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. Variance of X+Y
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Var(X) + Var(Y)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

2. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

3. Two requirements of OVB
Confidence set for two coefficients - two dimensional analog for the confidence interval
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
When one regressor is a perfect linear function of the other regressors

4. Economical(elegant)
Only requires two parameters = mean and variance
Variance reverts to a long run level
Based on an equation - P(A) = # of A/total outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Unbiased
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean of sampling distribution is the population mean
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

6. Marginal unconditional probability function
Does not depend on a prior event or information
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Based on an equation - P(A) = # of A/total outcomes
Least absolute deviations estimator - used when extreme outliers are not uncommon

7. Continuous representation of the GBM
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(x) + Variance(Y) + 2*covariance(XY)
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)
Regression can be non - linear in variables but must be linear in parameters

8. Unconditional vs conditional distributions
Special type of pooled data in which the cross sectional unit is surveyed over time
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

9. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
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)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

10. Simplified standard (un - weighted) variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance = (1/m) summation(u<n - i>^2)

11. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Transformed to a unit variable - Mean = 0 Variance = 1
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
E(mean) = mean

12. Block maxima
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Expected value of the sample mean is the population mean

13. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P(X=x - Y=y) = P(X=x) * P(Y=y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

14. Continuously compounded return equation
Based on an equation - P(A) = # of A/total outcomes
i = ln(Si/Si - 1)
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
Var(X) + Var(Y)

15. Test for unbiasedness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E(mean) = mean

16. Control variates technique
i = ln(Si/Si - 1)
Does not depend on a prior event or information
Model dependent - Options with the same underlying assets may trade at different volatilities
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

17. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Normal - Student's T - Chi - square - F distribution

18. P - value
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P(Z>t)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

19. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Population denominator = n - Sample denominator = n - 1
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

20. Historical std dev
Yi = B0 + B1Xi + ui
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

21. Variance of sampling distribution of means when n<N
Rxy = Sxy/(Sx*Sy)
When one regressor is a perfect linear function of the other regressors
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)

22. Logistic distribution
Variance(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)
Has heavy tails
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

23. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Choose parameters that maximize the likelihood of what observations occurring
(a^2)(variance(x)) + (b^2)(variance(y))
Based on an equation - P(A) = # of A/total outcomes

24. Variance(discrete)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean of sampling distribution is the population mean

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

26. Two ways to calculate historical volatility
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!)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

27. Importance sampling technique
Attempts to sample along more important paths
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance reverts to a long run level

28. Standard variable for non - normal distributions
Probability that the random variables take on certain values simultaneously
Least absolute deviations estimator - used when extreme outliers are not uncommon
Z = (Y - meany)/(stddev(y)/sqrt(n))
Attempts to sample along more important paths

29. Time series data
Returns over time for an individual asset
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
Normal - Student's T - Chi - square - F distribution
Expected value of the sample mean is the population mean

30. Four sampling distributions

31. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Regression can be non - linear in variables but must be linear in parameters

32. What does the OLS minimize?
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)
i = ln(Si/Si - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))
SSR

33. LFHS
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
More than one random variable
Low Frequency - High Severity events
Variance(y)/n = variance of sample Y

34. Key properties of linear regression
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Regression can be non - linear in variables but must be linear in parameters
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

35. Direction of OVB
Low Frequency - High Severity 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
Combine to form distribution with leptokurtosis (heavy tails)

36. Panel data (longitudinal or micropanel)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Statement of the error or precision of an estimate
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
Special type of pooled data in which the cross sectional unit is surveyed over time

37. Persistence
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Price/return tends to run towards a long - run level

38. Overall F - statistic
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

39. Inverse transform method
Price/return tends to run towards a long - run level
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

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

41. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Average return across assets on a given day
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

42. Kurtosis
Rxy = Sxy/(Sx*Sy)
Choose parameters that maximize the likelihood of what observations occurring
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

43. Square root rule
Variance(X) + Variance(Y) - 2*covariance(XY)
Probability that the random variables take on certain values simultaneously
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

44. Result of combination of two normal with same means
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Combine to form distribution with leptokurtosis (heavy tails)
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Variance of X - Y assuming dependence
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
Normal - Student's T - Chi - square - F distribution
Only requires two parameters = mean and variance

46. Efficiency
Combine to form distribution with leptokurtosis (heavy tails)
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Among all unbiased estimators - estimator with the smallest variance is efficient

47. Beta distribution
When one regressor is a perfect linear function of the other regressors
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

48. Multivariate Density Estimation (MDE)
Does not depend on a prior event or information
P(Z>t)
Rxy = Sxy/(Sx*Sy)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

49. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

50. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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!)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha