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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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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. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Use historical simulation approach but use the EWMA weighting system
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Application of mathematical statistics to economic data to lend empirical support to models

2. Law of Large Numbers
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases
Least absolute deviations estimator - used when extreme outliers are not uncommon
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!)

3. Sample covariance
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Transformed to a unit variable - Mean = 0 Variance = 1
Nonlinearity

4. EWMA
Use historical simulation approach but use the EWMA weighting system
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Low Frequency - High Severity events
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

5. Exact significance level
E(mean) = mean
P - value
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

6. Shortcomings of implied volatility
Variance(y)/n = variance of sample Y
Choose parameters that maximize the likelihood of what observations occurring
Model dependent - Options with the same underlying assets may trade at different volatilities
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

7. Multivariate probability
Based on a dataset
Regression can be non - linear in variables but must be linear in parameters
Mean = np - Variance = npq - Std dev = sqrt(npq)
More than one random variable

8. Mean reversion in asset dynamics
E(mean) = mean
More than one random variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Price/return tends to run towards a long - run level

9. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(X) + Variance(Y) - 2*covariance(XY)
Statement of the error or precision of an estimate

10. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Concerned with a single random variable (ex. Roll of a die)
We accept a hypothesis that should have been rejected
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

11. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
Among all unbiased estimators - estimator with the smallest variance is efficient
Easy to manipulate

12. Continuously compounded return equation
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
i = ln(Si/Si - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

13. LFHS
95% = 1.65 99% = 2.33 For one - tailed tests
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
Normal - Student's T - Chi - square - F distribution
Low Frequency - High Severity events

14. 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
Sampling distribution of sample means tend to be normal
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

15. LAD
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

16. Implications of homoscedasticity
P(X=x - Y=y) = P(X=x) * P(Y=y)
Regression can be non - linear in variables but must be linear in parameters
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

17. Standard variable for non - normal distributions
Variance(x)
Regression can be non - linear in variables but must be linear in parameters
Expected value of the sample mean is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))

18. Potential reasons for fat tails in return distributions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
P(Z>t)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

19. Two assumptions of square root rule
Only requires two parameters = mean and variance
Variance(X) + Variance(Y) - 2*covariance(XY)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Random walk (usually acceptable) - Constant volatility (unlikely)

20. Poisson distribution equations for mean variance and std deviation
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)

21. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Low Frequency - High Severity events
Variance(y)/n = variance of sample 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

22. P - value
P(Z>t)
Summation((xi - mean)^k)/n
Use historical simulation approach but use the EWMA weighting system
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

23. GPD
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

24. Poisson Distribution
Sample mean +/ - t*(stddev(s)/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)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

25. Binomial distribution
Mean of sampling distribution is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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!)

26. Empirical frequency
Based on a dataset
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Random walk (usually acceptable) - Constant volatility (unlikely)

27. Kurtosis
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Combine to form distribution with leptokurtosis (heavy tails)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

28. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Among all unbiased estimators - estimator with the smallest variance is efficient

29. Square root rule
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

30. Beta 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
SSR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Nonlinearity

31. Lognormal
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
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
Statement of the error or precision of an estimate

32. Four sampling distributions

33. Stochastic error term
We accept a hypothesis that should have been rejected
Contains variables not explicit in model - Accounts for randomness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

34. Central Limit Theorem
For n>30 - sample mean is approximately normal
Mean = np - Variance = npq - Std dev = sqrt(npq)
Based on a dataset
Variance reverts to a long run level

35. Simplified standard (un - weighted) variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = (1/m) summation(u<n - i>^2)
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
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!)

36. Discrete representation of the GBM
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
Var(X) + Var(Y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

37. 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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We accept a hypothesis that should have been rejected

38. Deterministic Simulation
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Price/return tends to run towards a long - run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

39. Multivariate Density Estimation (MDE)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters
Choose parameters that maximize the likelihood of what observations occurring
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

40. Logistic distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Has heavy tails
P - value
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

41. Maximum likelihood method
We accept a hypothesis that should have been rejected
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Choose parameters that maximize the likelihood of what observations occurring
Population denominator = n - Sample denominator = n - 1

42. Reliability
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient
Transformed to a unit variable - Mean = 0 Variance = 1

43. Standard error for Monte Carlo replications
Nonlinearity
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

44. Homoskedastic only F - stat
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)
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)

45. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Probability that the random variables take on certain values simultaneously
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

46. Overall F - statistic
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Transformed to a unit variable - Mean = 0 Variance = 1
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

47. Variance of aX + bY
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)) + (b^2)(variance(y))

48. Type I error
E(mean) = mean
Mean = np - Variance = npq - Std dev = sqrt(npq)
P - value
We reject a hypothesis that is actually true

49. What does the OLS minimize?
Variance reverts to a long run level
SSR
We reject a hypothesis that is actually true
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

50. Least squares estimator(m)
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!)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sampling distribution of sample means tend to be normal