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FRM Foundations Of Risk Management Quantitative Methods

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1. Cross - sectional
Only requires two parameters = mean and variance
E(mean) = mean
Average return across assets on a given day
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

2. Beta distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Independently and Identically Distributed

3. Biggest (and only real) drawback of GARCH mode
Price/return tends to run towards a long - run level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
For n>30 - sample mean is approximately normal
Nonlinearity

4. Non - parametric vs parametric calculation of VaR
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

5. Tractable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Easy to manipulate
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

6. Variance of X+Y assuming dependence
Sample mean will near the population mean as the sample size increases
Variance(x) + Variance(Y) + 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

7. Panel data (longitudinal or micropanel)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Special type of pooled data in which the cross sectional unit is surveyed over time

8. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Low Frequency - High Severity events

9. Test for statistical independence
(a^2)(variance(x)) + (b^2)(variance(y))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(X=x - Y=y) = P(X=x) * P(Y=y)
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Bootstrap method
Population denominator = n - Sample denominator = n - 1
Based on a dataset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

11. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Model dependent - Options with the same underlying assets may trade at different volatilities
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

12. Simplified standard (un - weighted) variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance = (1/m) summation(u<n - i>^2)

13. Gamma distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
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(x)
Probability that the random variables take on certain values simultaneously

14. Type I error
We reject a hypothesis that is actually true
More than one random variable
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

15. Heteroskedastic
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
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
If variance of the conditional distribution of u(i) is not constant
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

16. 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
Combine to form distribution with leptokurtosis (heavy tails)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

17. Two drawbacks of moving average series
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

18. Confidence ellipse
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Normal - Student's T - Chi - square - F distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

19. Central Limit Theorem
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
For n>30 - sample mean is approximately normal
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

20. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Population denominator = n - Sample denominator = n - 1
Regression can be non - linear in variables but must be linear in parameters
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

21. Covariance calculations using weight sums (lambda)
We reject a hypothesis that is actually true
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sampling distribution of sample means tend to be normal
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

22. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
We reject a hypothesis that is actually true

23. Conditional probability functions
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P(Z>t)

24. EWMA
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
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

25. Potential reasons for fat tails in return distributions
i = ln(Si/Si - 1)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Random walk (usually acceptable) - Constant volatility (unlikely)

26. Reliability
Variance reverts to a long run level
Contains variables not explicit in model - Accounts for randomness
Statement of the error or precision of an estimate
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

27. Variance of aX + bY
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
(a^2)(variance(x)) + (b^2)(variance(y))
Model dependent - Options with the same underlying assets may trade at different volatilities
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

28. Pooled data
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
More than one random variable

29. Variance - covariance approach for VaR of a portfolio
Average return across assets on a given day
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(x) + Variance(Y) + 2*covariance(XY)
Only requires two parameters = mean and variance

30. Exponential distribution
95% = 1.65 99% = 2.33 For one - tailed tests
i = ln(Si/Si - 1)
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

31. Cholesky factorization (decomposition)
Regression can be non - linear in variables but must be linear in parameters
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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
Yi = B0 + B1Xi + ui

32. Skewness
Transformed to a unit variable - Mean = 0 Variance = 1
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
Low Frequency - High Severity events
Sample mean +/ - t*(stddev(s)/sqrt(n))

33. Significance =1
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Based on a dataset
Confidence level
Attempts to sample along more important paths

34. Logistic distribution
Has heavy tails
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

35. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
If variance of the conditional distribution of u(i) is not constant

36. Continuous representation of the GBM
Independently and Identically Distributed
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)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

37. 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)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

38. Unstable return distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Use historical simulation approach but use the EWMA weighting system
(a^2)(variance(x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

39. Time series data
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Returns over time for an individual asset
We reject a hypothesis that is actually true

40. Discrete representation of the GBM
Summation((xi - mean)^k)/n
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Independently and Identically Distributed
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

41. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Does not depend on a prior event or information
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

42. i.i.d.
P(Z>t)
Population denominator = n - Sample denominator = n - 1
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Independently and Identically Distributed

43. Variance of X+b
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(x)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

44. Sample correlation
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Rxy = Sxy/(Sx*Sy)

45. Priori (classical) probability
(a^2)(variance(x)
Attempts to sample along more important paths
Based on an equation - P(A) = # of A/total outcomes
Variance(x) + Variance(Y) + 2*covariance(XY)

46. Efficiency
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Regression can be non - linear in variables but must be linear in parameters
Among all unbiased estimators - estimator with the smallest variance is efficient

47. K - th moment
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Summation((xi - mean)^k)/n
Variance reverts to a long run level

48. Poisson distribution equations for mean variance and std deviation
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Returns over time for an individual asset
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

49. Implications of homoscedasticity
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
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
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
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

50. Variance of X+Y
Variance(y)/n = variance of sample Y
Var(X) + Var(Y)
Concerned with a single random variable (ex. Roll of a die)
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

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