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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. Hazard rate of exponentially distributed random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Only requires two parameters = mean and variance
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

2. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

3. Panel data (longitudinal or micropanel)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(mean) = mean
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information

4. SER
(a^2)(variance(x)) + (b^2)(variance(y))
Low Frequency - High Severity events
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

5. Poisson Distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sampling distribution of sample means tend to be normal
P - value
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

6. Standard normal distribution
Yi = B0 + B1Xi + ui
Probability that the random variables take on certain values simultaneously
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1

7. Covariance calculations using weight sums (lambda)
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
Only requires two parameters = mean and variance
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

8. Logistic distribution
For n>30 - sample mean is approximately normal
Expected value of the sample mean is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Has heavy tails

9. Variance of X+Y assuming dependence
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean +/ - t*(stddev(s)/sqrt(n))
E(mean) = mean
Variance(x) + Variance(Y) + 2*covariance(XY)

10. Marginal unconditional probability function
Attempts to sample along more important paths
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Probability that the random variables take on certain values simultaneously
Does not depend on a prior event or information

11. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

12. BLUE
Among all unbiased estimators - estimator with the smallest variance is efficient
Sampling distribution of sample means tend to be normal
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

13. Hybrid method for conditional volatility
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Use historical simulation approach but use the EWMA weighting system
Summation((xi - mean)^k)/n

14. EWMA
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

15. i.i.d.
Independently and Identically Distributed
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Based on a dataset
When the sample size is large - the uncertainty about the value of the sample is very small
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

17. Covariance
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(XY) - E(X)E(Y)

18. Maximum likelihood method
Based on an equation - P(A) = # of A/total outcomes
Choose parameters that maximize the likelihood of what observations occurring
Concerned with a single random variable (ex. Roll of a die)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

19. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system
Among all unbiased estimators - estimator with the smallest variance is efficient
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

20. Four sampling distributions

21. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

22. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Does not depend on a prior event or information
E(XY) - E(X)E(Y)
Normal - Student's T - Chi - square - F distribution

23. Sample mean
Easy to manipulate
Probability that the random variables take on certain values simultaneously
Expected value of the sample mean is the population mean
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

24. Biggest (and only real) drawback of GARCH mode
Does not depend on a prior event or information
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When one regressor is a perfect linear function of the other regressors
Nonlinearity

25. Economical(elegant)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Probability that the random variables take on certain values simultaneously
Only requires two parameters = mean and variance

26. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Price/return tends to run towards a long - run level
Z = (Y - meany)/(stddev(y)/sqrt(n))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

27. Efficiency
Distribution with only two possible outcomes
Nonlinearity
Variance(y)/n = variance of sample Y
Among all unbiased estimators - estimator with the smallest variance is efficient

28. Variance of X+Y
Special type of pooled data in which the cross sectional unit is surveyed over time
Var(X) + Var(Y)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

29. Discrete representation of the GBM
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!)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence level

30. Cross - sectional
Average return across assets on a given day
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
For n>30 - sample mean is approximately normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

31. GEV
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Easy to manipulate

32. Type II Error
Sampling distribution of sample means tend to be normal
We accept a hypothesis that should have been rejected
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(y)/n = variance of sample Y

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

34. Sample correlation
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
Peaks over threshold - Collects dataset in excess of some threshold
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Rxy = Sxy/(Sx*Sy)

35. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Special type of pooled data in which the cross sectional unit is surveyed over time
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

36. Tractable
Easy to manipulate
Peaks over threshold - Collects dataset in excess of some threshold
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

37. Direction of OVB
Population denominator = n - Sample denominator = n - 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When the sample size is large - the uncertainty about the value of the sample is very small
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

38. Continuous representation of the GBM
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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)

39. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Has heavy tails
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

40. Continuously compounded return equation
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean of sampling distribution is the population mean
i = ln(Si/Si - 1)

41. GARCH
(a^2)(variance(x)) + (b^2)(variance(y))
P(Z>t)
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)
Only requires two parameters = mean and variance

42. Sample variance
Based on a dataset
Sampling distribution of sample means tend to be normal
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Continuous random variable
If variance of the conditional distribution of u(i) is not constant
95% = 1.65 99% = 2.33 For one - tailed tests
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
i = ln(Si/Si - 1)

44. Bootstrap method
If variance of the conditional distribution of u(i) is not constant
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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

45. ESS
If variance of the conditional distribution of u(i) is not constant
When the sample size is large - the uncertainty about the value of the sample is very small
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

46. Antithetic variable technique
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

47. Empirical frequency
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean
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!)
Based on a dataset

48. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Combine to form distribution with leptokurtosis (heavy tails)
Has heavy tails

49. Central Limit Theorem
We reject a hypothesis that is actually true
For n>30 - sample mean is approximately normal
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
Sample mean will near the population mean as the sample size increases

50. GPD
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Only requires two parameters = mean and variance
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