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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. Potential reasons for fat tails in return distributions
Choose parameters that maximize the likelihood of what observations occurring
Low Frequency - High Severity events
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

2. Overall F - statistic
Probability that the random variables take on certain values simultaneously
Variance(x) + Variance(Y) + 2*covariance(XY)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

3. Non - parametric vs parametric calculation of VaR
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
(a^2)(variance(x)
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

4. Single variable (univariate) probability
Var(X) + Var(Y)
Concerned with a single random variable (ex. Roll of a die)
SSR
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)

5. Unconditional vs conditional distributions
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
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

6. Mean(expected value)
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
Easy to manipulate
Z = (Y - meany)/(stddev(y)/sqrt(n))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

7. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Use historical simulation approach but use the EWMA weighting system
Statement of the error or precision of an estimate
Mean of sampling distribution is the population mean

8. Variance of X+Y
Among all unbiased estimators - estimator with the smallest variance is efficient
P(X=x - Y=y) = P(X=x) * P(Y=y)
Var(X) + Var(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

9. Simulation models
Regression can be non - linear in variables but must be linear in parameters
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

10. Variance of weighted scheme
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
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Model dependent - Options with the same underlying assets may trade at different volatilities

11. Variance of sampling distribution of means when n<N
We accept a hypothesis that should have been rejected
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Attempts to sample along more important paths
Has heavy tails

12. Covariance
More than one random variable
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
Normal - Student's T - Chi - square - F distribution
E(XY) - E(X)E(Y)

13. LFHS
Based on an equation - P(A) = # of A/total outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Low Frequency - High Severity events
Expected value of the sample mean is the population mean

14. Law of Large Numbers
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample mean will near the population mean as the sample size increases
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

15. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
i = ln(Si/Si - 1)

16. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P - value
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
Attempts to sample along more important paths

17. Heteroskedastic
P - value
SSR
If variance of the conditional distribution of u(i) is not constant
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

18. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Choose parameters that maximize the likelihood of what observations occurring
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P(X=x - Y=y) = P(X=x) * P(Y=y)

19. Gamma distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

20. ESS
We accept a hypothesis that should have been rejected
Confidence set for two coefficients - two dimensional analog for the confidence interval
For n>30 - sample mean is approximately normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

21. R^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Regression can be non - linear in variables but must be linear in parameters
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Statement of the error or precision of an estimate

22. Variance of aX + bY
Attempts to sample along more important paths
(a^2)(variance(x)) + (b^2)(variance(y))
Peaks over threshold - Collects dataset in excess of some threshold
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

23. Logistic distribution
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
When the sample size is large - the uncertainty about the value of the sample is very small
Nonlinearity
Has heavy tails

24. Standard variable for non - normal distributions
Statement of the error or precision of an estimate
Transformed to a unit variable - Mean = 0 Variance = 1
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

25. Weibul distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
If variance of the conditional distribution of u(i) is not constant

26. T distribution
Combine to form distribution with leptokurtosis (heavy tails)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

27. POT
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Peaks over threshold - Collects dataset in excess of some threshold
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

28. F distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Summation((xi - mean)^k)/n
Variance reverts to a long run level

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

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30. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Has heavy tails
Variance(y)/n = variance of sample Y
When one regressor is a perfect linear function of the other regressors

31. Standard normal distribution
Yi = B0 + B1Xi + ui
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Transformed to a unit variable - Mean = 0 Variance = 1
Application of mathematical statistics to economic data to lend empirical support to models

32. Maximum likelihood method
When the sample size is large - the uncertainty about the value of the sample is very small
Regression can be non - linear in variables but must be linear in parameters
Statement of the error or precision of an estimate
Choose parameters that maximize the likelihood of what observations occurring

33. Type I error
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)
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
We reject a hypothesis that is actually true
Regression can be non - linear in variables but must be linear in parameters

34. Implied standard deviation for options
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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!)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

35. P - value
P(Z>t)
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
Attempts to sample along more important paths
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

36. Control variates technique
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

37. GARCH
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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)
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

38. Shortcomings of implied volatility
We reject a hypothesis that is actually true
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

39. BLUE
P(X=x - Y=y) = P(X=x) * P(Y=y)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

40. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Application of mathematical statistics to economic data to lend empirical support to models
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

41. GEV
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Only requires two parameters = mean and variance
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Model dependent - Options with the same underlying assets may trade at different volatilities

42. Type II Error
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We accept a hypothesis that should have been rejected
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

43. Efficiency
If variance of the conditional distribution of u(i) is not constant
Variance(y)/n = variance of sample Y
Among all unbiased estimators - estimator with the smallest variance is efficient
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

44. Significance =1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample mean will near the population mean as the sample size increases
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Confidence level

45. Skewness
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
If variance of the conditional distribution of u(i) is not constant
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Independently and Identically Distributed

46. Statistical (or empirical) model
Based on a dataset
95% = 1.65 99% = 2.33 For one - tailed tests
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Yi = B0 + B1Xi + ui

47. Variance of sample mean
When one regressor is a perfect linear function of the other regressors
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
Variance(y)/n = variance of sample Y
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

48. Persistence
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
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

49. Reliability
Summation((xi - mean)^k)/n
Statement of the error or precision of an estimate
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

50. Continuous random variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Only requires two parameters = mean and variance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx