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Market Risk Analysis

Instructions

The take home will be a Group Work:

• The class will be split into groups of 4 students max

• Each group will prepare an excel file to illustrate the various computations and a pdf report (no word file allowed so please save your word file as a pdf before submission) with their comments and answers to the questions assigned below

• Justifications behind each answer’s reasoning must appear clearly in the answers:  no point will be granted without clear explanations

• Please send your pdf report to the following URKUND address to check for plagiarism: r.              .deguest   .              .comieseg@analyseurkund 

• Attempted plagiarism between groups will be severely punished and will lead to a disciplinary com- mittee hearing (for all team members equally)

Late submissions will receive a 0

1    Volatility Estimation with Moving Average Models (6 points)

a.  (1 point) Download the NASDAQ 100 equity index closing prices (from Bloomberg or Refinitiv or yahoo finance) over the period between 2005-12-30 and 2021-12-31 and compute its daily log-returns. Plot the price and log-returns for the NASDAQ 100 equity index.  What do you observe?  Please explain.

b.  (2 points) Use the moving average model (MA) to estimate the annual volatility of the NASDAQ 100 equity index with two different lags:  n = 20, and n = 60 over the period between 2018-12-31 and 2021-12-31. Plot both your results on the same graph and comment.

c.  (2 points) Use the exponentially weighted moving average model (EWMA) to estimate the annual volatility of the NASDAQ 100 equity index with two different parameters λ = 0.94 and λ = 0.97 over the period between 2018-12-31 and 2021-12-31.  You should implement the model with the recursive expression where the initial volatility estimation σ 2 (t = 0) is based on the MA model with n = 20. Plot both results on the same graph and comment.

d.  (1 point) Plot on the same graph the estimations using MA with n = 20, EWMA with λ = 0.94 together with the Implied Volatility of the NASDAQ 100 equity index (obtained from Bloomberg or Refinitiv or yahoo finance) over the period between 2018-12-31 and 2021-12-31, and comment.  In particular, what conclusion can you draw on the level of volatility if you use an estimation method based on historical data instead of an estimation based on implied volatility?

2    Risk Measurement Estimation (14 points)

You are the head of the risk unit of a small asset management company. The portfolio of the AM company is invested in two equity indices as follows: 40% in the S&P 500 equity index and 60% in the NASDAQ 100 equity index, both denominated in US dollars. You are concerned with risk measure estimation over a 1-day period and computed with levels equal to 5% and 1%. You will have to implement several risk measurement procedures to estimate the risk of your portfolio over a 1-day period.

2.1    Value-at-Risk (VaR) Estimation (6 points)

a.  (1 point) Download the S&P 500 equity index and NASDAQ  100 equity index  (from Bloomberg or Refintiv) over the period between 2005-12-30 and 2021-12-31 and compute their daily arithmetic returns. From these historical returns, build the hypothetical arithmetic returns of the AM company portfolio as if the current weights had applied to these historical returns. Plot the portfolio returns on a graph.

b.  (2 points) Compute the historical VaR of your portfolio returns based on a lag of size n  =  260 observations at level 5% and 1%.  Plot both historical VaR over the period between 2018-12-31 and 2021-12-31 on the same graph and comment your results.

c.  (2 points) Consider now a parametric approach to estimate the VaR based on the Gaussian distribution. Compute the parametric VaR of your portfolio returns by assuming that the mean of the returns is 0 (relative VaR) and its volatility is given by the moving average model with a lag of size n = 260 observations. Plot both Gaussian parametric VaR at level 5% and 1% over the period between 2018- 12-31 and 2021-12-31 on the same graph and comment your results.

d.  (1 points) Compare the historical and Gaussian VaR at level 1% on the same plot and comment your results.

2.2    Back-testing VaR Models (4 points)

a.  (2 points) Compute the time-series of exceedances for each VaR estimates computed previously over the period between 2018-12-31 and 2021-12-31.  Report in a table the total number of exceedances obtained over the period.

b.  (2 points) Run Kupiec’s test to check if you can validate each model with a confidence interval of 95%. Justify your conclusion.

2.3    Expected Shortfall Estimation (4 points)

a.  (2 points) Compute the historical Expected Shortfall (ES) of your portfolio returns based on a lag of size n = 260 observations at level 5% and 1%.  Plot both historical ES over the period between 2018-12-31 and 2021-12-31 on the same graph and comment your results.

b.  (2 points) Compare the historical VaR and historical ES at level 1% on the same plot.  Which risk measurement estimate is the largest and why? Is this conclusion always true or does it depend on the estimation method (here the historical method) used?