As discussed, the Value-at-Risk (VaR) measures the maximum potential loss that can occur on an investment under normal circumstances at a certain confidence level in a given time frame.

The parametric value-at-risk method, also known as the variance-covariance method or delta-normal method, is a statistical technique based on the assumption that returns follow a certain distribution (more specifically, a normal distribution) to calculate the potential loss of an investment portfolio. It requires historical time-series data of returns of the portfolio's constituents to estimate the statistical parameters such as the location (mean) and scale (standard deviation) of distribution (normal distribution), which represent the expected return and volatility, respectively.

Subsequently, the confidence level is to be decided, which indicates the probability of losses not exceeding the VaR. In other words, a confidence level of 1% represents that there is only a 1% chance that the portfolio will incur losses greater than the VaR.

For the purpose of VaR calculation using the parametric method, the z-score is determined (a statistical measurement that reflects the number of standard deviations from the mean a particular data point is). And to calculate the z-score, we use the standard normal distribution, whose statistical parameters such as mean is 0 and the standard deviation is 1.

Using the inverse of the standard normal distribution function, one can easily identify the z-score given the confidence level as probability as an argument to this function.

=NORM.S.INV(Probability)

=NORM.S.INV(99%) i.e. 2.3263. It means that 2.3263 times the standard deviation from the mean of the normal distribution would correspond to a confidence level of 99%

The formula for parametric VaR calculation is:

Value at Risk (VaR) = Mean - (z-Score * Standard Deviation)

### Let's understand the Parametric VaR method using an illustration-

As per the past return outcomes, the daily mean return on a portfolio of stocks having exposure of $5m is 0.1056% with a standard deviation of 0.9815%. Calculate 1-Day 99% VaR and 10-Day 99% VaR assuming 252 days a year.

1-Day Value at Risk (VaR) = Daily Mean - (z-Score * Daily Standard Deviation)

Where,

Daily Mean = 0.1024%

Daily Standard Deviation = 1.0457%

At 99% probability, the z-Score is 2.3263 approximately.

1-Day 99% Relative VaR = Mean - ( z-Score * Standard Deviation ) = 0.1024% - ( 2.3263 * 1.0457% ) = -2.3302%. Therefore, a 1-Day 99% Absolute VaR is -$0.1165m (-2.3302% of $5m) is the maximum potential loss that can occur in a single day with a 1% probability of being exceeded (in the 1% of the abnormal circumstances, the losses can exceed $0.1165m).

10-Day 99% VaR can be calculated as-

10-Day Value at Risk (VaR) = 10-Day Mean - (z-Score * 10-Day Standard Deviation)

Where,

10-Day Mean = Daily Mean * 10 = 0.1024% * 252 = 1.0243%

10-Day Standard Deviation = Daily Standard Deviation * √10 = 1.0457% * √250 = 3.3067%

At 99% probability, the z-Score is 2.3263 approximately.

10-Day 99% Relative VaR = Mean - ( z-Score * Standard Deviation ) = 1.0243% - ( 2.3263 * 3.3067% ) = -6.6683%. Therefore, a 10-Day 99% Absolute VaR is -$0.3334m (-6.6683% of $5m) is the maximum potential loss that can occur in 10 days with a 1% probability of being exceeded (in the 1% of the abnormal circumstances, the losses can exceed $0.3334m).

### Efficient Market Hypothesis (EMH) Theory

According to the weak form of the Efficient Market Hypothesis (EMH), prices reflect all publicly available information, including past prices and returns, and it is not possible to consistently beat the market by using this information. It implies that it is very difficult to predict the stock price at the close of the immediate next trading day, and the best prediction would be the current stock price, such that the expected mean return calculated on the basis of past returns would be 0.

If this theory holds good, the expected daily mean return would be 0, and thus,

1-Day 99% Relative VaR = Mean - ( z-Score * Standard Deviation ) = 0% - ( 2.3263 * 1.0457% ) = -2.4326%

and, 10-Day 99% Relative VaR = Mean - ( z-Score * Standard Deviation ) = 0% - ( 2.3263 * 3.3067% ) or 1-Day 99% Relative VaR * SQRT(10) = -2.4326% * SQRT(10) = -7.6926%.

Therefore, a 10-Day 99% Absolute VaR is -$0.3846m (-7.6926% of $5m) is the maximum potential loss that can occur in 10 days with a 1% probability of being exceeded (in the 1% of the abnormal circumstances, the losses can exceed $0.3846m).

## Drawbacks of the Parametric Value-at-Risk Method

The parametric value-at-risk method has some advantages over other VaR methods, such as its simplicity and ease of calculation, but it also has limitations that must be taken into account.

this method relies on the assumption that the returns follow a normal distribution. However, in reality, returns are often not normally distributed, and may exhibit skewness, kurtosis, and fat tails. Fat tails mean that there are a higher number of extreme events (both positive and negative) than would be expected in a normal distribution. This can result in an underestimate of the potential loss, which can be significant.

this method is sensitive to the choice of the mean and standard deviation estimates. If the estimates are not accurate, the VaR calculation will not reflect the true risk of the portfolio. This sensitivity to the choice of estimates can result in VaR calculations that are too high or too low. Also, this method is sensitive to outliers, meaning that an extreme event can have a significant impact on the VaR calculation.

this method assumes that returns are linear, meaning that a small change in the value of the portfolio results in a proportional change in the VaR. This is not always the case in financial markets, where returns can be highly non-linear and complex. The parametric VaR method cannot capture these non-linearities, which can result in inaccurate VaR calculations.