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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. Marginal unconditional probability function
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Does not depend on a prior event or information
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Statement of the error or precision of an estimate

2. Empirical frequency
Probability that the random variables take on certain values simultaneously
Based on a dataset
Easy to manipulate
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)

3. Type II Error
We accept a hypothesis that should have been rejected
Sample mean will near the population mean as the sample size increases
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Peaks over threshold - Collects dataset in excess of some threshold

4. Variance of X+Y
Var(X) + Var(Y)
Easy to manipulate
P - value
i = ln(Si/Si - 1)

5. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
(a^2)(variance(x)

6. Adjusted R^2
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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

7. Variance of sampling distribution of means when n<N
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

8. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval

9. Reliability
Peaks over threshold - Collects dataset in excess of some threshold
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
Statement of the error or precision of an estimate
Yi = B0 + B1Xi + ui

10. Consistent
Random walk (usually acceptable) - Constant volatility (unlikely)
When the sample size is large - the uncertainty about the value of the sample is very small
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
If variance of the conditional distribution of u(i) is not constant

11. Multivariate probability
We accept a hypothesis that should have been rejected
More than one random variable
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Low Frequency - High Severity events

12. Variance of sample mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Variance(y)/n = variance of sample Y
Attempts to sample along more important paths

13. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

14. Sample variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Price/return tends to run towards a long - run level
Random walk (usually acceptable) - Constant volatility (unlikely)

15. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
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

16. Two requirements of OVB
We accept a hypothesis that should have been rejected
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P - value

17. Priori (classical) probability
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Based on an equation - P(A) = # of A/total outcomes
Attempts to sample along more important paths
Confidence level

18. Mean reversion in variance
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
Variance reverts to a long run level
Sample variance = (1/(k - 1))Summation(Yi - mean)^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

19. Pooled data
Probability that the random variables take on certain values simultaneously
Variance(X) + Variance(Y) - 2*covariance(XY)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Average return across assets on a given day

20. K - th moment
Concerned with a single random variable (ex. Roll of a die)
Summation((xi - mean)^k)/n
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

21. Confidence ellipse
When one regressor is a perfect linear function of the other regressors
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Concerned with a single random variable (ex. Roll of a die)
Confidence set for two coefficients - two dimensional analog for the confidence interval

22. Sample correlation
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
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)
Rxy = Sxy/(Sx*Sy)

23. Exact significance level
More than one random variable
P - value
Does not depend on a prior event or information
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

24. Mean reversion
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

25. Statistical (or empirical) model
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
When the sample size is large - the uncertainty about the value of the sample is very small
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Yi = B0 + B1Xi + ui

26. Econometrics
For n>30 - sample mean is approximately normal
Application of mathematical statistics to economic data to lend empirical support to models
When the sample size is large - the uncertainty about the value of the sample is very small
Easy to manipulate

27. Significance =1
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
Confidence level
Application of mathematical statistics to economic data to lend empirical support to models
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

28. 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
E(mean) = mean
Variance(x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

29. Bernouli Distribution
(a^2)(variance(x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Distribution with only two possible outcomes
95% = 1.65 99% = 2.33 For one - tailed tests

30. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Independently and Identically Distributed
Average return across assets on a given day

31. Time series data
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
SSR
Returns over time for an individual asset

32. Variance - covariance approach for VaR of a portfolio
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Returns over time for an individual asset

33. Standard error for Monte Carlo replications
Statement of the error or precision of an estimate
Confidence level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

34. Efficiency
Does not depend on a prior event or information
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Among all unbiased estimators - estimator with the smallest variance is efficient

35. Sample mean
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Expected value of the sample mean is the population mean
Returns over time for an individual asset
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

36. WLS
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Sample mean will near the population mean as the sample size increases
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

37. Stochastic error term
Peaks over threshold - Collects dataset in excess of some threshold
Contains variables not explicit in model - Accounts for randomness
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
More than one random variable

38. Variance of X+Y assuming dependence
Confidence level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x) + Variance(Y) + 2*covariance(XY)

39. T distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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
Nonlinearity
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

40. Biggest (and only real) drawback of GARCH mode
Probability that the random variables take on certain values simultaneously
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
Confidence level
Nonlinearity

41. Monte Carlo Simulations
Peaks over threshold - Collects dataset in excess of some threshold
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sampling distribution of sample means tend to be normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

42. Multivariate Density Estimation (MDE)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

43. Continuous representation of the GBM
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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)

44. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Independently and Identically Distributed
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Does not depend on a prior event or information

45. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

46. Joint probability functions
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
Probability that the random variables take on certain values simultaneously
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

47. Two ways to calculate historical volatility
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Probability that the random variables take on certain values simultaneously

48. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Nonlinearity
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Use historical simulation approach but use the EWMA weighting system

49. Cross - sectional
Average return across assets on a given day
Returns over time for an individual asset
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Expected value of the sample mean is the population mean

50. Binomial distribution equations for mean variance and std dev
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Var(X) + Var(Y)
Choose parameters that maximize the likelihood of what observations occurring
Mean = np - Variance = npq - Std dev = sqrt(npq)