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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. Confidence ellipse
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Has heavy tails
Confidence set for two coefficients - two dimensional analog for the confidence interval

2. Mean(expected value)
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
P(X=x - Y=y) = P(X=x) * P(Y=y)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

3. Sample mean
Based on an equation - P(A) = # of A/total outcomes
Expected value of the sample mean is the population mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Low Frequency - High Severity events

4. GPD
Summation((xi - mean)^k)/n
Independently and Identically Distributed
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

5. Poisson distribution equations for mean variance and std deviation
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

6. Reliability
Least absolute deviations estimator - used when extreme outliers are not uncommon
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Statement of the error or precision of an estimate
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!)

7. Sample variance
Distribution with only two possible outcomes
Variance = (1/m) summation(u<n - i>^2)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
We accept a hypothesis that should have been rejected

8. Sample correlation
Rxy = Sxy/(Sx*Sy)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Choose parameters that maximize the likelihood of what observations occurring

9. GEV
Attempts to sample along more important paths
Contains variables not explicit in model - Accounts for randomness
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

10. Simplified standard (un - weighted) variance
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = (1/m) summation(u<n - i>^2)
i = ln(Si/Si - 1)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

11. Control variates technique
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
Only requires two parameters = mean and variance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

12. Unbiased
Mean of sampling distribution is the population mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

13. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

14. Importance sampling technique
Attempts to sample along more important paths
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Probability that the random variables take on certain values simultaneously

15. Antithetic variable technique
P(X=x - Y=y) = P(X=x) * P(Y=y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Type II Error
Returns over time for a combination of assets (combination of time series and cross - sectional data)
We accept a hypothesis that should have been rejected
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
Independently and Identically Distributed

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

18. Monte Carlo Simulations
Has heavy tails
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))

19. Overall F - statistic
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
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)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

20. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Does not depend on a prior event or information
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

21. Non - parametric vs parametric calculation of VaR
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
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. Bernouli Distribution
Distribution with only two possible outcomes
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

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

24. P - value
When one regressor is a perfect linear function of the other regressors
Contains variables not explicit in model - Accounts for randomness
P(Z>t)
Normal - Student's T - Chi - square - F distribution

25. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on an equation - P(A) = # of A/total outcomes
Contains variables not explicit in model - Accounts for randomness
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

26. Standard error for Monte Carlo replications
(a^2)(variance(x)) + (b^2)(variance(y))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

27. Shortcomings of implied volatility
Variance = (1/m) summation(u<n - i>^2)
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)
Model dependent - Options with the same underlying assets may trade at different volatilities

28. Binomial distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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!)
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

29. Mean reversion in variance
Variance reverts to a long run level
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

30. BLUE
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

31. Regime - switching volatility model
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

32. Hazard rate of exponentially distributed random variable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Yi = B0 + B1Xi + ui
For n>30 - sample mean is approximately normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

33. Two requirements of OVB
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(X) + Variance(Y) - 2*covariance(XY)

34. ESS
Based on an equation - P(A) = # of A/total outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

35. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Application of mathematical statistics to economic data to lend empirical support to models
E(mean) = mean
Regression can be non - linear in variables but must be linear in parameters

36. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Yi = B0 + B1Xi + ui
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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!)

37. Single variable (univariate) probability
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
More than one random variable
Concerned with a single random variable (ex. Roll of a die)

38. Beta 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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

39. Poisson 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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
i = ln(Si/Si - 1)
Sample mean will near the population mean as the sample size increases

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

41. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Regression can be non - linear in variables but must be linear in parameters
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Summation((xi - mean)^k)/n
More than one random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

43. Type I error
Sample mean will near the population mean as the sample size increases
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
We reject a hypothesis that is actually true

44. GARCH
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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)
Var(X) + Var(Y)

45. Variance of sample mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(y)/n = variance of sample Y
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

46. Least squares estimator(m)
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)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Statement of the error or precision of an estimate

47. Panel data (longitudinal or micropanel)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Special type of pooled data in which the cross sectional unit is surveyed over time
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

48. Discrete representation of the GBM
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Easy to manipulate
Statement of the error or precision of an estimate
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. Gamma distribution
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
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Has heavy tails

50. Continuously compounded return equation
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)