Yale Prep Risk Measurements

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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