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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. Exact significance level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
We accept a hypothesis that should have been rejected
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P - value

2. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Low Frequency - High Severity events

3. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Expected value of the sample mean is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean +/ - t*(stddev(s)/sqrt(n))

4. Consistent
If variance of the conditional distribution of u(i) is not constant
More than one random variable
We reject a hypothesis that is actually true
When the sample size is large - the uncertainty about the value of the sample is very small

5. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Price/return tends to run towards a long - run level
Combine to form distribution with leptokurtosis (heavy tails)

6. Lognormal
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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 Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

7. Stochastic error term
We reject a hypothesis that is actually true
Contains variables not explicit in model - Accounts for randomness
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Concerned with a single random variable (ex. Roll of a die)

8. Variance of X+Y
Var(X) + Var(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

9. Discrete representation of the GBM
i = ln(Si/Si - 1)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

10. Square root rule
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

11. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
If variance of the conditional distribution of u(i) is not constant
Probability that the random variables take on certain values simultaneously

12. Normal distribution
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
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(x)

13. Implied standard deviation for options
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
i = ln(Si/Si - 1)
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

14. K - th moment
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Summation((xi - mean)^k)/n

15. Homoskedastic
i = ln(Si/Si - 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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

16. Two assumptions of square root rule
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Random walk (usually acceptable) - Constant volatility (unlikely)
Among all unbiased estimators - estimator with the smallest variance is efficient
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

17. Variance of aX + bY
Peaks over threshold - Collects dataset in excess of some threshold
Variance(y)/n = variance of sample Y
(a^2)(variance(x)) + (b^2)(variance(y))
Transformed to a unit variable - Mean = 0 Variance = 1

18. Efficiency
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E(mean) = mean

19. Cross - sectional
Random walk (usually acceptable) - Constant volatility (unlikely)
Average return across assets on a given day
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
Choose parameters that maximize the likelihood of what observations occurring

20. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When one regressor is a perfect linear function of the other regressors
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

21. Pooled data
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!)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

22. Expected future variance rate (t periods forward)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

23. Control variates technique
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity

24. Variance of weighted scheme
Variance reverts to a long run level
More than one random variable
Only requires two parameters = mean and variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

25. Shortcomings of implied volatility
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Z = (Y - meany)/(stddev(y)/sqrt(n))
Model dependent - Options with the same underlying assets may trade at different volatilities
For n>30 - sample mean is approximately normal

26. Standard normal distribution
Normal - Student's T - Chi - square - F distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

27. Time series data
Only requires two parameters = mean and variance
Returns over time for an individual asset
SSR
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

28. Poisson Distribution
Distribution with only two possible outcomes
Among all unbiased estimators - estimator with the smallest variance is efficient
Use historical simulation approach but use the EWMA weighting system
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

29. Logistic distribution
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
Has heavy tails
Least absolute deviations estimator - used when extreme outliers are not uncommon
P - value

30. Mean reversion
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Distribution with only two possible outcomes
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

31. SER
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E(mean) = mean

32. Empirical frequency
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
Based on a dataset
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

33. Difference between population and sample variance
(a^2)(variance(x)) + (b^2)(variance(y))
For n>30 - sample mean is approximately normal
Population denominator = n - Sample denominator = n - 1
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

34. Unstable return distribution
Use historical simulation approach but use the EWMA weighting system
Expected value of the sample mean is the population mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

35. Kurtosis
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)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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

36. Sample covariance
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance(y)/n = variance of sample Y

37. Joint probability functions
P(Z>t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Probability that the random variables take on certain values simultaneously
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

38. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance reverts to a long run level
Choose parameters that maximize the likelihood of what observations occurring
Sample mean +/ - t*(stddev(s)/sqrt(n))

39. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

40. Perfect multicollinearity
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Population denominator = n - Sample denominator = n - 1
Special type of pooled data in which the cross sectional unit is surveyed over time
When one regressor is a perfect linear function of the other regressors

41. ESS
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

42. Limitations of R^2 (what an increase doesn't necessarily imply)

43. What does the OLS minimize?
SSR
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Model dependent - Options with the same underlying assets may trade at different volatilities

44. Two drawbacks of moving average series
Var(X) + Var(Y)
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Normal - Student's T - Chi - square - F distribution

45. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Probability that the random variables take on certain values simultaneously
E(XY) - E(X)E(Y)
Only requires two parameters = mean and variance

46. Standard error for Monte Carlo replications
Attempts to sample along more important paths
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
(a^2)(variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

47. Bernouli Distribution
Variance(x)
Regression can be non - linear in variables but must be linear in parameters
Distribution with only two possible outcomes
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

48. Variance - covariance approach for VaR of a portfolio
95% = 1.65 99% = 2.33 For one - tailed tests
Transformed to a unit variable - Mean = 0 Variance = 1
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)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. GEV
When one regressor is a perfect linear function of the other regressors
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Choose parameters that maximize the likelihood of what observations occurring
Variance(X) + Variance(Y) - 2*covariance(XY)

50. LFHS
Low Frequency - High Severity events
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment