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FRM Foundations Of Risk Management Quantitative Methods

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1. Persistence
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
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
Distribution with only two possible outcomes
Special type of pooled data in which the cross sectional unit is surveyed over time

2. Homoskedastic only F - stat
(a^2)(variance(x)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
If variance of the conditional distribution of u(i) is not constant

3. Chi - squared distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Nonlinearity
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
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

4. Time series data
Model dependent - Options with the same underlying assets may trade at different volatilities
Transformed to a unit variable - Mean = 0 Variance = 1
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Returns over time for an individual asset

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

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6. Confidence interval (from t)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Use historical simulation approach but use the EWMA weighting system

7. Two ways to calculate historical volatility
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
If variance of the conditional distribution of u(i) is not constant

8. Central Limit Theorem
Based on an equation - P(A) = # of A/total outcomes
Among all unbiased estimators - estimator with the smallest variance is efficient
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

9. Heteroskedastic
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
If variance of the conditional distribution of u(i) is not constant
i = ln(Si/Si - 1)

10. Two drawbacks of moving average series
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

11. Variance of sample mean
Variance(x)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Confidence level
Variance(y)/n = variance of sample Y

12. Mean reversion in variance
Yi = B0 + B1Xi + ui
Variance reverts to a long run level
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

13. Reliability
Statement of the error or precision of an estimate
P(Z>t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Yi = B0 + B1Xi + ui

14. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
When the sample size is large - the uncertainty about the value of the sample is very small
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

15. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
(a^2)(variance(x)
Variance reverts to a long run level
E(mean) = mean

16. Variance of X+b
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)
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
Sample mean will near the population mean as the sample size increases
Variance(x)

17. POT
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Peaks over threshold - Collects dataset in excess of some threshold
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
More than one random variable

18. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Probability that the random variables take on certain values simultaneously
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sample mean +/ - t*(stddev(s)/sqrt(n))

19. Unbiased
Mean of sampling distribution is the population mean
Distribution with only two possible outcomes
E(mean) = mean
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

20. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Among all unbiased estimators - estimator with the smallest variance is efficient
Only requires two parameters = mean and variance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

21. Continuously compounded return equation
Has heavy tails
We reject a hypothesis that is actually true
i = ln(Si/Si - 1)
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

22. GEV
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))
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

23. LFHS
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Low Frequency - High Severity events
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

24. Mean(expected value)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
If variance of the conditional distribution of u(i) is not constant
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

25. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
95% = 1.65 99% = 2.33 For one - tailed tests
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))

26. 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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

27. Type II Error
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
We accept a hypothesis that should have been rejected
Variance(y)/n = variance of sample Y
Summation((xi - mean)^k)/n

28. Standard variable for non - normal distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Z = (Y - meany)/(stddev(y)/sqrt(n))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

29. Least squares estimator(m)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Does not depend on a prior event or information

30. Sample variance
Distribution with only two possible outcomes
Special type of pooled data in which the cross sectional unit is surveyed over time
E(XY) - E(X)E(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

31. Exact significance level
For n>30 - sample mean is approximately normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
P - value
More than one random variable

32. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance reverts to a long run level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

33. Continuous representation of the GBM
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)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

34. Empirical frequency
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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)

35. Inverse transform method
Only requires two parameters = mean and variance
Independently and Identically Distributed
When the sample size is large - the uncertainty about the value of the sample is very small
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

36. Significance =1
Contains variables not explicit in model - Accounts for randomness
Transformed to a unit variable - Mean = 0 Variance = 1
Confidence level
Var(X) + Var(Y)

37. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)

38. Implied standard deviation for options
Variance(X) + Variance(Y) - 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

39. Efficiency
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
If variance of the conditional distribution of u(i) is not constant
When one regressor is a perfect linear function of the other regressors
Among all unbiased estimators - estimator with the smallest variance is efficient

40. BLUE
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
For n>30 - sample mean is approximately normal
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

41. Deterministic Simulation
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

42. Beta distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

43. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Price/return tends to run towards a long - run level
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Random walk (usually acceptable) - Constant volatility (unlikely)

44. Non - parametric vs parametric calculation of VaR
If variance of the conditional distribution of u(i) is not constant
Model dependent - Options with the same underlying assets may trade at different volatilities
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Only requires two parameters = mean and variance

45. Panel data (longitudinal or micropanel)
Expected value of the sample mean is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

46. Sample covariance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)

47. Type I error
Variance(y)/n = variance of sample Y
We reject a hypothesis that is actually true
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval

48. Difference between population and sample variance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

49. Sample correlation
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Rxy = Sxy/(Sx*Sy)
SSR

50. Kurtosis
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Sample mean will near the population mean as the sample size increases

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FRM Foundations Of Risk Management Quantitative Methods

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