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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. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Independently and Identically Distributed
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

2. GPD
Variance(y)/n = variance of sample Y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Rxy = Sxy/(Sx*Sy)
Combine to form distribution with leptokurtosis (heavy tails)

3. What does the OLS minimize?
Population denominator = n - Sample denominator = n - 1
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
SSR

4. Central Limit Theorem(CLT)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sampling distribution of sample means tend to be normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

5. Lognormal
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

6. Significance =1
Confidence level
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
Distribution with only two possible outcomes
Concerned with a single random variable (ex. Roll of a die)

7. GEV
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
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

8. Central Limit Theorem
For n>30 - sample mean is approximately normal
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. Extreme Value Theory
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

10. Weibul distribution
Low Frequency - High Severity events
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for a combination of assets (combination of time series and cross - sectional data)
i = ln(Si/Si - 1)

11. Economical(elegant)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Only requires two parameters = mean and variance
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

12. Persistence
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
When one regressor is a perfect linear function of the other regressors
Price/return tends to run towards a long - run level
Among all unbiased estimators - estimator with the smallest variance is efficient

13. i.i.d.
Application of mathematical statistics to economic data to lend empirical support to models
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)
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
Independently and Identically Distributed

14. Monte Carlo Simulations
For n>30 - sample mean is approximately normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Combine to form distribution with leptokurtosis (heavy tails)

15. Chi - squared distribution
When one regressor is a perfect linear function of the other regressors
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

16. Beta distribution
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

17. SER
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

18. Logistic distribution
Nonlinearity
Has heavy tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance(X) + Variance(Y) - 2*covariance(XY)

19. Tractable
Confidence set for two coefficients - two dimensional analog for the confidence interval
Easy to manipulate
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

20. Variance(discrete)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Rxy = Sxy/(Sx*Sy)
Confidence level

21. POT
Peaks over threshold - Collects dataset in excess of some threshold
Choose parameters that maximize the likelihood of what observations occurring
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

22. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Application of mathematical statistics to economic data to lend empirical support to models
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Peaks over threshold - Collects dataset in excess of some threshold

23. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Regression can be non - linear in variables but must be linear in parameters
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

24. Confidence interval (from t)
Sampling distribution of sample means tend to be normal
Combine to form distribution with leptokurtosis (heavy tails)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Probability that the random variables take on certain values simultaneously

25. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
i = ln(Si/Si - 1)
E(XY) - E(X)E(Y)

26. Variance of weighted scheme
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

27. K - th moment
Summation((xi - mean)^k)/n
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
(a^2)(variance(x)

28. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Concerned with a single random variable (ex. Roll of a die)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*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)

29. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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)
Among all unbiased estimators - estimator with the smallest variance is efficient

30. Hazard rate of exponentially distributed random variable
Average return across assets on a given day
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

31. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
We reject a hypothesis that is actually true

32. Time series data
Returns over time for an individual asset
Among all unbiased estimators - estimator with the smallest variance is efficient
Low Frequency - High Severity events
Variance(x) + Variance(Y) + 2*covariance(XY)

33. Variance of X+Y assuming dependence
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(Z>t)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x) + Variance(Y) + 2*covariance(XY)

34. Single variable (univariate) probability
Distribution with only two possible outcomes
Confidence level
Concerned with a single random variable (ex. Roll of a die)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

35. Unbiased
Choose parameters that maximize the likelihood of what observations occurring
SSR
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean

36. Marginal unconditional probability function
Use historical simulation approach but use the EWMA weighting system
Does not depend on a prior event or information
Has heavy tails
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

37. Stochastic error term
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Contains variables not explicit in model - Accounts for randomness
Variance(x) + Variance(Y) + 2*covariance(XY)
Yi = B0 + B1Xi + ui

38. Test for unbiasedness
We reject a hypothesis that is actually true
E(mean) = mean
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)

39. GARCH
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)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
95% = 1.65 99% = 2.33 For one - tailed tests

40. Simulating for VaR
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
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Has heavy tails

41. Poisson distribution equations for mean variance and std deviation
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean will near the population mean as the sample size increases

42. Homoskedastic
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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
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
Rxy = Sxy/(Sx*Sy)

43. Critical z values
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P - value
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
95% = 1.65 99% = 2.33 For one - tailed tests

44. Efficiency
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Among all unbiased estimators - estimator with the smallest variance is efficient
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

45. BLUE
Mean = np - Variance = npq - Std dev = sqrt(npq)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

46. Two ways to calculate historical volatility
Nonlinearity
Rxy = Sxy/(Sx*Sy)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Statistical (or empirical) model
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Summation((xi - mean)^k)/n
Random walk (usually acceptable) - Constant volatility (unlikely)
Yi = B0 + B1Xi + ui

48. Sample covariance
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

49. Bernouli Distribution
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
Distribution with only two possible outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

50. Variance of X+Y
Confidence level
When one regressor is a perfect linear function of the other regressors
Var(X) + Var(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