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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. Adjusted R^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

2. Simulation models
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
We reject a hypothesis that is actually true
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

3. K - th moment
More than one random variable
95% = 1.65 99% = 2.33 For one - tailed tests
Summation((xi - mean)^k)/n
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

4. Statistical (or empirical) model
Mean = np - Variance = npq - Std dev = sqrt(npq)
SSR
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Yi = B0 + B1Xi + ui

5. Sample variance
When the sample size is large - the uncertainty about the value of the sample is very small
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

6. Biggest (and only real) drawback of GARCH mode
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity

7. Poisson distribution equations for mean variance and std deviation
Among all unbiased estimators - estimator with the smallest variance is efficient
Price/return tends to run towards a long - run level
Nonlinearity
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

8. Continuous representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Returns over time for a combination of assets (combination of time series and cross - sectional data)

9. Perfect multicollinearity
Price/return tends to run towards a long - run level
(a^2)(variance(x)
When one regressor is a perfect linear function of the other regressors
Based on a dataset

10. ESS
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

11. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(mean) = mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

12. Confidence interval for sample mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
We accept a hypothesis that should have been rejected
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

13. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Nonlinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Expected value of the sample mean is the population mean

14. Unstable return distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Normal - Student's T - Chi - square - F distribution
Concerned with a single random variable (ex. Roll of a die)

15. Standard normal distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Distribution with only two possible outcomes

16. Econometrics
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
Expected value of the sample mean is the population mean
Application of mathematical statistics to economic data to lend empirical support to models
Mean = np - Variance = npq - Std dev = sqrt(npq)

17. Two requirements of OVB
When one regressor is a perfect linear function of the other regressors
Easy to manipulate
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

18. Regime - switching volatility model
Least absolute deviations estimator - used when extreme outliers are not uncommon
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E(XY) - E(X)E(Y)
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

19. LAD
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

20. Central Limit Theorem(CLT)
More than one random variable
Var(X) + Var(Y)
Rxy = Sxy/(Sx*Sy)
Sampling distribution of sample means tend to be normal

21. Panel data (longitudinal or micropanel)
More than one random variable
We accept a hypothesis that should have been rejected
Special type of pooled data in which the cross sectional unit is surveyed over time
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

22. Extreme Value Theory
Combine to form distribution with leptokurtosis (heavy tails)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

23. Potential reasons for fat tails in return distributions
We accept a hypothesis that should have been rejected
E(XY) - E(X)E(Y)
Nonlinearity
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Four sampling distributions

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25. Consistent
Combine to form distribution with leptokurtosis (heavy tails)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
When the sample size is large - the uncertainty about the value of the sample is very small
We accept a hypothesis that should have been rejected

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

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27. POT
Combine to form distribution with leptokurtosis (heavy tails)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Peaks over threshold - Collects dataset in excess of some threshold
Only requires two parameters = mean and variance

28. Simulating for VaR
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(y)/n = variance of sample Y
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

29. Marginal unconditional probability function
Does not depend on a prior event or information
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

30. Unbiased
Contains variables not explicit in model - Accounts for randomness
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
Mean of sampling distribution is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)

31. Variance - covariance approach for VaR of a portfolio
We accept a hypothesis that should have been rejected
Variance reverts to a long run level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Price/return tends to run towards a long - run level

32. Non - parametric vs parametric calculation of VaR
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
i = ln(Si/Si - 1)
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

33. Type I error
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
We reject a hypothesis that is actually true
Yi = B0 + B1Xi + ui
More than one random variable

34. Implied standard deviation for options
Based on an equation - P(A) = # of A/total outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

35. Joint probability functions
Least absolute deviations estimator - used when extreme outliers are not uncommon
Among all unbiased estimators - estimator with the smallest variance is efficient
Probability that the random variables take on certain values simultaneously
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

36. T distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Yi = B0 + B1Xi + ui

37. Importance sampling technique
Variance = (1/m) summation(u<n - i>^2)
Attempts to sample along more important paths
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

38. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Nonlinearity
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

39. Test for unbiasedness
E(mean) = mean
SSR
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Does not depend on a prior event or information

40. Variance of X+Y
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
i = ln(Si/Si - 1)
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
Var(X) + Var(Y)

41. Variance of X+Y assuming dependence
Special type of pooled data in which the cross sectional unit is surveyed over time
Attempts to sample along more important paths
Variance(x) + Variance(Y) + 2*covariance(XY)
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

42. Multivariate probability
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
More than one random variable

43. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to sample along more important paths
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

44. Covariance
Confidence level
Variance = (1/m) summation(u<n - i>^2)
E(XY) - E(X)E(Y)
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

45. Historical std dev
Statement of the error or precision of an estimate
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Model dependent - Options with the same underlying assets may trade at different volatilities
Does not depend on a prior event or information

46. Variance of X - Y assuming dependence
E(XY) - E(X)E(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
Variance(X) + Variance(Y) - 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

47. Continuously compounded return equation
P(X=x - Y=y) = P(X=x) * P(Y=y)
i = ln(Si/Si - 1)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

48. Conditional probability functions
(a^2)(variance(x)) + (b^2)(variance(y))
Concerned with a single random variable (ex. Roll of a die)
Independently and Identically Distributed
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

49. F distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

50. Bootstrap method
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P - value
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Returns over time for a combination of assets (combination of time series and cross - sectional data)

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FRM Foundations Of Risk Management Quantitative Methods

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