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
Statement of the error or precision of an estimate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance reverts to a long run level

2. Priori (classical) probability
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Use historical simulation approach but use the EWMA weighting system
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Based on an equation - P(A) = # of A/total outcomes

3. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Attempts to sample along more important paths
Mean of sampling distribution is the population mean

4. Stochastic error term
Does not depend on a prior event or information
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Contains variables not explicit in model - Accounts for randomness
Var(X) + Var(Y)

5. WLS
Sampling distribution of sample means tend to be normal
Expected value of the sample mean is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

6. Significance =1
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation((xi - mean)^k)/n
Based on an equation - P(A) = # of A/total outcomes
Confidence level

7. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Returns over time for an individual asset
Expected value of the sample mean is the population mean

8. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Yi = B0 + B1Xi + ui
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Only requires two parameters = mean and variance

9. Type II Error
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
We accept a hypothesis that should have been rejected
Variance(x) + Variance(Y) + 2*covariance(XY)
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Skewness
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
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
Returns over time for an individual asset
Z = (Y - meany)/(stddev(y)/sqrt(n))

11. Variance(discrete)
(a^2)(variance(x)) + (b^2)(variance(y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)
i = ln(Si/Si - 1)

12. Continuous representation of the GBM
Least absolute deviations estimator - used when extreme outliers are not uncommon
P - value
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)

13. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Peaks over threshold - Collects dataset in excess of some threshold
Confidence set for two coefficients - two dimensional analog for the confidence interval

14. Homoskedastic only F - stat
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Price/return tends to run towards a long - run level

15. Two requirements of OVB
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We reject a hypothesis that is actually true

16. Standard error for Monte Carlo replications
P(X=x - Y=y) = P(X=x) * P(Y=y)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths

17. Kurtosis
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
Combine to form distribution with leptokurtosis (heavy tails)
95% = 1.65 99% = 2.33 For one - tailed tests
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

18. Adjusted R^2
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
Low Frequency - High Severity events
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)

19. Mean reversion in variance
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Variance reverts to a long run level
If variance of the conditional distribution of u(i) is not constant

20. Antithetic variable technique
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
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

21. Time series data
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Average return across assets on a given day

22. Conditional probability functions
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(X) + Variance(Y) - 2*covariance(XY)
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

23. Discrete random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Does not depend on a prior event or information
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

24. Poisson distribution equations for mean variance and std deviation
Rxy = Sxy/(Sx*Sy)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Low Frequency - High Severity events

25. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Easy to manipulate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

26. Multivariate probability
Peaks over threshold - Collects dataset in excess of some threshold
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P - value
More than one random variable

27. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Contains variables not explicit in model - Accounts for randomness
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

28. Unstable return distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. Economical(elegant)
Transformed to a unit variable - Mean = 0 Variance = 1
Model dependent - Options with the same underlying assets may trade at different volatilities
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Only requires two parameters = mean and variance

30. 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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
SSR
P(Z>t)

31. Standard error
Application of mathematical statistics to economic data to lend empirical support to models
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
Based on a dataset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

32. Result of combination of two normal with same means
Concerned with a single random variable (ex. Roll of a die)
Combine to form distribution with leptokurtosis (heavy tails)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

33. Bootstrap method
Peaks over threshold - Collects dataset in excess of some threshold
P - value
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Confidence level

34. Pooled data
Sample mean will near the population mean as the sample size increases
Sample mean +/ - t*(stddev(s)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

35. What does the OLS minimize?
When the sample size is large - the uncertainty about the value of the sample is very small
SSR
Model dependent - Options with the same underlying assets may trade at different volatilities
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

36. Continuously compounded return equation
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
i = ln(Si/Si - 1)
Sample mean will near the population mean as the sample size increases
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

37. Normal 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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

38. Simulating for VaR
i = ln(Si/Si - 1)
Nonlinearity
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

39. Deterministic Simulation
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
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
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

40. Sample correlation
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Rxy = Sxy/(Sx*Sy)
Summation((xi - mean)^k)/n
Among all unbiased estimators - estimator with the smallest variance is efficient

41. Sample mean
Expected value of the sample mean is the population mean
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
Nonlinearity
(a^2)(variance(x)) + (b^2)(variance(y))

42. Unbiased
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Combine to form distribution with leptokurtosis (heavy tails)
Mean of sampling distribution is the population mean
Model dependent - Options with the same underlying assets may trade at different volatilities

43. Non - parametric vs parametric calculation of VaR
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
(a^2)(variance(x)) + (b^2)(variance(y))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(X) + Variance(Y) - 2*covariance(XY)

44. Joint probability functions
Peaks over threshold - Collects dataset in excess of some threshold
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability that the random variables take on certain values simultaneously
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

45. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Among all unbiased estimators - estimator with the smallest variance is efficient
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

46. Exponential distribution
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
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
Transformed to a unit variable - Mean = 0 Variance = 1

47. F distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean of sampling distribution is the population mean

48. SER
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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)

49. Variance of sample mean
Distribution with only two possible outcomes
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
Variance(y)/n = variance of sample Y
Based on a dataset

50. Regime - switching volatility model
Variance reverts to a long run level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon