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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. Lognormal
For n>30 - sample mean is approximately normal
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(x)
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

2. Standard error for Monte Carlo replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Expected value of the sample mean is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample mean +/ - t*(stddev(s)/sqrt(n))

3. Empirical frequency
Based on a dataset
Average return across assets on a given day
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(X) + Variance(Y) - 2*covariance(XY)

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

5. SER
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n

6. Mean reversion in asset dynamics
Nonlinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Price/return tends to run towards a long - run level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

7. Type II Error
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
We accept a hypothesis that should have been rejected
Sampling distribution of sample means tend to be normal
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

8. Single variable (univariate) probability
Distribution with only two possible outcomes
Concerned with a single random variable (ex. Roll of a die)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

9. Maximum likelihood method
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When one regressor is a perfect linear function of the other regressors
Choose parameters that maximize the likelihood of what observations occurring
Regression can be non - linear in variables but must be linear in parameters

10. Control variates technique
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Special type of pooled data in which the cross sectional unit is surveyed over time
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

11. Covariance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Confidence set for two coefficients - two dimensional analog for the confidence interval
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)

12. Key properties of linear regression
Transformed to a unit variable - Mean = 0 Variance = 1
Confidence level
Regression can be non - linear in variables but must be linear in parameters
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

13. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Application of mathematical statistics to economic data to lend empirical support to models
Special type of pooled data in which the cross sectional unit is surveyed over time

14. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Confidence level
Variance(x)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

15. GEV
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Low Frequency - High Severity events

16. Continuous random variable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

17. Time series data
Application of mathematical statistics to economic data to lend empirical support to models
P - value
Returns over time for an individual asset
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

18. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance = (1/m) summation(u<n - i>^2)

19. EWMA
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean +/ - t*(stddev(s)/sqrt(n))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

20. Historical std dev
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

21. Implied standard deviation for options
P(Z>t)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

22. Priori (classical) probability
Model dependent - Options with the same underlying assets may trade at different volatilities
Mean of sampling distribution is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Based on an equation - P(A) = # of A/total outcomes

23. Discrete representation of the GBM
Variance(X) + Variance(Y) - 2*covariance(XY)
Var(X) + Var(Y)
Rxy = Sxy/(Sx*Sy)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

24. Two assumptions of square root rule
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Random walk (usually acceptable) - Constant volatility (unlikely)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

25. Bernouli Distribution
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
E(XY) - E(X)E(Y)
Returns over time for an individual asset
Distribution with only two possible outcomes

26. Standard error
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
Attempts to sample along more important paths
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

27. Continuous representation of the GBM
Peaks over threshold - Collects dataset in excess of some threshold
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

28. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Use historical simulation approach but use the EWMA weighting system
Independently and Identically Distributed
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

29. Discrete random variable
Confidence level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Does not depend on a prior event or information
Use historical simulation approach but use the EWMA weighting system

30. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(Z>t)
Distribution with only two possible outcomes

31. Bootstrap method
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
When the sample size is large - the uncertainty about the value of the sample is very small

32. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Special type of pooled data in which the cross sectional unit is surveyed over time
Low Frequency - High Severity events

33. Chi - squared distribution
More than one random variable
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

34. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
If variance of the conditional distribution of u(i) is not constant
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Does not depend on a prior event or information

35. P - value
Peaks over threshold - Collects dataset in excess of some threshold
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P(Z>t)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

36. Marginal unconditional probability function
Least absolute deviations estimator - used when extreme outliers are not uncommon
Does not depend on a prior event or information
Expected value of the sample mean is the population mean
Variance(x)

37. Shortcomings of implied volatility
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
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

38. Logistic distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
P - value
Has heavy tails
i = ln(Si/Si - 1)

39. Two drawbacks of moving average series
Price/return tends to run towards a long - run level
Application of mathematical statistics to economic data to lend empirical support to models
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 an individual asset

40. Gamma distribution
Peaks over threshold - Collects dataset in excess of some threshold
Choose parameters that maximize the likelihood of what observations occurring
Has heavy tails
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

41. Variance of X+Y assuming dependence
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
Variance(x) + Variance(Y) + 2*covariance(XY)

42. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

43. Deterministic Simulation
E(XY) - E(X)E(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
Based on a dataset
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

44. Variance of X - Y assuming dependence
Variance = (1/m) summation(u<n - i>^2)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(X) + Variance(Y) - 2*covariance(XY)
P(Z>t)

45. Hazard rate of exponentially distributed random variable
When the sample size is large - the uncertainty about the value of the sample is very small
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

46. Joint probability functions
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Probability that the random variables take on certain values simultaneously

47. Difference between population and sample variance
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)
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes

48. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance reverts to a long run level
Choose parameters that maximize the likelihood of what observations occurring
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

49. Weibul distribution
Returns over time for an individual asset
P - value
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

50. Mean(expected value)
Returns over time for an individual asset
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))