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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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Match each statement with the correct term.
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1. Direction of OVB
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Easy to manipulate
Variance reverts to a long run level

2. Kurtosis
Transformed to a unit variable - Mean = 0 Variance = 1
Easy to manipulate
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

3. Simplified standard (un - weighted) variance
More than one random variable
Variance = (1/m) summation(u<n - i>^2)
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

4. SER
Yi = B0 + B1Xi + ui
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Contains variables not explicit in model - Accounts for randomness
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

5. Normal distribution
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)
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
Concerned with a single random variable (ex. Roll of a die)

6. Sample covariance
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
For n>30 - sample mean is approximately normal

7. Multivariate Density Estimation (MDE)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Regression can be non - linear in variables but must be linear in parameters

8. Bernouli Distribution
Based on a dataset
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
Distribution with only two possible outcomes

9. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Nonlinearity
Variance(x) + Variance(Y) + 2*covariance(XY)

10. Tractable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Combine to form distribution with leptokurtosis (heavy tails)
Easy to manipulate
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

11. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

12. Reliability
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Statement of the error or precision of an estimate
(a^2)(variance(x)) + (b^2)(variance(y))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

13. Stochastic error term
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Contains variables not explicit in model - Accounts for randomness
Transformed to a unit variable - Mean = 0 Variance = 1

14. Least squares estimator(m)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Average return across assets on a given day
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

15. Sample correlation
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Rxy = Sxy/(Sx*Sy)
Population denominator = n - Sample denominator = n - 1
Application of mathematical statistics to economic data to lend empirical support to models

16. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Yi = B0 + B1Xi + ui
Var(X) + Var(Y)
P - value

17. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

18. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Variance(x) + Variance(Y) + 2*covariance(XY)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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

19. Conditional probability functions
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

20. Exact significance level
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P - value
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

21. Type II Error
We accept a hypothesis that should have been rejected
Application of mathematical statistics to economic data to lend empirical support to models
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

22. Persistence
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
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
P - value

23. ESS
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

24. Economical(elegant)
Only requires two parameters = mean and variance
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)
Mean of sampling distribution is the population mean
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

25. Weibul distribution
Normal - Student's T - Chi - square - F distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Rxy = Sxy/(Sx*Sy)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

26. Confidence interval (from t)
Average return across assets on a given day
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Confidence set for two coefficients - two dimensional analog for the confidence interval

27. Expected future variance rate (t periods forward)
Has heavy tails
Normal - Student's T - Chi - square - F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

28. EWMA
Variance(x)
Rxy = Sxy/(Sx*Sy)
Expected value of the sample mean is the population mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

29. Antithetic variable technique
Among all unbiased estimators - estimator with the smallest variance is efficient
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P - value
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

30. Mean reversion
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)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Average return across assets on a given day
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

31. Continuous random variable
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

32. Two assumptions of square root rule
Distribution with only two possible outcomes
Random walk (usually acceptable) - Constant volatility (unlikely)
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

33. Mean(expected value)
We accept a hypothesis that should have been rejected
SSR
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
(a^2)(variance(x)

34. Mean reversion in asset dynamics
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
Price/return tends to run towards a long - run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

35. Cholesky factorization (decomposition)
Sample mean will near the population mean as the sample size increases
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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

36. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Among all unbiased estimators - estimator with the smallest variance is efficient
Independently and Identically Distributed
Attempts to sample along more important paths

37. Central Limit Theorem
Expected value of the sample mean is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
For n>30 - sample mean is approximately normal

38. Simulating for VaR
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Rxy = Sxy/(Sx*Sy)

39. Type I error
We reject a hypothesis that is actually true
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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)
Yi = B0 + B1Xi + ui

40. Continuously compounded return equation
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Price/return tends to run towards a long - run level
i = ln(Si/Si - 1)

41. Variance of aX
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
(a^2)(variance(x)
Distribution with only two possible outcomes

42. Hazard rate of exponentially distributed random variable
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

43. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(x) + Variance(Y) + 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)

44. Monte Carlo Simulations
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

45. Biggest (and only real) drawback of GARCH mode
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!)
Var(X) + Var(Y)
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
Nonlinearity

46. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

47. Panel data (longitudinal or micropanel)
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)
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
Nonlinearity

48. Critical z values
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean +/ - t*(stddev(s)/sqrt(n))

49. 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)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

50. Standard error for Monte Carlo replications
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P - value
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications