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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Skewness
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Normal - Student's T - Chi - square - F distribution

2. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(mean) = mean
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

3. Adjusted R^2
Sampling distribution of sample means tend to be normal
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
Summation((xi - mean)^k)/n

4. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

5. Two ways to calculate historical volatility
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Transformed to a unit variable - Mean = 0 Variance = 1
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

6. Homoskedastic
Application of mathematical statistics to economic data to lend empirical support to models
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests

7. Variance of X+Y
Expected value of the sample mean is the population mean
Var(X) + Var(Y)
Based on an equation - P(A) = # of A/total outcomes
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. POT
Statement of the error or precision of an estimate
Peaks over threshold - Collects dataset in excess of some threshold
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = (1/m) summation(u<n - i>^2)

9. Non - parametric vs parametric calculation of VaR
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared

10. Two requirements of OVB
Yi = B0 + B1Xi + ui
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

11. Sample covariance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

12. Confidence ellipse
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

13. Four sampling distributions

14. Kurtosis
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

15. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
95% = 1.65 99% = 2.33 For one - tailed tests
Variance = (1/m) summation(u<n - i>^2)

16. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

17. Implications of homoscedasticity
For n>30 - sample mean is approximately normal
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

18. Central Limit Theorem(CLT)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sampling distribution of sample means tend to be normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE

19. Lognormal
(a^2)(variance(x)
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
When one regressor is a perfect linear function of the other regressors
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

20. Variance of sample mean
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Concerned with a single random variable (ex. Roll of a die)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(y)/n = variance of sample Y

21. Poisson Distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Has heavy tails
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

22. Unconditional vs conditional distributions
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

23. Binomial distribution equations for mean variance and std dev
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
We reject a hypothesis that is actually true
Mean = np - Variance = npq - Std dev = sqrt(npq)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Extreme Value Theory
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
Variance(x)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

25. Block maxima
Rxy = Sxy/(Sx*Sy)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

26. Overall F - statistic
Sampling distribution of sample means tend to be normal
Low Frequency - High Severity events
Special type of pooled data in which the cross sectional unit is surveyed over time
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

27. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Low Frequency - High Severity events
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Normal - Student's T - Chi - square - F distribution

28. BLUE
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(y)/n = variance of sample Y
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

29. Direction of OVB
Does not depend on a prior event or information
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails

30. Variance(discrete)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

31. Hybrid method for conditional volatility
Var(X) + Var(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on an equation - P(A) = # of A/total outcomes
Use historical simulation approach but use the EWMA weighting system

32. P - value
Least absolute deviations estimator - used when extreme outliers are not uncommon
P(Z>t)
Independently and Identically Distributed
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

33. Poisson distribution equations for mean variance and std deviation
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

34. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Least absolute deviations estimator - used when extreme outliers are not uncommon

35. Simplified standard (un - weighted) variance
Population denominator = n - Sample denominator = n - 1
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance = (1/m) summation(u<n - i>^2)

36. Antithetic variable technique
Choose parameters that maximize the likelihood of what observations occurring
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

37. Binomial distribution
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)
Rxy = Sxy/(Sx*Sy)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

38. Potential reasons for fat tails in return distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true
Normal - Student's T - Chi - square - F distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

39. Simulating for VaR
Expected value of the sample mean is the population mean
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(X=x - Y=y) = P(X=x) * P(Y=y)

40. LFHS
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(Z>t)
Low Frequency - High Severity events

41. Efficiency
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Population denominator = n - Sample denominator = n - 1
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Among all unbiased estimators - estimator with the smallest variance is efficient

42. Exponential distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Independently and Identically Distributed
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

43. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)) + (b^2)(variance(y))
We reject a hypothesis that is actually true
Rxy = Sxy/(Sx*Sy)

44. i.i.d.
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))
When one regressor is a perfect linear function of the other regressors
Independently and Identically Distributed

45. Persistence
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
Rxy = Sxy/(Sx*Sy)
Probability that the random variables take on certain values simultaneously
If variance of the conditional distribution of u(i) is not constant

46. Maximum likelihood method
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Does not depend on a prior event or information
Choose parameters that maximize the likelihood of what observations occurring

47. Gamma distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(x) + Variance(Y) + 2*covariance(XY)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared

48. Multivariate probability
Variance(x) + Variance(Y) + 2*covariance(XY)
More than one random variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Regression can be non - linear in variables but must be linear in parameters

49. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
95% = 1.65 99% = 2.33 For one - tailed tests
Population denominator = n - Sample denominator = n - 1
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test

50. Biggest (and only real) drawback of GARCH mode
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Nonlinearity
Concerned with a single random variable (ex. Roll of a die)
Variance(x) + Variance(Y) + 2*covariance(XY)