AIM 1: List and discuss how asset return distributions deviate from the normal distribution, explain reasons for the existence of fat tails in a return distribution, and analyze the implications fat tails have on return distributions.

1、When comparing a fat-tailed distribution to an otherwise similar normal distribution, the fat-tailed distribution often has:

A) a different mean and standard deviation.

B) an equal probability mass close to the mean.

C) a lower probability mass at more than three standard deviations.

D) a lower probability mass at around one standard deviation.

The correct answer is B

Fat-tailed distributions typically have less probability mass in the intermediate range, around +/– one standard deviation, compared to the normal distribution. The first two moments (mean and variance) of the distributions are similar for the fat-tailed and normal distributions. Fat-tailed distributions have greater mass in the tails and a greater probability mass around the mean than the normal distribution.

2、A distribution of asset returns that has a significantly higher probability of obtaining large losses is described as:

A) right skewed.

B) fat tailed.

C) left skewed.

D) symmetrical.

The correct answer is C

A distribution is left skewed when the distribution is asymmetrical and there is a higher probability of large negative returns than there is for large positive returns.

3、All of the following are examples of why returns distributions can deviate from the normal distribution EXCEPT the distributions:

A) are symmetrical.

B) are fat tailed.

C) are skewed.

D) have unstable parameters.

The correct answer is A

Examples of common deviations from the normal distribution are fat tails and skewed and/or unstable parameters. The normal distribution is symmetrical.

4、Which of the following statements regarding fat-tail distributions is/are TRUE? A fat-tailed distribution: I. most likely results from time-varying volatility for the unconditional distribution. II. has a lower probability mass around one standard deviation from the mean than a normal distribution. III. has a lower probability mass around the mean than a normal distribution. IV. most likely results from time-varying means for the conditional distribution.

A) I only. 

B) I and II.

C) I and III.

D) II and IV.

The correct answer is B

The most likely explanation for “fat tails” is that the second moment or volatility is time-varying. For example, volatility changes in interest rates are observed prior to much anticipated Federal Reserve announcements. Examining a data sample at different points of time from the full sample could generate fat tails in the unconditional distribution even if the conditional distributions are normally distributed. The conditional mean is not expected to deviate over time. The first two moments (mean and variance) of the distributions are similar for the fat-tailed and normal distribution. However, fat-tailed distributions typically have less probability mass in the intermediate range, around +/–1 standard deviation, compared to the normal distribution. Fat-tailed distributions have greater mass in the tails and a greater probability mass around the mean than the normal distribution.

4、Which of the following deviations from normality leads to underestimating the distribution variance?

?            Higher probability of high returns.

?            Higher probability of mean returns.

?            The mean of the distribution is conditional on the economic environment.

?            The variance of the distribution is conditional on the economic environment.

A) II only.

B) III only.

C) I, II, and IV only.

D) III and IV only.

The correct answer is A

Statements I & III lead to overestimates of variance. Statement IV leads to over or under estimates of the variance.

AIM 3: Discuss the implications regime switching has on quantifying volatility.

A regime-switching volatility model of interest rates would assume all of the following EXCEPT:

A) the regime determines whether the volatility of interest rates is high or low.

B) the unconditional distribution of interest rates is normally distributed.

C) the mean is constant.

D) interest rates are conditionally normally distributed.

The correct answer is B

A regime-switching volatility model assumes different market regimes exist with high or low volatility.  The mean is assumed constant, but the volatility depends on the regime.  Conditional on the fact that interest rates are drawn from one regime, the distribution is normally distributed.  If interest rates are drawn from more than one regime, this unconditional distribution need not be normally distributed.

AIM 5: Compare and contrast parametric approaches for estimating conditional volatility, including the historical standard deviation approach, the RiskMetrics? approach and the GARCH approach, and discuss the advantages and disadvantages of nonparametric methods for volatility forecasting.

1、

The VAR measure for the fifth percentile using the historical simulation approach is closest to:

A)  –3.90%.

B)  –2.70%.

C)  –3.80%.

D)  –3.10%.

