Description
VaR for an exponential distribution
- Assume an asset value in 10 days follows an exponential distribution withmean W0. Derive a formula for the value-at-risk for confidence c and reference level W0 for the asset. Apply with W0 = 100 and c = 99%.
- Now assume you are short this asset. Compute your value-at-risk for confidence c and apply with the same numerical values.
- Comment on the similarity or difference between the results in the twoquestions.
- Repeat the previous questions with expected shortfall.
2 VaR for mixtures
You have to allocate $1bn in the stock market. You are discussing with your partner regarding the volatility of returns. She has a view that, in line with historical averages, the volatility of returns will be of σ1=10% in the next year. However, you believe that volatility will be higher, in the orders of σ2=15% for the next year. After discussing with your partner, you agree in the following way: stocks returns follow a normal distribution with mean µ and σ1 with probability π and normal distribution with mean µ and σ2 with probability 1 − π . For now assume that µ = 7% and π = 0.6.
- Compute the 1-year 99% VaR with your view, with your partner view,and with the common view. Compare results and provide a very brief explanation as if you were presenting to your manager.
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- To understand the role of π, plot a chart with π between 0 and 1 on the horizontal axiswhich and the corresponding VaR on the vertical axis. Comment on your results.
- More challenging. After presenting your common view to your manager, you are challenged with an alternative view about volatility: σ is timevarying. The volatility trader suggests that a sensible model for sigma is a gamma distribution. Explain in as many details as possible (either derive of formula or use a computer program) how to compute the VaR of your portfolio when returns have a normal distribution conditional on σ and σ is distributed according to a Gamma distribution.
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