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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. Type II Error
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Average return across assets on a given day
E(XY) - E(X)E(Y)
We accept a hypothesis that should have been rejected

2. Homoskedastic
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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)

3. Multivariate Density Estimation (MDE)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

4. Cholesky factorization (decomposition)
Returns over time for an individual asset
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

5. Single variable (univariate) probability
Low Frequency - High Severity events
Concerned with a single random variable (ex. Roll of a die)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

6. Mean reversion
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Regression can be non - linear in variables but must be linear in parameters

7. Unconditional vs conditional distributions
Attempts to sample along more important paths
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

8. Continuously compounded return equation
i = ln(Si/Si - 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!)
More than one random variable
Based on a dataset

9. Confidence interval for sample mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean will near the population mean as the sample size increases
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Least absolute deviations estimator - used when extreme outliers are not uncommon

10. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Concerned with a single random variable (ex. Roll of a die)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

11. Beta distribution
Yi = B0 + B1Xi + ui
Variance reverts to a long run level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Nonlinearity

12. Logistic distribution
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
Has heavy tails
Sample mean will near the population mean as the sample size increases
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

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

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14. Binomial distribution equations for mean variance and std dev
We reject a hypothesis that is actually true
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)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Skewness
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)
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
Easy to manipulate

16. Extreme Value Theory
Var(X) + Var(Y)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Peaks over threshold - Collects dataset in excess of some threshold
Variance(X) + Variance(Y) - 2*covariance(XY)

17. Econometrics
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Application of mathematical statistics to economic data to lend empirical support to models
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

18. Law of Large Numbers
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
95% = 1.65 99% = 2.33 For one - tailed tests
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

19. Perfect multicollinearity
Based on an equation - P(A) = # of A/total outcomes
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

20. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
95% = 1.65 99% = 2.33 For one - tailed tests
Distribution with only two possible outcomes

21. Normal distribution
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x)
Price/return tends to run towards a long - run level

22. Mean reversion in asset dynamics
Rxy = Sxy/(Sx*Sy)
Does not depend on a prior event or information
Price/return tends to run towards a long - run level
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

23. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

24. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

25. Homoskedastic only F - stat
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
(a^2)(variance(x)) + (b^2)(variance(y))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

26. Variance of sampling distribution of means when n<N
Returns over time for an individual asset
Variance(sample y) = (variance(y)/n)*(N - n/N - 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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

27. Statistical (or empirical) model
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Yi = B0 + B1Xi + ui
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
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

28. Weibul distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E(XY) - E(X)E(Y)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

29. Sample correlation
Transformed to a unit variable - Mean = 0 Variance = 1
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Rxy = Sxy/(Sx*Sy)

30. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Special type of pooled data in which the cross sectional unit is surveyed over time
We accept a hypothesis that should have been rejected
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

31. Continuous random variable
When the sample size is large - the uncertainty about the value of the sample is very small
Rxy = Sxy/(Sx*Sy)
Var(X) + Var(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

32. i.i.d.
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Independently and Identically Distributed

33. Tractable
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance reverts to a long run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Easy to manipulate

34. Test for unbiasedness
E(mean) = mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Sampling distribution of sample means tend to be normal

35. Mean(expected value)
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)
Expected value of the sample mean is the population mean
Model dependent - Options with the same underlying assets may trade at different volatilities

36. LFHS
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
Distribution with only two possible outcomes
Low Frequency - High Severity events

37. LAD
Variance = (1/m) summation(u<n - i>^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(x)
Least absolute deviations estimator - used when extreme outliers are not uncommon

38. Inverse transform method
i = ln(Si/Si - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Has heavy tails

39. Variance of X+b
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance(x)

40. Type I error
Low Frequency - High Severity events
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We reject a hypothesis that is actually true
Independently and Identically Distributed

41. Importance sampling technique
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
SSR
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Attempts to sample along more important paths

42. Monte Carlo Simulations
More than one random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance = (1/m) summation(u<n - i>^2)

43. Overall F - statistic
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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!)

44. Biggest (and only real) drawback of GARCH mode
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)
Population denominator = n - Sample denominator = n - 1
Distribution with only two possible outcomes
Nonlinearity

45. Standard variable for non - normal distributions
Model dependent - Options with the same underlying assets may trade at different volatilities
Z = (Y - meany)/(stddev(y)/sqrt(n))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Population denominator = n - Sample denominator = n - 1

46. Joint probability functions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Least absolute deviations estimator - used when extreme outliers are not uncommon
Easy to manipulate
Probability that the random variables take on certain values simultaneously

47. Persistence
Concerned with a single random variable (ex. Roll of a die)
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
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
Only requires two parameters = mean and variance

48. Unstable return distribution
Yi = B0 + B1Xi + ui
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. Central Limit Theorem
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance

50. Discrete representation of the GBM
Least absolute deviations estimator - used when extreme outliers are not uncommon
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)