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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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1. What does the OLS minimize?
Based on an equation - P(A) = # of A/total outcomes
Var(X) + Var(Y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SSR

2. Panel data (longitudinal or micropanel)
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

3. Standard variable for non - normal distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

4. LFHS
Statement of the error or precision of an estimate
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
Confidence level

5. Standard normal distribution
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
Transformed to a unit variable - Mean = 0 Variance = 1
Peaks over threshold - Collects dataset in excess of some threshold
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

6. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance reverts to a long run level
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
For n>30 - sample mean is approximately normal

7. Type I error
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
We reject a hypothesis that is actually true
(a^2)(variance(x)) + (b^2)(variance(y))

8. Block maxima
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Combine to form distribution with leptokurtosis (heavy tails)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. P - value
P(Z>t)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
More than one random variable

10. Square root rule
Based on an equation - P(A) = # of A/total outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

11. Perfect multicollinearity
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
When one regressor is a perfect linear function of the other regressors
Low Frequency - High Severity events
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

12. Test for statistical independence
Model dependent - Options with the same underlying assets may trade at different volatilities
P(X=x - Y=y) = P(X=x) * P(Y=y)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
E(mean) = mean

13. Importance sampling technique
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Attempts to sample along more important paths
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

14. Kurtosis
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P - value
Among all unbiased estimators - estimator with the smallest variance is efficient

15. Implications of homoscedasticity
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

16. Empirical frequency
P - value
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Population denominator = n - Sample denominator = n - 1
Based on a dataset

17. Direction of OVB
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Based on a dataset
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

18. Difference between population and sample variance
Sampling distribution of sample means tend to be normal
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Population denominator = n - Sample denominator = n - 1

19. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance reverts to a long run level
Independently and Identically Distributed
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

20. SER
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

21. Potential reasons for fat tails in return distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

22. Mean reversion in variance
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
Peaks over threshold - Collects dataset in excess of some threshold

23. Chi - squared distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
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

24. Stochastic error term
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Contains variables not explicit in model - Accounts for randomness
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

25. Joint probability functions
95% = 1.65 99% = 2.33 For one - tailed tests
P(X=x - Y=y) = P(X=x) * P(Y=y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Probability that the random variables take on certain values simultaneously

26. Multivariate Density Estimation (MDE)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Transformed to a unit variable - Mean = 0 Variance = 1
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Population denominator = n - Sample denominator = n - 1

27. Unstable return distribution
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
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)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

28. Marginal unconditional probability function
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
Confidence level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Does not depend on a prior event or information

29. F distribution
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
Low Frequency - High Severity events
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
Rxy = Sxy/(Sx*Sy)

30. Continuously compounded return equation
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
i = ln(Si/Si - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

31. GARCH
Combine to form distribution with leptokurtosis (heavy tails)
Use historical simulation approach but use the EWMA weighting system
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)

32. Heteroskedastic
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
If variance of the conditional distribution of u(i) is not constant

33. POT
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Peaks over threshold - Collects dataset in excess of some threshold

34. Economical(elegant)
Rxy = Sxy/(Sx*Sy)
More than one random variable
Nonlinearity
Only requires two parameters = mean and variance

35. Time series data
Returns over time for an individual asset
(a^2)(variance(x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Concerned with a single random variable (ex. Roll of a die)

36. Key properties of linear regression
(a^2)(variance(x)) + (b^2)(variance(y))
E(XY) - E(X)E(Y)
Application of mathematical statistics to economic data to lend empirical support to models
Regression can be non - linear in variables but must be linear in parameters

37. Variance of sampling distribution of means when n<N
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)

38. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

39. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

40. Skewness
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

41. Monte Carlo Simulations
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
For n>30 - sample mean is approximately normal

42. Central Limit Theorem
Use historical simulation approach but use the EWMA weighting system
For n>30 - sample mean is approximately normal
Summation((xi - mean)^k)/n
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

43. Central Limit Theorem(CLT)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Based on an equation - P(A) = # of A/total outcomes
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

44. GPD
Based on an equation - P(A) = # of A/total outcomes
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance(y)/n = variance of sample Y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

45. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
95% = 1.65 99% = 2.33 For one - tailed tests
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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. Discrete random variable
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Distribution with only two possible outcomes

47. Limitations of R^2 (what an increase doesn't necessarily imply)

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48. Single variable (univariate) probability
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Regression can be non - linear in variables but must be linear in parameters
Concerned with a single random variable (ex. Roll of a die)

49. Law of Large Numbers
Returns over time for an individual asset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Sample mean will near the population mean as the sample size increases

50. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Independently and Identically Distributed
Special type of pooled data in which the cross sectional unit is surveyed over time
Low Frequency - High Severity events