Learning Module 3: Measuring and Managing Market Risk

Learning Module 3 Measuring and Managing Market Risk Using the example given, it is correct to say any of the following: €2.2 million is the minimum loss we would expect 5% of the time. 5% of the time, losses would be at least €2.2 million. We would expect a loss of no more than €2.2 million 95% of the time. The last sentence is sometimes mistakenly phrased as "95% of the time we would expect to lose less than €2.2 million," but this statement could be taken to mean that 95% of the time we would incur losses, although those losses would be less than €2.2 million. In fact, a large percentage of the time we will make money. VaR and Standard Deviations The 16% VaR relates to a one standard deviation move as follows: In a normal distribution, 50% of the outcomes are to the right of the expected value and 50% are to the left. A one standard deviation interval implies that 68% of the outcomes lie within one standard deviation of the expected value; thus, 34% of the outcomes lie one standard deviation to the left of the expected value and 34% of the outcomes one standard deviation to the right. Adding the 50% of the outcomes that lie to the right of the expected value to the 34% of the outcomes that lie one standard deviation below the expected value means that 84% of all outcomes lie to the right of the point that is one standard deviation to the left of the expected value. Therefore, 16% of all outcomes lie below this point. Thus, a one standard deviation movement is equivalent to a 16% VaR (or an 84% level of confidence). EXAMPLE 1 Definition of VaR 1. Given a VaR of $12.5 million at 5% for one month, which of the following statements is correct? A. There is a 5% chance of losing $12.5 million over one month. B. There is a 95% chance that the expected loss over the next month is less than $12.5 million. C. The minimum loss that would be expected to occur over one month 5% of the time is $12.5 million. Solution to 1: C is correct because it is the only statement that accurately expresses the VaR. A is incorrect because VaR does not give the likelihood of losing a specific amount. B is incorrect because VaR is not an expected loss; rather, it is a minimum loss. 2. Which of the following statements is not correct? A. A 1% VaR implies a downward move of 1%. B. A one standard deviation downward move is equivalent to a 16% VaR. C. A 5% VaR implies a move of 1.65 standard deviations less than the expected value. Solution to 2: A is correct. A 1% VaR (99% confidence) is the point on the distribution 2.33 standard deviations below the expected value. Answers B and C correctly describe a 16% and 5% VaR, respectively. It is important to note that we cannot estimate a daily VaR and annualize it to arrive at an annual VaR estimate. First, to assume that a daily distribution of returns can be extrapolated to an annual distribution is a bold assumption. Second, annualizing the daily VaR is not the same as adjusting the expected return and the standard deviation to annual numbers and then calculating the annual VaR. The expected return is annualized by multiplying the daily return by 250, and the standard deviation is annualized by multiplying the daily standard deviation by the square root of 250. Thus, we can annualize the data and estimate an annual VaR, but we cannot estimate a daily VaR and annualize it without assuming a zero expected return. EXAMPLE 2 Parametric VaR 1. The parameters of normal distribution required to estimate parametric VaR A. expected value and standard deviation. R. skewness and kartosis C. standard deviation and skewness. Solution to 1: A is correct. The parameters of a normal distribution are the expected value and standard deviation. Skewness, as mentioned in Band C. and kurtosis, as mentioned in B, are characteristics used to describe a non normal distribu tion. 2. Assuming a daily expected return of 0.0354% and daily standard deviation of 10112% (as in the example in the text), which of the following is cinsest to the 1% VaR for a $150 million portfolio? Express your answer in dollars A. $3.5 million $2.4 million C$14 million Solution to 2: A is correct and is obtained as follows: Step 1 2.33 x 0.010112-0023561 Step 2 0.000384-0.023561-0023177 Step 3 Convert -0.023177 to 0.023177 Step 4 0.023177x5150 million $3,476,550 B is the estimated VaR at a 5% threshold, and C is the estimated VaR using a one standard deviation threshold 3. Assuming a daily expected return of 0.0384% and daily standard deviation of 10112% (as in the example in the text), the daily 5% parametric VaR is 52.445,150. Rounding the Vall to $2.4 million, which of the following values is closest to the annual 5% parametric VaR? Express your answer in dollars. A. 538 million 1. $25 million C $600 million Solution to 3: B is