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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. Gamma distribution
Based on an equation - P(A) = # of A/total outcomes
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
Var(X) + Var(Y)
SSR

2. Square root rule
Transformed to a unit variable - Mean = 0 Variance = 1
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Expected value of the sample mean is the population mean

3. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E(XY) - E(X)E(Y)
We accept a hypothesis that should have been rejected
Confidence level

4. Direction of OVB
Distribution with only two possible outcomes
Only requires two parameters = mean and variance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

5. Critical z values
Peaks over threshold - Collects dataset in excess of some threshold
95% = 1.65 99% = 2.33 For one - tailed tests
Does not depend on a prior event or information
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

6. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Mean of sampling distribution is the population mean

7. Consistent
Average return across assets on a given day
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)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
When the sample size is large - the uncertainty about the value of the sample is very small

8. Law of Large Numbers
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Sample mean will near the population mean as the sample size increases
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

9. Standard normal distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Transformed to a unit variable - Mean = 0 Variance = 1

10. Covariance calculations using weight sums (lambda)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Confidence set for two coefficients - two dimensional analog for the confidence interval

11. Variance of weighted scheme
(a^2)(variance(x)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence set for two coefficients - two dimensional analog for the confidence interval

12. Poisson distribution equations for mean variance and std deviation
Transformed to a unit variable - Mean = 0 Variance = 1
Var(X) + Var(Y)
Application of mathematical statistics to economic data to lend empirical support to models
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

13. Simulating for VaR
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Distribution with only two possible outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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

14. Extending the HS approach for computing value of a portfolio

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15. Variance of X+Y
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Var(X) + Var(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

16. Variance of sampling distribution of means when n<N
Statement of the error or precision of an estimate
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

17. Two drawbacks of moving average series
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Choose parameters that maximize the likelihood of what observations occurring
Price/return tends to run towards a long - run level

18. Mean reversion in asset dynamics
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Price/return tends to run towards a long - run level
Only requires two parameters = mean and variance
For n>30 - sample mean is approximately normal

19. P - value
P(Z>t)
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!)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

20. F distribution
We accept a hypothesis that should have been rejected
Concerned with a single random variable (ex. Roll of a die)
Model dependent - Options with the same underlying assets may trade at different volatilities
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

21. Conditional probability functions
Price/return tends to run towards a long - run level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

22. Tractable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Easy to manipulate

23. GEV
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
E(mean) = mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

24. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)) + (b^2)(variance(y))
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
Easy to manipulate

25. Beta distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean will near the population mean as the sample size increases
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

26. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(x) + Variance(Y) + 2*covariance(XY)
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

27. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance reverts to a long run level
Variance(x)

28. Perfect multicollinearity
E(mean) = mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and 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
When one regressor is a perfect linear function of the other regressors

29. Hybrid method for conditional volatility
Peaks over threshold - Collects dataset in excess of some threshold
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Use historical simulation approach but use the EWMA weighting system

30. Multivariate Density Estimation (MDE)
When one regressor is a perfect linear function of the other regressors
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
95% = 1.65 99% = 2.33 For one - tailed tests

31. What does the OLS minimize?
When the sample size is large - the uncertainty about the value of the sample is very small
SSR
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
(a^2)(variance(x)) + (b^2)(variance(y))

32. EWMA
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
SSR
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

33. SER
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 a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
P(X=x - Y=y) = P(X=x) * P(Y=y)
Based on an equation - P(A) = # of A/total outcomes

34. Deterministic Simulation
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Has heavy tails
Z = (Y - meany)/(stddev(y)/sqrt(n))

35. Standard error for Monte Carlo replications
Least absolute deviations estimator - used when extreme outliers are not uncommon
Choose parameters that maximize the likelihood of what observations occurring
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Average return across assets on a given day

36. Standard variable for non - normal distributions
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
When the sample size is large - the uncertainty about the value of the sample is very small
Z = (Y - meany)/(stddev(y)/sqrt(n))
P - value

37. Binomial distribution equations for mean variance and std dev
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

38. Hazard rate of exponentially distributed random variable
Summation((xi - mean)^k)/n
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

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

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40. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance reverts to a long run level

41. Test for statistical independence
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Yi = B0 + B1Xi + ui
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

42. Type II Error
Mean = np - Variance = npq - Std dev = sqrt(npq)
Contains variables not explicit in model - Accounts for randomness
Confidence set for two coefficients - two dimensional analog for the confidence interval
We accept a hypothesis that should have been rejected

43. Extreme Value Theory
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

44. Continuously compounded return equation
Variance(X) + Variance(Y) - 2*covariance(XY)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
i = ln(Si/Si - 1)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

45. GARCH
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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)

46. Four sampling distributions

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47. Biggest (and only real) drawback of GARCH mode
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Nonlinearity

48. Statistical (or empirical) model
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Yi = B0 + B1Xi + ui
Transformed to a unit variable - Mean = 0 Variance = 1

49. K - th moment
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Summation((xi - mean)^k)/n
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed

50. Overall F - statistic
Peaks over threshold - Collects dataset in excess of some threshold
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Special type of pooled data in which the cross sectional unit is surveyed over time