Module 7 var analytical method step by step
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This document explains how to calculate Value at Risk (VaR) using the analytical or variance-covariance method. It first converts annual return statistics to daily values. It then explains that the analytical method assumes returns are normally distributed. It describes converting the return to a standard normal variable to use probability tables. The document shows how to calculate 5% and 1% VaR by finding the critical z-value that corresponds to the bottom 5% and 1% of the normal distribution. It provides an example calculating the dollar value of daily VaR for a $10 million portfolio.
Module 7 var analytical method step by step
- 1. Risk Management: Value at Risk (VaR) Analytical Method (Variance/Covariance Method) Dr. Nivine Richie
- 2. Analytical Method FIRST, convert the annual statistics to daily values Daily expected return = annual expected return divided by 250 trading days in a year = (0.090/250) = 0.00036 or 0.036% per day Daily standard deviation = annual standard deviation divided by the square root of 250 trading days in a year = 0.155 250 = 0.00980 or 0.098% per day
- 3. The Normal Distribution The analytical method assumes that returns are normally distributed with a mean (i.e. an expected return) and a standard deviation (i.e. risk, or volatility)
- 4. The Normal Distribution To calculate the VaR, we must first convert the return (our random normal variable) into a standard normal variable (i.e. a “z” statistic) so that we can use standard normal probability tables
- 5. The Normal Distribution Any random normal variable X with a mean of μ and a standard deviation of 𝜎 can be converted to a standard normal variable, or Z, using this equation 𝑍 = 𝑋 − μ 𝜎
- 6. The Normal Distribution Converting to a standard normal variable is handy because the Z statistic has a mean of zero and a standard deviation of 1 and allows us to easily apply probability tables.
- 7. The Normal Distribution In our case, the variable X is the portfolio return, and the mean is the expected return. To find the 5% value at risk, we want to know what return would correspond to a 5% probability of losing at least that value at risk. In other words, we want to know the return that corresponds to the bottom 5% of the distribution
- 8. Understanding VaR Z=1Z= -1.65 Area under the curve to the left of this critical Z represents 5% probability
- 9. 5% Value at Risk To find the return that corresponds to the bottom 5% of the distribution, we are looking for the return that produces a –1.65 critical z-value 𝑍 = 𝑋 − μ 𝜎 – 1.65= 𝑉𝑎𝑅−𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
- 10. Value at Risk Solving for VaR at the 5% level, we have: Many times, VaR is converted to a dollar value 𝑉𝑎𝑅 = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 − 1.65(𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣) 𝑉𝑎𝑅 = 0.00036 − 1.65(0.0098) 𝑽𝒂𝑹 = −𝟎. 𝟎𝟏𝟓𝟖𝟏 𝒐𝒓 − 𝟏. 𝟓𝟖𝟏% 𝑉𝑎𝑅 = −0.01581 × $10,000,000 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 −$𝟏𝟓𝟖, 𝟏𝟎𝟎
- 11. Value at Risk To solve for VaR at the 1% level, we need to know that the critical Z-value is 2.33 (this would be given to you) 𝑉𝑎𝑅 = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 − 1.65(𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣) 𝑉𝑎𝑅 = 0.00036 − 2.33(0.0098) 𝑽𝒂𝑹 = −𝟎. 𝟎𝟐𝟐𝟓 𝒐𝒓 − 𝟐. 𝟐𝟓% 𝑉𝑎𝑅 = −0.0225 × $10,000,000 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 −$𝟐𝟐𝟓, 𝟎𝟎𝟎
- 12. Interpreting Value at Risk The portfolio faces a 5% probability of losing at least $158,100 in any one day The portfolio faces a 1% probability of losing at least $225,000 in any one day