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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.
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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. Law of Large Numbers
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
Expected value of the sample mean is the population mean
Sample mean will near the population mean as the sample size increases
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

2. Sample correlation
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
More than one random variable
Rxy = Sxy/(Sx*Sy)

3. Biggest (and only real) drawback of GARCH mode
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Nonlinearity
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
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

4. Statistical (or empirical) model
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Yi = B0 + B1Xi + ui
Variance(x) + Variance(Y) + 2*covariance(XY)

5. GPD
Contains variables not explicit in model - Accounts for randomness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

6. Discrete random variable
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Statement of the error or precision of an estimate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. Skewness
E(XY) - E(X)E(Y)
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
For n>30 - sample mean is approximately normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

8. Tractable
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
Easy to manipulate
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

9. Variance - covariance approach for VaR of a portfolio
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
When one regressor is a perfect linear function of the other regressors
Variance(x)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

10. Continuous random variable
Easy to manipulate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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

11. Econometrics
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
Concerned with a single random variable (ex. Roll of a die)
Application of mathematical statistics to economic data to lend empirical support to models
Contains variables not explicit in model - Accounts for randomness

12. Variance of X+Y
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Var(X) + Var(Y)
Easy to manipulate

13. Joint probability functions
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

14. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Does not depend on a prior event or information
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)

15. Implications of homoscedasticity
Based on an equation - P(A) = # of A/total outcomes
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x)

16. Significance =1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample mean will near the population mean as the sample size increases
Confidence level
Rxy = Sxy/(Sx*Sy)

17. Expected future variance rate (t periods forward)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance reverts to a long run level

18. Extreme Value Theory
Price/return tends to run towards a long - run level
E(XY) - E(X)E(Y)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean = np - Variance = npq - Std dev = sqrt(npq)

19. Cross - sectional
Average return across assets on a given day
SSR
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Contains variables not explicit in model - Accounts for randomness

20. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Transformed to a unit variable - Mean = 0 Variance = 1

21. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

22. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
95% = 1.65 99% = 2.33 For one - tailed tests
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

23. Normal distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Inverse transform method
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean will near the population mean as the sample size increases
More than one random variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

25. Stochastic error term
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Contains variables not explicit in model - Accounts for randomness
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Average return across assets on a given day

26. What does the OLS minimize?
SSR
Contains variables not explicit in model - Accounts for randomness
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Population denominator = n - Sample denominator = n - 1

27. Variance of X+b
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(x)

28. Sample covariance
Average return across assets on a given day
Yi = B0 + B1Xi + ui
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

29. Empirical frequency
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Based on a dataset
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
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

30. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Random walk (usually acceptable) - Constant volatility (unlikely)
For n>30 - sample mean is approximately normal
i = ln(Si/Si - 1)

31. Kurtosis
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

32. 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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

33. Confidence interval (from t)
E(XY) - E(X)E(Y)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Distribution with only two possible outcomes
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

34. F distribution
SSR
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(y)/n = variance of sample Y
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

35. 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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Easy to manipulate
Var(X) + Var(Y)

36. Weibul distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

37. Four sampling distributions

38. Poisson distribution equations for mean variance and std deviation
Sampling distribution of sample means tend to be normal
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

39. Variance of aX
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)

40. Non - parametric vs parametric calculation of VaR
Confidence level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

41. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean of sampling distribution is the population mean
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)

42. Variance of X+Y assuming dependence
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance(x) + Variance(Y) + 2*covariance(XY)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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!)

43. GEV
E(mean) = mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Attempts to sample along more important paths
If variance of the conditional distribution of u(i) is not constant

44. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Attempts to sample along more important paths
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

45. Variance of sampling distribution of means when n<N
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Price/return tends to run towards a long - run level
Use historical simulation approach but use the EWMA weighting system
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

46. Consistent
Random walk (usually acceptable) - Constant volatility (unlikely)
i = ln(Si/Si - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small

47. Square root rule
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!)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Low Frequency - High Severity events
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

48. Two assumptions of square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = (1/m) summation(u<n - i>^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
Random walk (usually acceptable) - Constant volatility (unlikely)

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

50. Key properties of linear regression
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Regression can be non - linear in variables but must be linear in parameters
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications