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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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
We reject a hypothesis that is actually true
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for an individual asset

2. Non - parametric vs parametric calculation of VaR
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
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)

3. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Sample mean will near the population mean as the sample size increases
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

4. Overall F - statistic
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
(a^2)(variance(x)) + (b^2)(variance(y))

5. Key properties of linear regression
When one regressor is a perfect linear function of the other regressors
Distribution with only two possible outcomes
Confidence level
Regression can be non - linear in variables but must be linear in parameters

6. Mean reversion
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

7. Persistence
Probability that the random variables take on certain values simultaneously
More than one random variable
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
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

8. Antithetic variable technique
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean will near the population mean as the sample size increases
Sampling distribution of sample means tend to be normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

9. Control variates technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many 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

10. Result of combination of two normal with same means
SSR
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)
E(XY) - E(X)E(Y)
Combine to form distribution with leptokurtosis (heavy tails)

11. Type II Error
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Yi = B0 + B1Xi + ui
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)

12. Biggest (and only real) drawback of GARCH mode
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Nonlinearity
Variance = (1/m) summation(u<n - i>^2)
Model dependent - Options with the same underlying assets may trade at different volatilities

13. Sample variance
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Does not depend on a prior event or information
Random walk (usually acceptable) - Constant volatility (unlikely)

14. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

15. Cross - sectional
Does not depend on a prior event or information
Average return across assets on a given day
Yi = B0 + B1Xi + ui
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

16. Continuous random variable
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

17. Confidence ellipse
Variance(x) + Variance(Y) + 2*covariance(XY)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Normal - Student's T - Chi - square - F distribution
Population denominator = n - Sample denominator = n - 1

18. SER
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample mean +/ - t*(stddev(s)/sqrt(n))

19. Chi - squared distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Based on an equation - P(A) = # of A/total outcomes

20. Mean reversion in variance
Sampling distribution of sample means tend to be normal
Variance reverts to a long run level
Summation((xi - mean)^k)/n
When one regressor is a perfect linear function of the other regressors

21. Hybrid method for conditional volatility
Variance reverts to a long run level
Use historical simulation approach but use the EWMA weighting system
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Special type of pooled data in which the cross sectional unit is surveyed over time

22. GARCH
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

23. POT
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Among all unbiased estimators - estimator with the smallest variance is efficient
Peaks over threshold - Collects dataset in excess of some threshold

24. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
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
Distribution with only two possible outcomes
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

25. Priori (classical) probability
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
Easy to manipulate
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

26. BLUE
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)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Expected value of the sample mean is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

27. Binomial distribution equations for mean variance and std dev
Sampling distribution of sample means tend to be normal
Variance reverts to a long run level
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

28. Conditional probability functions
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

29. Test for unbiasedness
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models

30. Standard variable for non - normal distributions
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Z = (Y - meany)/(stddev(y)/sqrt(n))
Choose parameters that maximize the likelihood of what observations occurring
Distribution with only two possible outcomes

31. Historical std dev
Peaks over threshold - Collects dataset in excess of some threshold
Price/return tends to run towards a long - run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

32. Poisson distribution equations for mean variance and std deviation
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance reverts to a long run level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
We reject a hypothesis that is actually true

33. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

34. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Low Frequency - High Severity events
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

35. i.i.d.
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
Independently and Identically Distributed
Model dependent - Options with the same underlying assets may trade at different volatilities
Transformed to a unit variable - Mean = 0 Variance = 1

36. LFHS
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
SSR
Distribution with only two possible outcomes
Low Frequency - High Severity events

37. Sample mean
Normal - Student's T - Chi - square - F distribution
Returns over time for an individual asset
Expected value of the sample mean is the population mean
Attempts to sample along more important paths

38. Poisson Distribution
Transformed to a unit variable - Mean = 0 Variance = 1
95% = 1.65 99% = 2.33 For one - tailed tests
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)

39. Variance(discrete)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Sample mean will near the population mean as the sample size increases

40. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance reverts to a long run level
We accept a hypothesis that should have been rejected

41. Sample correlation
Has heavy tails
Sample mean will near the population mean as the sample size increases
Rxy = Sxy/(Sx*Sy)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

42. Homoskedastic only F - stat
Mean of sampling distribution is the population mean
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

43. Variance of X+b
Mean of sampling distribution is the population mean
(a^2)(variance(x)
Variance(x)
Price/return tends to run towards a long - run level

44. Two drawbacks of moving average series
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

45. SER
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!)
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
Does not depend on a prior event or information
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

46. Implied standard deviation for options
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Normal - Student's T - Chi - square - F distribution

47. Perfect multicollinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))
When one regressor is a perfect linear function of the other regressors
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

48. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

49. Variance of aX + bY
Only requires two parameters = mean and variance
(a^2)(variance(x)) + (b^2)(variance(y))
Special type of pooled data in which the cross sectional unit is surveyed over time
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

50. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Concerned with a single random variable (ex. Roll of a die)
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)