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. Single variable (univariate) probability
Peaks over threshold - Collects dataset in excess of some threshold
E(XY) - E(X)E(Y)
Concerned with a single random variable (ex. Roll of a die)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

2. P - value
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
P(Z>t)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

3. Biggest (and only real) drawback of GARCH mode
Sample mean +/ - t*(stddev(s)/sqrt(n))
Nonlinearity
We accept a hypothesis that should have been rejected
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

4. What does the OLS minimize?
Nonlinearity
(a^2)(variance(x)
Does not depend on a prior event or information
SSR

5. Square root rule
Only requires two parameters = mean and variance
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

6. Poisson Distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. Type II Error
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
We accept a hypothesis that should have been rejected

8. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Transformed to a unit variable - Mean = 0 Variance = 1
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

9. Chi - squared distribution
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
Peaks over threshold - Collects dataset in excess of some threshold
(a^2)(variance(x)
Variance(X) + Variance(Y) - 2*covariance(XY)

10. Type I error
Based on an equation - P(A) = # of A/total outcomes
We reject a hypothesis that is actually true
Mean of sampling distribution is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time

11. T distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

12. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
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
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!)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

13. Homoskedastic
P - value
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
Distribution with only two possible outcomes
Application of mathematical statistics to economic data to lend empirical support to models

14. Standard normal distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Transformed to a unit variable - Mean = 0 Variance = 1
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

15. Multivariate Density Estimation (MDE)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
i = ln(Si/Si - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient

16. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Summation((xi - mean)^k)/n
i = ln(Si/Si - 1)

17. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

18. Multivariate probability
Price/return tends to run towards a long - run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
More than one random variable
Nonlinearity

19. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

20. Variance(discrete)
Transformed to a unit variable - Mean = 0 Variance = 1
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

21. K - th moment
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Confidence set for two coefficients - two dimensional analog for the confidence interval
Summation((xi - mean)^k)/n

22. Importance sampling technique
Attempts to sample along more important paths
Probability that the random variables take on certain values simultaneously
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Application of mathematical statistics to economic data to lend empirical support to models

23. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Statement of the error or precision of an estimate

24. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Nonlinearity
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

25. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Least absolute deviations estimator - used when extreme outliers are not uncommon

26. Variance of sample mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(y)/n = variance of sample 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
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!)

27. Implications of homoscedasticity
We accept a hypothesis that should have been rejected
Application of mathematical statistics to economic data to lend empirical support to models
Average return across assets on a given day
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

28. Unconditional vs conditional distributions
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance
Low Frequency - High Severity events
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

29. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
If variance of the conditional distribution of u(i) is not constant
P - value
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

30. Normal distribution
Independently and Identically Distributed
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

31. Variance of aX
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Statement of the error or precision of an estimate
(a^2)(variance(x)
Rxy = Sxy/(Sx*Sy)

32. Monte Carlo Simulations
More than one random variable
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
We reject a hypothesis that is actually true
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

33. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Average return across assets on a given day
E(XY) - E(X)E(Y)

34. Continuously compounded return equation
More than one random variable
i = ln(Si/Si - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

35. Test for statistical independence
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation((xi - mean)^k)/n
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

36. Standard error for Monte Carlo replications
Population denominator = n - Sample denominator = n - 1
Variance reverts to a long run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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)

37. Test for unbiasedness
E(mean) = mean
i = ln(Si/Si - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Rxy = Sxy/(Sx*Sy)

38. Economical(elegant)
Transformed to a unit variable - Mean = 0 Variance = 1
Only requires two parameters = mean and variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(mean) = mean

39. LFHS
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!)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Low Frequency - High Severity events
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

40. Marginal unconditional probability function
Regression can be non - linear in variables but must be linear in parameters
Does not depend on a prior event or information
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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

41. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui
We accept a hypothesis that should have been rejected
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Reliability
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

43. EWMA
Population denominator = n - Sample denominator = n - 1
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
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

44. Cross - sectional
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

45. Variance of aX + bY
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
Average return across assets on a given day

46. Binomial distribution equations for mean variance and std dev
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean = np - Variance = npq - Std dev = sqrt(npq)

47. Variance of X+b
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Mean of sampling distribution is the population mean
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

48. Binomial 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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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!)
i = ln(Si/Si - 1)

49. Unstable return distribution
Average return across assets on a given day
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Normal - Student's T - Chi - square - F 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

50. Beta distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)