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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. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
For n>30 - sample mean is approximately normal

2. Lognormal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Expected value of the sample mean is the population mean
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

3. Significance =1
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Confidence level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

4. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Attempts to sample along more important paths
Least absolute deviations estimator - used when extreme outliers are not uncommon

5. Result of combination of two normal with same means
Easy to manipulate
Based on an equation - P(A) = # of A/total outcomes
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Combine to form distribution with leptokurtosis (heavy tails)

6. Potential reasons for fat tails in return distributions
Distribution with only two possible outcomes
Variance(x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Normal - Student's T - Chi - square - F distribution

7. Consistent
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

8. Variance of weighted scheme
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Peaks over threshold - Collects dataset in excess of some threshold
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

9. Test for statistical independence
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(x) + Variance(Y) + 2*covariance(XY)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
(a^2)(variance(x)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(x) + Variance(Y) + 2*covariance(XY)

11. R^2
Contains variables not explicit in model - Accounts for randomness
We accept a hypothesis that should have been rejected
More than one random variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

12. Economical(elegant)
Nonlinearity
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Only requires two parameters = mean and variance
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

13. i.i.d.
Variance reverts to a long run level
Independently and Identically Distributed
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
We reject a hypothesis that is actually true

14. Covariance
E(XY) - E(X)E(Y)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(X) + Variance(Y) - 2*covariance(XY)
Price/return tends to run towards a long - run level

15. GARCH
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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Normal - Student's T - Chi - square - F distribution

16. GPD
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Normal - Student's T - Chi - square - F distribution
Price/return tends to run towards a long - run level
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

17. Central Limit Theorem
Application of mathematical statistics to economic data to lend empirical support to models
For n>30 - sample mean is approximately normal
P - value
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

18. Pooled data
Distribution with only two possible outcomes
Variance = (1/m) summation(u<n - i>^2)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(y)/n = variance of sample Y

19. Variance of aX
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
(a^2)(variance(x)

20. Unconditional vs conditional distributions
Based on a dataset
Statement of the error or precision of an estimate
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Confidence set for two coefficients - two dimensional analog for the confidence interval

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

22. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Transformed to a unit variable - Mean = 0 Variance = 1
Low Frequency - High Severity events
When the sample size is large - the uncertainty about the value of the sample is very small

23. Discrete representation of the GBM
Distribution with only two possible outcomes
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))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

24. Sample covariance
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(X) + Variance(Y) - 2*covariance(XY)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

25. Poisson distribution equations for mean variance and std deviation
We reject a hypothesis that is actually true
Variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

26. Cross - sectional
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Average return across assets on a given day
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Summation((xi - mean)^k)/n

27. ESS
Returns over time for an individual asset
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

28. Continuously compounded return equation
i = ln(Si/Si - 1)
Peaks over threshold - Collects dataset in excess of some threshold
Attempts to sample along more important paths
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

29. Variance of X+b
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)
Peaks over threshold - Collects dataset in excess of some threshold

30. Test for unbiasedness
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sampling distribution of sample means tend to be normal
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(mean) = mean

31. Poisson Distribution
E(mean) = mean
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)
Distribution with only two possible outcomes

32. Monte Carlo Simulations
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Confidence set for two coefficients - two dimensional analog for the confidence interval

33. Regime - switching volatility model
Sample mean will near the population mean as the sample size increases
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Choose parameters that maximize the likelihood of what observations occurring
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

34. Two requirements of OVB
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Average return across assets on a given day
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

35. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Var(X) + Var(Y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

36. LFHS
Low Frequency - High Severity events
Expected value of the sample mean is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
P(Z>t)

37. Variance of sample mean
P - value
Variance(y)/n = variance of sample Y
For n>30 - sample mean is approximately normal
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

38. Implications of homoscedasticity
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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

39. Expected future variance rate (t periods forward)
Returns over time for an individual asset
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Attempts to sample along more important paths
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

40. Law of Large Numbers
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Returns over time for a combination of assets (combination of time series and cross - sectional data)

41. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(y)/n = variance of sample Y
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
Application of mathematical statistics to economic data to lend empirical support to models

42. Exact significance level
Among all unbiased estimators - estimator with the smallest variance is efficient
P - value
Based on an equation - P(A) = # of A/total outcomes
E(mean) = mean

43. Discrete random variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
When one regressor is a perfect linear function of the other regressors
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

44. Confidence ellipse
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(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
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 set for two coefficients - two dimensional analog for the confidence interval

45. EWMA
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

46. Adjusted R^2
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Easy to manipulate
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. Heteroskedastic
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample mean will near the population mean as the sample size increases
If variance of the conditional distribution of u(i) is not constant
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

48. Sample correlation
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
95% = 1.65 99% = 2.33 For one - tailed tests
Rxy = Sxy/(Sx*Sy)

49. WLS
Confidence level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

50. Multivariate probability
Variance(X) + Variance(Y) - 2*covariance(XY)
Choose parameters that maximize the likelihood of what observations occurring
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
More than one random variable