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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Two ways to calculate historical volatility
Regression can be non - linear in variables but must be linear in parameters
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Price/return tends to run towards a long - run level
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

2. Cholesky factorization (decomposition)
If variance of the conditional distribution of u(i) is not constant
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
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

3. Empirical frequency
Based on a dataset
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
E(XY) - E(X)E(Y)

4. Four sampling distributions

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5. Direction of OVB
i = ln(Si/Si - 1)
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)
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

6. Difference between population and sample variance
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
Population denominator = n - Sample denominator = n - 1
Probability that the random variables take on certain values simultaneously
Has heavy tails

7. Confidence interval for sample mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
We reject a hypothesis that is actually true
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

8. Simulation models
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
Var(X) + Var(Y)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

9. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Var(X) + Var(Y)

10. LFHS
We accept a hypothesis that should have been rejected
Probability that the random variables take on certain values simultaneously
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Low Frequency - High Severity events

11. Adjusted R^2
E(XY) - E(X)E(Y)
Probability that the random variables take on certain values simultaneously
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

12. Continuous random variable
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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

13. Potential reasons for fat tails in return distributions
Confidence level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Contains variables not explicit in model - Accounts for randomness

14. What does the OLS minimize?
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SSR

15. Homoskedastic
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

16. Hazard rate of exponentially distributed random variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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)

17. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P(X=x - Y=y) = P(X=x) * P(Y=y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability that the random variables take on certain values simultaneously

18. K - th moment
Application of mathematical statistics to economic data to lend empirical support to models
Summation((xi - mean)^k)/n
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

19. Type II Error
We accept a hypothesis that should have been rejected
Confidence level
Yi = B0 + B1Xi + ui
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

20. Panel data (longitudinal or micropanel)
Expected value of the sample mean is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time
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!)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

21. Standard error
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
More than one random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

22. Heteroskedastic
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
If variance of the conditional distribution of u(i) is not constant
Independently and Identically Distributed
Returns over time for an individual asset

23. Regime - switching volatility model
Statement of the error or precision of an estimate
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

24. Shortcomings of implied volatility
Sampling distribution of sample means tend to be normal
Model dependent - Options with the same underlying assets may trade at different volatilities
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)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

25. Single variable (univariate) probability
Population denominator = n - Sample denominator = n - 1
Based on an equation - P(A) = # of A/total outcomes
Concerned with a single random variable (ex. Roll of a die)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

26. Binomial distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
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)
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!)

27. POT
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Peaks over threshold - Collects dataset in excess of some threshold
Concerned with a single random variable (ex. Roll of a die)
Combine to form distribution with leptokurtosis (heavy tails)

28. Standard error for Monte Carlo replications
(a^2)(variance(x)
Regression can be non - linear in variables but must be linear in parameters
Confidence level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

29. Gamma distribution
Variance reverts to a long run level
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

30. F distribution
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
More than one random variable
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

31. Law of Large Numbers
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
P(Z>t)
Sample mean will near the population mean as the sample size increases
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

32. Standard normal distribution
We reject a hypothesis that is actually true
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Transformed to a unit variable - Mean = 0 Variance = 1

33. Deterministic Simulation
Statement of the error or precision of an estimate
Rxy = Sxy/(Sx*Sy)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

34. 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
Easy to manipulate
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
(a^2)(variance(x)

35. R^2
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sampling distribution of sample means tend to be normal
Only requires two parameters = mean and variance
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

36. Kurtosis
Does not depend on a prior event or information
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Nonlinearity
We reject a hypothesis that is actually true

37. Historical std dev
Low Frequency - High Severity events
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Random walk (usually acceptable) - Constant volatility (unlikely)

38. Square root rule
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

39. Block maxima
Nonlinearity
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

40. Two assumptions of square root rule
Choose parameters that maximize the likelihood of what observations occurring
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Random walk (usually acceptable) - Constant volatility (unlikely)

41. Two requirements of OVB
SSR
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

42. Continuously compounded return equation
i = ln(Si/Si - 1)
Only requires two parameters = mean and variance
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

43. Variance of X - Y assuming dependence
Low Frequency - High Severity events
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Expected value of the sample mean is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)

44. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean of sampling distribution is the population mean

45. Type I error
We reject a hypothesis that is actually true
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

46. Inverse transform method
Expected value of the sample mean is the population mean
Based on an equation - P(A) = # of A/total outcomes
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Random walk (usually acceptable) - Constant volatility (unlikely)

47. Least squares estimator(m)
Population denominator = n - Sample denominator = n - 1
Choose parameters that maximize the likelihood of what observations occurring
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

48. P - value
P(Z>t)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

49. Chi - squared distribution
Has heavy tails
Based on a dataset
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
Variance = (1/m) summation(u<n - i>^2)

50. Unstable return distribution
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 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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Among all unbiased estimators - estimator with the smallest variance is efficient