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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. Gamma distribution
Attempts to sample along more important paths
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Concerned with a single random variable (ex. Roll of a die)

2. Variance of X+b
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(x)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sampling distribution of sample means tend to be normal

3. Simulation models
Confidence level
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
Choose parameters that maximize the likelihood of what observations occurring
We reject a hypothesis that is actually true

4. Variance of aX
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
(a^2)(variance(x)
i = ln(Si/Si - 1)

5. Covariance calculations using weight sums (lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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)

6. Difference between population and sample variance
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Population denominator = n - Sample denominator = n - 1
Sample mean +/ - t*(stddev(s)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

7. GARCH
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Random walk (usually acceptable) - Constant volatility (unlikely)
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)

8. Multivariate probability
Model dependent - Options with the same underlying assets may trade at different volatilities
Rxy = Sxy/(Sx*Sy)
Confidence level
More than one random variable

9. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Easy to manipulate
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

10. Least squares estimator(m)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

11. Standard normal distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
95% = 1.65 99% = 2.33 For one - tailed tests
Transformed to a unit variable - Mean = 0 Variance = 1
Has heavy tails

12. Central Limit Theorem(CLT)
If variance of the conditional distribution of u(i) is not constant
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Sampling distribution of sample means tend to be normal

13. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Distribution with only two possible outcomes
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

14. Single variable (univariate) probability
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We reject a hypothesis that is actually true
Concerned with a single random variable (ex. Roll of a die)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

15. Standard error
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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!)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

16. Sample correlation
Rxy = Sxy/(Sx*Sy)
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

17. Discrete random variable
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

18. Variance of sampling distribution of means when n<N
Returns over time for a combination of assets (combination of time series and cross - sectional data)
i = ln(Si/Si - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Law of Large Numbers
Population denominator = n - Sample denominator = n - 1
Price/return tends to run towards a long - run level
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
Sample mean will near the population mean as the sample size increases

20. SER
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)) + (b^2)(variance(y))
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
We reject a hypothesis that is actually true

21. Extreme Value Theory
Transformed to a unit variable - Mean = 0 Variance = 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x)

22. Bootstrap method
Concerned with a single random variable (ex. Roll of a die)
Independently and Identically Distributed
Contains variables not explicit in model - Accounts for randomness
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

23. Reliability
Rxy = Sxy/(Sx*Sy)
Based on a dataset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Statement of the error or precision of an estimate

24. Poisson distribution equations for mean variance and std deviation
Least absolute deviations estimator - used when extreme outliers are not uncommon
Distribution with only two possible outcomes
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Easy to manipulate

25. Normal distribution
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

26. GPD
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)
Summation((xi - mean)^k)/n
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

27. Discrete representation of the GBM
Least absolute deviations estimator - used when extreme outliers are not uncommon
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When one regressor is a perfect linear function of the other regressors
Peaks over threshold - Collects dataset in excess of some threshold

28. T distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

29. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Application of mathematical statistics to economic data to lend empirical support to models
Var(X) + Var(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

30. K - th moment
Summation((xi - mean)^k)/n
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Transformed to a unit variable - Mean = 0 Variance = 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

31. Priori (classical) probability
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
Based on an equation - P(A) = # of A/total outcomes
i = ln(Si/Si - 1)
Application of mathematical statistics to economic data to lend empirical support to models

32. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
More than one random variable
Normal - Student's T - Chi - square - F distribution

33. Adjusted R^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P(Z>t)
Price/return tends to run towards a long - run level
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

34. Cholesky factorization (decomposition)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
If variance of the conditional distribution of u(i) is not constant
Variance = (1/m) summation(u<n - i>^2)

35. Test for unbiasedness
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models

36. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

37. Persistence
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Returns over time for an individual asset
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

38. Overall F - statistic
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
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

39. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Nonlinearity
Independently and Identically Distributed
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

40. Significance =1
If variance of the conditional distribution of u(i) is not constant
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level
P - value

41. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E(mean) = mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Independently and Identically Distributed

42. Cross - sectional
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

43. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Contains variables not explicit in model - Accounts for randomness

44. Unconditional vs conditional distributions
Returns over time for an individual asset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(mean) = mean
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

45. Empirical frequency
P - value
When one regressor is a perfect linear function of the other regressors
Based on a dataset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

46. Logistic distribution
Has heavy tails
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance reverts to a long run level
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!)

47. Sample mean
Expected value of the sample mean is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Statement of the error or precision of an estimate
Probability that the random variables take on certain values simultaneously

48. Variance of aX + bY
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)) + (b^2)(variance(y))

49. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Summation((xi - mean)^k)/n
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Special type of pooled data in which the cross sectional unit is surveyed over time

50. Joint probability functions
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(y)/n = variance of sample Y
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Probability that the random variables take on certain values simultaneously