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# P1.T4.804. Value at risk (VaR) estimation approaches (Allen)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
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Learning objectives: Evaluate the various approaches for estimating VaR. Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility ... Explain long horizon volatility/VaR and the process of mean reversion according to an AR(1) model. Calculate conditional volatility with and without mean reversion. Describe the impact of mean reversion on long horizon conditional volatility estimation.

Questions:

804.1. Linda Allen introduces the following value at risk (VaR) estimation approaches:
• Historical simulation (HS)
• GARCH (1,1)
• Hybrid
• Multivariate density estimation (MDE)
• Historical standard deviation (STDEV)
Consider these six summary descriptions:

I. The simplest parametric approach whose weakness is sensitivity to window length and extreme observations
II. The most convenient and prominent non-parametric approach whose weakness is inefficient use of data
III. An interpretation of the exponentially weighted moving average (EWMA) that gives the risk manger a rule that can used to adapt prior beliefs about volatility in the face of news
IV. A parametric approach that assumes conditional returns are normal but unconditional tails are heavy; and that returns are not correlated but conditional variance is mean-reverting
V. An approach that weights past squared returns not by time but instead according to the difference between current and past states of the world
VI. An approach that modifies historical simulation by assigning exponentially declining weights to past data such that recent (distant) returns are assigned more (less) weight

Which sequence below correctly matches the VaR estimation approach with its summary description?

a. I = HS, II = AV, III = GARCH, IV = MDE, V = Hybrid, VI = STDEV
b. I = GARCH, II = MDE, III = Hybrid, IV = STDEV, V = HS, VI = AV
c. I = STDEV, II = HS, III = AV, IV = GARCH, V = MDE, VI = Hybrid
d. I = AV, II = Hybrid, III = STDEV, IV = HS, V = GARCH, VI = MDE

804.2. Dennis the Risk Analyst is calculating the 95.0% value at risk (VaR) under the hybrid approach which is a hybrid between the historical simulation (HS) and exponentially weighted moving average (EWMA). His historical window is only 90 days and he has set his smoothing parameter 0.860; that is, λ = 0.860 and K = 90 days. Below are displayed the (rounded) weights assigned under this approach to the three worst returns in the historical window (which occurred, respectively, 27, 13 and 15 days ago). In regard to the weight assigned to the 4th worst return, which was -6.0%, and the consequent 95.0% VaR, which of the following statements is TRUE?

a. The hybrid weight assigned to the 4th worst return (-6.0%) is 1.457% and the 95.0% VaR is (a worst expected loss of) 6.0%
b. The hybrid weight assigned to the 4th worst return (-6.0%) is 1.457% and the 95.0% VaR is (a worst expected loss of) 7.0%
c. The hybrid weight assigned to the 4th worst return (-6.0%) is 2.664% and the 95.0% VaR is (a worst expected loss of) 5.0%
d. The hybrid weight assigned to the 4th worst return (-6.0%) is 4.189% and the 95.0% VaR is (a worst expected loss of) 5.0%

804.3. Theresa the Risk Manager computed the daily volatility of her firm's $100.0 million equity portfolio. She determined the portfolio's standard deviation was 85 basis points per day. In her presentation to the Risk Committee of the Board of Directors, she includes the following statements: • The portfolio's volatility is about 13.44% per annum • The portfolio's one-year value at risk (VaR) with 99.0% confidence is about$22.11 million
Each of the following statements is TRUE except which is false?

a. She has assumed i.i.d.
b. She has assumed the return distribution is approximately normal
c. If the unconditional average (aka, long-run mean, LRM) daily return volatility is 1.0%, then 13.44% probably understates the per annum volatility
d. If the 1-lag autcovariance between returns is negative (aka, mean reversion in daily returns), then the 13.44% probably understates the per annum volatility