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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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1. Multivariate probability
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
(a^2)(variance(x)) + (b^2)(variance(y))
More than one random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

2. WLS
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

3. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

4. GARCH
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
We reject a hypothesis that is actually true
Mean = np - Variance = npq - Std dev = sqrt(npq)
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)

5. 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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(y)/n = variance of sample Y

6. Unstable return distribution
Expected value of the sample mean is the population mean
Statement of the error or precision of an estimate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

7. Confidence interval (from t)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean +/ - t*(stddev(s)/sqrt(n))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

8. Regime - switching volatility model
Probability that the random variables take on certain values simultaneously
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P - value
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

9. Sample covariance
We reject a hypothesis that is actually true
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

10. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

11. Stochastic error term
Sampling distribution of sample means tend to be normal
Contains variables not explicit in model - Accounts for randomness
95% = 1.65 99% = 2.33 For one - tailed tests
P - value

12. Four sampling distributions

13. Cross - sectional
Least absolute deviations estimator - used when extreme outliers are not uncommon
Average return across assets on a given day
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

14. Unbiased
Mean of sampling distribution is the population mean
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

15. Critical z values
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(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)
95% = 1.65 99% = 2.33 For one - tailed tests

16. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Independently and Identically Distributed
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

17. Significance =1
Confidence level
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
Among all unbiased estimators - estimator with the smallest variance is efficient

18. Sample variance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

19. Result of combination of two normal with same means
Variance(X) + Variance(Y) - 2*covariance(XY)
Price/return tends to run towards a long - run level
Combine to form distribution with leptokurtosis (heavy tails)
Variance(y)/n = variance of sample Y

20. Cholesky factorization (decomposition)
P - value
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

21. Discrete representation of the GBM
When one regressor is a perfect linear function of the other regressors
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Yi = B0 + B1Xi + ui
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

22. Implications of homoscedasticity
Random walk (usually acceptable) - Constant volatility (unlikely)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

23. Poisson Distribution
Based on a dataset
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Based on an equation - P(A) = # of A/total outcomes

24. Central Limit Theorem
For n>30 - sample mean is approximately normal
Variance(y)/n = variance of sample Y
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

25. F distribution
Peaks over threshold - Collects dataset in excess of some threshold
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

26. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Confidence level
Sample mean will near the population mean as the sample size increases

27. Sample correlation
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Rxy = Sxy/(Sx*Sy)

28. Central Limit Theorem(CLT)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sampling distribution of sample means tend to be normal
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

29. Monte Carlo Simulations
Peaks over threshold - Collects dataset in excess of some threshold
Confidence level
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

30. Standard variable for non - normal distributions
Expected value of the sample mean is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P(X=x - Y=y) = P(X=x) * P(Y=y)

31. SER
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Peaks over threshold - Collects dataset in excess of some threshold

32. Tractable
Easy to manipulate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent
Least absolute deviations estimator - used when extreme outliers are not uncommon

33. Chi - squared distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Distribution with only two possible outcomes
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

34. Adjusted R^2
Variance = (1/m) summation(u<n - i>^2)
Peaks over threshold - Collects dataset in excess of some threshold
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
E(mean) = mean

35. Efficiency
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean will near the population mean as the sample size increases
Among all unbiased estimators - estimator with the smallest variance is efficient

36. Continuously compounded return equation
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities

37. Variance of X+Y
Var(X) + Var(Y)
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
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. Type II Error
Variance(x)
We accept a hypothesis that should have been rejected
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
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

39. Statistical (or empirical) model
Normal - Student's T - Chi - square - F distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Yi = B0 + B1Xi + ui
Nonlinearity

40. Sample mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Expected value of the sample mean is the population mean
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

41. Beta distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to sample along more important paths
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Variance of X+Y assuming dependence
For n>30 - sample mean is approximately normal
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Contains variables not explicit in model - Accounts for randomness
Variance(x) + Variance(Y) + 2*covariance(XY)

43. EWMA
P - value
Use historical simulation approach but use the EWMA weighting system
Easy to manipulate
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

44. Perfect multicollinearity
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When one regressor is a perfect linear function of the other regressors
Yi = B0 + B1Xi + ui
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

45. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Use historical simulation approach but use the EWMA weighting system

46. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Rxy = Sxy/(Sx*Sy)
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
Population denominator = n - Sample denominator = n - 1

47. Direction of OVB
P - value
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

48. Priori (classical) probability
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

49. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
SSR
Based on a dataset
Random walk (usually acceptable) - Constant volatility (unlikely)

50. Kurtosis
If variance of the conditional distribution of u(i) is not constant
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
We reject a hypothesis that is actually true
Normal - Student's T - Chi - square - F distribution