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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. Binomial distribution equations for mean variance and std dev
Variance reverts to a long run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
E(mean) = mean
Variance = (1/m) summation(u<n - i>^2)

2. SER
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

3. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
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
E(XY) - E(X)E(Y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

4. Extreme Value Theory
Sample mean will near the population mean as the sample size increases
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance(X) + Variance(Y) - 2*covariance(XY)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

5. Economical(elegant)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Only requires two parameters = mean and variance
Independently and Identically Distributed

6. Mean reversion in variance
P - value
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance reverts to a long run level

7. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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)
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
Contains variables not explicit in model - Accounts for randomness

8. P - value
Peaks over threshold - Collects dataset in excess of some threshold
P(Z>t)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

9. Standard error for Monte Carlo replications
Regression can be non - linear in variables but must be linear in parameters
Does not depend on a prior event or information
Variance = (1/m) summation(u<n - i>^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

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

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11. Panel data (longitudinal or micropanel)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Special type of pooled data in which the cross sectional unit is surveyed over time

12. Discrete representation of the GBM
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Special type of pooled data in which the cross sectional unit is surveyed over time
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Peaks over threshold - Collects dataset in excess of some threshold

13. Sample variance
Distribution with only two possible outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

14. Importance sampling technique
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
Attempts to sample along more important paths
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

15. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Random walk (usually acceptable) - Constant volatility (unlikely)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

16. Poisson distribution equations for mean variance and std deviation
Returns over time for an individual asset
Peaks over threshold - Collects dataset in excess of some threshold
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

17. Sample correlation
Special type of pooled data in which the cross sectional unit is surveyed over time
Rxy = Sxy/(Sx*Sy)
Peaks over threshold - Collects dataset in excess of some threshold
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

18. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
i = ln(Si/Si - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

19. GPD
Easy to manipulate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Concerned with a single random variable (ex. Roll of a die)

20. Multivariate Density Estimation (MDE)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

21. Econometrics
When one regressor is a perfect linear function of the other regressors
Application of mathematical statistics to economic data to lend empirical support to models
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean = np - Variance = npq - Std dev = sqrt(npq)

22. Stochastic error term
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Contains variables not explicit in model - Accounts for randomness

23. 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)
Price/return tends to run towards a long - run level
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 = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

24. Hybrid method for conditional volatility
Returns over time for an individual asset
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Use historical simulation approach but use the EWMA weighting system
We accept a hypothesis that should have been rejected

25. Normal distribution
Confidence level
Sampling distribution of sample means tend to be normal
Dataset is parsed into blocks with greater length than the periodicity - Observations must be 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

26. T distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Nonlinearity
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

27. Perfect multicollinearity
When the sample size is large - the uncertainty about the value of the sample is very small
When one regressor is a perfect linear function of the other regressors
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Combine to form distribution with leptokurtosis (heavy tails)

28. What does the OLS minimize?
SSR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Use historical simulation approach but use the EWMA weighting system

29. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Population denominator = n - Sample denominator = n - 1
Special type of pooled data in which the cross sectional unit is surveyed over time
Random walk (usually acceptable) - Constant volatility (unlikely)

30. Tractable
Easy to manipulate
Low Frequency - High Severity events
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance(y)/n = variance of sample Y

31. Four sampling distributions

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32. Variance of sample 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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(y)/n = variance of sample Y
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

33. Significance =1
Confidence level
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Least absolute deviations estimator - used when extreme outliers are not uncommon

34. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

35. Sample mean
Variance(X) + Variance(Y) - 2*covariance(XY)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Expected value of the sample mean is the population mean

36. Unconditional vs conditional distributions
Statement of the error or precision of an estimate
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
(a^2)(variance(x)) + (b^2)(variance(y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

37. WLS
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Normal - Student's T - Chi - square - F distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

38. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Summation((xi - mean)^k)/n
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

39. Kurtosis
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
Probability that the random variables take on certain values simultaneously
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

40. Key properties of linear regression
Based on a dataset
Regression can be non - linear in variables but must be linear in parameters
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

41. Lognormal
We reject a hypothesis that is actually true
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(X) + Variance(Y) - 2*covariance(XY)
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

42. Continuously compounded return equation
Application of mathematical statistics to economic data to lend empirical support to models
Variance(x)
i = ln(Si/Si - 1)
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

43. Difference between population and sample variance
Variance(x)
Population denominator = n - Sample denominator = n - 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

44. Logistic distribution
95% = 1.65 99% = 2.33 For one - tailed tests
Has heavy tails
Nonlinearity
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

45. LAD
Mean = np - Variance = npq - Std dev = sqrt(npq)
SSR
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

46. 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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

47. Non - parametric vs parametric calculation of VaR
Confidence set for two coefficients - two dimensional analog for the confidence interval
Easy to manipulate
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sampling distribution of sample means tend to be normal

48. Weibul distribution
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. Multivariate probability
More than one random variable
P - value
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

50. Central Limit Theorem
Statement of the error or precision of an estimate
Normal - Student's T - Chi - square - F distribution
For n>30 - sample mean is approximately normal
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)