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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. Result of combination of two normal with same means
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Combine to form distribution with leptokurtosis (heavy tails)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

2. Potential reasons for fat tails in return distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

3. Key properties of linear regression
For n>30 - sample mean is approximately normal
Expected value of the sample mean is the population mean
Regression can be non - linear in variables but must be linear in parameters
Use historical simulation approach but use the EWMA weighting system

4. LFHS
Summation((xi - mean)^k)/n
Contains variables not explicit in model - Accounts for randomness
Low Frequency - High Severity events
Distribution with only two possible outcomes

5. Test for unbiasedness
SSR
Model dependent - Options with the same underlying assets may trade at different volatilities
Sample mean will near the population mean as the sample size increases
E(mean) = mean

6. T distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

7. Homoskedastic
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)
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
Average return across assets on a given day

8. Cross - sectional
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Average return across assets on a given day
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E(mean) = mean

9. Priori (classical) probability
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
Confidence level
Based on an equation - P(A) = # of A/total outcomes
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

10. Empirical frequency
P - value
Based on a dataset
Sample mean will near the population mean as the sample size increases
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

11. Variance of X+Y
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
Var(X) + Var(Y)
Transformed to a unit variable - Mean = 0 Variance = 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

12. Simulation models
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
(a^2)(variance(x)) + (b^2)(variance(y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

13. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Distribution with only two possible outcomes
Use historical simulation approach but use the EWMA weighting system
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

14. Variance - covariance approach for VaR of a portfolio
Average return across assets on a given day
Peaks over threshold - Collects dataset in excess of some threshold
Variance(x) + Variance(Y) + 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

15. Unbiased
Distribution with only two possible outcomes
Mean of sampling distribution is the population mean
Average return across assets on a given day
Independently and Identically Distributed

16. Normal distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
When the sample size is large - the uncertainty about the value of the sample is very small
Normal - Student's T - Chi - square - F distribution
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

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

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18. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Bernouli Distribution
Does not depend on a prior event or information
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Distribution with only two possible outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

20. Tractable
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

21. Direction of OVB
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
Average return across assets on a given day
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Combine to form distribution with leptokurtosis (heavy tails)

22. Consistent
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Distribution with only two possible outcomes
When the sample size is large - the uncertainty about the value of the sample is very small

23. Overall F - statistic
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We reject a hypothesis that is actually true
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

24. BLUE
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Independently and Identically Distributed

25. Implied standard deviation for options
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for an individual asset
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

26. Four sampling distributions

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27. Non - parametric vs parametric calculation of VaR
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

28. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Use historical simulation approach but use the EWMA weighting system

29. Stochastic error term
Nonlinearity
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Contains variables not explicit in model - Accounts for randomness

30. Joint probability functions
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Probability that the random variables take on certain values simultaneously
Yi = B0 + B1Xi + ui

31. Conditional probability functions
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

32. SER
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
Change in S = S<t - 1>(meanchange in time + stddev 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
Variance reverts to a long run level

33. Reliability
Expected value of the sample mean is the population mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Statement of the error or precision of an estimate

34. Hazard rate of exponentially distributed random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Does not depend on a prior event or information
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample mean will near the population mean as the sample size increases

35. Multivariate probability
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
More than one random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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. Control variates technique
Based on an equation - P(A) = # of A/total outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Transformed to a unit variable - Mean = 0 Variance = 1

37. Mean reversion
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Special type of pooled data in which the cross sectional unit is surveyed over time
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

38. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SSR

39. Confidence ellipse
Based on a dataset
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Confidence set for two coefficients - two dimensional analog for the confidence interval
Based on an equation - P(A) = # of A/total outcomes

40. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P - value

41. Variance of aX + bY
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(X) + Variance(Y) - 2*covariance(XY)

42. Cholesky factorization (decomposition)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Special type of pooled data in which the cross sectional unit is surveyed over 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

43. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Price/return tends to run towards a long - run level
Peaks over threshold - Collects dataset in excess of some threshold
Least absolute deviations estimator - used when extreme outliers are not uncommon

44. P - value
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Rxy = Sxy/(Sx*Sy)
P(Z>t)

45. Kurtosis
More than one random variable
Least absolute deviations estimator - used when extreme outliers are not uncommon
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Contains variables not explicit in model - Accounts for randomness

46. Covariance
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)

47. ESS
Variance(X) + Variance(Y) - 2*covariance(XY)
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
More than one random variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

48. Exponential distribution
Price/return tends to run towards a long - run level
We reject a hypothesis that is actually true
Sample mean +/ - t*(stddev(s)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

49. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P(Z>t)

50. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Returns over time for an individual asset
E(XY) - E(X)E(Y)
Does not depend on a prior event or information