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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. Sample mean
Application of mathematical statistics to economic data to lend empirical support to models
Expected value of the sample mean is the population mean
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

2. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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!)

3. Difference between population and sample variance
Probability that the random variables take on certain values simultaneously
Population denominator = n - Sample denominator = n - 1
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Price/return tends to run towards a long - run level

4. What does the OLS minimize?
Only requires two parameters = mean and variance
SSR
P(Z>t)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

5. Heteroskedastic
Among all unbiased estimators - estimator with the smallest variance is efficient
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
If variance of the conditional distribution of u(i) is not constant

6. Covariance calculations using weight sums (lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(mean) = mean
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

7. Two assumptions of square root rule
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)

8. Priori (classical) probability
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on an equation - P(A) = # of A/total outcomes
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

10. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Yi = B0 + B1Xi + ui
Returns over time for an individual asset
Variance = (1/m) summation(u<n - i>^2)

11. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Sample mean will near the population mean as the sample size increases
Based on an equation - P(A) = # of A/total outcomes

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

13. Bernouli Distribution
Distribution with only two possible outcomes
Random walk (usually acceptable) - Constant volatility (unlikely)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)

14. GEV
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Use historical simulation approach but use the EWMA weighting system
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance = (1/m) summation(u<n - i>^2)

15. Significance =1
Z = (Y - meany)/(stddev(y)/sqrt(n))
Confidence level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

16. Block maxima
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
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
Does not depend on a prior event or information
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

17. Non - parametric vs parametric calculation of VaR
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

18. Variance of aX + bY
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)) + (b^2)(variance(y))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

19. Implications of homoscedasticity
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Contains variables not explicit in model - Accounts for randomness
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
P(Z>t)

20. Multivariate Density Estimation (MDE)
Attempts to sample along more important paths
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

21. Confidence interval for sample mean
Nonlinearity
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Returns over time for an individual asset

22. Marginal unconditional probability function
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean +/ - t*(stddev(s)/sqrt(n))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Does not depend on a prior event or information

23. Beta distribution
We accept a hypothesis that should have been rejected
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Least absolute deviations estimator - used when extreme outliers are not uncommon

24. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
We accept a hypothesis that should have been rejected

25. Inverse transform method
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
Only requires two parameters = mean and variance
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

26. Square root rule
Summation((xi - mean)^k)/n
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)
(a^2)(variance(x)) + (b^2)(variance(y))

27. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Expected value of the sample mean is the population mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

28. Econometrics
Sampling distribution of sample means tend to be normal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Application of mathematical statistics to economic data to lend empirical support to models
E(XY) - E(X)E(Y)

29. 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
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

30. Type I error
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Combine to form distribution with leptokurtosis (heavy tails)
We reject a hypothesis that is actually true

31. Variance of X+Y
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Var(X) + Var(Y)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))

32. Key properties of linear regression
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Does not depend on a prior event or information
Distribution with only two possible outcomes
Regression can be non - linear in variables but must be linear in parameters

33. Variance of sampling distribution of means when n<N
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
(a^2)(variance(x)
Use historical simulation approach but use the EWMA weighting system
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

34. Test for unbiasedness
E(mean) = mean
(a^2)(variance(x)
Least absolute deviations estimator - used when extreme outliers are not uncommon
95% = 1.65 99% = 2.33 For one - tailed tests

35. Multivariate probability
More than one random variable
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Regression can be non - linear in variables but must be linear in parameters
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

36. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Regression can be non - linear in variables but must be linear in parameters
Combine to form distribution with leptokurtosis (heavy tails)

37. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on an equation - P(A) = # of A/total outcomes
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(y)/n = variance of sample Y

38. Simulating for VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

39. Control variates technique
We reject a hypothesis that is actually true
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

40. Extreme Value Theory
Population denominator = n - Sample denominator = n - 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

41. Two drawbacks of moving average series
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Statement of the error or precision of an estimate
Sample mean will near the population mean as the sample size increases

42. Stochastic error term
P - value
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
Contains variables not explicit in model - Accounts for randomness
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

43. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Regression can be non - linear in variables but must be linear in parameters
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

44. Two requirements of OVB
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

45. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Rxy = Sxy/(Sx*Sy)

46. Unstable return distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Does not depend on a prior event or information
Among all unbiased estimators - estimator with the smallest variance is efficient

47. Kurtosis
Sampling distribution of sample means tend to be normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Nonlinearity

48. Central Limit Theorem
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)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
For n>30 - sample mean is approximately normal

49. Tractable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Distribution with only two possible outcomes
Easy to manipulate

50. EWMA
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)