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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. Logistic distribution
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Has heavy tails
E(mean) = mean

2. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Among all unbiased estimators - estimator with the smallest variance is efficient
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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

3. Variance of aX
(a^2)(variance(x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Use historical simulation approach but use the EWMA weighting system
When the sample size is large - the uncertainty about the value of the sample is very small

4. Significance =1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Confidence level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

5. Panel data (longitudinal or micropanel)
Rxy = Sxy/(Sx*Sy)
Based on an equation - P(A) = # of A/total outcomes
Independently and Identically Distributed
Special type of pooled data in which the cross sectional unit is surveyed over time

6. Gamma distribution
Only requires two parameters = mean and variance
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean +/ - t*(stddev(s)/sqrt(n))

7. Overall F - statistic
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

8. Normal distribution
Population denominator = n - Sample denominator = n - 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

9. Empirical frequency
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Based on a dataset

10. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Transformed to a unit variable - Mean = 0 Variance = 1
Normal - Student's T - Chi - square - F distribution

11. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
We reject a hypothesis that is actually true

12. T distribution
SSR
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

13. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E(mean) = mean
Nonlinearity
More than one random variable

14. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
i = ln(Si/Si - 1)
Probability that the random variables take on certain values simultaneously
Variance(X) + Variance(Y) - 2*covariance(XY)

15. Unbiased
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
(a^2)(variance(x)
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
Mean of sampling distribution is the population mean

16. Critical z values
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Statement of the error or precision of an estimate
We accept a hypothesis that should have been rejected
95% = 1.65 99% = 2.33 For one - tailed tests

17. Simulation models
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
If variance of the conditional distribution of u(i) is not constant
Mean = np - Variance = npq - Std dev = sqrt(npq)
Choose parameters that maximize the likelihood of what observations occurring

18. Mean reversion in variance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

19. Control variates technique
Use historical simulation approach but use the EWMA weighting system
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P(Z>t)

20. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Use historical simulation approach but use the EWMA weighting system
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

21. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Based on a dataset
Low Frequency - High Severity events
Yi = B0 + B1Xi + ui

22. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Based on an equation - P(A) = # of A/total outcomes
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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

23. P - value
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P(Z>t)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

24. What does the OLS minimize?
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Mean of sampling distribution is the population mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR

25. Mean reversion
Based on a dataset
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Has heavy tails
We accept a hypothesis that should have been rejected

26. Result of combination of two normal with same means
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(x) + Variance(Y) + 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)
Variance(X) + Variance(Y) - 2*covariance(XY)

27. Time series data
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Returns over time for an individual asset
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

28. Variance(discrete)
Only requires two parameters = mean and variance
Confidence set for two coefficients - two dimensional analog for the confidence interval
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

29. Poisson Distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval

30. SER
Confidence set for two coefficients - two dimensional analog for the confidence interval
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Var(X) + Var(Y)

31. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

32. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

33. Multivariate Density Estimation (MDE)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Normal - Student's T - Chi - square - F distribution
Nonlinearity

34. Cross - sectional
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Average return across assets on a given day
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

35. Bernouli Distribution
Distribution with only two possible outcomes
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
Population denominator = n - Sample denominator = n - 1
Based on a dataset

36. Variance - covariance approach for VaR of a portfolio
Application of mathematical statistics to economic data to lend empirical support to models
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

37. Tractable
Easy to manipulate
Use historical simulation approach but use the EWMA weighting system
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

38. Square root rule
Based on a dataset
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sampling distribution of sample means tend to be normal
Confidence level

39. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Easy to manipulate
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

40. Heteroskedastic
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When one regressor is a perfect linear function of the other regressors
If variance of the conditional distribution of u(i) is not constant
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

41. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Average return across assets on a given day
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

42. Standard error for Monte Carlo replications
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

43. Difference between population and sample variance
Easy to manipulate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Population denominator = n - Sample denominator = n - 1

44. Type II Error
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance(y)/n = variance of sample Y
We accept a hypothesis that should have been rejected
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

45. Exponential distribution
Confidence level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

46. Variance of X+b
Confidence level
Variance(x)
Variance = (1/m) summation(u<n - i>^2)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

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

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48. Multivariate probability
Price/return tends to run towards a long - run level
95% = 1.65 99% = 2.33 For one - tailed tests
More than one random variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

49. Test for unbiasedness
Regression can be non - linear in variables but must be linear in parameters
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(mean) = mean

50. Monte Carlo Simulations
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
Var(X) + Var(Y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one