Estimating VaR

Parametric Method

The parametric method obtains a VaR estimate by using the formula below:

$$ VaR_p=\mu-\alpha_p\sigma $$

Where:

• \(VaR_p\) is the estimated VaR of portfolio \(p\).
• \(\mu\) is the mean of the portfolio or the expected return of the portfolio.
• \(\sigma\) is the standard deviation of the portfolio or the volatility of the portfolio.
• \(\alpha_p\) is determined by the confidence interval and the chosen theoretical distribution.

Based on a normal distribution, for example, the values for alpha are 1.282, 1.645, or 2.326 when fixing the probability of 90%, 95%, and 99%, in that order.

To obtain the variance and mean of portfolio P formed by combining two assets, A and B, with weights \(W_A\) and \(W_B\) respectively, we use the formula below:

$$ \begin{align*} \mu_p &=W_A\mu_A+W_B\mu_B \\ \sigma_p^2 &=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_A W_B Cov_{AB} \\ \end{align*} $$

Where:

• \(\mu_i\) is the mean of \(i\).
• \(\sigma_j^2\) is the variance of \(j\).
• \(Cov_{AB}\) is the covariance of assets A and B.

Example: Parametric Method

The standard deviation of the daily returns of asset A is given as 0.0231, and its mean as 0.0012. Estimate the 5% annual VaR for asset A, given that there are 250 trading days in a year, and the value of A is $200,000.

Solution

$$ \begin{align*} \text{Annual mean } (\mu) &=250\times0.0012=0.3 \\ \text{Annual standard deviation }\left(\sigma\right) & =\sqrt{250}\times0.0231=0.3637 \\ VaR_{A,250} &=200,000\left(0.3-1.645\times0.3637\right) \\ & =200,000\times0.2983=-$59,660, \end{align*} $$

Therefore, asset A has a 5% annual VaR of $59,660. This implies that there is a 5% probability that the asset will fall in value by more than $59,660 over one year if there is no trading.

Historical Simulation

Assume that we want to calculate the 1-day 5% VaR for an asset using 200 days of data. The 95^th percentile corresponds to the least bad of the worst 5% of returns. In this case, the VaR corresponds to the 10^th worst day.

Example: Historical Simulation

The following are the hypothetical ten worst returns for asset B from 120 days of data for 6 months.

{ -3.45%, -14.12%, -15.72%, -10.92%, -5.50%, -3.56%, -6.90%, -2.50%, -5.30%, -4.31% }

Find the 1-day 5% VaR for B.

Solution

First, we rearrange the given data starting with the worst day, to the least bad day, as shown below:

{ -15.72%, -14.12%, -10.92%, -6.90%, -5.50%, -5.30%, -4.31%, -3.56%, -3.45%, -2.50% }

The VaR corresponds to the \((5\%\times120)\) = 6^th worst day: -5.30%.

This implies that there is a 95% probability of getting at most a 5.3% loss.

Monte Carlo Simulation

Monte Carlo simulations generate random numbers that estimate the return of an asset at the end of the analysis horizon. It then uses the same procedure to obtain the VaR estimates as the historical simulation using the obtained returns rather than historical returns.

With powerful computing capacity, it is reasonably easy and fast to simulate very complex processes for portfolios with significant exposures.

Question 

Consider the following daily information about securities A, and B:

$$ \begin{array}{c|c|c|c} \textbf{Security} & \textbf{Standard deviation} & \textbf{Mean of} & \textbf{Covariance} \\ & \textbf{of returns} & \textbf{returns} & \\ \hline A & 0.0108 & 0.0011 & 0.0004 \\ B & 0.0131 & 0.0014 & \end{array} $$

Which of the following best estimates the 5% annual VaR for a portfolio that is 70% invested in security A, and 30% invested in security B, given that the total investment is $1 million and there are 300 trading days in a year?

1. $42,860.50
2. $52,860.50
3. $62,860.50

Solution

The correct answer is A.

Daily variance:

$$ \begin{align*} \sigma_p^2 &=W_A^2\sigma_A^2+W_B^2\sigma_B^2+2W_AW_BCov_{AB} \\ \sigma_p^2 &={0.7}^2\left({0.0108}^2\right)+{0.3}^2\left({0.0131}^2\right)+2\left(0.7\right)\left(0.3\right)\left(0.0004\right)=0.0002 \end{align*} $$

Therefore, the annual standard deviation of the portfolio is:

$$ \sigma_p=\sqrt{300(0.0002)}=0.2449 $$

Daily mean:

$$ \begin{align*} \mu_p & =W_A\mu_A+W_B\mu_B \\ \mu_p &=0.7\left(0.0011\right)+0.3\left(0.0014\right)=0.0012 \end{align*} $$

Therefore, the annual mean is:,

$$ 300\left(0.0012\right)=0.36 $$

The 5% annual VaR is then obtained by:

$$ VaR_p=0.36-1.645\times0.2449=-0.0429=-4.29\% $$

For a portfolio worth $1 million, the 5% annual VaR is:

$$ 4.29\%\times$1 \text{ million}=$42860.50 $$

Reading 41: Measuring and Managing Market Risk

LOS 41 (c) Estimate and interpret VaR under the parametric, historical simulation, and Monte Carlo simulation methods.