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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. Critical z values
Among all unbiased estimators - estimator with the smallest variance is efficient
95% = 1.65 99% = 2.33 For one - tailed tests
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

2. Antithetic variable technique
Contains variables not explicit in model - Accounts for randomness
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

3. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Average return across assets on a given day
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

4. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Peaks over threshold - Collects dataset in excess of some threshold
Average return across assets on a given day
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

5. Simulation models
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
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
We reject a hypothesis that is actually true

6. Unconditional vs conditional distributions
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When one regressor is a perfect linear function of the other regressors
i = ln(Si/Si - 1)

7. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Has heavy tails
SSR
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

8. Statistical (or empirical) model
Based on an equation - P(A) = # of A/total outcomes
Yi = B0 + B1Xi + ui
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

9. Pooled data
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Combine to form distribution with leptokurtosis (heavy tails)
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 a combination of assets (combination of time series and cross - sectional data)

10. Conditional probability functions
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
Z = (Y - meany)/(stddev(y)/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)

11. Simulating for VaR
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))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Attempts to sample along more important paths

12. P - value
P(Z>t)
Mean of sampling distribution is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Special type of pooled data in which the cross sectional unit is surveyed over time

13. Weibul distribution
E(mean) = mean
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sampling distribution of sample means tend to be normal
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

14. Deterministic Simulation
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

15. What does the OLS minimize?
SSR
If variance of the conditional distribution of u(i) is not constant
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

16. Variance of sample mean
Model dependent - Options with the same underlying assets may trade at different volatilities
Contains variables not explicit in model - Accounts for randomness
Variance(y)/n = variance of sample Y
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

17. LFHS
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Low Frequency - High Severity events
Contains variables not explicit in model - Accounts for randomness
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

18. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Confidence level
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

19. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sampling distribution of sample means tend to be normal
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

20. Poisson distribution equations for mean variance and std deviation
Variance = (1/m) summation(u<n - i>^2)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

21. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
SSR
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

22. Variance of aX
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
(a^2)(variance(x)
Combine to form distribution with leptokurtosis (heavy tails)
Distribution with only two possible outcomes

23. Consistent
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
When the sample size is large - the uncertainty about the value of the sample is very small
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval

24. Unbiased
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
When the sample size is large - the uncertainty about the value of the sample is very small
Mean of sampling distribution is the population mean

25. SER
Returns over time for an individual asset
Variance reverts to a long run level
Average return across assets on a given day
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

26. Multivariate Density Estimation (MDE)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
P(X=x - Y=y) = P(X=x) * P(Y=y)

27. Standard error
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
We reject a hypothesis that is actually true
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

28. LAD
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!)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for an individual asset

29. GARCH
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Summation((xi - mean)^k)/n
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)

30. Efficiency
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Has heavy tails
Among all unbiased estimators - estimator with the smallest variance is efficient
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

31. 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
i = ln(Si/Si - 1)
Variance reverts to a long run level

32. Bootstrap method
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to sample along more important paths
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

33. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Rxy = Sxy/(Sx*Sy)
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[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

34. i.i.d.
Independently and Identically Distributed
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

35. Mean reversion
Variance(x) + Variance(Y) + 2*covariance(XY)
Expected value of the sample mean is the population mean
Attempts to sample along more important paths
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

36. Shortcomings of implied volatility
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Yi = B0 + B1Xi + ui

37. Mean reversion in asset dynamics
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Price/return tends to run towards a long - run level
Attempts to sample along more important paths
Average return across assets on a given day

38. Mean(expected value)
Yi = B0 + B1Xi + ui
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

39. Logistic distribution
Summation((xi - mean)^k)/n
Variance(x) + Variance(Y) + 2*covariance(XY)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Has heavy tails

40. Variance of X+Y
Concerned with a single random variable (ex. Roll of a die)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Combine to form distribution with leptokurtosis (heavy tails)
Var(X) + Var(Y)

41. Empirical frequency
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on a dataset
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

42. Central Limit Theorem(CLT)
SSR
Sampling distribution of sample means tend to be normal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

43. EWMA
Distribution with only two possible outcomes
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

44. Panel data (longitudinal or micropanel)
E(XY) - E(X)E(Y)
Probability that the random variables take on certain values simultaneously
Special type of pooled data in which the cross sectional unit is surveyed over time
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

45. Time series data
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
When one regressor is a perfect linear function of the other regressors
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Returns over time for an individual asset

46. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Confidence level
Transformed to a unit variable - Mean = 0 Variance = 1
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Sample variance
(a^2)(variance(x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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)

48. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

49. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Independently and Identically Distributed
Choose parameters that maximize the likelihood of what observations occurring
When one regressor is a perfect linear function of the other regressors

50. Economical(elegant)
Variance = (1/m) summation(u<n - i>^2)
Normal - Student's T - Chi - square - F distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
Only requires two parameters = mean and variance