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Assignment 2

1. Backtesting of Value at Risk (VaR) in 1997.

We conducted a backtesting of the Value at Risk (VaR) model on a portfolio of 12 assets in 1997, we held the assets from the ten days before the first day of 1997 (17/12/1996), the calculated realised the losses every ten days thereafter for the entire year 1997. We invested $1,000,000 per stock, totalling $12,000,000 for the portfolio. According to the VaR model, we had 99% confidence level, which means there was a 1% chance that the portfolio loss could fall outside of the predicted range. Therefore, we could expect the portfolio to experience violations for approximately two or three days out of the 253 trading days, which represents approximately 1% of the total trading days.

Definition.

§ Number of violations > 2 or 3 days indicates underestimate the risk.

§ Number of violations< 2 or 3 days indicates overestimate the risk.

§ Number of violations = 2 or 3 days indicates meet expectations.

1.1 VaR Methodology under normality and historical simulation (1997).

We ran the code via Ox and created the .out files to obtain data for Excel. The files contain 253 rows, representing the 253 trading days in 1997. The first row shows the 99% VaR, and the second row indicates the realised loss. We also used the timept file to allocate the dates in 1997. To create a graph to see the violation, we added another row to see difference which calculates VaR minus the realised loss.

In the normality approach (see Graph 1), the blue line represents the 99% VaR and the orange line shows the actual losses incurred. The average VaR in 1997 was calculated to be $1,081,253, indicating a 99% confidence level that the maximum loss of the portfolio will not exceed this amount. There was only one instance of a violation, where the actual losses exceeded the VaR over a one-day period, during the entire year. However, based on the definition of violations, the range of violations should typically be between 2 to 3 days, indicating that the VaR may have overestimated the risk in this particular case.

Summarised the Group 1 violation table as below.

Graph 1. (overestimated)

In the historical simulation approach (see Graph 2), the blue line represents the 99% VaR and the orange line shows the actual losses incurred. The average VaR in 1997 was calculated to be $1,500,622.  However, no violations occurred under the historical simulation method, which overestimated the risk as well in this case.

Graph 2. (overestimated)

2. Backtesting for a portfolio of 6 stocks in 2020.

2.1 The Company

The following six U.S. hotel listed companies were selected for portfolio: Hilton Worldwide (HLT), Hyatt Hotels(H), Las Vegas Sands(LVS), MGM Resorts International(MGM), Vail Resorts(MTN), and Wynn Resorts(WYNN). A total of $1,000,000 was invested in each company, resulting in a portfolio worth $6,000,000. The assets were held from ten days before the first day of 2020 (17/12/2019) and the realised losses were calculated every ten days throughout the year 2020.

To evaluate the portfolio’s value at risk (VaR) and realised losses, two methods were used: normality and historical simulation. The 99% VaR and realised loss were graphed using these methods.

2.2  CRSP Data

The particular companies data used for this exercise was obtained from the CRSP, The raw data was selected for the period between 01/02/2016 and 31/12/2020. Since CRSP provides only raw and unadjusted data, we had to convert it to adjusted data. This was achieved by creating an additional roll and dividing the price by the cumulative factor to adjust price (cfacpr). Once the data for the six companies was downloaded, we combined the adjusted prices into a new single worksheet and saved it as “Q2merged.txt”. The necessary adjustments were also made to the ox code.

2.3 VaR Methodology under normality and historical simulation (2020).

We executed the code using Ox and generated .txt files to extract data for Excel. The files consist of 253 rows, representing the 253 trading days in 2020. In addition, we used the same approach to backtest data from 1997 and added an extra row to compare the difference between VaR and the actual loss incurred.

In Graph 3, the normality approach displays the 99% VaR with a blue line, while the actual losses are represented by an orange line. Based on our portfolio, we calculated the average VaR in 2020 to be $632,939, which indicates a 99% confidence level that the maximum loss of the portfolio will not exceed this amount. However, there were 24 days in 2020 where losses significantly exceeded the VaR, which is 10 times higher than the expected the 2 or 3 days. The sharp decline in stock prices was due to the global pandemic, and we will discuss the details further in the limitations of VaRs.

Summarised the Group 3 violation table as below.

Graph 3. (underestimated)

In Graph 4, the historical simulation approach representing the 99% VaR using a blue line, while the orange line represents the actual losses. After analysing our portfolio, we determined that the average VaR in 2020 was $696,702, Due to the global pandemic, we can see that there were 25 days in 2020 where losses significantly exceeded the expected VaR. This number is ten times higher than the usual 2 or 3 days.

     Graph 4. (underestimated)