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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. Least squares estimator(m)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Statement of the error or precision of an estimate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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

2. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
Population denominator = n - Sample denominator = n - 1
For n>30 - sample mean is approximately normal
Low Frequency - High Severity events

3. What does the OLS minimize?
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
SSR

4. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

5. Mean reversion in variance
When the sample size is large - the uncertainty about the value of the sample is very small
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!)
Regression can be non - linear in variables but must be linear in parameters

6. Consistent
Sampling distribution of sample means tend to be normal
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When the sample size is large - the uncertainty about the value of the sample is very small
Choose parameters that maximize the likelihood of what observations occurring

7. Single variable (univariate) probability
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Concerned with a single random variable (ex. Roll of a die)
Rxy = Sxy/(Sx*Sy)

8. Persistence
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Mean of sampling distribution is the population mean

9. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Sample mean will near the population mean as the sample size increases
For n>30 - sample mean is approximately normal
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

10. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Random walk (usually acceptable) - Constant volatility (unlikely)

11. Standard variable for non - normal distributions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Z = (Y - meany)/(stddev(y)/sqrt(n))

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

13. Logistic distribution
More than one random variable
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
(a^2)(variance(x)) + (b^2)(variance(y))
Has heavy tails

14. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
For n>30 - sample mean is approximately normal
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
Normal - Student's T - Chi - square - F distribution

15. Cholesky factorization (decomposition)
Variance(y)/n = variance of sample Y
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
P - value
Easy to manipulate

16. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(X) + Variance(Y) - 2*covariance(XY)

17. Potential reasons for fat tails in return distributions
Based on a dataset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Confidence level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

18. Two assumptions of square root rule
Does not depend on a prior event or information
Sample mean will near the population mean as the sample size increases
Random walk (usually acceptable) - Constant volatility (unlikely)
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

19. Limitations of R^2 (what an increase doesn't necessarily imply)

20. Simulation models
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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)

21. Test for unbiasedness
E(mean) = mean
Confidence level
Only requires two parameters = mean and variance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

22. Pooled data
Sample mean will near the population mean as the sample size increases
Does not depend on a prior event or information
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Returns over time for a combination of assets (combination of time series and cross - sectional data)

23. EWMA
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When one regressor is a perfect linear function of the other regressors
Concerned with a single random variable (ex. Roll of a die)

24. K - th moment
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
Summation((xi - mean)^k)/n
More than one random variable
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

25. Tractable
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Easy to manipulate
Mean of sampling distribution is the population mean

26. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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(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

27. Skewness
Returns over time for an individual asset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

28. Standard error for Monte Carlo replications
Population denominator = n - Sample denominator = n - 1
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Probability that the random variables take on certain values simultaneously
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

29. Continuously compounded return equation
i = ln(Si/Si - 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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

30. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

31. GARCH
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)
Based on an equation - P(A) = # of A/total outcomes
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
Attempts to sample along more important paths

32. Chi - squared distribution
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
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
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

33. F distribution
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(y)/n = variance of sample Y
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

34. Direction of OVB
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

35. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
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

36. Type I error
Sample mean +/ - t*(stddev(s)/sqrt(n))
We reject a hypothesis that is actually true
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

37. Implications of homoscedasticity
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

38. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Independently and Identically Distributed
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

39. Key properties of linear regression
Variance = (1/m) summation(u<n - i>^2)
Does not depend on a prior event or information
Regression can be non - linear in variables but must be linear in parameters
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

40. Beta distribution
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(y)/n = variance of sample Y
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

41. Variance(discrete)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

42. Multivariate Density Estimation (MDE)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sampling distribution of sample means tend to be normal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Attempts to sample along more important paths

43. Variance of X+Y
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small
Combine to form distribution with leptokurtosis (heavy tails)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

44. Econometrics
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
E(mean) = mean
Application of mathematical statistics to economic data to lend empirical support to models

45. Variance of sample mean
Variance(y)/n = variance of sample Y
SSR
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
Mean of sampling distribution is the population mean

46. Test for statistical independence
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Use historical simulation approach but use the EWMA weighting system
P(X=x - Y=y) = P(X=x) * P(Y=y)

47. Type II Error
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected
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
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

48. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample mean +/ - t*(stddev(s)/sqrt(n))
E(mean) = mean
Population denominator = n - Sample denominator = n - 1

49. Standard error
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
i = ln(Si/Si - 1)

50. Lognormal
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Z = (Y - meany)/(stddev(y)/sqrt(n))