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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

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. Confidence interval for sample mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Price/return tends to run towards a long - run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

2. Mean(expected value)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Regression can be non - linear in variables but must be linear in parameters
Concerned with a single random variable (ex. Roll of a die)

3. Simulation models
We reject a hypothesis that is actually true
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

4. Gamma distribution
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
SSR
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

5. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P - value
Summation((xi - mean)^k)/n
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

6. Bootstrap method
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

7. Unstable return distribution
SSR
Use historical simulation approach but use the EWMA weighting system
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

8. POT
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Peaks over threshold - Collects dataset in excess of some threshold
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 exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

9. Central Limit Theorem(CLT)
Statement of the error or precision of an estimate
Sampling distribution of sample means tend to be normal
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

10. Type II Error
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
More than one random variable
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
We accept a hypothesis that should have been rejected

11. Variance of X+b
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x)

12. Beta distribution
Variance(x) + Variance(Y) + 2*covariance(XY)
Sampling distribution of sample means tend to be normal
Attempts to sample along more important paths
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

13. Two ways to calculate historical volatility
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)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

14. Tractable
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Easy to manipulate
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

15. Potential reasons for fat tails in return distributions
Contains variables not explicit in model - Accounts for randomness
Probability that the random variables take on certain values simultaneously
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

16. Standard error
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Use historical simulation approach but use the EWMA weighting system
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)

17. R^2
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

18. T distribution
Application of mathematical statistics to economic data to lend empirical support to models
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
i = ln(Si/Si - 1)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

19. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Least absolute deviations estimator - used when extreme outliers are not uncommon

20. Regime - switching volatility model
If variance of the conditional distribution of u(i) is not constant
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x) + Variance(Y) + 2*covariance(XY)

21. Poisson distribution equations for mean variance and std deviation
Based on an equation - P(A) = # of A/total outcomes
When one regressor is a perfect linear function of the other regressors
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sampling distribution of sample means tend to be normal

22. Hazard rate of exponentially distributed random variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
SSR
Price/return tends to run towards a long - run level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

23. Maximum likelihood method
Variance(X) + Variance(Y) - 2*covariance(XY)
Price/return tends to run towards a long - run level
Choose parameters that maximize the likelihood of what observations occurring
Statement of the error or precision of an estimate

24. Two drawbacks of moving average series
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
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We reject a hypothesis that is actually true

25. Panel data (longitudinal or micropanel)
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Has heavy tails
Special type of pooled data in which the cross sectional unit is surveyed over time

26. Unbiased
Mean of sampling distribution is the population mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Model dependent - Options with the same underlying assets may trade at different volatilities
Independently and Identically Distributed

27. Variance of X+Y assuming dependence
Contains variables not explicit in model - Accounts for randomness
We accept a hypothesis that should have been rejected
Variance(x) + Variance(Y) + 2*covariance(XY)
(a^2)(variance(x)

28. Lognormal
Probability that the random variables take on certain values simultaneously
Variance(y)/n = variance of sample Y
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

29. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
P(Z>t)

30. Statistical (or empirical) model
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Yi = B0 + B1Xi + ui
Rxy = Sxy/(Sx*Sy)

31. Central Limit Theorem
Does not depend on a prior event or information
Rxy = Sxy/(Sx*Sy)
For n>30 - sample mean is approximately normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

32. K - th moment
Summation((xi - mean)^k)/n
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

33. Exponential 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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence level

34. Confidence interval (from t)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
When one regressor is a perfect linear function of the other regressors
Sample mean +/ - t*(stddev(s)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

35. Adjusted R^2
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
Expected value of the sample mean is the population mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Based on an equation - P(A) = # of A/total outcomes

36. Normal distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Probability that the random variables take on certain values simultaneously
Attempts to sample along more important paths
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

37. WLS
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
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

38. Exact significance level
If variance of the conditional distribution of u(i) is not constant
P - value
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

39. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
E(mean) = mean
Contains variables not explicit in model - Accounts for randomness
When one regressor is a perfect linear function of the other regressors

40. Variance - covariance approach for VaR of a portfolio
Only requires two parameters = mean and variance
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)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

41. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
For n>30 - sample mean is approximately normal
Mean of sampling distribution is the population mean
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Bernouli Distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
When the sample size is large - the uncertainty about the value of the sample is very small
Distribution with only two possible outcomes

44. Type I error
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We reject a hypothesis that is actually true
Based on a dataset

45. Binomial distribution
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!)
Variance(x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We reject a hypothesis that is actually true

46. Discrete random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

47. Importance sampling technique
Average return across assets on a given day
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to sample along more important paths
Yi = B0 + B1Xi + ui

48. Covariance
Confidence level
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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

49. LFHS
Random walk (usually acceptable) - Constant volatility (unlikely)
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
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
Low Frequency - High Severity events

50. Test for unbiasedness
i = ln(Si/Si - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
E(mean) = mean
Low Frequency - High Severity events