LO 1.2: Estimate VaR using a parametric approach for both normal and lognormal

LO 1.2: Estimate VaR using a parametric approach for both normal and lognormal return distributions.
In contrast to the historical simulation method, the parametric approach (e.g., the delta- normal approach) explicitly assumes a distribution for the underlying observations. For this LO, we will analyze two cases: (1) VaR for returns that follow a normal distribution, and (2) VaR for returns that follow a lognormal distribution.
Norm al VaR
Intuitively, the VaR for a given confidence level denotes the point that separates the tail losses from the remaining distribution. The VaR cutoff will be in the left tail of the returns distribution. Flence, the calculated value at risk is negative, but is typically reported as a positive value since the negative amount is implied (i.e., it is the value that is at risk). In equation form, the VaR at significance level a is:
VaR(a% ) = pp/L + CTP/L x za
where p and a denote the mean and standard deviation of the profit/loss distribution and 2: denotes the critical value (i.e., quantile) of the standard normal. In practice, the population parameters (i and a are not likely known, in which case the researcher will use the sample mean and standard deviation.
Example: Computing VaR (normal distribution)
Assume that the profit/loss distribution for XYZ is normally distributed with an annual mean of $ 13 million and a standard deviation of $ 10 million. Calculate the VaR at the 93% and 99% confidence levels using a parametric approach.
Answer:
VaR(5%) = -$15 million + $10 million x 1.65 = $1.5 million. Therefore, XYZ expects to lose at most $1.5 million over the next year with 95% confidence. Equivalently, XYZ expects to lose more than $1.5 million with a 5% probability.
VaR(l%) = -$15 million + $10 million x 2.33 = $8.3 million. Note that the VaR (at 99% confidence) is greater than the VaR (at 95% confidence) as follows from the definition of value at risk.
Now suppose that the data you are using is arithmetic return data rather than profit/loss data. The arithmetic returns follow a normal distribution as well. As you would expect, because of the relationship between prices, profits/losses, and returns, the corresponding VaR is very similar in format:
VaR(a%) = (pr + CTr x za ) x Pt-1
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2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Example: Computing VaR (arithmetic returns)
A portfolio has a beginning period value of $100. The arithmetic returns follow a normal distribution with a mean of 10% and a standard deviation of 20%. Calculate VaR at both the 95% and 99% confidence levels.
Answer:
VaR(l%) = (-10% + 2.33 x 20%) x 100 = $36.6
Lognormal VaR
The lognormal distribution is right-skewed with positive outliers and bounded below by zero. As a result, the lognormal distribution is commonly used to counter the possibility of negative asset prices (P ). Technically, if we assume that geometric returns follow a normal distribution (jiR, crR), then the natural logarithm of asset prices follows a normal distribution and P follows a lognormal distribution. After some algebraic manipulation, we can derive the following expression for lognormal VaR:
VaR(a%) = Pt1 x (l – e^R -^R *^)
Example: Computing VaR (lognormal distribution)
A diversified portfolio exhibits a normally distributed geometric return with mean and standard deviation of 10% and 20%, respectively. Calculate the 5% and 1% lognormal VaR assuming the beginning period portfolio value is $100.
Answer:
Lognormal VaR(5%) = 100 x (1 exp[0.1 0.2 x 1.65])
= 100 x (1 – exp[-0.23]) = $20.55
Lognormal VaR(l%) = 100 x (1 exp[0.1 0.2 x 2.33])
= 100 x (1 exp[0.366]) = $30.65
2018 Kaplan, Inc.
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Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Note that the calculation of lognormal VaR (geometric returns) and normal VaR (arithmetic returns) will be similar when we are dealing with short-time periods and practical return estimates.
E x p e c t e d S h o r t f a l l

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