LO 1.7: Interpret Q Q plots to identify the characteristics o f a distribution.
A natural question to ask in the course of our analysis is, From what distribution is the data drawn? The truth is that you will never really know since you only observe the realizations from random draws of an unknown distribution. However, visual inspection can be a very simple but powerful technique.
In particular, the quantile-quantile (QQ) plot is a straightforward way to visually examine if empirical data fits the reference or hypothesized theoretical distribution (assume standard normal distribution for this discussion). The process graphs the quantiles at regular confidence intervals for the empirical distribution against the theoretical distribution. As an example, if both the empirical and theoretical data are drawn from the same distribution, then the median (confidence level = 30%) of the empirical distribution would plot very close to zero, while the median of the theoretical distribution would plot exactly at zero.
Continuing in this fashion for other quantiles (40%, 60%, and so on) will map out a function. If the two distributions are very similar, the resulting Q Q plot will be linear.
Let us compare a theoretical standard normal distribution relative to an empirical ^-distribution (assume that the degrees of freedom for the ^-distribution are sufficiently small and that there are noticeable differences from the normal distribution). We know that both distributions are symmetric, but the ^-distribution will have fatter tails. Hence, the quantiles near zero (confidence level = 50%) will match up quite closely. As we move further into the tails, the quantiles between the ^-distribution and the normal will diverge (see Figure 3). For example, at a confidence level of 95%, the critical .z-value is 1.65, but for the ^-distribution, it is closer to 1.68 (degrees of freedom of approximately 40). At 97.5% confidence, the difference is even larger, as the .z-value is equal to 1.96 and the r-stat is equal to 2.02. More generally, if the middles of the QQplot match up, but the tails do not, then the empirical distribution can be interpreted as symmetric with tails that differ from a normal distribution (either fatter or thinner).
2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
Figure 3: Q Q Plot
2018 Kaplan, Inc.
Topic 1 Cross Reference to GARP Assigned Reading – Dowd, Chapter 3
K e y C o n c e p t s
Historical simulation is the easiest method to estimate value at risk. All that is required is to reorder the profit/loss observations in increasing magnitude of losses and identify the breakpoint between the tail region and the remainder of distribution.
Parametric estimation of VaR requires a specific distribution of prices or equivalently, returns. This method can be used to calculate VaR with either a normal distribution or a lognormal distribution.
Under the assumption of a normal distribution, VaR (i.e., delta-normal VaR) is calculated as follows:
VaR = |ip/L + tfp/L x za
Under the assumption of a lognormal distribution, lognormal VaR is calculated as follows:
VaR = Pt_! x (l – e^R –<7RXza ) LO 1.3 VaR identifies the lower bound of the profit/loss distribution, but it does not estimate the expected tail loss. Expected shortfall overcomes this deficiency by dividing the tail region into equal probability mass slices and averaging their corresponding VaRs. LO 1.4 A more general risk measure than either VaR or ES is known as a coherent risk measure. LO 1.3 A coherent risk measure is a weighted average of the quantiles of the loss distribution where the weights are user-specific based on individual risk aversion. A coherent risk measure will assign each quantile (not just tail quantiles) a weight. The average of the weighted VaRs is the estimated loss. 2018 Kaplan, Inc. Page 11 Topic 1 Cross Reference to GARP Assigned Reading - Dowd, Chapter 3 LO 1.6 Sound risk management requires the computation of the standard error of a coherent risk measure to estimate the precision of the risk measure itself. The simplest method creates a confidence interval around the quantile in question. To compute standard error, it is necessary to find the variance of the quantile, which will require estimates from the underlying distribution. LO 1.7 The quantile-quantile (QQ) plot is a visual inspection of an empirical quantile relative to a hypothesized theoretical distribution. If the empirical distribution closely matches the theoretical distribution, the QQplot would be linear. Page 12 2018 Kaplan, Inc. Topic 1 Cross Reference to GARP Assigned Reading - Dowd, Chapter 3 C o n c e p t C h e c k e r s The VaR at a 93% confidence level is estimated to be 1.36 from a historical simulation of 1,000 observations. Which of the following statements is most likely true? A. The parametric assumption of normal returns is correct. B. The parametric