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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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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. POT
Mean of sampling distribution is the population mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Peaks over threshold - Collects dataset in excess of some threshold
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

2. Direction of OVB
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

3. Exact significance level
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P - value
(a^2)(variance(x)) + (b^2)(variance(y))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

4. Variance of aX
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
(a^2)(variance(x)
Variance(X) + Variance(Y) - 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

5. Bernouli Distribution
We accept a hypothesis that should have been rejected
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Distribution with only two possible outcomes

6. Block maxima
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample mean will near the population mean as the sample size increases
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)

7. Result of combination of two normal with same means
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)
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
Combine to form distribution with leptokurtosis (heavy tails)
Variance(x) + Variance(Y) + 2*covariance(XY)

8. Statistical (or empirical) model
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
Yi = B0 + B1Xi + ui
Based on an equation - P(A) = # of A/total outcomes

9. Variance of X+Y
Var(X) + Var(Y)
Based on a dataset
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!)
P - value

10. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample mean will near the population mean as the sample size increases

11. Mean(expected value)
Yi = B0 + B1Xi + ui
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Low Frequency - High Severity events
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

12. Implications of homoscedasticity
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Confidence level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

13. Empirical frequency
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related
Based on a dataset
Model dependent - Options with the same underlying assets may trade at different volatilities

14. GPD
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

15. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Mean of sampling distribution is the population mean
Rxy = Sxy/(Sx*Sy)
Normal - Student's T - Chi - square - F distribution

16. Exponential distribution
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

17. Heteroskedastic
P(Z>t)
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related
If variance of the conditional distribution of u(i) is not constant
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

18. Confidence ellipse
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Confidence set for two coefficients - two dimensional analog for the confidence interval
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia

19. Persistence
P - value
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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

20. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Rxy = Sxy/(Sx*Sy)
If variance of the conditional distribution of u(i) is not constant
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

21. Variance of weighted scheme
Variance(sample y) = (variance(y)/n)*(N - n/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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E(mean) = mean

22. Poisson distribution equations for mean variance and std deviation
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

23. Type II Error
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We accept a hypothesis that should have been rejected
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases

24. Hybrid method for conditional volatility
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sample mean +/ - t*(stddev(s)/sqrt(n))
Use historical simulation approach but use the EWMA weighting system

25. Cholesky factorization (decomposition)
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
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
95% = 1.65 99% = 2.33 For one - tailed tests
Has heavy tails

26. Priori (classical) probability
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Based on an equation - P(A) = # of A/total outcomes
Regression can be non - linear in variables but must be linear in parameters
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

27. Test for statistical independence
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Combine to form distribution with leptokurtosis (heavy tails)
Variance(x)

28. Extending the HS approach for computing value of a portfolio

29. Variance of X+b
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

30. Simulation models
95% = 1.65 99% = 2.33 For one - tailed tests
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

31. 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!)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Var(X) + Var(Y)
Contains variables not explicit in model - Accounts for randomness

32. Discrete random variable
i = ln(Si/Si - 1)
Only requires two parameters = mean and variance
Nonlinearity
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

33. Inverse transform method
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

34. Potential reasons for fat tails in return distributions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

35. Skewness
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Low Frequency - High Severity events
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

36. Least squares estimator(m)
When one regressor is a perfect linear function of the other regressors
Based on a dataset
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

37. Test for unbiasedness
Easy to manipulate
Low Frequency - High Severity events
E(mean) = mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

38. Joint probability functions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability that the random variables take on certain values simultaneously
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

39. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Expected value of the sample mean is the population mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Normal - Student's T - Chi - square - F distribution

40. Continuous random variable
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
When one regressor is a perfect linear function of the other regressors
Peaks over threshold - Collects dataset in excess of some threshold

41. Multivariate probability
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!)
More than one random variable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

42. P - value
P(Z>t)
Special type of pooled data in which the cross sectional unit is surveyed over time
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

43. Expected future variance rate (t periods forward)
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Contains variables not explicit in model - Accounts for randomness

44. Extreme Value Theory
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!)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Returns over time for an individual asset

45. Continuously compounded return equation
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
i = ln(Si/Si - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

46. Covariance calculations using weight sums (lambda)
Yi = B0 + B1Xi + ui
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Var(X) + Var(Y)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

47. 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)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

48. F distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
For n>30 - sample mean is approximately normal
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
We accept a hypothesis that should have been rejected

49. Continuous representation of the GBM
Price/return tends to run towards a long - run level
(a^2)(variance(x)
Combine to form distribution with leptokurtosis (heavy tails)
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)

50. Non - parametric vs parametric calculation of VaR
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Special type of pooled data in which the cross sectional unit is surveyed over time
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events