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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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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Unstable return distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Has heavy tails

2. Variance of sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(y)/n = variance of sample Y
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

3. Standard error
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Mean = np - Variance = npq - Std dev = sqrt(npq)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

4. Inverse transform method
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
P(X=x - Y=y) = P(X=x) * P(Y=y)
Statement of the error or precision of an estimate

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

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6. Normal 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
Has heavy tails
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!)
Choose parameters that maximize the likelihood of what observations occurring

7. Persistence
Low Frequency - High Severity events
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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

8. Sample correlation
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Rxy = Sxy/(Sx*Sy)
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
Price/return tends to run towards a long - run level

9. Econometrics
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR
Use historical simulation approach but use the EWMA weighting system
Application of mathematical statistics to economic data to lend empirical support to models

10. Heteroskedastic
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
If variance of the conditional distribution of u(i) is not constant
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

11. WLS
Yi = B0 + B1Xi + ui
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Population denominator = n - Sample denominator = n - 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

12. Square root rule
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Least absolute deviations estimator - used when extreme outliers are not uncommon
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

13. Shortcomings of implied volatility
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
E(XY) - E(X)E(Y)
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.

14. Simulating for VaR
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Concerned with a single random variable (ex. Roll of a die)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

15. SER
i = ln(Si/Si - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(x)
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

16. R^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
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

17. Variance of aX + bY
Regression can be non - linear in variables but must be linear in parameters
(a^2)(variance(x)) + (b^2)(variance(y))
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

18. Binomial distribution
Confidence level
(a^2)(variance(x)) + (b^2)(variance(y))
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!)
Only requires two parameters = mean and variance

19. Direction of OVB
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
For n>30 - sample mean is approximately normal
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

20. Importance sampling technique
Attempts to sample along more important paths
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

21. Continuously compounded return equation
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
E(XY) - E(X)E(Y)
i = ln(Si/Si - 1)

22. Variance of X - Y assuming dependence
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(X) + Variance(Y) - 2*covariance(XY)
Average return across assets on a given day
Easy to manipulate

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

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24. Pooled data
Price/return tends to run towards a long - run level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Rxy = Sxy/(Sx*Sy)
Peaks over threshold - Collects dataset in excess of some threshold

25. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
i = ln(Si/Si - 1)
We accept a hypothesis that should have been rejected

26. Two requirements of OVB
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

27. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
If variance of the conditional distribution of u(i) is not constant
Variance reverts to a long run level
Use historical simulation approach but use the EWMA weighting system

28. Multivariate probability
More than one random variable
Rxy = Sxy/(Sx*Sy)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Statement of the error or precision of an estimate

29. Four sampling distributions

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30. Deterministic Simulation
More than one random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

31. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When one regressor is a perfect linear function of the other regressors
Returns over time for an individual asset

32. Continuous representation of the GBM
Attempts to sample along more important paths
Transformed to a unit variable - Mean = 0 Variance = 1
Population denominator = n - Sample denominator = n - 1
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)

33. Tractable
Application of mathematical statistics to economic data to lend empirical support to models
Sampling distribution of sample means tend to be normal
Easy to manipulate
Confidence set for two coefficients - two dimensional analog for the confidence interval

34. Expected future variance rate (t periods forward)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

35. Joint probability functions
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously

36. EWMA
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

37. Poisson distribution equations for mean variance and std deviation
Use historical simulation approach but use the EWMA weighting system
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Covariance calculations using weight sums (lambda)
Low Frequency - High Severity events
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
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)

39. Simulation models
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
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
Special type of pooled data in which the cross sectional unit is surveyed over time

40. Efficiency
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Price/return tends to run towards a long - run level
Among all unbiased estimators - estimator with the smallest variance is efficient

41. Sample covariance
P(X=x - Y=y) = P(X=x) * P(Y=y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
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

42. Biggest (and only real) drawback of GARCH mode
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance reverts to a long run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Nonlinearity

43. P - value
(a^2)(variance(x)) + (b^2)(variance(y))
P(Z>t)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

44. Least squares estimator(m)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
Use historical simulation approach but use the EWMA weighting system

45. Mean reversion in asset dynamics
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Price/return tends to run towards a long - run level

46. Simplified standard (un - weighted) variance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Independently and Identically Distributed
Variance = (1/m) summation(u<n - i>^2)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

47. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Based on a dataset
(a^2)(variance(x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

48. Adjusted R^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

49. Cholesky factorization (decomposition)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Variance(x)

50. Bootstrap method
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Distribution with only two possible outcomes