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. Covariance
Combine to form distribution with leptokurtosis (heavy tails)
Choose parameters that maximize the likelihood of what observations occurring
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)

2. Heteroskedastic
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
If variance of the conditional distribution of u(i) is not constant
Contains variables not explicit in model - Accounts for randomness
Application of mathematical statistics to economic data to lend empirical support to models

3. Econometrics
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to sample along more important paths
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

4. Deterministic Simulation
Choose parameters that maximize the likelihood of what observations occurring
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Among all unbiased estimators - estimator with the smallest variance is efficient

5. Exact significance level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Attempts to sample along more important paths
P - value
Among all unbiased estimators - estimator with the smallest variance is efficient

6. Efficiency
Random walk (usually acceptable) - Constant volatility (unlikely)
Choose parameters that maximize the likelihood of what observations occurring
Among all unbiased estimators - estimator with the smallest variance is efficient
Concerned with a single random variable (ex. Roll of a die)

7. Expected future variance rate (t periods forward)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

8. SER
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
If variance of the conditional distribution of u(i) is not constant
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

9. Exponential 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
Mean of sampling distribution is the population mean
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

10. Sample covariance
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Statement of the error or precision of an estimate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

11. Unstable return distribution
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
P - value

12. Pooled data
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E(mean) = mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on an equation - P(A) = # of A/total outcomes

13. Implications of homoscedasticity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Easy to manipulate
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

14. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(y)/n = variance of sample Y
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

15. Reliability
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Statement of the error or precision of an estimate
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

16. What does the OLS minimize?
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
SSR

17. Direction of OVB
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Returns over time for an individual asset
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

18. Maximum likelihood method
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Var(X) + Var(Y)
Average return across assets on a given day
Choose parameters that maximize the likelihood of what observations occurring

19. Continuously compounded return equation
E(mean) = mean
Confidence level
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

20. Binomial distribution equations for mean variance and std dev
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 and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

21. Cross - sectional
Statement of the error or precision of an estimate
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Average return across assets on a given day
P - value

22. 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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Expected value of the sample mean is the population mean
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

23. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(x) + Variance(Y) + 2*covariance(XY)
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!)

24. POT
Peaks over threshold - Collects dataset in excess of some threshold
Low Frequency - High Severity events
Variance = (1/m) summation(u<n - i>^2)
P(Z>t)

25. Mean(expected value)
Average return across assets on a given day
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

26. GARCH
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)
Transformed to a unit variable - Mean = 0 Variance = 1
Price/return tends to run towards a long - run level
Based on a dataset

27. Gamma distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Use historical simulation approach but use the EWMA weighting system
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

28. Statistical (or empirical) model
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
Variance(y)/n = variance of sample Y
Yi = B0 + B1Xi + ui
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

29. Significance =1
If variance of the conditional distribution of u(i) is not constant
Confidence level
P - value
Confidence set for two coefficients - two dimensional analog for the confidence interval

30. BLUE
For n>30 - sample mean is approximately normal
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

31. Mean reversion in asset dynamics
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
Attempts to sample along more important paths
Price/return tends to run towards a long - run level
Variance reverts to a long run level

32. Empirical frequency
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Based on a dataset

33. Key properties of linear regression
More than one random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Regression can be non - linear in variables but must be linear in parameters

34. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

35. Bernouli Distribution
Variance reverts to a long run level
Distribution with only two possible outcomes
We accept a hypothesis that should have been rejected
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

36. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Rxy = Sxy/(Sx*Sy)

37. Single variable (univariate) probability
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Concerned with a single random variable (ex. Roll of a die)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Transformed to a unit variable - Mean = 0 Variance = 1

38. Variance of aX + bY
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)
(a^2)(variance(x)) + (b^2)(variance(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
(a^2)(variance(x)

39. Monte Carlo Simulations
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

40. Central Limit Theorem(CLT)
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!)
Sampling distribution of sample means tend to be normal
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Z = (Y - meany)/(stddev(y)/sqrt(n))

41. Consistent
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When the sample size is large - the uncertainty about the value of the sample is very small
Returns over time for an individual asset

42. Binomial distribution
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 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)
Sample mean will near the population mean as the sample size increases
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

43. Variance of weighted scheme
Variance(x)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Model dependent - Options with the same underlying assets may trade at different volatilities
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

44. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Statement of the error or precision of an estimate
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

45. Block maxima
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Model dependent - Options with the same underlying assets may trade at different volatilities
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample mean +/ - t*(stddev(s)/sqrt(n))

46. Standard error for Monte Carlo replications
When the sample size is large - the uncertainty about the value of the sample is very small
Normal - Student's T - Chi - square - F distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

47. K - th moment
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
Rxy = Sxy/(Sx*Sy)
Summation((xi - mean)^k)/n
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

48. Covariance calculations using weight sums (lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. Skewness
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
SSR
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

50. Confidence ellipse
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
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Confidence set for two coefficients - two dimensional analog for the confidence interval