The correct answer is B

Under the historical simulation approach, all returns in the estimation window are equally weighted. In this example, K = 150; therefore, each return has a weight of 1 / 150 = .666667%, as shown in the following table. The fifth percentile is somewhere between –2.80% and –2.60%. The midpoint –2.70% has a cumulative weight of 5.00% (5.00% = (4.67% + 5.33%) / 2). If the midpoint did not have a cumulative weight of exactly 5.00%, interpolation would be necessary to find the fifth percentile.

2、The VAR measure for the fifth percentile using the hybrid approach is closest to:

A) –3.82%.

B) –4.10%.

C) –3.80%.

D) –3.10%.

The correct answer is

The lowest and second lowest returns have cumulative weights of 2.68% and 5.21%, respectively. The point halfway between the two lowest returns is interpolated as –3.95% with a cumulative weight of 3.945%, calculated as follows: (2.68% + 5.21%) / 2. Further interpolation is required to find the fifth percentile VAR level with a return somewhere between –3.80% and –3.95%. The 5 percent VAR using the hybrid approach is calculated as:

3.95% – (3.95% – 3.80%)[(0.05 – 0.03945) / (0.0521 – 0.03945)] = 3.95% – 0.15%[0.8340] = 3.8249%

Notice that the answer has to be between –3.8% and –3.95%, so –3.82 is the only possible answer.

3、Which of the following approaches is the most restrictive regarding the underlying assumption of the asset return distribution?

A) nonparametric.

B) parametric.

C) hybrid.

D) multivariate density estimation.

The correct answer is B

A parametric model typically assumes asset returns are normally or lognormally distributed with time-varying volatility. The other approaches do not require assumptions regarding the underlying asset return distribution.

4、Which of the following derivative instruments could be classified as linear or approximately linear?

I.           Swaption

II.         Forward on commodity

III.        Interest rate cap

IV.      Futures on equity index

V.        Currency swap

A) II and IV.

B) I and III.

C) II, IV, and V.

D) II, III, and IV.

The correct answer is C

The value of a linear derivative has a constant linear relationship with the underlying asset.  The relationship does not need to be one-to-one but it must be constant (or approximately constant) and linear.  Forwards, futures, and swaps are generally linear.  The value of a nonlinear derivative is a function of the change in the underlying asset and depends on the state of the underlying asset.  Options generally are nonlinear.

5、The historical standard deviation approach differs from the RiskMetricsTM and GARCH approaches for estimating conditional volatility, because it:

A) is a parametric method.

B) places a lower weight on more recent data.

C) uses recent historical data.

D) applies a set of weights to past squared returns.

The correct answer is B

All three methods are parametric, use historical data, and apply weights to past squared returns. The historical standard deviation approach weighs all returns in the estimation window equally. The RiskMetricsTM and GARCH approaches are exponential smoothing approaches that place a higher weight on more recent data.

6、Which of the following statements regarding volatility in VAR models are TRUE? I. The RiskMetricsTM approach is very similar to the GARCH model. II. The historical standard deviation approach creates a variance-covariance matrix that is estimated under the assumption that all asset returns are normally distributed. III. The parametric approach typically assumes asset returns are normally or lognormally distributed with constant volatility. IV. Exponential smoothing methods and the historical standard deviation methods both apply a set of weights to recent past squared returns.

A) I, III, and IV.

B) I, II, and III.

C) I, II, and IV. 

D) II, III, and IV.

The correct answer is C

The third statement is false. The parametric approach typically assumes asset returns are normally or lognormally distributed with time-varying volatility. The RiskMetricsTM approach is actually a special case of the GARCH model. Both the exponential and historical standard deviation approaches create a variance-covariance matrix that is estimated under the assumption that all asset returns are normally distributed. Exponential smoothing methods and the historical standard deviation methods both apply a set of weights to recent past squared returns. The difference is that in the historical standard deviation method all weights are equal whereas more recent returns are weighted more heavily in exponential methods.

