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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Stochastic error term
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Contains variables not explicit in model - Accounts for randomness

2. Central Limit Theorem(CLT)
Only requires two parameters = mean and variance
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Sampling distribution of sample means tend to be normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

3. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance reverts to a long run level
Independently and Identically Distributed
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

4. Continuous random variable
Has heavy tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Mean of sampling distribution is the population mean

5. Persistence
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Var(X) + Var(Y)

6. Monte Carlo Simulations
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
Yi = B0 + B1Xi + ui
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

7. Historical std dev
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

8. Cholesky factorization (decomposition)
Choose parameters that maximize the likelihood of what observations occurring
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Variance(X) + Variance(Y) - 2*covariance(XY)

9. Variance(discrete)
Probability that the random variables take on certain values simultaneously
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Low Frequency - High Severity events

10. Inverse transform method
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
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

11. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
i = ln(Si/Si - 1)

12. EWMA
Among all unbiased estimators - estimator with the smallest variance is efficient
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on a dataset
When one regressor is a perfect linear function of the other regressors

13. Covariance
Variance(x) + Variance(Y) + 2*covariance(XY)
E(XY) - E(X)E(Y)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Var(X) + Var(Y)

14. Pooled data
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Rxy = Sxy/(Sx*Sy)
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

15. Two ways to calculate historical volatility
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Easy to manipulate
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Average return across assets on a given day

16. P - value
Rxy = Sxy/(Sx*Sy)
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

17. Time series data
Returns over time for an individual asset
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(y)/n = variance of sample Y

18. Discrete random variable
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sample mean will near the population mean as the sample size increases
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

19. Extreme Value Theory
Nonlinearity
E(XY) - E(X)E(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

20. Result of combination of two normal with same means
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
When the sample size is large - the uncertainty about the value of the sample is very small

21. Four sampling distributions

22. Exact significance level
P - value
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Random walk (usually acceptable) - Constant volatility (unlikely)

23. Importance sampling technique
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Attempts to sample along more important paths
Based on an equation - P(A) = # of A/total outcomes
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

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

25. Variance of X - Y assuming dependence
When the sample size is large - the uncertainty about the value of the sample is very small
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(X) + Variance(Y) - 2*covariance(XY)
i = ln(Si/Si - 1)

26. LAD
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Population denominator = n - Sample denominator = n - 1
Nonlinearity

27. Multivariate probability
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
More than one random variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

28. Hazard rate of exponentially distributed random variable
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
More than one random variable
Mean of sampling distribution is the population mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

29. Binomial distribution equations for mean variance and std dev
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Expected value of the sample mean is the population mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

30. Simulation models
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

31. Conditional probability functions
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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!)

32. GARCH
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)
Regression can be non - linear in variables but must be linear in parameters
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

33. 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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Confidence level

34. Implications of homoscedasticity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
(a^2)(variance(x)
Sampling distribution of sample means tend to be normal
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

35. Perfect multicollinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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 one regressor is a perfect linear function of the other regressors

36. Binomial distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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!)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

37. Mean reversion in variance
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Variance reverts to a long run level

38. Type II Error
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
We accept a hypothesis that should have been rejected
Concerned with a single random variable (ex. Roll of a die)
We reject a hypothesis that is actually true

39. Exponential distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence set for two coefficients - two dimensional analog for the confidence interval
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

40. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
E(XY) - E(X)E(Y)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

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

42. Variance of X+b
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

43. Variance of X+Y
(a^2)(variance(x)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Var(X) + Var(Y)

44. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E(XY) - E(X)E(Y)
Var(X) + Var(Y)

45. Confidence interval (from t)
Regression can be non - linear in variables but must be linear in parameters
Sample mean +/ - t*(stddev(s)/sqrt(n))
i = ln(Si/Si - 1)
Sample mean will near the population mean as the sample size increases

46. Simplified standard (un - weighted) variance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Independently and Identically Distributed
Variance = (1/m) summation(u<n - i>^2)

47. K - th moment
Distribution with only two possible outcomes
Summation((xi - mean)^k)/n
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

48. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(X) + Variance(Y) - 2*covariance(XY)

49. Efficiency
Sample mean +/ - t*(stddev(s)/sqrt(n))
Among all unbiased estimators - estimator with the smallest variance is efficient
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

50. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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)
Returns over time for an individual asset
Variance reverts to a long run level