FRM Foundations Of Risk Management Quantitative Methods

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1. Maximum likelihood method
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
SSR
Choose parameters that maximize the likelihood of what observations occurring


2. Mean reversion
Only requires two parameters = mean and variance
Returns over time for an individual asset
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution


3. Stochastic error term
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Contains variables not explicit in model - Accounts for randomness


4. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
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
Easy to manipulate
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)


5. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
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


6. Joint probability functions
Probability that the random variables take on certain values simultaneously
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Special type of pooled data in which the cross sectional unit is surveyed over time


7. Gamma distribution
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
Combine to form distribution with leptokurtosis (heavy tails)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)


8. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
More than one random 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


9. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Based on a dataset
Variance reverts to a long run level
95% = 1.65 99% = 2.33 For one - tailed tests


10. Multivariate probability
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
More than one random variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d


11. GARCH
For n>30 - sample mean is approximately normal
Expected value of the sample mean is the population mean
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)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx


12. Simulating for VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
More than one random variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))


13. Unbiased
Confidence level
Independently and Identically Distributed
Distribution with only two possible outcomes
Mean of sampling distribution is the population mean


14. Simulation models
Dataset is parsed into blocks with greater length than the periodicity - Observations must be 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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sampling distribution of sample means tend to be normal


15. Confidence ellipse
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(x)


16. Persistence
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Based on a dataset
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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


17. WLS
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample variance = (1/(k - 1))Summation(Yi - mean)^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


18. Binomial distribution
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!)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda


19. Consistent
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
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!)
When the sample size is large - the uncertainty about the value of the sample is very small
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


20. Cholesky factorization (decomposition)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
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


21. Sample variance
Based on an equation - P(A) = # of A/total outcomes
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2


22. Bernouli Distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)


23. K - th moment
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Summation((xi - mean)^k)/n


24. Variance of X+Y
Random walk (usually acceptable) - Constant volatility (unlikely)
Var(X) + Var(Y)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P - value


25. Poisson Distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Among all unbiased estimators - estimator with the smallest variance is efficient
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)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)


26. Exponential distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())


27. Discrete representation of the GBM
Among all unbiased estimators - estimator with the smallest variance is efficient
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Contains variables not explicit in model - Accounts for randomness
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps


28. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Rxy = Sxy/(Sx*Sy)
Easy to manipulate


29. Two assumptions of square root rule
Statement of the error or precision of an estimate
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)


30. Efficiency
Mean = np - Variance = npq - Std dev = sqrt(npq)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Among all unbiased estimators - estimator with the smallest variance is efficient


31. Variance of X+b
Attempts to sample along more important paths
Population denominator = n - Sample denominator = n - 1
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
Variance(x)


32. Binomial distribution equations for mean variance and std dev
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Based on a dataset
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Mean = np - Variance = npq - Std dev = sqrt(npq)


33. Adjusted R^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"


34. Unstable return distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sampling distribution of sample means tend to be normal
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Expected value of the sample mean is the population mean


35. Variance(discrete)
If variance of the conditional distribution of u(i) is not constant
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Confidence level
Price/return tends to run towards a long - run level


36. Bootstrap method
Only requires two parameters = mean and variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on a dataset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions


37. EWMA
E(mean) = mean
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean = np - Variance = npq - Std dev = sqrt(npq)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda


38. Weibul distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Mean of sampling distribution is the population mean
Choose parameters that maximize the likelihood of what observations occurring
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha


39. Standard variable for non - normal distributions
Summation((xi - mean)^k)/n
Z = (Y - meany)/(stddev(y)/sqrt(n))
Confidence set for two coefficients - two dimensional analog for the confidence interval
When one regressor is a perfect linear function of the other regressors


40. Mean reversion in asset dynamics
P - value
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha


41. Deterministic Simulation
Z = (Y - meany)/(stddev(y)/sqrt(n))
Only requires two parameters = mean and variance
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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


42. Covariance
Attempts to sample along more important paths
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Normal - Student's T - Chi - square - F distribution
E(XY) - E(X)E(Y)


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


44. Sample mean
Statement of the error or precision of an estimate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Only requires two parameters = mean and variance
Expected value of the sample mean is the population mean


45. Control variates technique
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2


46. Extreme Value Theory
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window


47. Time series data
Returns over time for an individual asset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Yi = B0 + B1Xi + ui


48. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
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
Does not depend on a prior event or information


49. Variance of sampling distribution of means when n<N
We accept a hypothesis that should have been rejected
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Normal - Student's T - Chi - square - F distribution
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


50. Overall F - statistic
Variance(x) + Variance(Y) + 2*covariance(XY)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Z = (Y - meany)/(stddev(y)/sqrt(n))