correct. It is found by annualizing the daily return and standard de viation and using these figures in the calculation. The annual return and standard deviation are, respectively, 0.096000 (0.000384x250) and 0159885 (0.010112250) Step 1 0.159885 1.65-0263810 Step 2 0.096000-0263810-0.167810 Convert-0.167810 to 0.167810 Step 3 0.167810 $150 million $25,171,500 Step 4 A incorrectly multiplies the daily VaR by the square root of the number of trading days in a year (V250), and Cincorrectly multiplies the daily VaR by the approximate number of trading days in a year (250). Neither A nor C make the appropriate adjustment to annualize the standard deviation. EXAMPLE 3 Historical Simulation VaR 1. Which of the following statements about the historical simulation method of estimating VaR is most correct? A. A 5% historical simulation VaR is the value that is 5% to the left of the expected value. B. A 5% historical simulation VaR is the value that is 1.65 standard deviations to the left of the expected value. C. A 5% historical simulation VaR is the fifth percentile, meaning the point on the distribution beyond which 5% of the outcomes result in larger losses. Solution to 1: C is correct. In the historical method, the portfolio returns are arrayed lowest to highest and the observation at the fifth percentile (95% of the outcomes are better than this outcome) is the VaR. A is not correct because it draws a point on the distribution relative to the expected value rather than using the 5% of the outcomes that are in the left-most of the distribution. B confuses the parametric and historical methods. In the parametric method, the 5% VaR lies 1.65 standard deviations below the mean. 2. Which of the following is a limitation of the historical simulation method? A. The past may not repeat itself. B. There is a reliance on the normal distribution. C. Estimates of the mean and variance could be biased. Solution to 2: A is correct. The historical simulation method estimates VaR based on the historical distribution of the risk factors. B is not correct; the historical simulation method does not rely on any particular distribution because it simply uses whatever distribution applied in the past. C is not correct because the historical distribution does not formally estimate the mean and variance. EXAMPLE 4 Monte Carlo Simulation VaR 1. When will the Monte Carlo method of estimating VaR produce virtually the same results as the parametric method? A. When the Monte Carlo method assumes a non-normal distribution. B. When the Monte Carlo method uses the historical return and distribution parameters. C. When the parameters and the distribution used in the parametric method are the same as those used in the Monte Carlo method and the Monte Carlo method uses a sufficiently large sample. Solution to 1: C is correct. The Monte Carlo method simulates outcomes using whatever distribution is specified by the user. If a normal distribution is used and a sufficiently large number of simulations are run, the parameters of the Monte Carlo sample will converge with those used in the parametric method and the overall VaR should be very close to that of the parametric method. A is incorrect because the parametric method is not well-adapted to a non-normal distribution. B is incorrect because neither the Monte Carlo method nor the parametric method focuses on historical outcomes. 2. Which of the following is an advantage of the Monte Carlo method? A. The VaR is easy to calculate with a simple formula. B. It is flexible enough to accommodate many types of distributions. C. The number of necessary simulations is determined by the parameters. Solution to 2: B is correct. The method can handle any distribution. A is incorrect because Monte Carlo simulation is not a simple formula. C is incorrect; there is no industry-wide agreement as to the necessary number of simulations. EXAMPLE 5 Advantages and Limitations of VaR 1. Which of the following is not an advantage of VaR? A. It is a simple concept to communicate. B. There is widespread agreement on how to calculate it. C. It can be used to compare risk across portfolios or trading units. Solution to 1: B is correct. There is no consensus on how to calculate VaR. A and Care both advantages of VaR, as we noted that VaR is fairly simple to communicate and it can show the contribution of each unit to the overall VaR. 2. Which of the following is a limitation of VaR? A. It requires the use of the normal distribution. B. The maximum VaR is prescribed by federal securities regulators. C. It focuses exclusively on potential losses, without considering potential gains. Solution to 2: C is correct. VaR deals exclusively with left-tail or adverse events. A is wrong because although parametric VaR does generally use the normal distribution, the historical simulation method uses whatever distribution occurred in the past and Monte Carlo simulation uses whatever distribution the user chooses. B is incorrect because regulators do not specify maximum VaRs, although they may encourage and require that the measure be used. EXAMPLE 6 Extensions of VaR 1. Conditional VaR measures the: A. VaR over all possible losses. B. VaR under normal market conditions. C. average loss, given that VaR is exceeded. Solution to 1: C is correct. Conditional VaR is the average loss conditional on exceeding the VaR. A is not correct because CVaR is not concerned with losses that do not exceed the VaR threshold, and B is incorrect because VaR does not distinguish between normal and non-normal markets. 