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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. Stochastic error term
Distribution with only two possible outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities
Contains variables not explicit in model - Accounts for randomness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

2. Variance of X+b
Returns over time for an individual asset
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(y)/n = variance of sample Y
Variance(x)

3. Test for unbiasedness
If variance of the conditional distribution of u(i) is not constant
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(mean) = mean

4. Efficiency
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
Among all unbiased estimators - estimator with the smallest variance is efficient
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
P(X=x - Y=y) = P(X=x) * P(Y=y)

5. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
P - value
Does not depend on a prior event or information
More than one random variable

6. SER
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
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)
i = ln(Si/Si - 1)

7. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Expected value of the sample mean is the population mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We reject a hypothesis that is actually true

8. Lognormal
We reject a hypothesis that is actually true
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
95% = 1.65 99% = 2.33 For one - tailed tests
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

9. Variance(discrete)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P(Z>t)

10. Mean reversion in variance
Variance reverts to a long run level
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

11. Implications of homoscedasticity
Easy to manipulate
Variance(sample y) = (variance(y)/n)*(N - n/N - 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

12. Simulating for VaR
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

13. Standard error for Monte Carlo replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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)
We reject a hypothesis that is actually true

14. Sample mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Expected value of the sample mean is the population mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Z = (Y - meany)/(stddev(y)/sqrt(n))

15. Monte Carlo Simulations
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

16. Variance of sample mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Variance(y)/n = variance of sample Y

17. Marginal unconditional probability function
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Summation((xi - mean)^k)/n

18. F distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

19. Exact significance level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Independently and Identically Distributed
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
P - value

20. LFHS
Has heavy tails
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!)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Low Frequency - High Severity events

21. What does the OLS minimize?
SSR
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

22. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Nonlinearity
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

23. Variance of aX + bY
Model dependent - Options with the same underlying assets may trade at different volatilities
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Contains variables not explicit in model - Accounts for randomness
(a^2)(variance(x)) + (b^2)(variance(y))

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

25. Binomial distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Var(X) + Var(Y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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!)

26. Expected future variance rate (t periods forward)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

27. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Least absolute deviations estimator - used when extreme outliers are not uncommon

28. Bernouli Distribution
Based on an equation - P(A) = # of A/total outcomes
Confidence level
Normal - Student's T - Chi - square - F distribution
Distribution with only two possible outcomes

29. Variance of X+Y assuming dependence
E(XY) - E(X)E(Y)
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x) + Variance(Y) + 2*covariance(XY)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

30. Sample covariance
Low Frequency - High Severity events
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

31. Non - parametric vs parametric calculation of VaR
When the sample size is large - the uncertainty about the value of the sample is very small
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters

32. Multivariate probability
More than one random variable
Attempts to sample along more important paths
Confidence level
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

33. Difference between population and sample variance
Regression can be non - linear in variables but must be linear in parameters
Sampling distribution of sample means tend to be normal
Population denominator = n - Sample denominator = n - 1
Nonlinearity

34. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

35. Importance sampling technique
Attempts to sample along more important paths
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

36. Poisson distribution equations for mean variance and std deviation
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Confidence level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Choose parameters that maximize the likelihood of what observations occurring

37. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
When one regressor is a perfect linear function of the other regressors
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Yi = B0 + B1Xi + ui

38. Normal distribution
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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

39. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Price/return tends to run towards a long - run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

40. R^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
P(Z>t)

41. LAD
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

42. Empirical frequency
Variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset
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. i.i.d.
We reject a hypothesis that is actually true
Independently and Identically Distributed
Var(X) + Var(Y)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

44. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Choose parameters that maximize the likelihood of what observations occurring
Variance(x) + Variance(Y) + 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests

45. Overall F - statistic
Returns over time for an individual asset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(x)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean of sampling distribution is the population mean
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability that the random variables take on certain values simultaneously

47. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Does not depend on a prior event or information
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
95% = 1.65 99% = 2.33 For one - tailed tests

48. Continuous random variable
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
When the sample size is large - the uncertainty about the value of the sample is very small
Sample mean will near the population mean as the sample size increases

49. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Combine to form distribution with leptokurtosis (heavy tails)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample mean will near the population mean as the sample size increases

50. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Only requires two parameters = mean and variance
We accept a hypothesis that should have been rejected
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx