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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. Variance of aX
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
(a^2)(variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Model dependent - Options with the same underlying assets may trade at different volatilities

2. Beta 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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

3. Implied standard deviation for options
Independently and Identically Distributed
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample mean +/ - t*(stddev(s)/sqrt(n))
(a^2)(variance(x)

4. Expected future variance rate (t periods forward)
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n
Use historical simulation approach but use the EWMA weighting system
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

5. Block maxima
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
When the sample size is large - the uncertainty about the value of the sample is very small
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

6. Law of Large Numbers
Based on an equation - P(A) = # of A/total outcomes
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
Sample mean will near the population mean as the sample size increases
We reject a hypothesis that is actually true

7. Panel data (longitudinal or micropanel)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Special type of pooled data in which the cross sectional unit is surveyed over time

8. ESS
Rxy = Sxy/(Sx*Sy)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

9. Direction of OVB
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Choose parameters that maximize the likelihood of what observations occurring

10. Consistent
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
When the sample size is large - the uncertainty about the value of the sample is very small

11. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

12. Stochastic error term
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
Contains variables not explicit in model - Accounts for randomness
Confidence set for two coefficients - two dimensional analog for the confidence interval
Least absolute deviations estimator - used when extreme outliers are not uncommon

13. Multivariate probability
More than one random variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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

14. Covariance calculations using weight sums (lambda)
(a^2)(variance(x)
Normal - Student's T - Chi - square - F distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

15. Empirical frequency
Based on a dataset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Sample mean will near the population mean as the sample size increases
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

16. Test for statistical independence
Concerned with a single random variable (ex. Roll of a die)
E(mean) = mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Normal - Student's T - Chi - square - F distribution

17. Chi - squared distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
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
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

18. Lognormal
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
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Regression can be non - linear in variables but must be linear in parameters

19. Variance of aX + bY
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
(a^2)(variance(x)) + (b^2)(variance(y))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sampling distribution of sample means tend to be normal

20. Importance sampling technique
Attempts to sample along more important paths
Price/return tends to run towards a long - run level
Contains variables not explicit in model - Accounts for randomness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

21. Variance - covariance approach for VaR of a portfolio
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
(a^2)(variance(x)) + (b^2)(variance(y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

22. Sample correlation
Rxy = Sxy/(Sx*Sy)
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)
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
Variance reverts to a long run level

23. Simulation models
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Price/return tends to run towards a long - run level
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
Rxy = Sxy/(Sx*Sy)

24. Mean reversion in variance
Variance reverts to a long run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

25. SER
Variance reverts to a long run level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

26. Confidence ellipse
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Yi = B0 + B1Xi + ui
Confidence set for two coefficients - two dimensional analog for the confidence interval
Concerned with a single random variable (ex. Roll of a die)

27. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

28. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Statement of the error or precision of an estimate

29. Limitations of R^2 (what an increase doesn't necessarily imply)

30. 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
Choose parameters that maximize the likelihood of what observations occurring
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

31. Hazard rate of exponentially distributed random variable
We accept a hypothesis that should have been rejected
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

32. Marginal unconditional probability function
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Does not depend on a prior event or information
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
Variance(y)/n = variance of sample Y

33. Mean(expected value)
Based on an equation - P(A) = # of A/total outcomes
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Has heavy tails
Only requires two parameters = mean and variance

34. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Distribution with only two possible outcomes
Expected value of the sample mean is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)

35. Homoskedastic only F - stat
Attempts to sample along more important paths
(a^2)(variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

36. Implications of homoscedasticity
Contains variables not explicit in model - Accounts for randomness
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
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

37. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
For n>30 - sample mean is approximately normal
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)) + (b^2)(variance(y))

38. GPD
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Based on a dataset
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

39. Square root rule
i = ln(Si/Si - 1)
Distribution with only two possible outcomes
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = np - Variance = npq - Std dev = sqrt(npq)

40. Antithetic variable technique
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Low Frequency - High Severity events
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

41. Adjusted R^2
Least absolute deviations estimator - used when extreme outliers are not uncommon
For n>30 - sample mean is approximately normal
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
Combine to form distribution with leptokurtosis (heavy tails)

42. Variance of weighted scheme
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Var(X) + Var(Y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

43. Variance of X+Y
Var(X) + Var(Y)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Combine to form distribution with leptokurtosis (heavy tails)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

44. Continuous random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
(a^2)(variance(x)
Combine to form distribution with leptokurtosis (heavy tails)

45. Four sampling distributions

46. Type I error
We reject a hypothesis that is actually true
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sample mean will near the population mean as the sample size increases
More than one random variable

47. Standard error for Monte Carlo replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

48. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Use historical simulation approach but use the EWMA weighting system
Sampling distribution of sample means tend to be normal
Variance = (1/m) summation(u<n - i>^2)

49. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

50. GEV
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Summation((xi - mean)^k)/n