2. Which of the following correctly identifies incremental VaR? A. The change in VaR from increasing a position in an asset. B. The increase in VaR that might occur during extremely volatile markets. C. The difference between the asset with the highest VaR and the asset with the second highest VaR. Solution to 2: A correctly defines incremental VaR. Incremental VaR is the change in VaR from increasing a position in an asset, not a change in VaR from an increase in volatility. B is not correct because incremental volatility reflects the results of intentional changes in exposure, not uncontrollable market volatility. C is not correct because incremental VaR is not the difference in the VaRs of the assets with the greatest and second greatest VaRs. 3. Which of the following statements is correct about marginal Vak? A. The marginal VaR is the same as the incremental VaR. B. The marginal VaR is the VaR required to meet margin calls. C. Marginal VaR estimates the change in VaR for a small change in a given portfolio holding. Solution to 3: C is correct. In A, marginal VaR is a similar concept to incremental VaR in that they both deal with the effect of changes in VaR, but they are not the same concept. B is incorrect because marginal VaR has nothing to do with margin calls. EXAMPLE 7 Sensitivity Risk Measures 1. Which of the following most accurately characterizes duration and convexity? A. Sensitivity of bond prices to interest rates B. First and second-order effects of yield changes on bond prices C. Weighted-average time to maturity based on the coupon payments and principal Solution to 1: B is correct. Duration is the first-order effect and convexity the second-order effect of a change in interest rates on the value of a bond. A and Care are correct with respect to duration, but not for convexity. 2. Which of the following statements about the delta of a call option is not correct? A. It ranges between 0 and 1. B. It precisely captures the change in the call value when the underlying changes. C. It approaches 1 for an in-the-money option and 0 for an out-of-the-money option. Solution to 2: B is correct. A and C correctly characterize delta, whereas B states that delta is precise, which is incorrect because it gives an approximate relationship. 3. Which of the following statements about gamma and vega are correct? A. Gamma is a second-order effect, and vega is a first-order effect. 8. Gamma is the effect of volatility, and vega is the effect of changes in volatility. C. Gamma is a second-order effect arising from changes in the sensitivity of volatility to the underlying price. Solution to 3: A is correct. B is not correct because gamma does not capture the effect of volatility. Vega is the effect of volatility, but it relates to the level and not the change in volatility. C is incorrect because although gamma is a second-order effect on the option value, it is not related to the sensitivity of volatility to the underlying price. EXAMPLE 8 Scenario Analysis 1. Which of the following is an example of a reverse stress test? A. Identify the top 10 exposures in the portfolio, and then generate a hypothetical stress that could adversely affect all 10 simultaneously. B. Find the worst single day's performance that could have occurred for the current portfolio had it been held throughout the past five years. Find the returns that occurred in all risk factors in the 2008 global financial crisis, reverse the sign on these, and apply them to today's portfolio. Solution to 1: A is correct. B is not a reverse stress test because reverse stress tests focus more narrowly on trouble spots for a specific portfolio. C would illustrate how the portfolio would have performed in an extremely strong market, quite unlike what occurred in 2008. 2. Which kind of market participant is least likely to use scenario analysis as a pass/fail stress test? A. Bank B. Long-only asset manager C. Hedge fund using leverage Solution to 2: B is correct. Long-only asset managers do not typically use leverage and are thus less likely to become insolvent, making a pass/fail test for solvency less relevant to them. A and C are not correct because parties that use leverage, such as hedge funds and banks, are likely to use stress tests to determine what market movements could impair their capital and lead to insolvency. 