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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Panel data (longitudinal or micropanel)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Sample mean +/ - t*(stddev(s)/sqrt(n))
Special type of pooled data in which the cross sectional unit is surveyed over time
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

2. Biggest (and only real) drawback of GARCH mode
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E(mean) = mean
Nonlinearity

3. Cholesky factorization (decomposition)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
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))

4. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Use historical simulation approach but use the EWMA weighting system

5. Mean reversion
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

6. Continuous random variable
(a^2)(variance(x)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

7. Single variable (univariate) probability
Average return across assets on a given day
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Concerned with a single random variable (ex. Roll of a die)

8. Two assumptions of square root rule
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Random walk (usually acceptable) - Constant volatility (unlikely)

9. K - th moment
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
Summation((xi - mean)^k)/n
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

10. Monte Carlo Simulations
E(XY) - E(X)E(Y)
Sample mean will near the population mean as the sample size increases
Distribution with only two possible outcomes
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

11. Weibul distribution
We reject a hypothesis that is actually true
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(y)/n = variance of sample Y
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

12. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Among all unbiased estimators - estimator with the smallest variance is efficient
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Population denominator = n - Sample denominator = n - 1

13. Perfect multicollinearity
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
When one regressor is a perfect linear function of the other regressors
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P(X=x - Y=y) = P(X=x) * P(Y=y)

14. Antithetic variable technique
Yi = B0 + B1Xi + ui
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Low Frequency - High Severity events

15. Confidence ellipse
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Normal - Student's T - Chi - square - F distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

16. GARCH
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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)
Mean of sampling distribution is the population mean

17. i.i.d.
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Independently and Identically Distributed
Random walk (usually acceptable) - Constant volatility (unlikely)

18. Econometrics
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Application of mathematical statistics to economic data to lend empirical support to models
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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Simulating for VaR
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

20. Square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Contains variables not explicit in model - Accounts for randomness
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Least absolute deviations estimator - used when extreme outliers are not uncommon

21. Direction of OVB
Mean = np - Variance = npq - Std dev = sqrt(npq)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

22. Beta distribution
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

23. Homoskedastic
We reject a hypothesis that is actually true
Variance = (1/m) summation(u<n - i>^2)
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!)
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

24. Importance sampling technique
Attempts to sample along more important paths
Rxy = Sxy/(Sx*Sy)
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

25. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

26. Type I error
We reject a hypothesis that is actually true
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

27. Economical(elegant)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Only requires two parameters = mean and variance
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for an individual asset

28. Shortcomings of implied volatility
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Model dependent - Options with the same underlying assets may trade at different volatilities
Probability that the random variables take on certain values simultaneously

29. Binomial distribution equations for mean variance and std dev
Regression can be non - linear in variables but must be linear in parameters
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
E(XY) - E(X)E(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)

30. Historical std dev
P(Z>t)
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)
Regression can be non - linear in variables but must be linear in parameters
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

31. Variance of aX
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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)
When one regressor is a perfect linear function of the other regressors
(a^2)(variance(x)

32. Bernouli Distribution
Use historical simulation approach but use the EWMA weighting system
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Distribution with only two possible outcomes

33. Logistic distribution
Only requires two parameters = mean and variance
Has heavy tails
(a^2)(variance(x)
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)

34. Poisson Distribution
More than one random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

35. SER
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
Population denominator = n - Sample denominator = n - 1
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Attempts to sample along more important paths

36. Variance of X - Y assuming dependence
Statement of the error or precision of an estimate
Variance(X) + Variance(Y) - 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(X=x - Y=y) = P(X=x) * P(Y=y)

37. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
When one regressor is a perfect linear function of the other regressors
Based on a dataset
Variance(x) + Variance(Y) + 2*covariance(XY)

38. Discrete representation of the GBM
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

39. Block maxima
If variance of the conditional distribution of u(i) is not constant
Low Frequency - High Severity events
Based on a dataset
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

40. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Distribution with only two possible outcomes

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

42. Tractable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Easy to manipulate
Distribution with only two possible outcomes

43. Maximum likelihood method
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Choose parameters that maximize the likelihood of what observations occurring

44. Mean reversion in asset dynamics
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Price/return tends to run towards a long - run level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

45. Persistence
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
When one regressor is a perfect linear function of the other regressors
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Average return across assets on a given day

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

47. EWMA
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)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

48. Efficiency
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Among all unbiased estimators - estimator with the smallest variance is efficient
Average return across assets on a given day
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

49. Covariance calculations using weight sums (lambda)
Rxy = Sxy/(Sx*Sy)
Transformed to a unit variable - Mean = 0 Variance = 1
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

50. Type II Error
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
We accept a hypothesis that should have been rejected
Variance(X) + Variance(Y) - 2*covariance(XY)