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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. LAD
P(X=x - Y=y) = P(X=x) * P(Y=y)
When the sample size is large - the uncertainty about the value of the sample is very small
Least absolute deviations estimator - used when extreme outliers are not uncommon
Low Frequency - High Severity events

2. Hazard rate of exponentially distributed random variable
Application of mathematical statistics to economic data to lend empirical support to models
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
We reject a hypothesis that is actually true
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

3. SER
Probability that the random variables take on certain values simultaneously
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
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
Expected value of the sample mean is the population mean

4. Economical(elegant)
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
When one regressor is a perfect linear function of the other regressors
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

5. Two drawbacks of moving average series
Mean = lambda - Variance = lambda - Std dev = sqrt(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
Peaks over threshold - Collects dataset in excess of some threshold
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

6. Non - parametric vs parametric calculation of VaR
(a^2)(variance(x)
Expected value of the sample mean is the population mean
Population denominator = n - Sample denominator = n - 1
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

7. Pooled data
Has heavy tails
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset
Returns over time for a combination of assets (combination of time series and cross - sectional data)

8. Cross - sectional
Nonlinearity
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))
Average return across assets on a given day

9. Unconditional vs conditional distributions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Rxy = Sxy/(Sx*Sy)
(a^2)(variance(x)) + (b^2)(variance(y))

10. Priori (classical) probability
Random walk (usually acceptable) - Constant volatility (unlikely)
Var(X) + Var(Y)
Based on an equation - P(A) = # of A/total outcomes
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

11. Variance of sample mean
Normal - Student's T - Chi - square - F distribution
Variance(y)/n = variance of sample Y
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

12. Econometrics
Sampling distribution of sample means tend to be normal
Peaks over threshold - Collects dataset in excess of some threshold
Application of mathematical statistics to economic data to lend empirical support to models
Mean = np - Variance = npq - Std dev = sqrt(npq)

13. Exact significance level
Confidence level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
E(XY) - E(X)E(Y)
P - value

14. Panel data (longitudinal or micropanel)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
We reject a hypothesis that is actually true
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Special type of pooled data in which the cross sectional unit is surveyed over time

15. Significance =1
Confidence level
Transformed to a unit variable - Mean = 0 Variance = 1
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
Concerned with a single random variable (ex. Roll of a die)

16. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Based on an equation - P(A) = # of A/total outcomes
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

17. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(x)
Variance(X) + Variance(Y) - 2*covariance(XY)

18. Expected future variance rate (t periods forward)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
(a^2)(variance(x)
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

19. Binomial distribution
Variance reverts to a long run level
For n>30 - sample mean is approximately normal
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!)
Nonlinearity

20. Covariance calculations using weight sums (lambda)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

21. 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
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

22. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
When the sample size is large - the uncertainty about the value of the sample is very small

23. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We accept a hypothesis that should have been rejected
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E(mean) = mean

24. Law of Large Numbers
i = ln(Si/Si - 1)
Sample mean will near the population mean as the sample size increases
More than one random variable
P(X=x - Y=y) = P(X=x) * P(Y=y)

25. Difference between population and sample variance
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
Population denominator = n - Sample denominator = n - 1
(a^2)(variance(x)) + (b^2)(variance(y))
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

26. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Z = (Y - meany)/(stddev(y)/sqrt(n))
Distribution with only two possible outcomes
Yi = B0 + B1Xi + ui

27. POT
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Peaks over threshold - Collects dataset in excess of some threshold

28. Time series data
We accept a hypothesis that should have been rejected
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
(a^2)(variance(x)) + (b^2)(variance(y))
Returns over time for an individual asset

29. Tractable
Easy to manipulate
P(X=x - Y=y) = P(X=x) * P(Y=y)
Among all unbiased estimators - estimator with the smallest variance is efficient
Random walk (usually acceptable) - Constant volatility (unlikely)

30. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
(a^2)(variance(x)) + (b^2)(variance(y))
Model dependent - Options with the same underlying assets may trade at different volatilities

31. Persistence
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
E(XY) - E(X)E(Y)
Easy to manipulate

32. Logistic distribution
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
SSR
P(X=x - Y=y) = P(X=x) * P(Y=y)
Has heavy tails

33. Square root rule
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sampling distribution of sample means tend to be normal
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

34. Marginal unconditional probability function
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Does not depend on a prior event or information
Mean of sampling distribution is the population mean
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

35. Importance sampling technique
If variance of the conditional distribution of u(i) is not constant
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to sample along more important paths
i = ln(Si/Si - 1)

36. Empirical frequency
Variance = (1/m) summation(u<n - i>^2)
When the sample size is large - the uncertainty about the value of the sample is very small
Based on a dataset
(a^2)(variance(x)

37. Implied standard deviation for options
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
Variance(x)
Variance = (1/m) summation(u<n - i>^2)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

38. Antithetic variable technique
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Price/return tends to run towards a long - run level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

39. Maximum likelihood method
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Choose parameters that maximize the likelihood of what observations occurring

40. Variance of aX
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
Population denominator = n - Sample denominator = n - 1
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
(a^2)(variance(x)

41. R^2
Use historical simulation approach but use the EWMA weighting system
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
E(mean) = mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

42. Standard variable for non - normal distributions
If variance of the conditional distribution of u(i) is not constant
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Nonlinearity
Z = (Y - meany)/(stddev(y)/sqrt(n))

43. GPD
Mean of sampling distribution is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

44. Lognormal
Based on a dataset
Sample mean +/ - t*(stddev(s)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

45. Mean reversion in variance
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Has heavy tails
Variance reverts to a long run level
Nonlinearity

46. Efficiency
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Sampling distribution of sample means tend to be normal

47. i.i.d.
Independently and Identically Distributed
(a^2)(variance(x)) + (b^2)(variance(y))
Based on a dataset
Has heavy tails

48. Test for unbiasedness
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E(mean) = mean
Has heavy tails
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

49. Joint probability functions
Probability that the random variables take on certain values simultaneously
When the sample size is large - the uncertainty about the value of the sample is very small
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
Based on an equation - P(A) = # of A/total outcomes

50. GARCH
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Does not depend on a prior event or information
Peaks over threshold - Collects dataset in excess of some threshold
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)