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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Bootstrap method
Has heavy tails
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

2. Variance of sampling distribution of means when n<N
Based on an equation - P(A) = # of A/total outcomes
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
Confidence level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

3. Adjusted R^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

4. Type II Error
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
We accept a hypothesis that should have been rejected

5. Multivariate probability
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
More than one random variable
Price/return tends to run towards a long - run level
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

6. POT
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Peaks over threshold - Collects dataset in excess of some threshold
Transformed to a unit variable - Mean = 0 Variance = 1

7. Simulation models
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

8. R^2
Choose parameters that maximize the likelihood of what observations occurring
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

9. Econometrics
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Statement of the error or precision of an estimate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Application of mathematical statistics to economic data to lend empirical support to models

10. Mean reversion in asset dynamics
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Rxy = Sxy/(Sx*Sy)
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!)
Price/return tends to run towards a long - run level

11. Empirical frequency
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on a dataset
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

12. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
(a^2)(variance(x)
Regression can be non - linear in variables but must be linear in parameters
When the sample size is large - the uncertainty about the value of the sample is very small

13. Exact significance level
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

14. Central Limit Theorem
Attempts to sample along more important paths
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
For n>30 - sample mean is approximately normal
Distribution with only two possible outcomes

15. F distribution
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance reverts to a long run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Confidence ellipse
Nonlinearity
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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!)
Confidence set for two coefficients - two dimensional analog for the confidence interval

17. Continuous random variable
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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

18. Panel data (longitudinal or micropanel)
i = ln(Si/Si - 1)
Sampling distribution of sample means tend to be normal
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
Special type of pooled data in which the cross sectional unit is surveyed over time

19. Unconditional vs conditional distributions
Least absolute deviations estimator - used when extreme outliers are not uncommon
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Low Frequency - High Severity events

20. Biggest (and only real) drawback of GARCH mode
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Nonlinearity
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)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

21. Two drawbacks of moving average series
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Based on an equation - P(A) = # of A/total outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

22. Discrete random variable
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Var(X) + Var(Y)
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

23. Standard error
Random walk (usually acceptable) - Constant volatility (unlikely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Low Frequency - High Severity events

24. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Expected value of the sample mean is the population mean
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Returns over time for an individual asset

25. Two requirements of OVB
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

26. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Use historical simulation approach but use the EWMA weighting system

27. Sample correlation
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x)
E(mean) = mean
Rxy = Sxy/(Sx*Sy)

28. Multivariate Density Estimation (MDE)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
i = ln(Si/Si - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

29. Maximum likelihood method
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Concerned with a single random variable (ex. Roll of a die)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Choose parameters that maximize the likelihood of what observations occurring

30. SER
Distribution with only two possible outcomes
Variance reverts to a long run level
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

31. SER
Concerned with a single random variable (ex. Roll of a die)
Sampling distribution of sample means tend to be normal
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

32. 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)
For n>30 - sample mean is approximately normal
Population denominator = n - Sample denominator = n - 1
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

33. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Choose parameters that maximize the likelihood of what observations occurring
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)

34. What does the OLS minimize?
SSR
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui

35. Potential reasons for fat tails in return distributions
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
Nonlinearity
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

36. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
We accept a hypothesis that should have been rejected
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
Summation((xi - mean)^k)/n

37. Variance - covariance approach for VaR of a portfolio
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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)
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

38. Central Limit Theorem(CLT)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sampling distribution of sample means tend to be normal

39. Antithetic variable technique
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

40. Normal distribution
Contains variables not explicit in model - Accounts for randomness
Among all unbiased estimators - estimator with the smallest variance is efficient
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

41. Block maxima
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

42. K - th moment
Statement of the error or precision of an estimate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation((xi - mean)^k)/n
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

43. 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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

44. Variance of sample mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(y)/n = variance of sample Y
Normal - Student's T - Chi - square - F distribution
Contains variables not explicit in model - Accounts for randomness

45. Historical std dev
Based on a dataset
Attempts to sample along more important paths
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Use historical simulation approach but use the EWMA weighting system

46. Variance of X+Y
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We reject a hypothesis that is actually true
Var(X) + Var(Y)
Variance = (1/m) summation(u<n - i>^2)

47. Extreme Value Theory
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Rxy = Sxy/(Sx*Sy)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

48. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Nonlinearity
Expected value of the sample mean is the population mean

49. Unbiased
Has heavy tails
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Mean of sampling distribution is the population mean

50. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
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