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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. Joint probability functions
Probability that the random variables take on certain values simultaneously
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests
Z = (Y - meany)/(stddev(y)/sqrt(n))

2. R^2
95% = 1.65 99% = 2.33 For one - tailed tests
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Rxy = Sxy/(Sx*Sy)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

3. Two requirements of OVB
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
(a^2)(variance(x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

4. Difference between population and sample variance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Does not depend on a prior event or information
Population denominator = n - Sample denominator = n - 1
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

5. T distribution
Concerned with a single random variable (ex. Roll of a die)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

6. Homoskedastic
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

7. Kurtosis
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

8. 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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Attempts to sample along more important paths
Among all unbiased estimators - estimator with the smallest variance is efficient

9. Bernouli Distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Distribution with only two possible outcomes
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
Variance(x)

10. Econometrics
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Low Frequency - High Severity events
Application of mathematical statistics to economic data to lend empirical support to models
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

11. Economical(elegant)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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!)
Only requires two parameters = mean and variance
More than one random variable

12. Discrete random variable
Regression can be non - linear in variables but must be linear in parameters
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 a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

13. LAD
Nonlinearity
95% = 1.65 99% = 2.33 For one - tailed tests
Normal - Student's T - Chi - square - F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon

14. P - value
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Based on an equation - P(A) = # of A/total outcomes
Price/return tends to run towards a long - run level
P(Z>t)

15. Discrete representation of the GBM
Sample mean will near the population mean as the sample size increases
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

16. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

17. Overall F - statistic
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Yi = B0 + B1Xi + ui

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

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19. LFHS
Confidence set for two coefficients - two dimensional analog for the confidence interval
(a^2)(variance(x)
Low Frequency - High Severity events
Sample mean will near the population mean as the sample size increases

20. Variance of sampling distribution of means when n<N
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

21. Poisson distribution equations for mean variance and std deviation
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(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

22. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Summation((xi - mean)^k)/n

23. Sample mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Expected value of the sample mean is the population mean

24. i.i.d.
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance reverts to a long run level

25. What does the OLS minimize?
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance reverts to a long run level
Independently and Identically Distributed
SSR

26. Significance =1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
Confidence level
More than one random variable

27. WLS
Rxy = Sxy/(Sx*Sy)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean +/ - t*(stddev(s)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

28. Unbiased
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Mean of sampling distribution is the population mean
Summation((xi - mean)^k)/n

29. Multivariate Density Estimation (MDE)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Attempts to sample along more important paths

30. Deterministic Simulation
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Var(X) + Var(Y)

31. Central Limit Theorem
When the sample size is large - the uncertainty about the value of the sample is very small
Regression can be non - linear in variables but must be linear in parameters
For n>30 - sample mean is approximately normal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

32. Logistic distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Has heavy tails
Variance = (1/m) summation(u<n - i>^2)

33. Biggest (and only real) drawback of GARCH mode
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Confidence level
Nonlinearity

34. Perfect multicollinearity
Sample mean will near the population mean as the sample size increases
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

35. Monte Carlo Simulations
Nonlinearity
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Mean = np - Variance = npq - Std dev = sqrt(npq)

36. Test for statistical independence
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(X=x - Y=y) = P(X=x) * P(Y=y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

37. Multivariate probability
For n>30 - sample mean is approximately normal
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
More than one random variable
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

38. Variance of X+b
Variance(x)
Mean of sampling distribution is the population mean
More than one random variable
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

39. Inverse transform method
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
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

40. POT
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)
Peaks over threshold - Collects dataset in excess of some threshold

41. Weibul distribution
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Normal - Student's T - Chi - square - F distribution

42. Type II Error
Concerned with a single random variable (ex. Roll of a die)
Choose parameters that maximize the likelihood of what observations occurring
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
We accept a hypothesis that should have been rejected

43. Cross - sectional
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Average return across assets on a given day
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
When one regressor is a perfect linear function of the other regressors

44. Variance of sample mean
Variance(y)/n = variance of sample Y
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x)
P - value

45. Stochastic error term
Var(X) + Var(Y)
Based on a dataset
Contains variables not explicit in model - Accounts for randomness
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

46. Sample variance
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Does not depend on a prior event or information
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

47. Direction of OVB
Nonlinearity
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(X) + Variance(Y) - 2*covariance(XY)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

48. F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

49. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
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
For n>30 - sample mean is approximately normal

50. Heteroskedastic
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
If variance of the conditional distribution of u(i) is not constant
Regression can be non - linear in variables but must be linear in parameters
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