assumption of lognormal returns is correct. C. The historical distribution has fatter tails than a normal distribution. D. The historical distribution has thinner tails than a normal distribution. Assume the profit/loss distribution for XYZ is normally distributed with an annual mean of $20 million and a standard deviation of $10 million. The 5% VaR is calculated and interpreted as which of the following statements? A. 5% probability of losses of at least $3.50 million. B. 5% probability of earnings of at least $3.50 million. C. 95% probability of losses of at least $3.50 million. D. 95% probability of earnings of at least $3.50 million. Which of the following statements about expected shortfall estimates and coherent risk measures are true? A. Expected shortfall and coherent risk measures estimate quantiles for the entire loss distribution. B. Expected shortfall and coherent risk measures estimate quantiles for the tail region. C. Expected shortfall estimates quantiles for the tail region and coherent risk measures estimate quantiles for the non-tail region only. D. Expected shortfall estimates quantiles for the entire distribution and coherent risk measures estimate quantiles for the tail region only. Which of the following statements most likely increases standard errors from coherent risk measures? A. Increasing sample size and increasing the left tail probability. B. Increasing sample size and decreasing the left tail probability. C. Decreasing sample size and increasing the left tail probability. D. Decreasing sample size and decreasing the left tail probability. The quantile-quantile plot is best used for what purpose? A. Testing an empirical distribution from a theoretical distribution. B. Testing a theoretical distribution from an empirical distribution. C. Identifying an empirical distribution from a theoretical distribution. D. Identifying a theoretical distribution from an empirical distribution. 2. 3. 4. 5. 2018 Kaplan, Inc. Page 13 Topic 1 Cross Reference to GARP Assigned Reading - Dowd, Chapter 3 C o n c e p t C h e c k e r An s w e r s 1. D The historical simulation indicates that the 5% tail loss begins at 1.56, which is less than the 1.65 predicted by a standard normal distribution. Therefore, the historical simulation has thinner tails than a standard normal distribution. 2. D The value at risk calculation at 95% confidence is: -20 million + 1.65 x 10 million = -$3.50 million. Since the expected loss is negative and VaR is an implied negative amount, the interpretation is that XYZ will earn less than +$3.50 million with 5% probability, which is equivalent to XYZ earning at least $3.50 million with 95% probability. 3. B ES estimates quantiles for n - 1 equal probability masses in the tail region only. The coherent risk measure estimates quantiles for the entire distribution including the tail region. 4. C Decreasing sample size clearly increases the standard error of the coherent risk measure given that standard error is defined as: V pQ p)/n f(q) As the left tail probability, p, increases, the probability of tail events increases, which also increases the standard error. Mathematically, p(l - p) increases asp increases untilp = 0.5. Small values ofp imply smaller standard errors. 5. C Once a sample is obtained, it can be compared to a reference distribution for possible identification. The QQ plot maps the quantiles one to one. If the relationship is close to linear, then a match for the empirical distribution is found. The QQ plot is used for visual inspection only without any formal statistical test. Page 14 2018 Kaplan, Inc. The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in: N o n -p a r a m e t r ic A p p r o a c h e s Topic 2 E x a m F o c u s This topic introduces non-parametric estimation and bootstrapping (i.e., resampling). The key difference between these approaches and parametric approaches discussed in the previous topic is that with non-parametric approaches the underlying distribution is not specified, and it is a data driven, not assumption driven, analysis. For example, historical simulation is limited by the discreteness of the data, but non-parametric analysis smoothes the data points to allow for any VaR confidence level between observations. For the exam, pay close attention to the description of the bootstrap historical simulation approach as well as the various weighted historical simulations approaches. Non-parametric estimation does not make restrictive assumptions about the underlying distribution like parametric methods, which assume very specific forms such as normal or lognormal distributions. Non-parametric estimation lets the data drive the estimation. The flexibility of these methods makes them excellent candidates for VaR estimation, especially if tail events are sparse. B o o t s t r a p H i s t o r i c a l S i m u l a t i o n A p p r o a c h