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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
Based on a dataset
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(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
SSR

2. Conditional probability functions
Var(X) + Var(Y)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

3. Empirical frequency
95% = 1.65 99% = 2.33 For one - tailed tests
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Based on a dataset

4. Poisson Distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Only requires two parameters = mean and variance
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

5. Non - parametric vs parametric calculation of VaR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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)
We accept a hypothesis that should have been rejected

6. Significance =1
More than one random variable
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Confidence level
Concerned with a single random variable (ex. Roll of a die)

7. Reliability
Statement of the error or precision of an estimate
Easy to manipulate
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

8. Confidence ellipse
Random walk (usually acceptable) - Constant volatility (unlikely)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean +/ - t*(stddev(s)/sqrt(n))
Regression can be non - linear in variables but must be linear in parameters

9. Standard error for Monte Carlo replications
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

10. Pooled data
Special type of pooled data in which the cross sectional unit is surveyed over time
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

11. Mean reversion in asset dynamics
E(mean) = mean
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Price/return tends to run towards a long - run level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

12. What does the OLS minimize?
SSR
We accept a hypothesis that should have been rejected
More than one random variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

13. SER
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

14. Type I error
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
We reject a hypothesis that is actually true
E(XY) - E(X)E(Y)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

15. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample mean +/ - t*(stddev(s)/sqrt(n))

16. Econometrics
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models
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

17. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)

18. Variance - covariance approach for VaR of a portfolio
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When one regressor is a perfect linear function of the other regressors
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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)

19. Type II Error
E(XY) - E(X)E(Y)
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)
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
We accept a hypothesis that should have been rejected

20. Implications of homoscedasticity
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Yi = B0 + B1Xi + ui
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

21. Continuous representation of the GBM
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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)
E(mean) = mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. Skewness
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Independently and Identically Distributed
Distribution with only two possible outcomes

23. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
For n>30 - sample mean is approximately normal

24. Variance of aX + bY
Variance reverts to a long run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
(a^2)(variance(x)) + (b^2)(variance(y))
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

25. Simulation models
If variance of the conditional distribution of u(i) is not constant
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(x) + Variance(Y) + 2*covariance(XY)

26. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Combine to form distribution with leptokurtosis (heavy tails)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

27. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Among all unbiased estimators - estimator with the smallest variance is efficient
Distribution with only two possible outcomes
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

28. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Regression can be non - linear in variables but must be linear in parameters

29. Exact significance level
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
P - value
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

30. Perfect multicollinearity
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When one regressor is a perfect linear function of the other regressors

31. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Probability that the random variables take on certain values simultaneously
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
(a^2)(variance(x)) + (b^2)(variance(y))

32. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

33. Deterministic Simulation
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
More than one random variable

34. Two drawbacks of moving average series
E(XY) - E(X)E(Y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Has heavy tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

35. Biggest (and only real) drawback of GARCH mode
Least absolute deviations estimator - used when extreme outliers are not uncommon
E(XY) - E(X)E(Y)
Nonlinearity
Special type of pooled data in which the cross sectional unit is surveyed over time

36. F distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Model dependent - Options with the same underlying assets may trade at different volatilities

37. Simplified standard (un - weighted) 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)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance = (1/m) summation(u<n - i>^2)
SSR

38. Variance(discrete)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Concerned with a single random variable (ex. Roll of a die)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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

39. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When the sample size is large - the uncertainty about the value of the sample is very small
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

40. i.i.d.
Independently and Identically Distributed
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

41. Multivariate probability
More than one random variable
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

42. Variance of X+b
Var(X) + Var(Y)
Variance(x)
95% = 1.65 99% = 2.33 For one - tailed tests
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

43. Mean reversion in variance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance reverts to a long run level
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

44. Difference between population and sample variance
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Population denominator = n - Sample denominator = n - 1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Yi = B0 + B1Xi + ui

45. Importance sampling technique
SSR
Distribution with only two possible outcomes
Mean of sampling distribution is the population mean
Attempts to sample along more important paths

46. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

47. Variance of aX
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
(a^2)(variance(x)
Returns over time for an individual asset

48. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample mean +/ - t*(stddev(s)/sqrt(n))
We accept a hypothesis that should have been rejected

49. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Easy to manipulate
Contains variables not explicit in model - Accounts for randomness
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

50. Bootstrap method
Probability that the random variables take on certain values simultaneously
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on a dataset