Test your basic knowledge |

FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Yi = B0 + B1Xi + ui
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

2. Variance of aX + bY
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Peaks over threshold - Collects dataset in excess of some threshold
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
(a^2)(variance(x)) + (b^2)(variance(y))

3. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Combine to form distribution with leptokurtosis (heavy tails)
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

4. Lognormal
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
Variance(y)/n = variance of sample Y
Peaks over threshold - Collects dataset in excess of some threshold
Mean of sampling distribution is the population mean

5. Mean reversion
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Var(X) + Var(Y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

6. Standard error
Variance(x) + Variance(Y) + 2*covariance(XY)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(y)/n = variance of sample Y

7. Marginal unconditional probability function
Has heavy tails
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Does not depend on a prior event or information
Choose parameters that maximize the likelihood of what observations occurring

8. Regime - switching volatility model
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Probability that the random variables take on certain values simultaneously
We reject a hypothesis that is actually true

9. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
We accept a hypothesis that should have been rejected
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Has heavy tails

10. What does the OLS minimize?
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
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
Returns over time for an individual asset
SSR

11. ESS
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
(a^2)(variance(x)

12. Logistic distribution
Sample mean will near the population mean as the sample size increases
Has heavy tails
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

13. Poisson distribution equations for mean variance and std deviation
Combine to form distribution with leptokurtosis (heavy tails)
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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

14. Adjusted R^2
(a^2)(variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Independently and Identically Distributed
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

15. Efficiency
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Attempts to sample along more important paths
Among all unbiased estimators - estimator with the smallest variance is efficient
Nonlinearity

16. GARCH
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)
Choose parameters that maximize the likelihood of what observations occurring
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)
Mean of sampling distribution is the population mean

17. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Sample mean will near the population mean as the sample size increases
Returns over time for an individual asset

18. Joint probability functions
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Probability that the random variables take on certain values simultaneously
Does not depend on a prior event or information

19. Sample mean
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on an equation - P(A) = # of A/total outcomes
We reject a hypothesis that is actually true

20. 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
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
(a^2)(variance(x)

21. Standard variable for non - normal distributions
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Summation((xi - mean)^k)/n
Z = (Y - meany)/(stddev(y)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

22. Square root rule
Independently and Identically Distributed
Yi = B0 + B1Xi + ui
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

23. Mean reversion in variance
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance reverts to a long run level
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

24. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Regression can be non - linear in variables but must be linear in parameters
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

25. Continuously compounded return equation
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
i = ln(Si/Si - 1)

26. Pooled data
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Returns over time for a combination of assets (combination of time series and cross - sectional data)

27. i.i.d.
When the sample size is large - the uncertainty about the value of the sample is very small
Independently and Identically Distributed
SSR
For n>30 - sample mean is approximately normal

28. Significance =1
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
P - value
Confidence level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

29. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)

30. Result of combination of two normal with same means
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)
Combine to form distribution with leptokurtosis (heavy tails)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

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

32. K - th moment
Among all unbiased estimators - estimator with the smallest variance is efficient
When one regressor is a perfect linear function of the other regressors
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
Summation((xi - mean)^k)/n

33. Bernouli Distribution
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Distribution with only two possible outcomes

34. Exact significance level
P - value
Expected value of the sample mean is the population mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

35. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
More than one random variable
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

36. Monte Carlo Simulations
(a^2)(variance(x)) + (b^2)(variance(y))
Low Frequency - High Severity events
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

37. Multivariate Density Estimation (MDE)
For n>30 - sample mean is approximately normal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sampling distribution of sample means tend to be normal
E(XY) - E(X)E(Y)

38. WLS
(a^2)(variance(x)
Easy to manipulate
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

39. Law of Large Numbers
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean will near the population mean as the sample size increases
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

40. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Variance(X) + Variance(Y) - 2*covariance(XY)
We accept a hypothesis that should have been rejected
Combine to form distribution with leptokurtosis (heavy tails)

41. Single variable (univariate) probability
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Confidence set for two coefficients - two dimensional analog for the confidence interval
Concerned with a single random variable (ex. Roll of a die)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

42. GPD
(a^2)(variance(x)) + (b^2)(variance(y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean = np - Variance = npq - Std dev = sqrt(npq)

43. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
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
i = ln(Si/Si - 1)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

44. Sample correlation
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Rxy = Sxy/(Sx*Sy)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

45. Direction of OVB
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Nonlinearity
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

46. Least squares estimator(m)
Probability that the random variables take on certain values simultaneously
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

47. Time series data
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Mean of sampling distribution is the population mean
Returns over time for an individual asset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

48. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Among all unbiased estimators - estimator with the smallest variance is efficient
Expected value of the sample mean is the population mean

49. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Contains variables not explicit in model - Accounts for randomness
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
SSR

50. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Based on an equation - P(A) = # of A/total outcomes