Calculate the rolling standard deviation for a sample of 21

Question

1. Calculate the 99%/10day Value at Risk for an investment In the market index using a' sample standard deviation.

• Calculate the rolling standard deviation for a sample of 21 days, e.g., using days 1-21, 2-22...

• Project standard deviation from 1D to 10D period using the additivity of variance σ_10D = √10 x σ_t² and scale an average daily return as μ_10D = μ x 10, where p is a mean of all data.

• Calculate Value at Risk as follows:

VaR = μ_10D + σ_10D x Factor

where Factor is a percentile of the Standard Normal Distribution that 'cuts' 1% on the tail.

2. Calculate the 99%/10 day Value at Risk for an investment in the market index using a GARCH-filtered standard deviation. An example of GARCH calculation is provided in the ARCH Lecture.

• Assume a GARCH(1,1) model has been estimated from the long-term data as follows:

σ_t² = 0.000001 + 0.047μ²_t-1 + 0.9466σ²_t-1

where ω = 0.000001, a = 0.047, and β = 0.9466.

• Set the initial value of σ₀² equal to the variance for the whole dataset. 'Then, recursively use the past variance and return from the row above to set up a GARCH calculation.

• Use GARCH-filtered standard deviation to calculate Val according to f. Remember to scale.

3. Calculate the percentage of VaR breaches for each measure. Are the breaches independent in time of the level of volatility?

• VaR Is fixed at time t and compared to the realised return at time t + 10. A breach occurs when a realised 10-day index return In (S_t+10/S_t) is below the VaR quantity (negative scale).

• 20/08/2009 is the first day on which VaR computation. Is available for both. VaR_SD and VaR_GARCH. Total number of comparisons is N = 978. Number of breaches divided by number of comparisons will give the percentage of breaches.

• Optionally, calculate the conditional probability of breach in VaR for the next period, given that a breach was observed for the previous period.

Methodology Note: In Excel, use 1F() function to compare.

The question about VaR breaches being an iid process can be answered either by a visual analysis (location-dispersion, Q-Q plot) or more formal statistical testing of Normality, autocorrelation, or Poisson process fit.

Think about the application of techniques that you are familiar with.

Attachment:- Data.xlsx