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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. Binomial distribution
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!)
P - value
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

2. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Easy to manipulate
Variance = (1/m) summation(u<n - i>^2)
Average return across assets on a given day

3. Lognormal
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Probability that the random variables take on certain values simultaneously
If variance of the conditional distribution of u(i) is not constant

4. Two drawbacks of moving average series
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
(a^2)(variance(x)
Only requires two parameters = mean and variance

5. K - th moment
Summation((xi - mean)^k)/n
If variance of the conditional distribution of u(i) is not constant
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

6. T distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean of sampling distribution is the population mean

7. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
When the sample size is large - the uncertainty about the value of the sample is very small
If variance of the conditional distribution of u(i) is not constant
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

8. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Rxy = Sxy/(Sx*Sy)
Distribution with only two possible outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

9. Key properties of linear regression
Has heavy tails
Regression can be non - linear in variables but must be linear in parameters
E(mean) = mean
Variance reverts to a long run level

10. Sample mean
Expected value of the sample mean is the population mean
If variance of the conditional distribution of u(i) is not constant
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^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

11. Joint probability functions
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Probability that the random variables take on certain values simultaneously

12. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases

13. Overall F - statistic
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Returns over time for an individual asset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

14. Skewness
If variance of the conditional distribution of u(i) is not constant
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
Average return across assets on a given day
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

15. Deterministic Simulation
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
Application of mathematical statistics to economic data to lend empirical support to models
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
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

16. Sample variance
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(x) + Variance(Y) + 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

17. Priori (classical) probability
Only requires two parameters = mean and variance
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Based on an equation - P(A) = # of A/total outcomes
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

18. WLS
Distribution with only two possible outcomes
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Does not depend on a prior event or information

19. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Average return across assets on a given day
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

20. Confidence interval (from t)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Easy to manipulate
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Sample mean +/ - t*(stddev(s)/sqrt(n))

21. Exponential distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence set for two coefficients - two dimensional analog for the confidence interval
Probability that the random variables take on certain values simultaneously
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

22. Four sampling distributions

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23. Heteroskedastic
(a^2)(variance(x)
If variance of the conditional distribution of u(i) is not constant
P - value
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

24. Difference between population and sample variance
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
Population denominator = n - Sample denominator = n - 1
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Returns over time for an individual asset

25. Confidence ellipse
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Random walk (usually acceptable) - Constant volatility (unlikely)

26. Hazard rate of exponentially distributed random variable
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

27. Extreme Value Theory
Variance(y)/n = variance of sample Y
If variance of the conditional distribution of u(i) is not constant
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

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

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29. Variance - covariance approach for VaR of a portfolio
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

30. Conditional probability functions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
E(XY) - E(X)E(Y)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

31. F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sample mean +/ - t*(stddev(s)/sqrt(n))

32. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Variance(x) + Variance(Y) + 2*covariance(XY)
Random walk (usually acceptable) - Constant volatility (unlikely)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

33. Biggest (and only real) drawback of GARCH mode
Mean = np - Variance = npq - Std dev = sqrt(npq)
Summation((xi - mean)^k)/n
Nonlinearity
Concerned with a single random variable (ex. Roll of a die)

34. Economical(elegant)
Price/return tends to run towards a long - run level
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Only requires two parameters = mean and variance

35. Logistic distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Distribution with only two possible outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Has heavy tails

36. Variance of sampling distribution of means when n<N
Rxy = Sxy/(Sx*Sy)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Peaks over threshold - Collects dataset in excess of some threshold

37. R^2
Based on an equation - P(A) = # of A/total outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

38. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Random walk (usually acceptable) - Constant volatility (unlikely)

39. Gamma distribution
Expected value of the sample mean is the population mean
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
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

40. Continuous random variable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sampling distribution of sample means tend to be normal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

41. i.i.d.
Independently and Identically Distributed
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Based on an equation - P(A) = # of A/total outcomes
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

42. Reliability
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Statement of the error or precision of an estimate
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time

43. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
We accept a hypothesis that should have been rejected
Among all unbiased estimators - estimator with the smallest variance is efficient
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

44. Two assumptions of square root rule
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(X) + Variance(Y) - 2*covariance(XY)
Random walk (usually acceptable) - Constant volatility (unlikely)

45. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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)
Var(X) + Var(Y)

46. Homoskedastic only F - stat
95% = 1.65 99% = 2.33 For one - tailed tests
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

47. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(y)/n = variance of sample Y
P(X=x - Y=y) = P(X=x) * P(Y=y)

48. Historical std dev
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
Does not depend on a prior event or information
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
(a^2)(variance(x)) + (b^2)(variance(y))

49. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Does not depend on a prior event or information
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

50. Variance of weighted scheme
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Statement of the error or precision of an estimate
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one