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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. T distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

2. Historical std dev
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
P - value

3. Variance - covariance approach for VaR of a portfolio
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

4. Variance of sample mean
P - value
Variance(y)/n = variance of sample Y
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Random walk (usually acceptable) - Constant volatility (unlikely)

5. R^2
Based on an equation - P(A) = # of A/total outcomes
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Choose parameters that maximize the likelihood of what observations occurring
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

6. F distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Based on an equation - P(A) = # of A/total outcomes
More than one random variable

7. Cholesky factorization (decomposition)
Combine to form distribution with leptokurtosis (heavy tails)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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!)
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

8. Variance of sampling distribution of means when n<N
Confidence set for two coefficients - two dimensional analog for the confidence interval
Distribution with only two possible outcomes
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

9. Block maxima
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

10. Four sampling distributions

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11. Unbiased
Mean of sampling distribution is the population mean
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Only requires two parameters = mean and variance

12. Key properties of linear regression
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Regression can be non - linear in variables but must be linear in parameters
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

13. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
For n>30 - sample mean is approximately normal
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)

14. Poisson Distribution
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

15. Variance of X+b
When one regressor is a perfect linear function of the other regressors
Variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Least absolute deviations estimator - used when extreme outliers are not uncommon

16. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
When the sample size is large - the uncertainty about the value of the sample is very small
Confidence set for two coefficients - two dimensional analog for the confidence interval
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

17. Continuous representation of the GBM
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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 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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

18. Deterministic Simulation
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
Variance = (1/m) summation(u<n - i>^2)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Expected value of the sample mean is the population mean

19. Test for unbiasedness
E(mean) = mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
When one regressor is a perfect linear function of the other regressors
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

20. Extreme Value Theory
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
For n>30 - sample mean is approximately normal
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

21. Homoskedastic
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Expected value of the sample mean is the population mean
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

22. Marginal unconditional probability function
P(X=x - Y=y) = P(X=x) * P(Y=y)
Expected value of the sample mean is the population mean
Does not depend on a prior event or information
Confidence set for two coefficients - two dimensional analog for the confidence interval

23. Kurtosis
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!)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Population denominator = n - Sample denominator = n - 1

24. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Price/return tends to run towards a long - run level
Application of mathematical statistics to economic data to lend empirical support to models

25. Simulation models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Price/return tends to run towards a long - run level

26. Hazard rate of exponentially distributed random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
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)
Independently and Identically Distributed

27. Time series data
Peaks over threshold - Collects dataset in excess of some threshold
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Use historical simulation approach but use the EWMA weighting system
Returns over time for an individual asset

28. Panel data (longitudinal or micropanel)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Based on an equation - P(A) = # of A/total outcomes
Variance reverts to a long run level
Special type of pooled data in which the cross sectional unit is surveyed over time

29. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence level
Rxy = Sxy/(Sx*Sy)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

30. Control variates technique
(a^2)(variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
95% = 1.65 99% = 2.33 For one - tailed tests

31. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Among all unbiased estimators - estimator with the smallest variance is efficient
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

32. Central Limit Theorem
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample mean +/ - t*(stddev(s)/sqrt(n))

33. Shortcomings of implied volatility
Combine to form distribution with leptokurtosis (heavy tails)
Model dependent - Options with the same underlying assets may trade at different volatilities
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
If variance of the conditional distribution of u(i) is not constant

34. Variance of X+Y
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Var(X) + Var(Y)
Least absolute deviations estimator - used when extreme outliers are not uncommon

35. Significance =1
Expected value of the sample mean is the population mean
Rxy = Sxy/(Sx*Sy)
Confidence level
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

36. Type I error
Contains variables not explicit in model - Accounts for randomness
If variance of the conditional distribution of u(i) is not constant
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We reject a hypothesis that is actually true

37. i.i.d.
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
Expected value of the sample mean is the population mean
Concerned with a single random variable (ex. Roll of a die)
Independently and Identically Distributed

38. Covariance
E(XY) - E(X)E(Y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Based on an equation - P(A) = # of A/total outcomes
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

39. Two ways to calculate historical volatility
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Use historical simulation approach but use the EWMA weighting system
Price/return tends to run towards a long - run level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

40. Variance of weighted scheme
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Summation((xi - mean)^k)/n
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Normal - Student's T - Chi - square - F distribution

41. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance reverts to a long run level

42. GPD
Combine to form distribution with leptokurtosis (heavy tails)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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 Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

43. Covariance calculations using weight sums (lambda)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)

44. Sample covariance
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
Application of mathematical statistics to economic data to lend empirical support to models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
For n>30 - sample mean is approximately normal

45. Statistical (or empirical) model
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Yi = B0 + B1Xi + ui
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

46. Law of Large Numbers
Has heavy tails
Rxy = Sxy/(Sx*Sy)
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

47. Lognormal
Concerned with a single random variable (ex. Roll of a die)
Mean of sampling distribution is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

48. Two requirements of OVB
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

49. Bernouli Distribution
Distribution with only two possible outcomes
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
When one regressor is a perfect linear function of the other regressors
Model dependent - Options with the same underlying assets may trade at different volatilities

50. 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
Variance(X) + Variance(Y) - 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())