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.

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