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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. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
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

2. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Nonlinearity
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

3. What does the OLS minimize?
We accept a hypothesis that should have been rejected
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Among all unbiased estimators - estimator with the smallest variance is efficient
SSR

4. Chi - squared distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)
Concerned with a single random variable (ex. Roll of a die)
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

5. T distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Confidence set for two coefficients - two dimensional analog for the confidence interval
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

6. Lognormal
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 of sampling distribution is the population mean
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
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

7. Extending the HS approach for computing value of a portfolio

8. Confidence interval (from t)
Choose parameters that maximize the likelihood of what observations occurring
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

9. Mean reversion in asset dynamics
Attempts to sample along more important paths
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Price/return tends to run towards a long - run level

10. Critical z values
Low Frequency - High Severity events
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
95% = 1.65 99% = 2.33 For one - tailed tests
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

11. i.i.d.
Sampling distribution of sample means tend to be normal
Low Frequency - High Severity events
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
Independently and Identically Distributed

12. 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)
Distribution with only two possible outcomes
i = ln(Si/Si - 1)
When the sample size is large - the uncertainty about the value of the sample is very small

13. Unstable return distribution
Statement of the error or precision of an estimate
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

14. Historical std dev
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

15. Central Limit Theorem(CLT)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Easy to manipulate
Sampling distribution of sample means tend to be normal
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

16. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Attempts to sample along more important paths

17. Sample covariance
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
SSR

18. Law of Large Numbers
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Sample mean will near the population mean as the sample size increases
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

19. Binomial distribution equations for mean variance and std dev
Does not depend on a prior event or information
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

20. Test for statistical independence
Average return across assets on a given day
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
P(X=x - Y=y) = P(X=x) * P(Y=y)

21. Mean reversion in variance
Variance reverts to a long run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Use historical simulation approach but use the EWMA weighting system

22. Standard error
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Has heavy tails
Variance reverts to a long run level

23. Two assumptions of square root rule
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Price/return tends to run towards a long - run level
Random walk (usually acceptable) - Constant volatility (unlikely)
Summation((xi - mean)^k)/n

24. Variance of X+Y
Confidence level
Var(X) + Var(Y)
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

25. Unbiased
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Mean of sampling distribution is the population mean
Among all unbiased estimators - estimator with the smallest variance is efficient

26. Variance - covariance approach for VaR of a portfolio
P - value
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

27. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

28. Significance =1
E(XY) - E(X)E(Y)
Var(X) + Var(Y)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Confidence level

29. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Average return across assets on a given day

30. Two ways to calculate historical volatility
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Summation((xi - mean)^k)/n
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

31. Simulation models
Returns over time for a combination of assets (combination of time series and cross - sectional data)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

32. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)
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)

33. Reliability
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Statement of the error or precision of an estimate
Only requires two parameters = mean and variance
Does not depend on a prior event or information

34. P - value
Distribution with only two possible outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
i = ln(Si/Si - 1)
P(Z>t)

35. Variance of aX
(a^2)(variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

36. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Population denominator = n - Sample denominator = n - 1
Sample mean will near the population mean as the sample size increases

37. Importance sampling technique
Attempts to sample along more important paths
Does not depend on a prior event or information
P - value
Variance(y)/n = variance of sample Y

38. Economical(elegant)
Model dependent - Options with the same underlying assets may trade at different volatilities
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
SSR
Only requires two parameters = mean and variance

39. Key properties of linear regression
Variance(y)/n = variance of sample Y
Regression can be non - linear in variables but must be linear in parameters
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

40. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Regression can be non - linear in variables but must be linear in parameters

41. Binomial distribution
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!)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
When one regressor is a perfect linear function of the other regressors

42. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Price/return tends to run towards a long - run level
Concerned with a single random variable (ex. Roll of a die)
Least absolute deviations estimator - used when extreme outliers are not uncommon

43. Skewness
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
Easy to manipulate
We accept a hypothesis that should have been rejected
Based on an equation - P(A) = # of A/total outcomes

44. Control variates technique
Independently and Identically Distributed
Combine to form distribution with leptokurtosis (heavy tails)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

45. Efficiency
Contains variables not explicit in model - Accounts for randomness
Among all unbiased estimators - estimator with the smallest variance is efficient
i = ln(Si/Si - 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)

46. Time series data
95% = 1.65 99% = 2.33 For one - tailed tests
Special type of pooled data in which the cross sectional unit is surveyed over time
Returns over time for an individual asset
Choose parameters that maximize the likelihood of what observations occurring

47. K - th moment
We accept a hypothesis that should have been rejected
Summation((xi - mean)^k)/n
Statement of the error or precision of an estimate
We reject a hypothesis that is actually true

48. Confidence interval for sample mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When the sample size is large - the uncertainty about the value of the sample is very small

49. LFHS
Low Frequency - High Severity events
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

50. Variance of sample mean
Only requires two parameters = mean and variance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(y)/n = variance of sample Y
Attempts to sample along more important paths