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. Unstable return distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Only requires two parameters = mean and variance
Average return across assets on a given day
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

2. Variance of X+Y assuming dependence
Distribution with only two possible outcomes
Mean of sampling distribution is the population mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x) + Variance(Y) + 2*covariance(XY)

3. LFHS
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Low Frequency - High Severity events

4. Normal distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
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
We accept a hypothesis that should have been rejected

5. F distribution
P - value
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We reject a hypothesis that is actually true

6. Economical(elegant)
Only requires two parameters = mean and variance
Variance(x) + Variance(Y) + 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Random walk (usually acceptable) - Constant volatility (unlikely)

7. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Normal - Student's T - Chi - square - F distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

8. Potential reasons for fat tails in return distributions
Contains variables not explicit in model - Accounts for randomness
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient

9. What does the OLS minimize?
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(y)/n = variance of sample Y
SSR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

10. Key properties of linear regression
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
Special type of pooled data in which the cross sectional unit is surveyed over time
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

11. Adjusted R^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

12. Difference between population and sample variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Concerned with a single random variable (ex. Roll of a die)
Independently and Identically Distributed
Population denominator = n - Sample denominator = n - 1

13. Unconditional vs conditional distributions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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!)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

14. Sample correlation
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)
Z = (Y - meany)/(stddev(y)/sqrt(n))

15. Logistic distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean will near the population mean as the sample size increases
Has heavy tails
Population denominator = n - Sample denominator = n - 1

16. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E(mean) = mean
Use historical simulation approach but use the EWMA weighting system
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

17. Continuously compounded return equation
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
i = ln(Si/Si - 1)
Average return across assets on a given day
When one regressor is a perfect linear function of the other regressors

18. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

19. GEV
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Has heavy tails
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

20. Standard error for Monte Carlo replications
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability that the random variables take on certain values simultaneously
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Summation((xi - mean)^k)/n

21. Variance of sampling distribution of means when n<N
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation((xi - mean)^k)/n
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Summation((xi - mean)^k)/n
Expected value of the sample mean is the population mean

23. Conditional probability functions
Mean = np - Variance = npq - Std dev = sqrt(npq)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

24. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Has heavy tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
When one regressor is a perfect linear function of the other regressors

25. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
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
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

26. Test for unbiasedness
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
If variance of the conditional distribution of u(i) is not constant
E(mean) = mean

27. Tractable
Mean of sampling distribution is the population mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Easy to manipulate
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

28. T distribution
Variance(y)/n = variance of sample Y
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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Application of mathematical statistics to economic data to lend empirical support to models

29. Poisson Distribution
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Rxy = Sxy/(Sx*Sy)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

30. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Contains variables not explicit in model - Accounts for randomness
Mean of sampling distribution is the population mean
Mean = np - Variance = npq - Std dev = sqrt(npq)

31. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P - value

32. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Normal - Student's T - Chi - square - F distribution

33. Variance of X - Y assuming dependence
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
Variance reverts to a long run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(X) + Variance(Y) - 2*covariance(XY)

34. Joint probability functions
Probability that the random variables take on certain values simultaneously
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

35. Two assumptions of square root rule
Sampling distribution of sample means tend to be normal
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Statement of the error or precision of an estimate
Random walk (usually acceptable) - Constant volatility (unlikely)

36. Mean(expected value)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E(XY) - E(X)E(Y)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

37. Simulating for VaR
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E(XY) - E(X)E(Y)
Confidence set for two coefficients - two dimensional analog for the confidence interval

38. Four sampling distributions

39. Homoskedastic
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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 and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

40. Variance of aX
95% = 1.65 99% = 2.33 For one - tailed tests
(a^2)(variance(x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

41. Binomial distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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!)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(X=x - Y=y) = P(X=x) * P(Y=y)

42. Variance(discrete)
(a^2)(variance(x)) + (b^2)(variance(y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(X) + Variance(Y) - 2*covariance(XY)

43. Variance of sample mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
E(mean) = mean
SSR
Variance(y)/n = variance of sample Y

44. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Random walk (usually acceptable) - Constant volatility (unlikely)

45. SER
We reject a hypothesis that is actually true
More than one random variable
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
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

46. Chi - squared distribution
Choose parameters that maximize the likelihood of what observations occurring
P - value
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
Regression can be non - linear in variables but must be linear in parameters

47. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

48. Lognormal
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
(a^2)(variance(x)

49. Bootstrap method
Use historical simulation approach but use the EWMA weighting system
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

50. Exponential distribution
Variance reverts to a long run level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Independently and Identically Distributed
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)