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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. Test for unbiasedness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean
Regression can be non - linear in variables but must be linear in parameters
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

2. Econometrics
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Based on a dataset
Application of mathematical statistics to economic data to lend empirical support to models
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

3. POT
Among all unbiased estimators - estimator with the smallest variance is efficient
Does not depend on a prior event or information
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Peaks over threshold - Collects dataset in excess of some threshold

4. Test for statistical independence
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
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)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

5. Binomial distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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!)

6. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
(a^2)(variance(x)
Var(X) + Var(Y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

7. Result of combination of two normal with same means
P(Z>t)
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))
Combine to form distribution with leptokurtosis (heavy tails)

8. Standard normal distribution
Distribution with only two possible outcomes
Transformed to a unit variable - Mean = 0 Variance = 1
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Has heavy tails

9. 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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Rxy = Sxy/(Sx*Sy)

10. Two drawbacks of moving average series
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E(XY) - E(X)E(Y)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

11. Variance of X - Y assuming dependence
Choose parameters that maximize the likelihood of what observations occurring
Variance(X) + Variance(Y) - 2*covariance(XY)
Application of mathematical statistics to economic data to lend empirical support to models
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

12. Adjusted R^2
Choose parameters that maximize the likelihood of what observations occurring
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
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

13. Regime - switching volatility model
Based on a dataset
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

14. Significance =1
Confidence level
Combine to form distribution with leptokurtosis (heavy tails)
Has heavy tails
Returns over time for an individual asset

15. Consistent
Special type of pooled data in which the cross sectional unit is surveyed over time
Normal - Student's T - Chi - square - F distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Sample mean +/ - t*(stddev(s)/sqrt(n))

16. Sample variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))

17. T distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

18. EWMA
Returns over time for an individual asset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

19. Economical(elegant)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Only requires two parameters = mean and variance
Variance(y)/n = variance of sample Y
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

20. Simulation models
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
(a^2)(variance(x)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

21. What does the OLS minimize?
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
SSR

22. Perfect multicollinearity
Contains variables not explicit in model - Accounts for randomness
When one regressor is a perfect linear function of the other regressors
Distribution with only two possible outcomes
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

23. 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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

24. Reliability
Statement of the error or precision of an estimate
Combine to form distribution with leptokurtosis (heavy tails)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability that the random variables take on certain values simultaneously

25. Importance sampling technique
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(y)/n = variance of sample Y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Attempts to sample along more important paths

26. Variance of X+Y
When one regressor is a perfect linear function of the other regressors
P(X=x - Y=y) = P(X=x) * P(Y=y)
Var(X) + Var(Y)
Variance(y)/n = variance of sample Y

27. Hybrid method for conditional volatility
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Use historical simulation approach but use the EWMA weighting system
E(mean) = mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

28. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
We accept a hypothesis that should have been rejected
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Normal - Student's T - Chi - square - F distribution

29. Mean reversion in variance
Variance reverts to a long run level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

30. Confidence ellipse
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We accept a hypothesis that should have been rejected
Confidence set for two coefficients - two dimensional analog for the confidence interval
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

31. GPD
Model dependent - Options with the same underlying assets may trade at different volatilities
Probability that the random variables take on certain values simultaneously
Has heavy tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

32. Two ways to calculate historical volatility
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

33. Cholesky factorization (decomposition)
Variance(x) + Variance(Y) + 2*covariance(XY)
Normal - Student's T - Chi - square - F distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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

34. Covariance
Independently and Identically Distributed
E(XY) - E(X)E(Y)
When the sample size is large - the uncertainty about the value of the sample is very small
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

35. Four sampling distributions

36. 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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P - value
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

37. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Tractable
Easy to manipulate
Transformed to a unit variable - Mean = 0 Variance = 1
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

39. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

40. Sample covariance
Special type of pooled data in which the cross sectional unit is surveyed over time
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Distribution with only two possible outcomes
P(X=x - Y=y) = P(X=x) * P(Y=y)

41. Lognormal
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

42. Stochastic error term
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Contains variables not explicit in model - Accounts for randomness
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

43. GEV
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)
Returns over time for an individual asset
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

44. Marginal unconditional probability function
Among all unbiased estimators - estimator with the smallest variance is efficient
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Does not depend on a prior event or information
E(XY) - E(X)E(Y)

45. WLS
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!)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Expected value of the sample mean is the population mean

46. Time series data
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Returns over time for an individual asset
We reject a hypothesis that is actually true

47. Critical z values
Combine to form distribution with leptokurtosis (heavy tails)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Confidence set for two coefficients - two dimensional analog for the confidence interval
95% = 1.65 99% = 2.33 For one - tailed tests

48. Cross - sectional
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Average return across assets on a given day
Statement of the error or precision of an estimate
E(mean) = mean

49. Conditional probability functions
Population denominator = n - Sample denominator = n - 1
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Returns over time for an individual asset

50. Sample mean
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean
Returns over time for an individual asset