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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

2. Simplified standard (un - weighted) variance
i = ln(Si/Si - 1)
Variance = (1/m) summation(u<n - i>^2)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

3. Reliability
Statement of the error or precision of an estimate
We accept a hypothesis that should have been rejected
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Independently and Identically Distributed

4. Two drawbacks of moving average series
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!)
Use historical simulation approach but use the EWMA weighting system
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

5. K - th moment
Summation((xi - mean)^k)/n
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Rxy = Sxy/(Sx*Sy)
Combine to form distribution with leptokurtosis (heavy tails)

6. Homoskedastic
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Mean = np - Variance = npq - Std dev = sqrt(npq)

7. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Yi = B0 + B1Xi + ui
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Contains variables not explicit in model - Accounts for randomness

8. Hybrid method for conditional volatility
Low Frequency - High Severity events
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Use historical simulation approach but use the EWMA weighting system
Least absolute deviations estimator - used when extreme outliers are not uncommon

9. Unbiased
Mean of sampling distribution is the population mean
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

10. Cross - sectional
Mean of sampling distribution is the population mean
Sampling distribution of sample means tend to be normal
Average return across assets on a given day
(a^2)(variance(x)) + (b^2)(variance(y))

11. Persistence
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
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
Based on a dataset
When the sample size is large - the uncertainty about the value of the sample is very small

12. Biggest (and only real) drawback of GARCH mode
Normal - Student's T - Chi - square - F distribution
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
Nonlinearity
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

13. Tractable
Peaks over threshold - Collects dataset in excess of some threshold
Among all unbiased estimators - estimator with the smallest variance is efficient
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Easy to manipulate

14. Simulation models
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
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

15. Variance of X+Y
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Regression can be non - linear in variables but must be linear in parameters
Var(X) + Var(Y)

16. Central Limit Theorem
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
For n>30 - sample mean is approximately normal
Model dependent - Options with the same underlying assets may trade at different volatilities
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

17. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

18. Homoskedastic only F - stat
Sample mean +/ - t*(stddev(s)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sampling distribution of sample means tend to be normal

19. Single variable (univariate) probability
P(Z>t)
Sample mean will near the population mean as the sample size increases
Concerned with a single random variable (ex. Roll of a die)
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!)

20. Poisson distribution equations for mean variance and std deviation
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
Independently and Identically Distributed
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

21. Priori (classical) probability
Easy to manipulate
Based on an equation - P(A) = # of A/total outcomes
Sample mean +/ - t*(stddev(s)/sqrt(n))
(a^2)(variance(x)) + (b^2)(variance(y))

22. Statistical (or empirical) model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Yi = B0 + B1Xi + ui

23. Heteroskedastic
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(Z>t)
If variance of the conditional distribution of u(i) is not constant

24. Weibul distribution
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. Consistent
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
When the sample size is large - the uncertainty about the value of the sample is very small
Statement of the error or precision of an estimate

26. Central Limit Theorem(CLT)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
We reject a hypothesis that is actually true
Sampling distribution of sample means tend to be normal
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

27. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
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
95% = 1.65 99% = 2.33 For one - tailed tests
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

28. Binomial distribution equations for mean variance and std dev
Attempts to sample along more important paths
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Combine to form distribution with leptokurtosis (heavy tails)
Mean = np - Variance = npq - Std dev = sqrt(npq)

29. Variance of sample mean
Does not depend on a prior event or information
Combine to form distribution with leptokurtosis (heavy tails)
Variance(y)/n = variance of sample Y
Application of mathematical statistics to economic data to lend empirical support to models

30. GARCH
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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)
Nonlinearity
Random walk (usually acceptable) - Constant volatility (unlikely)

31. Extreme Value Theory
Concerned with a single random variable (ex. Roll of a die)
Confidence level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for a combination of assets (combination of time series and cross - sectional data)

32. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Peaks over threshold - Collects dataset in excess of some threshold
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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

33. POT
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Concerned with a single random variable (ex. Roll of a die)
Yi = B0 + B1Xi + ui
Peaks over threshold - Collects dataset in excess of some threshold

34. Expected future variance rate (t periods forward)
Random walk (usually acceptable) - Constant volatility (unlikely)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(Z>t)

35. Maximum likelihood method
E(XY) - E(X)E(Y)
Average return across assets on a given day
Choose parameters that maximize the likelihood of what observations occurring
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

36. Covariance
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on a dataset
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E(XY) - E(X)E(Y)

37. Variance of X+b
Only requires two parameters = mean and variance
Variance(x)
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!)
We reject a hypothesis that is actually true

38. Lognormal
Distribution with only two possible outcomes
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

39. EWMA
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

40. Covariance calculations using weight sums (lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Attempts to sample along more important paths
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time

41. GPD
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
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

42. Continuous random variable
Variance(y)/n = variance of sample Y
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Independently and Identically Distributed

43. Bernouli Distribution
Distribution with only two possible outcomes
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
(a^2)(variance(x)
Normal - Student's T - Chi - square - F distribution

44. Variance of sampling distribution of means when n<N
P(Z>t)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

45. Result of combination of two normal with same means
Expected value of the sample mean is the population mean
Combine to form distribution with leptokurtosis (heavy tails)
Easy to manipulate
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

46. Perfect multicollinearity
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(X=x - Y=y) = P(X=x) * P(Y=y)
When one regressor is a perfect linear function of the other regressors
Low Frequency - High Severity events

47. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Independently and Identically Distributed
Confidence set for two coefficients - two dimensional analog for the confidence interval
Among all unbiased estimators - estimator with the smallest variance is efficient

48. T distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Low Frequency - High Severity events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

49. Deterministic Simulation
Concerned with a single random variable (ex. Roll of a die)
Combine to form distribution with leptokurtosis (heavy tails)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

50. R^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Based on an equation - P(A) = # of A/total outcomes