7、Which of the following is/are (an) advantage(s) of nonparametric methods compared to parametric methods for quantifying volatility? I. Nonparametric models require assumptions regarding the entire distribution of returns. II. Data is used more efficiently with nonparametric methods than parametric methods. III. Fat tails, skewness and other deviations from some assumed distribution are no longer a concern in the estimation process for nonparametric methods. IV. Multivariate density estimation (MDE) allows for weights to vary based on how relevant the data is to the current market environment by weighting the most recent data more heavily.

A) I and II.

B) I and III.

C) III only.

D) III and IV.

The correct answer is C

Fat tails, skewness, and other deviations from some assumed distribution are no longer a concern in the estimation process for nonparametric methods. The other statements are false for the following reasons. Nonparametric models do not require assumptions regarding the entire distribution of returns. Data is used more efficiently with parametric methods than nonparametric methods. Multivariate density estimation (MDE) allows for weights to vary based on how relevant the data is to the current market environment, regardless of the timing of the most relevant data. MDE is also very flexible in introducing dependence on state variables.

8、Consider the following four GARCH equations:

Equation 1: σ2n = 0.83 + 0.05μ2n-1 + 0.93σ2n-1

Equation 2: σ2n = 0.06 + 0.04μ2n-1 + 0.95σ2n-1

Equation 3: σ2n = 0.60 + 0.10μ2n-1 + 0.94σ2n-1

Equation 4: σ2n = 0.03 + 0.03μ2n-1 + 0.93σ2n-1

Which of the following statements regarding these equations is (are) CORRECT?

I.Equation 1 is a stationary model.

II.Equation 2 shows no mean reversion

III.Volatility will revert to a long run mean level faster with Equation 1 than it will for Equation 4.

IV.Volatility will revert to a long run mean level faster with Equation 3 than it will for Equation 2.

A) I only.

B) III only.

C) II and III only.

D) II and IV only.

The correct answer is A

The format of the GARCH equation is σ2n = ω + αμ2n-1 + βσ2n-1, where (α + β) = persistence. For a model to be stationary over time, the persistence must be less than one. A persistence of one means there is no mean reversion and the higher the persistence, the longer it will take for volatility to revert to a long run mean level following a large shock or movement. The persistence for Equation 2 is (0.04 + 0.95) = 0.99, which is less than one meaning there is mean reversion. The persistence for Equation 1 is higher than that of Equation 3, meaning mean reversion will take longer for Equation 1. Because the persistence for Equation 1 is less than one, Equation 1 is a stationary model. Equation 3 has a persistence greater than one, which mean the model shows no mean reversion. Only Statement III is correct.

9、GARCH(1,1) (generalized autoregressive conditional heteroskedastic) models of volatility may be useful for option traders because they:

A) provide efficient estimates of past volatility.

B) are useful in forecasting future volatility

C) are the simplest volatility models to estimate.

D) are used in the Black-Scholes option pricing model.

The correct answer is B

GARCH(1,1) models have been shown to be useful in forecasting future volatility, which may indicate to an option trader the relative value of an option price.

10、You estimate the following GARCH model:

σ2n = 0.04 + 0.30μ2t-1 + 0.50σ2n-1

If the most recent volatility estimate and error term are 0.15 and 0.02, respectively, the long-run average volatility is closest to:

A) 0.16.

B) 0.23.

C) 0.20.

D) 0.04.

The correct answer is C

11、A portfolio manager is using an exponentially weighted moving average (EWMA) model to forecast volatility for a particular market parameter. What is the implication of an EWMA weighting parameter value of 0.84?

A) A greater weight is placed on the most recent change in parameter value than on the previous volatility estimate.

B) An equal weight is placed on the previous volatility estimate as on the most recent change in parameter value.

C) A greater weight is placed on the previous volatility estimate than on the most recent change in parameter value.

D) More information is required to determine the implication.

The correct answer is C

The EWMA weighting parameter of 0.84 indicates that a weighting of 0.84 will be placed on the previous volatility estimate and a weighting of 0.16 will be placed on the most recent change in the parameter value.

