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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. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Low Frequency - High Severity events
Variance = (1/m) summation(u<n - i>^2)

2. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

3. Multivariate probability
Average return across assets on a given day
More than one random variable
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

4. Square root rule
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset
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)

5. GPD
Rxy = Sxy/(Sx*Sy)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Special type of pooled data in which the cross sectional unit is surveyed over time
Among all unbiased estimators - estimator with the smallest variance is efficient

6. POT
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean will near the population mean as the sample size increases
Peaks over threshold - Collects dataset in excess of some threshold

7. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Low Frequency - High Severity events
Returns over time for a combination of assets (combination of time series and cross - sectional data)

8. K - th moment
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
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Summation((xi - mean)^k)/n

9. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(x) + Variance(Y) + 2*covariance(XY)

10. Type II Error
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Easy to manipulate
We accept a hypothesis that should have been rejected
Sample mean +/ - t*(stddev(s)/sqrt(n))

11. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Random walk (usually acceptable) - Constant volatility (unlikely)
(a^2)(variance(x)

12. Time series data
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Returns over time for an individual asset

13. Bernouli Distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Normal - Student's T - Chi - square - F distribution
Nonlinearity
Distribution with only two possible outcomes

14. WLS
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
95% = 1.65 99% = 2.33 For one - tailed tests

15. Lognormal
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(x) + Variance(Y) + 2*covariance(XY)

16. Four sampling distributions

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17. Logistic distribution
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Has heavy tails
Yi = B0 + B1Xi + ui

18. Economical(elegant)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Only requires two parameters = mean and variance
P - value
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

19. Variance of aX + bY
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)) + (b^2)(variance(y))
For n>30 - sample mean is approximately normal
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

20. Heteroskedastic
Price/return tends to run towards 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)
Var(X) + Var(Y)
If variance of the conditional distribution of u(i) is not constant

21. Homoskedastic only F - stat
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Choose parameters that maximize the likelihood of what observations occurring
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

22. Bootstrap method
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
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

23. Exact significance level
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

24. Reliability
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Statement of the error or precision of an estimate
Combine to form distribution with leptokurtosis (heavy tails)

25. LFHS
Peaks over threshold - Collects dataset in excess of some threshold
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
SSR
Low Frequency - High Severity events

26. Importance sampling technique
Attempts to sample along more important paths
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
Population denominator = n - Sample denominator = n - 1
Among all unbiased estimators - estimator with the smallest variance is efficient

27. What does the OLS minimize?
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
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)
SSR

28. Efficiency
P - value
Confidence level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Among all unbiased estimators - estimator with the smallest variance is efficient

29. ESS
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

30. Single variable (univariate) probability
Special type of pooled data in which the cross sectional unit is surveyed over time
Concerned with a single random variable (ex. Roll of a die)
Mean of sampling distribution is the population mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

31. Sample variance
Only requires two parameters = mean and variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

32. Consistent
Special type of pooled data in which the cross sectional unit is surveyed over time
When the sample size is large - the uncertainty about the value of the sample is very small
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

33. Variance of X+Y
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E(XY) - E(X)E(Y)
Attempts to sample along more important paths
Var(X) + Var(Y)

34. Gamma distribution
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
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
Rxy = Sxy/(Sx*Sy)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

35. Result of combination of two normal with same means
Regression can be non - linear in variables but must be linear in parameters
Mean of sampling distribution is the population mean
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Combine to form distribution with leptokurtosis (heavy tails)

36. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
If variance of the conditional distribution of u(i) is not constant
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

37. Test for unbiasedness
Peaks over threshold - Collects dataset in excess of some threshold
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(mean) = mean

38. Central Limit Theorem
For n>30 - sample mean is approximately normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

39. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

40. Standard error
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Transformed to a unit variable - Mean = 0 Variance = 1
Price/return tends to run towards a long - run level
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

41. Non - parametric vs parametric calculation of VaR
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)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. Variance(discrete)
Nonlinearity
When one regressor is a perfect linear function of the other regressors
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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)

43. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

44. Persistence
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
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
P(X=x - Y=y) = P(X=x) * P(Y=y)

45. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Random walk (usually acceptable) - Constant volatility (unlikely)
Nonlinearity

46. Standard variable for non - normal distributions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Var(X) + Var(Y)
Z = (Y - meany)/(stddev(y)/sqrt(n))

47. Two drawbacks of moving average series
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

48. Beta distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Returns over time for a combination of assets (combination of time series and cross - sectional data)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

49. Perfect multicollinearity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When one regressor is a perfect linear function of the other regressors

50. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Variance reverts to a long run level
SSR
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!)