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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. SER
Normal - Student's T - Chi - square - F distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Independently and Identically Distributed

2. GARCH
Population denominator = n - Sample denominator = n - 1
Easy to manipulate
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)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

3. Simulating for VaR
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Choose parameters that maximize the likelihood of what observations occurring
Yi = B0 + B1Xi + ui

4. R^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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!)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Low Frequency - High Severity events

6. Adjusted 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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
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

7. Perfect multicollinearity
Variance(x) + Variance(Y) + 2*covariance(XY)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
When one regressor is a perfect linear function of the other regressors

8. Overall F - statistic
Easy to manipulate
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. Variance - covariance approach for VaR of a portfolio
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
95% = 1.65 99% = 2.33 For one - tailed tests
E(mean) = mean

10. T distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Yi = B0 + B1Xi + ui
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

11. POT
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance = (1/m) summation(u<n - i>^2)
Normal - Student's T - Chi - square - F distribution
Peaks over threshold - Collects dataset in excess of some threshold

12. ESS
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Independently and Identically Distributed
Variance(y)/n = variance of sample Y

13. Normal distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
(a^2)(variance(x)) + (b^2)(variance(y))
Yi = B0 + B1Xi + ui
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

14. Variance(discrete)
Least absolute deviations estimator - used when extreme outliers are not uncommon
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

15. Unstable return distribution
Var(X) + Var(Y)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

16. Biggest (and only real) drawback of GARCH mode
Summation((xi - mean)^k)/n
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Average return across assets on a given day
Nonlinearity

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

18. Two assumptions of square root rule
Normal - Student's T - Chi - square - F distribution
(a^2)(variance(x)
Variance(x)
Random walk (usually acceptable) - Constant volatility (unlikely)

19. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

20. P - value
P(Z>t)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance reverts to a long run level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

21. Variance of aX
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)
(a^2)(variance(x)
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

22. Standard normal distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(y)/n = variance of sample Y
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

23. Continuous random variable
Variance reverts to a long run level
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

24. Variance of X+Y assuming dependence
We reject a hypothesis that is actually true
Variance(x) + Variance(Y) + 2*covariance(XY)
Yi = B0 + B1Xi + ui
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

25. Hazard rate of exponentially distributed random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Low Frequency - High Severity events
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

26. Four sampling distributions

27. Consistent
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When the sample size is large - the uncertainty about the value of the sample is very small

28. Exact significance level
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
P - value
SSR

29. Chi - squared distribution
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Has heavy tails

30. i.i.d.
Independently and Identically Distributed
Based on an equation - P(A) = # of A/total outcomes
Transformed to a unit variable - Mean = 0 Variance = 1
Peaks over threshold - Collects dataset in excess of some threshold

31. Logistic distribution
Has heavy tails
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance reverts to a long run level
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

32. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance reverts to a long run level
Var(X) + Var(Y)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

33. Potential reasons for fat tails in return distributions
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
Based on a dataset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

34. Mean(expected value)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

35. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Rxy = Sxy/(Sx*Sy)
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small

36. Regime - switching volatility model
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance = (1/m) summation(u<n - i>^2)
Does not depend on a prior event or information

37. Pooled data
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Rxy = Sxy/(Sx*Sy)

38. Variance of X+Y
i = ln(Si/Si - 1)
Attempts to sample along more important paths
Var(X) + Var(Y)
Transformed to a unit variable - Mean = 0 Variance = 1

39. Marginal unconditional probability function
Has heavy tails
Rxy = Sxy/(Sx*Sy)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Does not depend on a prior event or information

40. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Only requires two parameters = mean and variance
Nonlinearity
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

41. Homoskedastic
Var(X) + Var(Y)
Easy to manipulate
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

42. Gamma distribution
Choose parameters that maximize the likelihood of what observations occurring
Low Frequency - High Severity events
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
Price/return tends to run towards a long - run level

43. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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 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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

44. Priori (classical) probability
Distribution with only two possible outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
If variance of the conditional distribution of u(i) is not constant
Based on an equation - P(A) = # of A/total outcomes

45. Simulation models
Low Frequency - High Severity events
Variance(X) + Variance(Y) - 2*covariance(XY)
Returns over time for an individual asset
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

46. K - th moment
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance reverts to a long run level
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Summation((xi - mean)^k)/n

47. Variance of X+b
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Attempts to sample along more important paths

48. Efficiency
Choose parameters that maximize the likelihood of what observations occurring
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)
Among all unbiased estimators - estimator with the smallest variance is efficient
More than one random variable

49. LFHS
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!)
Low Frequency - High Severity events
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Price/return tends to run towards a long - run level

50. Sample correlation
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
Rxy = Sxy/(Sx*Sy)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)