12、When using an EWMA model to estimate portfolio volatility, the method used to estimate the portfolio covariance matrix should be the:

A) historical method.

B) EWMA method.

C) GARCH(1,1).

D) hyper geometric method.

The correct answer is B

For internal consistency, the covariance consistency condition requires that the method that is used to estimate portfolio volatility also has to be the method used to estimate the portfolio covariance matrix.

13、Which of the following is/are a shortcoming(s) when using the normal distribution for estimating GARCH(1,1) parameters?

I. Parameters such as volatility often exhibit mean reverting characteristics. 

II. Maximum likelihood estimation with a normal distribution is not tractable.

III. Financial and economic data series often do not follow a normal distribution.

IV. Maximum likelihood estimation with a normal distribution will never generate a local maxima.

A) III only.

B) I, III, and IV only.

C) II and IV only.

D) I and II only.

The correct answer is A

Using a normal distribution as the assumed probability generating process is only appropriate if the underlying data follow a normally distributed process. Most financial and economic time series exhibit substantial deviations from normally distributed processes. Note that maximum likelihood estimators select values for model parameters that maximize the likelihood that observed data will occur in a sample. The maximum likelihood method can be used with any type of probability distribution. Mean reversion characteristics of parameters would not be considered a shortcoming for using the normal distribution.

14、RiskMetrics uses the following value for the decay factor of daily data:

A) 0.92.

B) 0.94.

C) 0.95.

D) 0.97.

The correct answer is B

RiskMetrics uses a decay factor of 0.94 for daily data and 0.97 for monthly data.

15、Which of the following are true about the RiskMetrics, GARCH, and historical standard deviation approaches to estimate conditional volatility?

I.           RiskMetrics and historical standard deviation assume equal weights on all observations.

II.         RiskMetrics and GARCH are parametric models: historical standard deviation is not.

III.        Increasing λ suggests a higher relative weight on the most recent data for exponential smoothing models.

IV.      The most recent weight for GARCH exceeds the most recent weight for historical standard deviation, assuming the same high number of observations.

A) II, III, and IV only.

B) III and IV only.

C) II and III only.

D) I, II, and IV only.

The correct answer is B

RiskMetrics does not assign equal weights across observations. Historical standard deviation is a parametric model.

16、Using both RiskMetrics and historical standard deviation, calculate the K-value that equates the most recent weight between the two models. Assume λ is 0.98.

A) K = 30.

B) K = 50.

C) K = 51.

D) K = 98.

The correct answer is B

(1 ? λ) λt = (1 ? 0.98)(0.98)0 = 0.02; 1/K = 0.02, K = 50.

17、How many of the following statements about VAR methodologies is (are) TRUE?

I.           The parametric approach is typically defined by the calculation of the distribution mean and variance.

II.         The nonparametric approach has the advantage of no required asset distribution.

III.        The implied-volatility based approach estimates volatility using current market prices.

IV.      The GARCH approach is a parametric model that uses time varying weights on historic returns to calculate distribution parameters.

A) Three statements are true.

B) Two statements are true.

C) One statment is true.

D) All statements are true.

The correct answer is D

All of the statements are true.

18、Consider the following EWMA models that are used to estimate daily return volatility. Which model’s volatility estimates will have the most day-to-day volatility, and which model will be the slowest to respond to new data, respectively?

Model 1: σ_n² = 0.04μ_{n ? 1}² + 0.96σ_{n ? 1}²

Model 2: σ_n² = 0.02μ_{n ? 1}² + 0.98σ_{n ? 1}²

Model 3: σ_n² = 0.20μ_{n ? 1}² + 0.80σ_{n ? 1}²

Model 4: σ_n² = 0.10μ_{n ? 1}² + 0.90σ_{n ? 1}²

       Greatest day-to-day volatility    Slowest to respond to new data

A)        Model 2                     Model 2

B)        Model 3                                  Model 2

C)        Model 2                                 Model 3

D)        Model 1                                 Model 4

The correct answer is B

The form of the basic EWMA model is σn2 = (λ ? 1)μn ? 12 + λσn ? 12, where λ is the weight on the previous volatility estimate. EWMA models with a low value for λ (Model 3) will put more weight on the previous day's return and will lead to volatility estimates that in themselves are highly volatile from day to day. EWMA models with a high value for λ (close to 1, such as Model 2) will put less weight on the previous day's return, and the model will respond more slowly to new data.

