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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. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E(XY) - E(X)E(Y)

2. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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!)

3. Reliability
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Statement of the error or precision of an estimate
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

4. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(x)
When one regressor is a perfect linear function of the other regressors

5. R^2
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample mean will near the population mean as the sample size increases

6. Two drawbacks of moving average series
Mean = np - Variance = npq - Std dev = sqrt(npq)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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!)

7. Efficiency
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Among all unbiased estimators - estimator with the smallest variance is efficient

8. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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
Attempts to sample along more important paths

9. Variance of aX
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
(a^2)(variance(x)

10. POT
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Peaks over threshold - Collects dataset in excess of some threshold

11. F distribution
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
95% = 1.65 99% = 2.33 For one - tailed tests

12. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

13. GARCH
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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 set for two coefficients - two dimensional analog for the confidence interval
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

14. Type II Error
E(mean) = mean
Model dependent - Options with the same underlying assets may trade at different volatilities
We accept a hypothesis that should have been rejected
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

15. BLUE
Returns over time for an individual asset
Use historical simulation approach but use the EWMA weighting system
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

16. Direction of OVB
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sampling distribution of sample means tend to be normal
Concerned with a single random variable (ex. Roll of a die)

17. Implied standard deviation for options
We accept a hypothesis that should have been rejected
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)

18. Logistic distribution
Distribution with only two possible outcomes
Has heavy tails
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
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)

19. Deterministic Simulation
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample mean will near the population mean as the sample size increases
Returns over time for an individual asset

20. GPD
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes

21. Critical z values
We accept a hypothesis that should have been rejected
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
95% = 1.65 99% = 2.33 For one - tailed tests

22. SER
Among all unbiased estimators - estimator with the smallest variance is efficient
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
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
Model dependent - Options with the same underlying assets may trade at different volatilities

23. Block maxima
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
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

24. Bernouli Distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Distribution with only two possible outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

25. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Use historical simulation approach but use the EWMA weighting system
More than one random variable
i = ln(Si/Si - 1)

26. Tractable
Easy to manipulate
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
More than one random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

27. Sample correlation
Price/return tends to run towards a long - run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
P - value
Rxy = Sxy/(Sx*Sy)

28. What does the OLS minimize?
SSR
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

29. Antithetic variable technique
Choose parameters that maximize the likelihood of what observations occurring
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Variance of X - Y assuming dependence
Choose parameters that maximize the likelihood of what observations occurring
When the sample size is large - the uncertainty about the value of the sample is very small
Low Frequency - High Severity events
Variance(X) + Variance(Y) - 2*covariance(XY)

31. LAD
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
If variance of the conditional distribution of u(i) is not constant

32. Variance - covariance approach for VaR of a portfolio
Variance = (1/m) summation(u<n - i>^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

33. Confidence ellipse
When one regressor is a perfect linear function of the other regressors
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

34. Historical std dev
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance reverts to a long run level
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

35. Kurtosis
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
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

36. ESS
Has heavy tails
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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. Standard error for Monte Carlo replications
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(x) + Variance(Y) + 2*covariance(XY)
P(X=x - Y=y) = P(X=x) * P(Y=y)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

38. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Nonlinearity
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

39. Inverse transform method
If variance of the conditional distribution of u(i) is not constant
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
i = ln(Si/Si - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

40. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

41. Variance of X+Y
If variance of the conditional distribution of u(i) is not constant
P(X=x - Y=y) = P(X=x) * P(Y=y)
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

42. Joint probability functions
We reject a hypothesis that is actually true
Probability that the random variables take on certain values simultaneously
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Use historical simulation approach but use the EWMA weighting system

43. Marginal unconditional probability function
Does not depend on a prior event or information
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

44. Least squares estimator(m)
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

45. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
For n>30 - sample mean is approximately normal
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
Peaks over threshold - Collects dataset in excess of some threshold

46. Variance of aX + bY
Returns over time for a combination of assets (combination of time series and cross - sectional data)
(a^2)(variance(x)) + (b^2)(variance(y))
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
Based on a dataset

47. Stochastic error term
Statement of the error or precision of an estimate
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
Contains variables not explicit in model - Accounts for randomness

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

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49. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean

50. Perfect multicollinearity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
When one regressor is a perfect linear function of the other regressors
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)