Questions 1. Conceptual Questions about Risk Measures. Before VaR and ES have become important risk measures to manage portfolio risk, the volatility of the portfolio returns was the main risk measure. In this context, explain what VaR is and why it is an interesting alternative measure of risk when compared to the traditional volatility benchmark. …

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**Questions**

**1. Conceptual Questions about Risk Measures.**

- Before VaR and ES have become important risk measures to manage portfolio risk, the volatility of the portfolio returns was the main risk measure. In this context, explain what VaR is and why it is an interesting alternative measure of risk when compared to the traditional volatility benchmark.
- Explain the main conceptual di erences between the Riskmetrics VaR and the VaR obtained with a traditional Historical Simulation.
- Explain what Expected Shortfall is and why it is a more appropriate measure of risk than VaR. Illustrate the appropriateness of ES as compared to VaR with an example.

**2. Properties of Measures of Risk and their use.**

Consider the following DGP for the returns: R_{t+1} = _{t}_{+1}z_{t+1}, with z_{t+1} i.i.d. F_{z}.

- Show that VaR satis es the positive homogeneity axiom of coherent measures of risk. Then, obtain an explicit formula for the V aR
_{t}^{p}_{+1}of R_{t+1}. Finally, explain the importance of the positive homogeneity axiom in your calculation of V aR_{t}^{p}_{+1}. - Let X be a random variable representing the potential losses of a portfolio, with existing moment generating function M
_{X}( ) = E[e^{X}]; 8 0. De ne the following risk measure, that we denominate the moment generating risk measure, for a con dence level 1 p:

p(X) = inf >0 ln( ) . Show that (:) is a Coherent Measure of Risk. p

Note: If you can’t show sub-additivity with general X_{i}’s you can assume that the X_{i}’s are independent to show the validity of this axiom under this simpler hypothesis. In this case, I will discount 1 point from the 10.

Knowing that the DGP for the returns is given by R_{t+1} = _{t+1}z_{t+1}, we will estimate the risk measure of 2.2 for the shocks’ losses X = z_{t+1} in two ways. First, non-parametrically directly from data. Second, parametrically, assuming that F_{z} N( ; ^{2}), but with the Gaussian parameters estimated from data.

- Use the leveraged GARCH(1,1) model for the volatility:
_{t}^{2}_{+1}= ! + (R_{t t})^{2}+_{t}^{2}. Take the AppleMidtermFall2020.xls worksheet (adjusted close prices), calculate the historical returns fR_{t+1}g^{T}_{t=1}, and normalize those returns by the leveraged GARCH(1,1) estimated volatility_{t+1}to obtain the sequence of shocks fz_{t+1}g. Show the values of the estimated parameters f!; ;; g and plot a graph of the leveraged GARCH(1,1) volatility. - Using the innovation terms z
_{t+1}’s obtained in 2.3, create a grid with 1000 observations ranging from 0.01 to 100 and, for each_{j}, estimate the sample value of the moment generating function of X: M^X ( j) = Pi=1T. Plot the values of against 1ln( ) . p Estimate ^p(X) by minimizing this function, for p = 2:5% (that is, the 2:5% highest losses of your n o portfolio). - Obtain an analytical solution for ^
_{p}(X) when X N( ;^{2}) as a function of p; ; . Compare the results of this item with those in 2.4 and also with a leveraged GARCH(1,1) Gaussian V aR_{p}with p = 2:5%. Comment on your comparisons. In other words, I am asking you to compare the results of three risk measures here: the nonparametric moment generating risk measure of item 2.4, the parametric moment generating risk measure of this item, and the leveraged GARCH(1,1) Gaussian VaR risk measure.

Note: A leveraged GARCH(1,1) Gaussian VaR that considers the following DGP for the returns: _{t}^{2}_{+1} = ! + (R_{t t})^{2} + _{t}^{2 }is simply a variation of the RiskMetrics model R_{t+1} = _{t+1}z_{t+1}, with z_{t+1} i.i.d. N(0; 1) and t2+1 = ! + (Rt t)2 + t2.

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**3. VaR and ES with the t-student(d) distribution and with Filtered Historical Simulation.**

- Consider a t-student(d) random variable with density ft(d)(x; d) = x2 (1+d) d ) . Show that its variance is given by .
- Fit a standard t-student to the shocks z
_{t}’s of the IBM stock. To that end, rst estimate the variance of the returns with a t-student GARCH(1,1) model and normalize the returns R_{t}’s with the square root of this variance, that is: zt =^{R}^{t}. Then, if the GARCH software hasn’t_{t}GARCH given you the degrees of freedom of the implied t-student distribution for the z’s, use the obtained z’s to estimate the degrees of freedom parameter d of the t distribution, either by Maximum Likelihood or by the trick of matching moments learned in class. Finally, use the QQ-plot to inform you about the quality of the approximation of the distribution of the z’s by the standard t-student distribution.

Note: The GARCH can alternatively be estimated with QMLE (assuming that the z’s are Gaussian even though they are not).

- Calculate, for three di erent values of p = 1%; 2:5%; 5%, the ratio of ES to VaR for the tted standard t-distribution of item 3.2 using the formulas from pages 24 and 25 of the slides of Lectures 9 10 and 11.pdf” on Non-normal distributions. Compare them to the ratio that you would obtain by using the corresponding nonparametric estimates of ES and VaR for the shocks fz
_{t}g’s (that is, using the Filtered Historical Simulation). If you wanted to minimize the chances of underestimating the risk of your portfolio, which method would you choose, the GARCH with parametric standard t or the Filtered Historical Simulation (GARCH with nonparametric distribution for z)?

**4. Risk estimation based on the GPD via EVT**

Let L_{t+1} be a random variable representing the portfolio loss for tomorrow, and F_{L}_{t+1} (y) its cumula-tive distribution function. In class we saw that the tail of the distribution F_{L}_{t+1} (y) (observations that exceed a certain threshold u), here represented by the conditional distribution F_{u}(y) = P (L_{t+1}__1__yjL_{t+1} > u), is well approximated by a Pareto Distribution (PD) F_{u}(y) = 1 cy , with y > u and > 0.

Use the Excel le spxlongFall2020.xls” that contains a long dataset of S&P 500 prices and sup-pose that you are long one unit of the index. Calculate log returns, normalize the returns using a GARCH(1,1) variance to obtain the sequence of shocks fz_{t}g, and switch the sign of these shocks to obtain the corresponding sequence of losses y_{t} = z_{t}. You should set the threshold u in two ways and then compare the results of each solution: First, use the 95% quantile of the loss distribution. Second, use a QQ-plot against a standard Gaussian distribution and choose the threshold such that the positive losses that deviate from the 45-degree line should appear in the tail. For each case, use the Hill estimator described in the slides of Lectures 9, 10, and 11.pdf” to estimate the parameter and the Equation on page 43 to estimate the parameter c.

Estimate VaR and ES for = 95; 99; 99:5; 99:9% using the PD and compare to the standard Risk-Metrics VaR and ES. The goal of this exercise is to observe and contrast what happens on a usual region of the tail (95%) compared to a very extreme area of the tail (99:9%). Comment your results.

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