AIM 7: Explain, in the context of volatility forecasting methods, the process of return aggregation.

All of the following are appropriate methods for addressing return aggregation in volatility forecasting methods EXCEPT:

A) the historical standard deviation approach creates a variance-covariance matrix that is estimated under the assumption that all asset returns are normally distributed.

B) the historical simulation approach weights returns based on market values today, regardless of the actual allocation of positions K days ago.

C) the RiskMetricsTM approach creates a variance-covariance matrix that is estimated under the assumption that volatility is constant over time.

D) for well-diversified portfolios, the strong law of large numbers is required to estimate the volatility of the vector of aggregated returns.

The correct answer is C

Both the RiskMetricsTM and the historical standard deviation approach create variance-covariance matrices that are estimated under the assumption that all asset returns are normally distributed. A major disadvantage of this approach is the number of calculations required to estimate VAR.

AIM 8: Explain how implied volatility can predict future volatility and discuss its advantages and disadvantages.

Approaches for estimating value-at-risk (VAR) can be based on the history of past returns or on current market data. The approach that focuses exclusively on current market data is:

A) the parametric approach.

B) the implied-volatility-based approach.

C) the hybrid approach.

D) the nonparametric approach. 

The correct answer is B

The implied-volatility-based approach uses a derivatives pricing model such as the Black-Scholes option pricing model to estimate implied volatility based on current market data rather than historical data.

AIM 9: Explain the implications of mean reversion in returns and return volatility, respectively have on VAR forecasts over long time horizons.

Which of the following statements regarding mean reversion is FALSE?

A) The single period conditional variance of the rate of change is 2.

B) The 2-period variance is calculated as (1+b2)2, where b is the rate of change in mean reversion. 

C) The long horizon risk is smaller than the square root volatility if mean reversion exists.

D) If the rate of change of mean reversion, b, is greater than one, the process is mean reverting.

The correct answer is D

Under the context of mean reversion the single-period conditional variance of the rate of change is σ2. The 2-period variance is (1 + b2)σ2. If b = 1, the typical variance would occur as this represents a random walk. If b < 1, the process is mean reverting.

AIM 10: Discuss the effects non-synchronous data has on estimating correlation and describe approaches that mitigate the impact of non-synchronous data on risk estimates.

1、Assume that the flow of information is constant and that returns are independent of one another. By what amount should the covariance term be inflated to adjust for nonsynchronous data between the New York Stock Exchange and the London Stock Exchange if there is a 6-hour lag between the market closing times?

A) 2.66.

B) 3.00.

C) 6.00.

D) 1.33.

The correct answer is D

For a 24-hour period, there are 6 hours that overlap for the NYSE and London Stock Exchange. If we assume information is constant, we need to inflate the covariance by multiplying it by 1.33, calculated as follows:

AIM 6: Compare and contrast the use of historic simulation, multivariate density estimation, and hybrid methods for volatility forecasting.

The value at risk measure for the fifth percentile using the hybrid approach is closest to:

A)  –3.90%.

B)  –4.30%.

C)  –4.04%.

D)  –4.10%.

The correct answer is C

The lowest and second lowest returns have cumulative weights of 3.18% and 6.00%, respectively. The point halfway between the two lowest returns is interpolated as –4.10% with a cumulative weight of 4.59% calculated as follows: –4.10% = (–4.30%+ –3.90%)/2; 4.59% = (3.18%+6.00%)/2. Further interpolation is required to find the fifth percentile VAR level with a return somewhere between –3.90% and –4.10%. The 5% VAR using the hybrid approach is calculated as: 4.10% – (4.10% – 3.90%)[(0.05 – 0.0459)/(0.06 – 0.0459)] = 4.10% – 0.20%[0.2908] = 4.04%.

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