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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Application of mathematical statistics to economic data to lend empirical support to models

2. Exponential distribution
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Contains variables not explicit in model - Accounts for randomness

3. Sample correlation
Rxy = Sxy/(Sx*Sy)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Sample mean will near the population mean as the sample size increases

4. Variance of X - Y assuming dependence
For n>30 - sample mean is approximately normal
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(X) + Variance(Y) - 2*covariance(XY)

5. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Normal - Student's T - Chi - square - F distribution
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
95% = 1.65 99% = 2.33 For one - tailed tests

6. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Contains variables not explicit in model - Accounts for randomness
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance reverts to a long run level

7. Critical z values
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
95% = 1.65 99% = 2.33 For one - tailed tests
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

8. Cholesky factorization (decomposition)
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
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
(a^2)(variance(x)) + (b^2)(variance(y))

9. Monte Carlo Simulations
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Based on an equation - P(A) = # of A/total outcomes
More than one random variable

10. Four sampling distributions

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11. Stochastic error term
Based on an equation - P(A) = # of A/total outcomes
Contains variables not explicit in model - Accounts for randomness
Model dependent - Options with the same underlying assets may trade at different volatilities
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

12. Variance of X+b
Variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Population denominator = n - Sample denominator = n - 1
Has heavy tails

13. Mean reversion
Mean of sampling distribution is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

14. K - th moment
Returns over time for an individual asset
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Summation((xi - mean)^k)/n

15. Result of combination of two normal with same means
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Combine to form distribution with leptokurtosis (heavy tails)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

16. Unstable return distribution
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
Based on a dataset

17. Statistical (or empirical) model
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
P - value
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

18. Bernouli Distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Distribution with only two possible outcomes
Transformed to a unit variable - Mean = 0 Variance = 1
Returns over time for an individual asset

19. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

20. Kurtosis
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We reject a hypothesis that is actually true
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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)

21. Discrete random variable
SSR
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))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

22. WLS
Contains variables not explicit in model - Accounts for randomness
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
We reject a hypothesis that is actually true
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

23. Difference between population and sample variance
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Regression can be non - linear in variables but must be linear in parameters
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

24. Weibul distribution
i = ln(Si/Si - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))

25. Mean(expected value)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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)
(a^2)(variance(x)

26. Implications of homoscedasticity
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!)
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 upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Population denominator = n - Sample denominator = n - 1

27. Poisson Distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
i = ln(Si/Si - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

28. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Has heavy tails
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

29. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P(Z>t)
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
i = ln(Si/Si - 1)

30. Significance =1
Confidence level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval

31. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Application of mathematical statistics to economic data to lend empirical support to models
Combine to form distribution with leptokurtosis (heavy tails)

32. EWMA
Variance(y)/n = variance of sample Y
Variance(X) + Variance(Y) - 2*covariance(XY)
More than one random variable
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

33. Hazard rate of exponentially distributed random variable
Price/return tends to run towards a long - run level
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(X) + Variance(Y) - 2*covariance(XY)

34. 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
SSR
Variance(x) + Variance(Y) + 2*covariance(XY)
Var(X) + Var(Y)

35. Type I error
We reject a hypothesis that is actually true
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
For n>30 - sample mean is approximately normal
Independently and Identically Distributed

36. Type II Error
Price/return tends to run towards a long - run level
We accept a hypothesis that should have been rejected
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

37. Test for statistical independence
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

38. Least squares estimator(m)
E(mean) = mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

39. Multivariate Density Estimation (MDE)
Yi = B0 + B1Xi + ui
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Var(X) + Var(Y)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

40. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

41. Covariance
Variance reverts to a long run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
E(XY) - E(X)E(Y)
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

42. Standard error for Monte Carlo replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
We reject a hypothesis that is actually true
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

43. ESS
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)
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

44. LFHS
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
Application of mathematical statistics to economic data to lend empirical support to models
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Low Frequency - High Severity events

45. Single variable (univariate) probability
For n>30 - sample mean is approximately normal
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Concerned with a single random variable (ex. Roll of a die)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

46. Continuously compounded return equation
Expected value of the sample mean is the population mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
i = ln(Si/Si - 1)

47. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Transformed to a unit variable - Mean = 0 Variance = 1

48. R^2
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. i.i.d.
Sampling distribution of sample means tend to be normal
Choose parameters that maximize the likelihood of what observations occurring
Independently and Identically Distributed
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

50. Cross - sectional
i = ln(Si/Si - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Average return across assets on a given day