3. What is the most accurate approach to scenario analysis for a portfolio that uses options? A. Apply the scenario to option delta. B. Apply the scenario to option delta + gamma. C. Fully reprice the options using the market returns specified under the scenario. Solution to 3: C is correct. Both A and B risk misestimating the actual results of the scenario because both delta and gamma estimate how an option's value might change for a small move in the underlying asset, not the large movements typically used in a scenario analysis. Granularity: Too Much or Too Little? Sensitivity measures are aggregated in categories or "buckets." (A bucket is a risk factor description such as "one- to five-year French sovereign debt.") When a number of fixed-income positions are assigned to the same bucket, the effect is an assumption of perfect correlation across the risks encompassed by that bucket. For the "one-to five-year French sovereign debt risk factor, a short duration position in four-year French sovereign debt will be assumed to fully offset a long duration position in two-year French sovereign debt. However, this may not be true in the case of a non-parallel interest rate change; these points on the yield curve do not have a correlation coefficient of 1 to one another. The broader the buckets used, the more they can hide this kind of correlation risk; but the narrower the buckets used, the greater the complexity and thus the more difficult to portray portfolios in simple, accessible ways. The width or the narrowness of the risk factor buckets used to portray sensitivity measures is referred to as granularity. EXAMPLE 10 Creating Constraints with Risk Measures 1. Which of the following is not an example of risk budgeting? A. Giving a foreign exchange trading desk a VaR limit of $10 million B. Allowing a portfolio manager to have an ex-ante tracking error up to 5% in a given portfolio C. Reducing the positions in a portfolio after a loss of a 5% of capital has occurred in a single month Solution to 1: C is correct. This is an example of a stop-loss limit, not risk budgeting. The other choices are both examples of risk budgeting. 2. Which statement is true regarding risk budgeting in cases in which marginal VaR is used? A. The total risk budget is never equal to the sum of the individual sub-portfolios' risk budgets. B. The total risk budget is always equal to the sum of the individual sub-portfolios' risk budgets. C. If the total risk budget is equal to the sum of the individual sub-portfolios' risk budgets, there is a risk that this approach may cause capital to be underutilized. Solution to 2: B is correct. When using marginal VaR, the total risk budget will be equal to the sum of the individual risk budgets. Choice A is not correct. C is also incorrect; it would be correct if each sub-portfolio's individual VaR measure, not adjusted for its marginal contribution, were used, which could lead to underutilization of capital. EXAMPLE 11 Uses of Risk Measures by Market Participants 1. Which type of market participant is most likely to consistently express risk measures as a percentage of assets and relative to a benchmark? A. Banks B. Corporations CLong-only asset managers Solution to 1: C is correct. Long-only asset managers most commonly express risk measures in percentage terms and relative to a benchmark, whereas the entities in answers A and B measure risk more commonly in currency units and in absolute terms (not relative to a benchmark). Banks occasionally express risk measures, such as economic capital, as a percentage of assets or other balance sheet measures, but bank risk measures are typically expressed in currency units. 2. How does an ex-ante tracking error differ from an ex-post tracking error? A. Ex ante tracking error takes into account the behavior of options, whereas ex-post tracking error does not. B. Ex post tracking error uses a more accurate forecast of future markets than the forecast used for ex-ante tracking error. C. Ex ante tracking error uses current portfolio holdings exposed to the variability of historical markets, whereas ex-post tracking error measures the variability of historical portfolio holdings in historical markets. Solution to 2: C is correct. A is incorrect because although ex-post tracking error accounts for the options that were in the portfolio in the past, ex-ante tracking error might actually misstate the risk of options if it is computed using the para-metric method. B is incorrect because ex post tracking error is not aiming to forecast the future; it is only measuring the variability of past results.