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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 30 questions in 15 minutes. 1 minute extra for reading the instructions.
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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. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
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 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
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

2. GPD
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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

3. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Concerned with a single random variable (ex. Roll of a die)
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

4. Result of combination of two normal with same means
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Combine to form distribution with leptokurtosis (heavy tails)
Variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

5. Confidence ellipse
Independently and Identically Distributed
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(x) + Variance(Y) + 2*covariance(XY)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

6. Continuous representation of the GBM
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance = (1/m) summation(u<n - i>^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)

7. Inverse transform method
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

8. Binomial distribution equations for mean variance and std dev
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

9. Unbiased
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean of sampling distribution is the population mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

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

11. Homoskedastic
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean will near the population mean as the sample size increases
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

12. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Confidence set for two coefficients - two dimensional analog for the confidence interval
Only requires two parameters = mean and variance
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

13. Shortcomings of implied volatility
Choose parameters that maximize the likelihood of what observations occurring
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Model dependent - Options with the same underlying assets may trade at different volatilities
Combine to form distribution with leptokurtosis (heavy tails)

14. Panel data (longitudinal or micropanel)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Special type of pooled data in which the cross sectional unit is surveyed over time
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Distribution with only two possible outcomes

15. Statistical (or empirical) model
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Yi = B0 + B1Xi + ui
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

16. Standard error
Returns over time for an individual asset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

17. Difference between population and sample variance
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Population denominator = n - Sample denominator = n - 1
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Only requires two parameters = mean and variance

18. GARCH
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
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. Variance of aX + bY
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
(a^2)(variance(x)) + (b^2)(variance(y))
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 level

20. F distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

21. Deterministic Simulation
SSR
E(mean) = mean
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
Normal - Student's T - Chi - square - F distribution

22. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
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
For n>30 - sample mean is approximately normal
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

23. Skewness
Variance reverts to a long run level
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
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

24. Empirical frequency
Based on a dataset
Statement of the error or precision of an estimate
Has heavy tails
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

25. Variance of aX
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
(a^2)(variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Combine to form distribution with leptokurtosis (heavy tails)

26. Single variable (univariate) probability
Yi = B0 + B1Xi + ui
Concerned with a single random variable (ex. Roll of a die)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sampling distribution of sample means tend to be normal

27. Central Limit Theorem
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
95% = 1.65 99% = 2.33 For one - tailed tests
For n>30 - sample mean is approximately normal

28. Econometrics
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
(a^2)(variance(x)
Application of mathematical statistics to economic data to lend empirical support to models
Has heavy tails

29. Simplified standard (un - weighted) variance
E(XY) - E(X)E(Y)
Sample mean will near the population mean as the sample size increases
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance = (1/m) summation(u<n - i>^2)

30. Square root rule
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric