Parametric vs Non-Parametric VaR

Value at Risk (VaR) is a common measure of tail risk in terms of a financial loss event.

Understanding Returns

The convention when computing these returns (as well as VaR) is to quote return loss- es as positive values. For example, if a portfolio is expected to decrease in value by $2 million, the commonly used terminology is "expected loss is $2 million" rather than "expected profit is –$2 million."

Profit/loss data: Change in value of asset/portfolio, Pt, at the end of period “t” plus any interim payments, Dt.

Arithmetic return: Assumption is that interim payments do not earn a return (i.e., no reinvestment). Hence, this approach is not appropriate for long investment horizons.

Geometric return: Assumption is that interim payments are continuously reinvested. Note that this approach ensures that asset price can never be negative.

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Historical Simulation VaR

Historical Simulation VaR (HSVar) relies on a historical sample of past returns, ordered by magnitude of loss, where the appropriate quantile is calculated (i.e.: 95% or 99%). This differs from the empirical calculation of VaR using mathematical parameters (i.e.: Parametric VaR). The observation that follows the threshold loss level denotes the VaR limit. We are essentially searching for the observation that separates the tail from the body of the distribution. More generally, the observation that determines VaR for n observations at the (1 − α) confidence level would be: (α × n) + 1.

One of the distinct advantages of Historical Simulation VaR is that there is no need to make any assumptions about the exact probability distribution of the returns. This is different from parametric VaR, where you need to make an assumption on whether profit/losses are normally distributed and whether returns are geometric.

To illustrate this VaR method, assume you have gathered 1,000 monthly returns for a security and produced the distribution shown in Histogram of Monthly Returns. You decide that you want to compute the monthly VaR for this security at a confidence lev- el of 95%. At a 95% confidence level, the lower tail displays the lowest 5% of the un- derlying distribution's returns. For this distribution, the value associated with a 95% confidence level is a return of –15.5%. If you have $1,000,000 invested in this security, the one-month VaR is $155,000 (–15.5% × $1,000,000).

Since VaR is to be estimated at 95% confidence, this means that 5% (i.e., 50) of the ordered observations would fall in the tail of the distribution. Therefore, the 51st or-

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dered loss observation would separate the 5% of largest losses from the remaining 95% of returns.

The VaR limit will be at the observation that separates the tail loss with area equal to 5% from the remainder of the distribution.

Assumptions & restrictions of HSVar

• -  Future returns follow historical return process

• -  Does not require specification of probability distribution

• -  Unable to account for changes in parameter values

  Method of HSVaR Calculation

- Historical returns are ordered from low to high

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- Identify observation corresponding to specific confidence level

Parametric Value at Risk VaR (Delta Normal VaR)

Unlike the historical simulation method, the parametric approach (e.g., the delta-nor- mal approach) explicitly assumes a probability distribution for the underlying observa- tions.

There are generally two cases of parametric VaR calculations:

1. VaR for returns that follow a normal distribution, and

2. VaR for returns that follow a lognormal distribution.

(a) Normal Parametric VaR (using P/L)

Using P/L (Profit/Loss) data, the VaR for a given confidence level (i.e.: 95%, 99%) de- notes the point that separates the tail losses from the remaining distribution. The VaR cutoff will be in the left tail of the returns distribution. As such, 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 of a loss). I

n equation form, the VaR at significance level α is shown below.

VaR can be calculated both with and without a mean (u). Generally speaking, if you are given one it should be used in the calculation using the formula below.

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Where μ and σ denote the mean and standard deviation of the profit/loss distribution and z denotes the critical value (i.e., quantile) of the standard normal. In practice, the population parameters μ and σ are not likely known, in which case the researcher will use the sample mean and sample standard deviation.

Now suppose that the data you are using is arithmetic return data rather than prof- it/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:

Note: If no mean data is available, then use the slightly modified version as fol- lows:

(b) Normal Parametric VaR (using arithmetic return)

If the data you being used is arithmetic return data rather than profit/loss (P/L) data, the formula is slightly different to incorporate the initial value. It’s important to not that the arithmetic returns follow a normal distribution as well. Because of the relationship between prices, profits/losses, and returns, the corresponding VaR formula is very sim- ilar in format:

Example: Computing VaR (arithmetic returns)

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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(5%) = (−10% + 1.65 × 20%) × 100 = $23.0 VaR(1%) = (−10% + 2.33 × 20%) × 100 = $36.6

(c) Lognormal VaR

The lognormal distribution is right-skewed with positive outliers and, most importantly in terms of modeling asset prices, bounded below by zero. Given this inability to as- sume a negative value, the lognormal distribution is commonly used to counter the possibility of negative asset prices (Pt). Technically, if we assume that geometric re- turns follow a normal distribution (μR, σR), then the natural logarithm of asset prices follows a normal distribution and Pt follows a lognormal distribution. The following ex- pression is for the calculation of lognormal VaR:

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% log- normal VaR assuming the beginning period portfolio value is $100.

Ans- wer:

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Rule of Thumb: when you have arithmetic return data, you would use the “normal” VaR equation . When you have geometric return data, you would use the “lognormal” version.

Example VaR Problem

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Recalling the two equations:

Parametric VaR - Changing Time Period

In calculating VaR for a different time basis than the one the mean and standard devia- tion are based on (typically annual), some adaptation of these input figures are re- quired. For instance, to get 1-day VaR given annual mean & standard deviations, the mean would be divided by 252, and the standard deviation would be divided by the square root of 252 as shown above.

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Stressed VaR

Normal VaR is typically measured using a 10-day period.
In a crisis period, you cannot rely on the assumption of liquid markets. The idea

was created of adding both 99% VaR and 99% SVar (Stressed VaR) together to calcu- late a more general market-risk capital figure.

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