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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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sample mean will near the population mean as the sample size increases

2. Pooled data
We reject a hypothesis that is actually true
Mean of sampling distribution is the population mean
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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

3. Four sampling distributions

4. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

5. P - value
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
P(Z>t)

6. Empirical frequency
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

7. Variance of sample mean
Variance(y)/n = variance of sample Y
Based on a dataset
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
When one regressor is a perfect linear function of the other regressors

8. R^2
Confidence set for two coefficients - two dimensional analog for the confidence interval
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When one regressor is a perfect linear function of the other regressors
Price/return tends to run towards a long - run level

9. Skewness
Based on an equation - P(A) = # of A/total outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P - value
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

10. Historical std dev
Special type of pooled data in which the cross sectional unit is surveyed over time
(a^2)(variance(x)) + (b^2)(variance(y))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

11. Test for statistical independence
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
(a^2)(variance(x)
P(X=x - Y=y) = P(X=x) * P(Y=y)

12. Confidence interval (from t)
95% = 1.65 99% = 2.33 For one - tailed tests
Sample mean +/ - t*(stddev(s)/sqrt(n))
P - value
We accept a hypothesis that should have been rejected

13. LAD
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Choose parameters that maximize the likelihood of what observations occurring
Least absolute deviations estimator - used when extreme outliers are not uncommon
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

14. Sample correlation
For n>30 - sample mean is approximately normal
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Rxy = Sxy/(Sx*Sy)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

15. Central Limit Theorem
i = ln(Si/Si - 1)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Low Frequency - High Severity events
For n>30 - sample mean is approximately normal

16. Test for unbiasedness
E(mean) = 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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Average return across assets on a given day

17. Variance of weighted scheme
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Population denominator = n - Sample denominator = n - 1

18. Persistence
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
SSR
Returns over time for an individual asset
Mean = np - Variance = npq - Std dev = sqrt(npq)

19. Confidence interval for sample mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample variance = (1/(k - 1))Summation(Yi - mean)^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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

20. LFHS
Low Frequency - High Severity events
If variance of the conditional distribution of u(i) is not constant
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

21. Control variates technique
Rxy = Sxy/(Sx*Sy)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

22. Covariance calculations using weight sums (lambda)
P(Z>t)
Nonlinearity
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(X) + Variance(Y) - 2*covariance(XY)

23. Cholesky factorization (decomposition)
We reject a hypothesis that is actually true
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean will near the population mean as the sample size increases
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

24. ESS
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Var(X) + Var(Y)
Statement of the error or precision of an estimate

25. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system

26. Maximum likelihood method
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
P(Z>t)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Choose parameters that maximize the likelihood of what observations occurring

27. Least squares estimator(m)
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Easy to manipulate

28. Multivariate Density Estimation (MDE)
Rxy = Sxy/(Sx*Sy)
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
[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)

29. Implications of homoscedasticity
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Model dependent - Options with the same underlying assets may trade at different volatilities
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

30. Variance - covariance approach for VaR of a portfolio
Contains variables not explicit in model - Accounts for randomness
Concerned with a single random variable (ex. Roll of a die)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Random walk (usually acceptable) - Constant volatility (unlikely)

31. Two requirements of OVB
P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = (1/m) summation(u<n - i>^2)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

32. Simulating for VaR
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

33. Unconditional vs conditional distributions
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
95% = 1.65 99% = 2.33 For one - tailed tests

34. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Price/return tends to run towards a long - run level
Var(X) + Var(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

35. Hazard rate of exponentially distributed random variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

36. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Nonlinearity
Based on a dataset
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

37. Monte Carlo Simulations
Among all unbiased estimators - estimator with the smallest variance is efficient
Peaks over threshold - Collects dataset in excess of some threshold
Special type of pooled data in which the cross sectional unit is surveyed over time
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

38. Time series data
SSR
Returns over time for an individual asset
Use historical simulation approach but use the EWMA weighting system
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

39. Consistent
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
When the sample size is large - the uncertainty about the value of the sample is very small
Concerned with a single random variable (ex. Roll of a die)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

40. Variance of X+Y
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(XY) - E(X)E(Y)
Var(X) + Var(Y)
Based on a dataset

41. 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
Peaks over threshold - Collects dataset in excess of some threshold
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

42. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Sampling distribution of sample means tend to be normal
When one regressor is a perfect linear function of the other regressors
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

43. POT
Peaks over threshold - Collects dataset in excess of some threshold
Model dependent - Options with the same underlying assets may trade at different volatilities
Rxy = Sxy/(Sx*Sy)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

44. Variance of X - Y assuming dependence
Low Frequency - High Severity events
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(X) + Variance(Y) - 2*covariance(XY)

45. Significance =1
Least absolute deviations estimator - used when extreme outliers are not uncommon
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence level
Does not depend on a prior event or information

46. Expected future variance rate (t periods forward)
Attempts to sample along more important paths
Has heavy tails
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When the sample size is large - the uncertainty about the value of the sample is very small

47. Poisson distribution equations for mean variance and std deviation
Random walk (usually acceptable) - Constant volatility (unlikely)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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)
Var(X) + Var(Y)

48. Multivariate probability
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))

49. Hybrid method for conditional volatility
Among all unbiased estimators - estimator with the smallest variance is efficient
Does not depend on a prior event or information
Use historical simulation approach but use the EWMA weighting system
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

50. i.i.d.
Independently and Identically Distributed
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
For n>30 - sample mean is approximately normal