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. Mean reversion
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

2. Persistence
P(Z>t)
Price/return tends to run towards a long - run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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

3. Confidence interval (from t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Expected value of the sample mean is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)
Sample mean +/ - t*(stddev(s)/sqrt(n))

4. K - th moment
We accept a hypothesis that should have been rejected
Summation((xi - mean)^k)/n
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Probability that the random variables take on certain values simultaneously

5. Variance of aX
Variance(y)/n = variance of sample Y
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

6. Conditional probability functions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

7. Single variable (univariate) probability
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability that the random variables take on certain values simultaneously
Based on a dataset
Concerned with a single random variable (ex. Roll of a die)

8. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
When one regressor is a perfect linear function of the other regressors
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
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)

9. P - value
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)
Variance(x)
P(Z>t)
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. Two assumptions of square root rule
Has heavy tails
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Random walk (usually acceptable) - Constant volatility (unlikely)

11. Non - parametric vs parametric calculation of VaR
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to sample along more important paths
When the sample size is large - the uncertainty about the value of the sample is very small
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

12. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Returns over time for an individual asset
P - value
Expected value of the sample mean is the population mean

13. GEV
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

14. Variance of aX + bY
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
(a^2)(variance(x)) + (b^2)(variance(y))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

15. Standard error
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Least absolute deviations estimator - used when extreme outliers are not uncommon
P - value
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

16. Mean reversion in asset dynamics
When one regressor is a perfect linear function of the other regressors
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Price/return tends to run towards a long - run level

17. Mean reversion in variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance reverts to a long run level
Sample mean +/ - t*(stddev(s)/sqrt(n))

18. Panel data (longitudinal or micropanel)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Special type of pooled data in which the cross sectional unit is surveyed over time
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance reverts to a long run level

19. Exponential distribution
If variance of the conditional distribution of u(i) is not constant
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

20. Gamma distribution
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(y)/n = variance of sample Y
Variance(X) + Variance(Y) - 2*covariance(XY)

21. Shortcomings of implied volatility
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
Average return across assets on a given day
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Model dependent - Options with the same underlying assets may trade at different volatilities

22. SER
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P(Z>t)

23. Variance - covariance approach for VaR of a portfolio
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
Normal - Student's T - Chi - square - F distribution
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

24. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

25. Bernouli Distribution
Distribution with only two possible outcomes
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

26. Simulation models
Distribution with only two possible outcomes
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Mean = np - Variance = npq - Std dev = sqrt(npq)

27. EWMA
Based on an equation - P(A) = # of A/total outcomes
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
i = ln(Si/Si - 1)

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

29. Multivariate probability
Yi = B0 + B1Xi + ui
More than one random variable
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

30. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset
Choose parameters that maximize the likelihood of what observations occurring
95% = 1.65 99% = 2.33 For one - tailed tests

31. Lognormal
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

32. Critical z values
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
95% = 1.65 99% = 2.33 For one - tailed tests

33. Heteroskedastic
Low Frequency - High Severity events
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
If variance of the conditional distribution of u(i) is not constant
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

34. Empirical frequency
For n>30 - sample mean is approximately normal
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on a dataset
We reject a hypothesis that is actually true

35. Homoskedastic
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
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
Combine to form distribution with leptokurtosis (heavy tails)

36. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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
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
(a^2)(variance(x)) + (b^2)(variance(y))

37. Unstable return distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample mean +/ - t*(stddev(s)/sqrt(n))
We accept a hypothesis that should have been rejected
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

38. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sampling distribution of sample means tend to be normal

39. Variance of sample mean
Var(X) + Var(Y)
Variance(y)/n = variance of sample Y
When the sample size is large - the uncertainty about the value of the sample is very small
Concerned with a single random variable (ex. Roll of a die)

40. Covariance calculations using weight sums (lambda)
Based on a dataset
Combine to form distribution with leptokurtosis (heavy tails)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

41. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Does not depend on a prior event or information
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

42. Significance =1
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(y)/n = variance of sample Y
Confidence level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

43. Test for unbiasedness
E(mean) = mean
Distribution with only two possible outcomes
SSR
For n>30 - sample mean is approximately normal

44. Implications of homoscedasticity
Only requires two parameters = mean and variance
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
More than one random variable

45. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E(XY) - E(X)E(Y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

46. Standard normal distribution
Concerned with a single random variable (ex. Roll of a die)
Transformed to a unit variable - Mean = 0 Variance = 1
Average return across assets on a given day
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

47. Statistical (or empirical) model
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Yi = B0 + B1Xi + ui

48. Two ways to calculate historical volatility
P(Z>t)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

49. Time series data
Special type of pooled data in which the cross sectional unit is surveyed over time
P(Z>t)
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
Returns over time for an individual asset

50. Normal distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Yi = B0 + B1Xi + ui
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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