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# P1.T4.806. Putting value at risk (VaR) to work (Allen Ch.3)

#### Nicole Seaman

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Learning objectives: Describe the limitations of the delta-normal method. Explain the full revaluation method for computing VaR. Compare delta-normal and full revaluation approaches for computing VaR. Explain structured Monte Carlo, stress testing, and scenario analysis methods for computing VaR, and identify strengths and weaknesses of each approach. Describe the implications of correlation breakdown for scenario analysis. Describe Worst-Case Scenario (WCS) analysis and compare WCS to VaR.

Questions:

806.1. Patricia the Portfolio Manager owns an option portfolio that contains both long and short positions in European call and put options. The position gamma of the portfolio is negative, specifically it is -19,500. Recall that position gamma is the product of option quantity and each option's percentage gamma. For example, a long position in 10,000 call options that each have a percentage (per option) gamma of 0.050 has a position gamma of +500; on the other hand, a short position in the same number of options has a position gamma of -500. Position gamma can be summed across the option portfolio.

Patricia initially estimates her option portfolio's value at risk (VaR) based on a delta approximation. Her analysis quotes VaR in Loss(+)/Profit(-) format; aka, L/P units. In this way, losses and VaR are expressed as positive values. However, she realizes that such an estimate omits the portfolio's negative position gamma, so she re-computes the portfolio's VaR by using a DELTA-GAMMA approximation. How does the revised estimate compare to the first delta-only estimate?

a. The delta-gamma L/P VaR is higher
b. The delta-gamma L/P VaR is lower
c. If the position delta is positive, the L/P VaR is lower; but if the position delta is negative, the L/P VaR is higher.

806.2. Peter the Portfolio Manager has a long position in a bond with ten (10) years to maturity. The bond happens to be currently priced at par, such that its yield (aka, yield to maturity) is equal to its semi-annual coupon rate of 12.0%. The bond's modified duration is 6.08 years and its convexity is ~ 50.0 years-squared (note these can each be retrieved analytically in the case of a par bond). Peter wants to estimate the 99.0% value at risk (VaR) of the bond. He assumes a yield volatility of 100 basis points. Therefore, his duration-based estimate of VaR is $1,000.00 * 6.08 years * 1.0% volatility * 2.33 =$141.66. Note this 99.0% assumes a worst expected yield change of 2.33%.

However, he realizes that he forgot to account for the bond's convexity. If Peter revised his VaR estimate to include convexity, which of the following is nearest to the ADJUSTMENT in his VaR estimate (just difference due to the inclusion of convexity's effect)?

a. It will decrease by about $0.54 to$141.12
b. It will decrease by about $13.57 to$128.09
c. It will increase by about $11.65 to$153.31
d. It will increase by about $27.14 to$168.81

806.3. Linda Allen explains that "the calculation of value at risk (VaR) can be an easy task if the portfolio consists of linear securities," but non-linear securities pose a special challenge. Tools for coping with non-linear securities include structured Monte Carlo, stress testing and scenario analysis. In regard to these tools, according to Linda Allen, which of the following statements is TRUE?

a. To model the value at risk (VaR) of an option straddle, Monte Carlo simulation is superior to delta-normal approximation
b. The best and simplest method for modeling correlation breakdown is to increase the number of simulations and "go further out" in the loss tail
c. A key drawback of the worst-case scenario (WCS) metric is that it cannot generate a distribution but only a single point estimate
d. To model correlation breakdown, the correlation matrix can be stressed but not beyond the point at which it becomes invertible; an invertible matrix should be avoided due to the increased model risk