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 in variance
Variance reverts to a long run level
Returns over time for an individual asset
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

2. Continuously compounded return equation
Variance = (1/m) summation(u<n - i>^2)
We reject a hypothesis that is actually true
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
i = ln(Si/Si - 1)

3. Tractable
Easy to manipulate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

4. K - th moment
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean of sampling distribution is the population mean
(a^2)(variance(x)) + (b^2)(variance(y))
Summation((xi - mean)^k)/n

5. Cholesky factorization (decomposition)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

6. P - value
Confidence level
P(Z>t)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Based on a dataset

7. Variance of aX
(a^2)(variance(x)
Transformed to a unit variable - Mean = 0 Variance = 1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

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

9. Confidence interval (from t)
Has heavy tails
Sample mean +/ - t*(stddev(s)/sqrt(n))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

11. Single variable (univariate) probability
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Concerned with a single random variable (ex. Roll of a die)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

12. Covariance calculations using weight sums (lambda)
Attempts to sample along more important paths
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

13. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Population denominator = n - Sample denominator = n - 1
Application of mathematical statistics to economic data to lend empirical support to models

14. Mean reversion
Probability that the random variables take on certain values simultaneously
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Distribution with only two possible outcomes
Application of mathematical statistics to economic data to lend empirical support to models

15. Key properties of linear regression
More than one random variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Regression can be non - linear in variables but must be linear in parameters
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

16. Two ways to calculate historical volatility
Yi = B0 + B1Xi + ui
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Does not depend on a prior event or information
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

17. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Regression can be non - linear in variables but must be linear in parameters
Contains variables not explicit in model - Accounts for randomness
Random walk (usually acceptable) - Constant volatility (unlikely)

18. Standard error for Monte Carlo replications
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

19. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Choose parameters that maximize the likelihood of what observations occurring
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

20. Heteroskedastic
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
If variance of the conditional distribution of u(i) is not constant

21. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(X) + Variance(Y) - 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)

22. Bernouli Distribution
Distribution with only two possible outcomes
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Random walk (usually acceptable) - Constant volatility (unlikely)

23. Perfect multicollinearity
Yi = B0 + B1Xi + ui
Mean = np - Variance = npq - Std dev = sqrt(npq)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
When one regressor is a perfect linear function of the other regressors

24. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
i = ln(Si/Si - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

25. Significance =1
Model dependent - Options with the same underlying assets may trade at different volatilities
Confidence level
Statement of the error or precision of an estimate
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

26. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample mean will near the population mean as the sample size increases
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

27. Maximum likelihood method
Variance = (1/m) summation(u<n - i>^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Choose parameters that maximize the likelihood of what observations occurring
Mean = np - Variance = npq - Std dev = sqrt(npq)

28. Standard normal distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
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

29. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

30. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Price/return tends to run towards a long - run level

31. Shortcomings of implied volatility
Sampling distribution of sample means tend to be normal
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Attempts to sample along more important paths

32. Direction of OVB
Mean = np - Variance = npq - Std dev = sqrt(npq)
P - value
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

33. Continuous representation of the GBM
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)
Independently and Identically Distributed
(a^2)(variance(x)) + (b^2)(variance(y))
If variance of the conditional distribution of u(i) is not constant

34. Central Limit Theorem
Confidence set for two coefficients - two dimensional analog for the confidence interval
For n>30 - sample mean is approximately normal
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

35. Simplified standard (un - weighted) variance
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance = (1/m) summation(u<n - i>^2)
For n>30 - sample mean is approximately normal
Confidence set for two coefficients - two dimensional analog for the confidence interval

36. Law of Large Numbers
Mean = np - Variance = npq - Std dev = sqrt(npq)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample mean will near the population mean as the sample size increases
Mean of sampling distribution is the population mean

37. Variance of X+Y
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

38. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Low Frequency - High Severity events

39. Type I error
Mean = np - Variance = npq - Std dev = sqrt(npq)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
We reject a hypothesis that is actually true
Sampling distribution of sample means tend to be normal

40. Economical(elegant)
Only requires two parameters = mean and variance
SSR
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Statement of the error or precision of an estimate

41. Discrete representation of the GBM
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
We accept a hypothesis that should have been rejected
Only requires two parameters = mean and variance

42. LFHS
We reject a hypothesis that is actually true
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Low Frequency - High Severity events
Independently and Identically Distributed

43. Critical z values
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
P - value
95% = 1.65 99% = 2.33 For one - tailed tests

44. T distribution
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
Distribution with only two possible outcomes
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Only requires two parameters = mean and variance

45. Weibul distribution
Price/return tends to run towards a long - run level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

46. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Confidence level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)

47. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and 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

48. Stochastic error term
Price/return tends to run towards a long - run level
Contains variables not explicit in model - Accounts for randomness
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

49. Adjusted R^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Only requires two parameters = mean and variance
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

50. Regime - switching volatility model
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
We accept a hypothesis that should have been rejected
Var(X) + Var(Y)