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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. Statistical (or empirical) model
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[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Yi = B0 + B1Xi + ui
Does not depend on a prior event or information

2. Non - parametric vs parametric calculation of VaR
Z = (Y - meany)/(stddev(y)/sqrt(n))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
SSR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

3. Two requirements of OVB
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance reverts to a long run level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

4. Unconditional vs conditional distributions
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Rxy = Sxy/(Sx*Sy)

5. Type I error
Least absolute deviations estimator - used when extreme outliers are not uncommon
Expected value of the sample mean is the population mean
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
We reject a hypothesis that is actually true

6. 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
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
Choose parameters that maximize the likelihood of what observations occurring
We accept a hypothesis that should have been rejected

7. Confidence interval for sample mean
More than one random variable
When one regressor is a perfect linear function of the other regressors
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

8. Variance of X+Y assuming dependence
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(x) + Variance(Y) + 2*covariance(XY)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

9. Overall F - statistic
Attempts to sample along more important paths
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
Rxy = Sxy/(Sx*Sy)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

10. K - th moment
Statement of the error or precision of an estimate
Based on an equation - P(A) = # of A/total outcomes
Summation((xi - mean)^k)/n
Application of mathematical statistics to economic data to lend empirical support to models

11. Two ways to calculate historical volatility
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
We reject a hypothesis that is actually true
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

12. GARCH
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information

13. Test for unbiasedness
Variance(x) + Variance(Y) + 2*covariance(XY)
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

14. Standard error for Monte Carlo replications
P - value
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance = (1/m) summation(u<n - i>^2)
Variance(x)

15. Confidence ellipse
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Summation((xi - mean)^k)/n

16. Two drawbacks of moving average series
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Yi = B0 + B1Xi + ui
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

17. Shortcomings of implied volatility
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

18. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Random walk (usually acceptable) - Constant volatility (unlikely)
Contains variables not explicit in model - Accounts for randomness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

19. Block maxima
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

20. Variance of X+Y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Based on an equation - P(A) = # of A/total outcomes
Concerned with a single random variable (ex. Roll of a die)
Var(X) + Var(Y)

21. Result of combination of two normal with same means
Nonlinearity
Distribution with only two possible outcomes
Combine to form distribution with leptokurtosis (heavy tails)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

22. Central Limit Theorem
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sampling distribution of sample means tend to be normal
For n>30 - sample mean is approximately normal

23. i.i.d.
Independently and Identically Distributed
Variance(X) + Variance(Y) - 2*covariance(XY)
i = ln(Si/Si - 1)
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

24. Stochastic error term
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Based on a dataset
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Contains variables not explicit in model - Accounts for randomness

25. Extreme Value Theory
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Special type of pooled data in which the cross sectional unit is surveyed over time

26. Covariance
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
If variance of the conditional distribution of u(i) is not constant
E(XY) - E(X)E(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

27. Economical(elegant)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Only requires two parameters = mean and variance
For n>30 - sample mean is approximately normal

28. Sample variance
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^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
Confidence set for two coefficients - two dimensional analog for the confidence interval

29. Standard normal distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Transformed to a unit variable - Mean = 0 Variance = 1
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

30. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Returns over time for an individual asset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

31. Importance sampling technique
Attempts to sample along more important paths
Confidence level
We accept a hypothesis that should have been rejected
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

32. Multivariate probability
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
(a^2)(variance(x)) + (b^2)(variance(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
More than one random variable

33. Persistence
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Random walk (usually acceptable) - Constant volatility (unlikely)

34. Control variates technique
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

35. Econometrics
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Application of mathematical statistics to economic data to lend empirical support to models
Sampling distribution of sample means tend to be normal
i = ln(Si/Si - 1)

36. Beta distribution
Contains variables not explicit in model - Accounts for randomness
Regression can be non - linear in variables but must be linear in parameters
Variance(x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

37. Binomial distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Normal - Student's T - Chi - square - F distribution
Only requires two parameters = mean and variance
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!)

38. Chi - squared distribution
i = ln(Si/Si - 1)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

39. Priori (classical) probability
Mean of sampling distribution is the population mean
Based on an equation - P(A) = # of A/total outcomes
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Special type of pooled data in which the cross sectional unit is surveyed over time

40. Single variable (univariate) probability
Variance(x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Concerned with a single random variable (ex. Roll of a die)
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

41. Gamma distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
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

42. ESS
SSR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

43. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

44. Skewness
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When one regressor is a perfect linear function of the other regressors

45. Empirical frequency
Based on a dataset
Contains variables not explicit in model - Accounts for randomness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Only requires two parameters = mean and variance

46. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Normal - Student's T - Chi - square - F distribution
If variance of the conditional distribution of u(i) is not constant
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

47. Two assumptions of square root rule
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)
Choose parameters that maximize the likelihood of what observations occurring
E(XY) - E(X)E(Y)

48. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Rxy = Sxy/(Sx*Sy)
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
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

49. Marginal unconditional probability function
Does not depend on a prior event or information
Variance(x)
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

50. Hazard rate of exponentially distributed random variable
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance reverts to a long run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)