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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. Standard error for Monte Carlo replications
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

2. Economical(elegant)
95% = 1.65 99% = 2.33 For one - tailed tests
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
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance

3. Cholesky factorization (decomposition)
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)
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
Nonlinearity
Variance = (1/m) summation(u<n - i>^2)

4. Potential reasons for fat tails in return distributions
Sampling distribution of sample means tend to be normal
Based on a dataset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

5. Four sampling distributions

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

7. Sample correlation
Rxy = Sxy/(Sx*Sy)
Mean = np - Variance = npq - Std dev = sqrt(npq)
E(mean) = mean
Has heavy tails

8. Variance of aX
(a^2)(variance(x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean of sampling distribution is the population mean

9. Sample mean
Sampling distribution of sample means tend to be normal
Statement of the error or precision of an estimate
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Expected value of the sample mean is the population mean

10. Normal distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Low Frequency - High Severity events
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

11. Confidence ellipse
(a^2)(variance(x)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

12. Direction of OVB
Variance(y)/n = variance of sample Y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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!)
Variance(x)

13. Hazard rate of exponentially distributed random variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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)

14. What does the OLS minimize?
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
SSR
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

15. Consistent
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
When the sample size is large - the uncertainty about the value of the sample is very small
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Variance of X - Y assuming dependence
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(X) + Variance(Y) - 2*covariance(XY)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

17. Homoskedastic
(a^2)(variance(x)) + (b^2)(variance(y))
Does not depend on a prior event or information
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)

18. Exact significance level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Does not depend on a prior event or information
P - value

19. K - th moment
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Summation((xi - mean)^k)/n

20. 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
Application of mathematical statistics to economic data to lend empirical support to models
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

21. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
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)

22. Binomial distribution equations for mean variance and std dev
Var(X) + Var(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean = np - Variance = npq - Std dev = sqrt(npq)

23. Unstable return distribution
Variance(y)/n = variance of sample Y
Easy to manipulate
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

24. Heteroskedastic
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean +/ - t*(stddev(s)/sqrt(n))
If variance of the conditional distribution of u(i) is not constant

25. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Regression can be non - linear in variables but must be linear in parameters

26. Block maxima
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Var(X) + Var(Y)
Independently and Identically Distributed
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

27. WLS
Contains variables not explicit in model - Accounts for randomness
Use historical simulation approach but use the EWMA weighting system
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

28. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

29. Deterministic Simulation
P - value
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

30. Sample covariance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability that the random variables take on certain values simultaneously
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

31. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Model dependent - Options with the same underlying assets may trade at different volatilities
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

32. Multivariate Density Estimation (MDE)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x) + Variance(Y) + 2*covariance(XY)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

33. Historical std dev
Regression can be non - linear in variables but must be linear in parameters
P(X=x - Y=y) = P(X=x) * P(Y=y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

34. Bootstrap method
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance = (1/m) summation(u<n - i>^2)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance reverts to a long run level

35. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Easy to manipulate
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 that the random variables take on certain values simultaneously

36. Standard variable for non - normal distributions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Z = (Y - meany)/(stddev(y)/sqrt(n))
Use historical simulation approach but use the EWMA weighting system
Independently and Identically Distributed

37. Variance of X+Y assuming dependence
Variance = (1/m) summation(u<n - i>^2)
Population denominator = n - Sample denominator = n - 1
Variance(x) + Variance(Y) + 2*covariance(XY)
Z = (Y - meany)/(stddev(y)/sqrt(n))

38. Chi - squared distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

39. Square root rule
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

40. Law of Large Numbers
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Sample mean will near the population mean as the sample size increases

41. Lognormal
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
i = ln(Si/Si - 1)
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
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

42. Single variable (univariate) probability
Does not depend on a prior event or information
When the sample size is large - the uncertainty about the value of the sample is very small
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Concerned with a single random variable (ex. Roll of a die)

43. Exponential distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
Choose parameters that maximize the likelihood of what observations occurring
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

44. R^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Yi = B0 + B1Xi + ui
Z = (Y - meany)/(stddev(y)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

45. Efficiency
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!)
Variance(x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Among all unbiased estimators - estimator with the smallest variance is efficient

46. Kurtosis
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more 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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

47. Type I error
Based on a dataset
We reject a hypothesis that is actually true
Variance(x)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

48. ESS
Expected value of the sample mean is the population mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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!)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

49. Discrete random variable
Based on a dataset
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Transformed to a unit variable - Mean = 0 Variance = 1

50. Covariance calculations using weight sums (lambda)
We accept a hypothesis that should have been rejected
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period