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FRM Foundations Of Risk Management Quantitative Methods

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Answer 50 questions in 15 minutes.
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1. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Expected value of the sample mean is the population mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

2. BLUE
Average return across assets on a given day
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
i = ln(Si/Si - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)

3. Weibul distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

4. R^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Based on an equation - P(A) = # of A/total outcomes
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. Variance(discrete)
Average return across assets on a given day
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

6. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Rxy = Sxy/(Sx*Sy)

7. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

8. EWMA
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

9. ESS
Based on an equation - P(A) = # of A/total outcomes
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

10. Antithetic variable technique
When one regressor is a perfect linear function of the other regressors
Easy to manipulate
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Yi = B0 + B1Xi + ui

11. Significance =1
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
When the sample size is large - the uncertainty about the value of the sample is very small
Sample mean +/ - t*(stddev(s)/sqrt(n))

12. What does the OLS minimize?
Z = (Y - meany)/(stddev(y)/sqrt(n))
SSR
Nonlinearity
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

13. Adjusted R^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)

14. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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 = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y

15. LFHS
Low Frequency - High Severity events
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Special type of pooled data in which the cross sectional unit is surveyed over time

16. Statistical (or empirical) model
Variance(x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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)
Yi = B0 + B1Xi + ui

17. Beta distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Least absolute deviations estimator - used when extreme outliers are not uncommon
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(X) + Variance(Y) - 2*covariance(XY)

18. Standard error
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value

19. Pooled data
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

20. Tractable
Var(X) + Var(Y)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Easy to manipulate
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

21. Hazard rate of exponentially distributed random variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

22. Continuously compounded return equation
i = ln(Si/Si - 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)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Application of mathematical statistics to economic data to lend empirical support to models

23. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SSR
(a^2)(variance(x)

24. Continuous random variable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Probability that the random variables take on certain values simultaneously
(a^2)(variance(x)) + (b^2)(variance(y))

25. Lognormal
When the sample size is large - the uncertainty about the value of the sample is very small
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample mean will near the population mean as the sample size increases
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

26. Two drawbacks of moving average series
We accept a hypothesis that should have been rejected
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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)
Least absolute deviations estimator - used when extreme outliers are not uncommon

27. Kurtosis
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
(a^2)(variance(x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

28. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Population denominator = n - Sample denominator = n - 1
Returns over time for an individual asset
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

29. Mean reversion
We reject a hypothesis that is actually true
Random walk (usually acceptable) - Constant volatility (unlikely)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

30. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to sample along more important paths
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Concerned with a single random variable (ex. Roll of a die)

31. P - value
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SSR
Nonlinearity
P(Z>t)

32. Sample correlation
Rxy = Sxy/(Sx*Sy)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

33. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance = (1/m) summation(u<n - i>^2)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

34. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Does not depend on a prior event or information
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

35. 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
Distribution with only two possible outcomes
SSR
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

36. Multivariate probability
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Combine to form distribution with leptokurtosis (heavy tails)
More than one random variable
Model dependent - Options with the same underlying assets may trade at different volatilities

37. 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
Sampling distribution of sample means tend to be normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

38. Unconditional vs conditional distributions
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

39. Confidence ellipse
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)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Confidence set for two coefficients - two dimensional analog for the confidence interval
Z = (Y - meany)/(stddev(y)/sqrt(n))

40. Heteroskedastic
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
Use historical simulation approach but use the EWMA weighting system
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

41. Test for unbiasedness
E(mean) = mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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

42. Homoskedastic only F - stat
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

43. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When the sample size is large - the uncertainty about the value of the sample is very small
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

44. Importance sampling technique
Easy to manipulate
Attempts to sample along more important paths
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

45. Central Limit Theorem(CLT)
Peaks over threshold - Collects dataset in excess of some threshold
Sampling distribution of sample means tend to be normal
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
i = ln(Si/Si - 1)

46. Implications of homoscedasticity
Application of mathematical statistics to economic data to lend empirical support to models
We accept a hypothesis that should have been rejected
Only requires two parameters = mean and variance
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

47. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Model dependent - Options with the same underlying assets may trade at different volatilities

48. Limitations of R^2 (what an increase doesn't necessarily imply)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
We accept a hypothesis that should have been rejected
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

49. Homoskedastic
Probability that the random variables take on certain values simultaneously
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
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
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

50. Least squares estimator(m)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

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FRM Foundations Of Risk Management Quantitative Methods

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