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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Econometrics
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases
Application of mathematical statistics to economic data to lend empirical support to models
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

2. Variance of X+Y
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Var(X) + Var(Y)

3. Poisson distribution equations for mean variance and std deviation
P(Z>t)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Returns over time for an individual asset
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

4. Non - parametric vs parametric calculation of VaR
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample mean will near the population mean as the sample size increases
Only requires two parameters = mean and variance
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

5. Variance of X+b
Variance(x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E(mean) = mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

6. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (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)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

7. Variance - covariance approach for VaR of a portfolio
Confidence level
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

8. Marginal unconditional probability function
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Does not depend on a prior event or information
Yi = B0 + B1Xi + ui
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

9. Law of Large Numbers
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)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Does not depend on a prior event or information
Sample mean will near the population mean as the sample size increases

10. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
95% = 1.65 99% = 2.33 For one - tailed tests
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
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

11. Pooled data
Confidence level
Returns over time for a combination of assets (combination of time series and cross - sectional data)
For n>30 - sample mean is approximately normal
P(Z>t)

12. Exponential distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
If variance of the conditional distribution of u(i) is not constant
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected

13. WLS
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Contains variables not explicit in model - Accounts for randomness

14. Time series data
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Returns over time for an individual asset
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

15. Weibul distribution
Regression can be non - linear in variables but must be linear in parameters
Nonlinearity
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

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

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17. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - 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!)
E(mean) = mean

18. Covariance
Probability that the random variables take on certain values simultaneously
Sample mean will near the population mean as the sample size increases
Rxy = Sxy/(Sx*Sy)
E(XY) - E(X)E(Y)

19. Sample covariance
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Summation((xi - mean)^k)/n
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

20. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Does not depend on a prior event or information
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

21. Statistical (or empirical) model
Expected value of the sample mean is the population mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Yi = B0 + B1Xi + ui

22. Importance sampling technique
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
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)
We accept a hypothesis that should have been rejected
Attempts to sample along more important paths

23. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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

24. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Based on a dataset
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We reject a hypothesis that is actually true

25. LAD
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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!)
Has heavy tails
Least absolute deviations estimator - used when extreme outliers are not uncommon

26. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Independently and Identically Distributed

27. Gamma distribution
Price/return tends to run towards 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
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
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

28. GPD
(a^2)(variance(x)) + (b^2)(variance(y))
P - value
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Statement of the error or precision of an estimate

29. Exact significance level
Variance reverts to a long run level
Has heavy tails
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P - value

30. Multivariate probability
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample mean will near the population mean as the sample size increases
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
More than one random variable

31. Two assumptions of square root rule
Sampling distribution of sample means tend to be normal
Random walk (usually acceptable) - Constant volatility (unlikely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Nonlinearity

32. Standard normal 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
Rxy = Sxy/(Sx*Sy)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Transformed to a unit variable - Mean = 0 Variance = 1

33. Homoskedastic only F - stat
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!)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

34. Four sampling distributions

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35. Test for statistical independence
Average return across assets on a given day
P(X=x - Y=y) = P(X=x) * P(Y=y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Confidence set for two coefficients - two dimensional analog for the confidence interval

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!)
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
Use historical simulation approach but use the EWMA weighting system

37. Simplified standard (un - weighted) variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Independently and Identically Distributed
Confidence level
Variance = (1/m) summation(u<n - i>^2)

38. Test for unbiasedness
E(mean) = mean
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

39. Perfect multicollinearity
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When one regressor is a perfect linear function of the other regressors
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

40. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

41. ESS
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)

42. Covariance calculations using weight sums (lambda)
Contains variables not explicit in model - Accounts for randomness
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

43. Confidence interval (from t)
If variance of the conditional distribution of u(i) is not constant
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

44. Two requirements of OVB
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Rxy = Sxy/(Sx*Sy)
Yi = B0 + B1Xi + ui
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

45. 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
(a^2)(variance(x)) + (b^2)(variance(y))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Contains variables not explicit in model - Accounts for randomness

46. Variance of sampling distribution of means when n<N
Model dependent - Options with the same underlying assets may trade at different volatilities
We reject a hypothesis that is actually true
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

47. Skewness
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
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))

48. Multivariate Density Estimation (MDE)
P - value
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

49. Sample correlation
Confidence level
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
We accept a hypothesis that should have been rejected
Rxy = Sxy/(Sx*Sy)

50. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(y)/n = variance of sample Y
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)