A trading desk is holding a portfolio of various options on 2 different underlying assets. As the risk manager, you want to report a 10-day 99% Value-at-Risk (VaR) for the portfolio.

1. Describe how you would compute the VaR under (i) the historical simulation approach, (ii) the linear model and (iii) Monte Carlo simulation.

1. Which one is your preferred method? Explain.

1.

• VAR is the minimum loss that an organization would expect over a certain period of time. It can be expressed either in currency or as a certain percentage of portfolio value. The different methods of computing VAR are as follows - 

I) Historical simulation - This technique helps to estimate VAR based on actual periodic changes in risk factors over a lookback period, for example in the course of the most recent 5 years and applying current weights to the historical asset returns. This return doesn't speak to a real portfolio yet rather remakes the historical backdrop of a theoretical portfolio utilizing the current position.

II) Linear model - This assumes that the returns of the assets are normally distributed. Portfolio returns are also normally distributed as the return of the portfolio is a linear combination of normal variables.

III) Monte Carlo simulation - This technique includes building up a model for future stock value returns and running numerous theoretical preliminaries through the model. 

 1.

• The preferred approach of estimating VAR is the Linear model because of its simplicity and straightforwardness. Unlike the other approaches, this model requires only two parameters i.e. mean and standard deviation.

Step-by-step explanation

• Different methods of estimating VAR are as follows -

I) Historical Simulation approach -

The Historical simulation approach of estimating VAR utilizes the current portfolio and reprices it using the actual historical changes in the risk components experienced during the lookback period. We start by breaking down the portfolio into risk factors and assembling the verifiable returns of each risk factor from the picked lookback period. We reprice the current portfolio given the profits that happened on every day of the chronicled lookback period and sort the outcomes from biggest loss to the greatest gain. To assess a 10-day VAR at a 99% confidence interval, we pick the point on the subsequent appropriation past which 1% of the results result in bigger losses.

II) Linear model -

The linear model of estimating VAR is also referred to as the analytical method and sometimes the variance-covariance method. Similar to other methods this model also starts with a risk decomposition of portfolio holdings. It generally assumes that the return distributions of risk factors in the portfolio are normally distributed. It then uses the expected return and standard deviation of return of each risk factor to estimate the VAR. It is not compulsorily required to use only the normal distribution. Other distributions can also be used. But as the normal distribution requires only two parameters i.e. mean and standard deviation, it is typically used.

 In estimating VAR, we need to identify a VAR threshold i.e. a point in the left tail of the distribution, typically either the 5% left tail or the 1% left tail. If the portfolio is characterized by normally distributed returns and the expected return and standard deviation are known, it is easy to identify any point on the distribution. A normal distribution with an expected return (μ) and standard deviation (σ) can be converted to a standard normal distribution, which is a special case of the normal distribution in which the expected return is zero and the standard deviation is one. Standard normal distribution is also known as a z-distribution. If we have observed a return R from a normal distribution, we can convert to its equivalent z-distribution value by the transformation:

Z = (R - μ) / σ

Note - 

In a standard normal (z) distribution, a 5% VAR is 1.65 standard deviations below the expected value of zero, and 1% VAR is 2.33 standard deviations below the expected value of zero.

For example, To estimate a 99% VAR(or 1% VAR), we need to know the return that is 2.33 standard deviations to the left of the expected return.

Suppose the return and standard deviation of the portfolio are 8% and 10% respectively and the value of the portfolio is $100,000

The steps for estimating 99% VAR are as follows -

Step 1 Multiply the standard deviation of the portfolio by 2.33

10*2.33 = 23.30%

Step 2 Subtract the answer of step 1 from the expected return 

8% - 23.3% = -15.3%

Step 3 Because VAR expresses an absolute value. Change the negative sign of the value in step 2 i.e. 15.3%

Step 4 Multiply the value of step 3 with the value of the portfolio to get the VAR.

15.3% of $100,000 = $15,300

Thus, using the linear model, our estimate of VAR is $15,300 meaning that on 1% of trading days the portfolio would be expected to incur a loss of at least $15,300.

III) Monte Carlo simulation -

Monte Carlo simulation is a technique of assessing the VAR wherein the client builds up his own presumptions about the measurable qualities of the distribution and utilizes those attributes to produce random results that speak about the hypothetical returns of a portfolio with the specific attributes. It avoids the complexity of the linear model when the portfolio has a greater number of assets. It does not need to be constrained by the assumption that of normal distributions.

Monte Carlo simulation requires the generation of random values of the underlying unknowns. For example, here the unknowns are the returns on the two risk factors, represented by the IWM and SPY ETFs. We assumed that the two securities represent the risk factors. We now decompose the portfolio holdings into these risk factors. First, we simulate the returns of these two risk factors, and then we re-price our exposures to the risk factors under the range of simulated returns and then recording the outcomes. We then sort the outcomes from worst to best. 

A 99% VAR under Monte Carlo simulation would simply be the one percentile of the simulated values. 

1.

• Linear model - The greatest advantage of using the linear model is its easy calculation and straightforwardness. The assumptions of the normal distribution allow the user to easily estimate the parameters using the historical data although the judgment is needed in case of misleading historical data. It is best utilized in circumstances in which one is certain that the normal distribution the best approximation of true distribution.