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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. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We reject a hypothesis that is actually true
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

2. Deterministic Simulation
More than one random variable
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Average return across assets on a given day

3. Priori (classical) probability
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(XY) - E(X)E(Y)
Based on an equation - P(A) = # of A/total outcomes
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

4. Maximum likelihood method
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
Choose parameters that maximize the likelihood of what observations occurring
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors

5. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sample mean will near the population mean as the sample size increases
For n>30 - sample mean is approximately normal
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

6. Economical(elegant)
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
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

7. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Only requires two parameters = mean and variance
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

8. Confidence ellipse
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

9. Simulation models
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Var(X) + Var(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
Summation((xi - mean)^k)/n

10. Law of Large Numbers
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Nonlinearity
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean will near the population mean as the sample size increases

11. SER
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Among all unbiased estimators - estimator with the smallest variance is efficient
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
(a^2)(variance(x)) + (b^2)(variance(y))

12. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

13. GPD
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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!)

14. Direction of OVB
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

15. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

16. Square root rule
Contains variables not explicit in model - Accounts for randomness
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

17. Test for unbiasedness
Population denominator = n - Sample denominator = n - 1
P - value
E(mean) = mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

18. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
SSR
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

19. Covariance calculations using weight sums (lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
P(Z>t)
Mean of sampling distribution is the population mean

20. What does the OLS minimize?
When one regressor is a perfect linear function of the other regressors
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR
Variance reverts to a long run level

21. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

23. Normal distribution
Confidence level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
Combine to form distribution with leptokurtosis (heavy tails)

24. Beta distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation((xi - mean)^k)/n

25. Unbiased
Mean of sampling distribution is the population mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information

26. F 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
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
More than one random variable
(a^2)(variance(x)

27. LAD
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
P(Z>t)
Has heavy tails

28. Homoskedastic only F - stat
(a^2)(variance(x)) + (b^2)(variance(y))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

29. Conditional probability functions
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Var(X) + Var(Y)

30. Test for statistical independence
Variance(x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P(X=x - Y=y) = P(X=x) * P(Y=y)
Combine to form distribution with leptokurtosis (heavy tails)

31. Homoskedastic
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

32. 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
P - value
Regression can be non - linear in variables but must be linear in parameters
Normal - Student's T - Chi - square - F distribution

33. Binomial distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Returns over time for an individual asset
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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!)

34. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Easy to manipulate
Special type of pooled data in which the cross sectional unit is surveyed over time

35. Gamma distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
95% = 1.65 99% = 2.33 For one - tailed tests
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
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)

36. Sample covariance
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
For n>30 - sample mean is approximately normal
Sampling distribution of sample means tend to be normal
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

37. Stochastic error term
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

38. Discrete representation of the GBM
Expected value of the sample mean is the population mean
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

39. Empirical frequency
Normal - Student's T - Chi - square - F distribution
Based on a dataset
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

40. Simulating for VaR
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

41. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Random walk (usually acceptable) - Constant volatility (unlikely)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

42. Mean reversion in asset dynamics
Has heavy tails
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
Price/return tends to run towards a long - run level
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. 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)
We reject a hypothesis that is actually true
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

44. K - th moment
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Model dependent - Options with the same underlying assets may trade at different volatilities
Summation((xi - mean)^k)/n

45. Reliability
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Contains variables not explicit in model - Accounts for randomness
Statement of the error or precision of an estimate

46. Biggest (and only real) drawback of GARCH mode
Contains variables not explicit in model - Accounts for randomness
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Nonlinearity
For n>30 - sample mean is approximately normal

47. Adjusted R^2
Application of mathematical statistics to economic data to lend empirical support to models
Low Frequency - High Severity events
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)
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. Efficiency
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(y)/n = variance of sample Y
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

49. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Yi = B0 + B1Xi + ui
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

50. Continuously compounded return equation
i = ln(Si/Si - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Peaks over threshold - Collects dataset in excess of some threshold
Variance(X) + Variance(Y) - 2*covariance(XY)