FRM Foundations Of Risk Management Quantitative Methods

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1. Key properties of linear regression
For n>30 - sample mean is approximately normal
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
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed


2. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Expected value of the sample mean is the population mean
Confidence level


3. Exponential distribution
When the sample size is large - the uncertainty about the value of the sample is very small
Expected value of the sample mean is the population mean
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta


4. Weibul distribution
Peaks over threshold - Collects dataset in excess of some threshold
P(Z>t)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))


5. Regime - switching volatility model
Least absolute deviations estimator - used when extreme outliers are not uncommon
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Attempts to sample along more important paths
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for


6. Persistence
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
More than one random variable


7. Simulating for VaR
E(XY) - E(X)E(Y)
Variance(y)/n = variance of sample Y
Variance(X) + Variance(Y) - 2*covariance(XY)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution


8. Adjusted R^2
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
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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)


9. Priori (classical) probability
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P - value
Based on an equation - P(A) = # of A/total outcomes


10. SER
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
P(Z>t)
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(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


11. Poisson distribution equations for mean variance and std deviation
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
(a^2)(variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions


12. Four sampling distributions


13. Logistic distribution
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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


14. Discrete representation of the GBM
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)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))


15. Variance of sample mean
Based on an equation - P(A) = # of A/total outcomes
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(y)/n = variance of sample Y
Population denominator = n - Sample denominator = n - 1


16. Beta distribution
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates


17. Tractable
Summation((xi - mean)^k)/n
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
P(Z>t)
Easy to manipulate


18. Variance - covariance approach for VaR of a portfolio
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Normal - Student's T - Chi - square - F distribution


19. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Returns over time for an individual asset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2


20. Homoskedastic only F - stat
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)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)


21. Shortcomings of implied volatility
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"
Model dependent - Options with the same underlying assets may trade at different volatilities
(a^2)(variance(x)


22. Binomial distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean of sampling distribution is the population mean
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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!)


23. Variance of sampling distribution of means when n<N
Probability that the random variables take on certain values simultaneously
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean of sampling distribution is the population mean


24. Unstable return distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)


25. Implications of homoscedasticity
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))
Random walk (usually acceptable) - Constant volatility (unlikely)
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


26. Continuous random variable
Average return across assets on a given day
Special type of pooled data in which the cross sectional unit is surveyed over time
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Does not depend on a prior event or information


27. Direction of OVB
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Concerned with a single random variable (ex. Roll of a die)


28. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Mean of sampling distribution is the population mean


29. Two assumptions of square root rule
Only requires two parameters = mean and variance
Sample mean will near the population mean as the sample size increases
Random walk (usually acceptable) - Constant volatility (unlikely)
When the sample size is large - the uncertainty about the value of the sample is very small


30. Discrete random variable
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)


31. Statistical (or empirical) model
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Yi = B0 + B1Xi + ui
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y


32. Implied standard deviation for options
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(x) + Variance(Y) + 2*covariance(XY)


33. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Only requires two parameters = mean and variance
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(X) + Variance(Y) - 2*covariance(XY)


34. Lognormal
Application of mathematical statistics to economic data to lend empirical support to models
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda


35. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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)


36. GEV
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)


37. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
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!)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Concerned with a single random variable (ex. Roll of a die)


38. Maximum likelihood method
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Choose parameters that maximize the likelihood of what observations occurring
Distribution with only two possible outcomes


39. Covariance
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E(XY) - E(X)E(Y)
Confidence level
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state


40. Variance of X+Y
When one regressor is a perfect linear function of the other regressors
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Var(X) + Var(Y)


41. Block maxima
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.


42. Variance of X+Y assuming dependence
Variance reverts to a long run level
Variance(x) + Variance(Y) + 2*covariance(XY)
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE


43. Variance of aX + bY
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))


44. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Nonlinearity
Distribution with only two possible outcomes


45. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Model dependent - Options with the same underlying assets may trade at different volatilities
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)


46. Test for statistical independence
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(X=x - Y=y) = P(X=x) * P(Y=y)
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


47. Efficiency
Does not depend on a prior event or information
E(XY) - E(X)E(Y)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient


48. Skewness
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
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


49. Variance of X - Y assuming dependence
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
Based on an equation - P(A) = # of A/total outcomes
Mean of sampling distribution is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)


50. i.i.d.
Price/return tends to run towards a long - run level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Independently and Identically Distributed
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha