(s)r(s)
s
p(s) = probability of a state r(s) = return if a state occurs s = state
Variance (VAR)
Example of Variance
Returns for a stock are 10% in year 1, 20% in year 2 and -15% in year 3.
The average of these three returns is 5%.
The differences between each return and the average are 5%, 15%, and -20% for each consecutive year.
Squaring these devi
Standard Deviation (Stdev)
square root of the variance
StDev of risky asset A with risk-free asset
??P = ??A??A
Standard Deviation of two risky assets
SD Market Returns Expectations
approximately 68% of the returns will fall within a range that is within 1 standard deviation of the mean, 95% of the returns will fall within a range that is within 2 standard deviations of the mean, and 99% of the returns will fall between within a rang
Variance
Variance measures how far things are spread out
Covariance
Measures how much two random variables move or change together
??????A,B = ??A,B ??A??B
Covariance and Correlation
� Portfolio risk depends on the correlation between the returns of the assets in the portfolio
� Covariance and the correlation coefficient provide a measure of the way returns of two assets vary
Correlation Coefficients: Possible Values
If p = 1.0, the securities are perfectly positively correlated
If p= - 1.0, the securities are perfectly negatively correlated
Standard Deviation and downside risk
� We commonly use standard deviation to measure risk, but it really measures volatility (the variability of price movements).
� Since SD is a measure of total risk it includes deviations to the upside and downside.
� Investors don't see "upside" variation
Downside risk can be measured
downside deviation, information ratio, and value-at-risk.
Semi-variance
downside deviation
Semi variance is a measure of the dispersion of all observations that fall below the mean or target value of a data set. Semi variance is an average of the squared deviations of values that are less than the mean.
Sortino ratio
The Sortino ratio is the excess return over the risk-free rate divided by the downside semi-variance, and so it measures the return to "bad" volatility. (Volatility caused by negative returns is considered bad or undesirable by an investor, while volatili
Value-at-Risk (VaR)
Value at Risk (VaR) is a measure of the risk of investments. It estimates how much a set of investments might lose, given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry
Skewness
describes asymmetry of data points from a normal distribution; if data points are skewed to the left it is described as negative skew and if data points are skewed to the right it is described as positive skew
Kurtosis
measures the peakedness of a probability distribution or normal distribution curve; if kurtosis is positive (leptokurtic), the chart will show fat tails and a low, even distribution; if kurtosis is low (platykurtic), a chart will show skinny tails and dis
Systematic Risk
SystematicRisk - Beta
- Un-diversifiable
Unsystematic Risk
- Idiosyncratic
- Diversifiable
Beta
Remember
� Beta is used in Capital Asset Pricing Model (CAPM)
� Measures systematic (market) risk
� Represented by the Greek letter �
� Beta is calculated using regression analysis
� Beta indicates an asset's likelihood of moving up or down with the marke
beta formula
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