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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. K - th moment
Summation((xi - mean)^k)/n
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance = (1/m) summation(u<n - i>^2)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

2. Two ways to calculate historical volatility
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Probability that the random variables take on certain values simultaneously
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

3. Sample variance
Nonlinearity
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
Confidence level
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

4. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
More than one random variable
Easy to manipulate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

5. Variance of X+Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Choose parameters that maximize the likelihood of what observations occurring
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Var(X) + Var(Y)

6. Hybrid method for conditional volatility
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Use historical simulation approach but use the EWMA weighting system
Variance = (1/m) summation(u<n - i>^2)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

7. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Expected value of the sample mean is the population mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Confidence level

8. Variance of X+b
(a^2)(variance(x)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Use historical simulation approach but use the EWMA weighting system
Variance(x)

9. Variance of X - Y assuming dependence
More than one random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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)
Variance(X) + Variance(Y) - 2*covariance(XY)

10. Continuously compounded return equation
Choose parameters that maximize the likelihood of what observations occurring
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
i = ln(Si/Si - 1)

11. EWMA
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
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
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!)

12. Standard normal distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Probability that the random variables take on certain values simultaneously
Transformed to a unit variable - Mean = 0 Variance = 1

13. Non - parametric vs parametric calculation of VaR
Independently and Identically Distributed
Regression can be non - linear in variables but must be linear in parameters
Based on a dataset
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

14. Two requirements of OVB
Distribution with only two possible outcomes
Price/return tends to run towards a long - run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

15. Sample correlation
Rxy = Sxy/(Sx*Sy)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
i = ln(Si/Si - 1)

16. Bernouli Distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sampling distribution of sample means tend to be normal
Returns over time for an individual asset
Distribution with only two possible outcomes

17. Stochastic error term
Model dependent - Options with the same underlying assets may trade at different volatilities
Contains variables not explicit in model - Accounts for randomness
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Peaks over threshold - Collects dataset in excess of some threshold

18. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors

19. Econometrics
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Expected value of the sample mean is the population mean
Variance reverts to a long run level
Application of mathematical statistics to economic data to lend empirical support to models

20. F distribution
Rxy = Sxy/(Sx*Sy)
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 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
Has heavy tails

21. Economical(elegant)
Returns over time for an individual asset
SSR
Only requires two parameters = mean and variance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

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

23. Statistical (or empirical) model
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
We accept a hypothesis that should have been rejected
Yi = B0 + B1Xi + ui
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

24. Standard error for Monte Carlo replications
Confidence level
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

25. 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)
Contains variables not explicit in model - Accounts for randomness
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on an equation - P(A) = # of A/total outcomes

26. Logistic distribution
Has heavy tails
Mean of sampling distribution is the population mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

27. Monte Carlo Simulations
Peaks over threshold - Collects dataset in excess of some threshold
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

28. Panel data (longitudinal or micropanel)
Population denominator = n - Sample denominator = n - 1
Special type of pooled data in which the cross sectional unit is surveyed over time
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
i = ln(Si/Si - 1)

29. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean = np - Variance = npq - Std dev = sqrt(npq)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

30. GEV
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Least absolute deviations estimator - used when extreme outliers are not uncommon
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

31. BLUE
Combine to form distribution with leptokurtosis (heavy tails)
i = ln(Si/Si - 1)
Choose parameters that maximize the likelihood of what observations occurring
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

32. Marginal unconditional probability function
Does not depend on a prior event or information
Variance reverts to a long run level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

33. Antithetic variable technique
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Yi = B0 + B1Xi + ui
Returns over time for an individual asset

34. Maximum likelihood method
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Choose parameters that maximize the likelihood of what observations occurring
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
P(Z>t)

35. Direction of OVB
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Combine to form distribution with leptokurtosis (heavy tails)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

36. Single variable (univariate) probability
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean will near the population mean as the sample size increases
Concerned with a single random variable (ex. Roll of a die)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

37. SER
P(Z>t)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
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

38. GARCH
95% = 1.65 99% = 2.33 For one - tailed tests
Average return across assets on a given day
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)
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

39. Binomial distribution equations for mean variance and std dev
Mean of sampling distribution is the population mean
When the sample size is large - the uncertainty about the value of the sample is very small
Attempts to sample along more important paths
Mean = np - Variance = npq - Std dev = sqrt(npq)

40. Exact significance level
P - value
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
i = ln(Si/Si - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)

41. Binomial distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
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!)
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

42. Central Limit Theorem(CLT)
Model dependent - Options with the same underlying assets may trade at different volatilities
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sampling distribution of sample means tend to be normal
Based on an equation - P(A) = # of A/total outcomes

43. Poisson distribution equations for mean variance and std deviation
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Does not depend on a prior event or information
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Confidence level

44. Lognormal
When one regressor is a perfect linear function of the other regressors
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

45. Conditional probability functions
Use historical simulation approach but use the EWMA weighting system
Concerned with a single random variable (ex. Roll of a die)
Probability that the random variables take on certain values simultaneously
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

46. Test for unbiasedness
E(mean) = mean
Sampling distribution of sample means tend to be normal
Variance(x)
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. Deterministic Simulation
Summation((xi - mean)^k)/n
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
95% = 1.65 99% = 2.33 For one - tailed tests
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

48. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

49. WLS
Summation((xi - mean)^k)/n
Mean of sampling distribution is the population mean
Independently and Identically Distributed
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

50. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E(XY) - E(X)E(Y)
Based on a dataset
(a^2)(variance(x)) + (b^2)(variance(y))